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+ (S A P). (d) P -> (R --> S). (c) P -> (R V S). 2.2. Construct the truth table for each of the following statements. (a) P-->(P-'Q). (d) (P-'Q)<-- -1PV Q. (e) (P -4 Q A R) V (-,P A Q). (b) P V Q,(-+ Q V P. (f) P A Q - (Q A -, Q --> R A Q). (c) P --> -, (Q A R). 2.3. Suppose the value of P -> Q is T; what can be said about the value of (P V -,Q) V R -+ (S A -,S). -iPAQHPVQ? 2.4. (a) Suppose the value of P 4-, Q is T; what can be said about the values of P- -1 Q and --, 1' <- Q? (b) Suppose the value of P <-* Q is F; what can be said about the values of P <- --IQ and -1 P 4-4 Q? 2.5. For each of the following determine whether the information given is sufficient to decide the truth value of the statement. If the information is enough, state the truth value. If it is insufficient, show that both truth values are possible. (a) (P - Q) - R. T (b) P A (Q'R). T (c) P V (Q --> R). T (d) -1 (P V Q) -. -1P A -, Q. T (e) (P - Q) ---> (-,Q --* -,P). T (f) (P A Q) -> (P V S). T F 2.6. In Example 1.3 we symbolized the statement If either labor or management is stubborn, then the strike will be settled if the government obtains an injunction, but troops are not sent into the mills as L V Al - ( S -,R). By a truth-value analysis, determine whether this statement is true or false under each of the following assumptions. (a) Labor is stubborn, management is not, the strike will be settled, the government obtains an injunction, and troops are sent into the mills. (b) Both labor and management are stubborn, the strike |
will not be settled, the government fails to obtain an injunction, and troops are sent into the mills. 2.7. Referring to the statement in the preceding exercise, suppose it is agreed that 4.3 I T /w Statement Calculus. Validity 169 If the government obtains an injunction, then troops will be sent into the mills. If troops are sent into the mills, then the strike will not be settled. The strike will be settled. Management is stubborn. Determine whether the statement in question is true or not. 3. The Statement Calculus. Validity The foregoing is intended to suggest the nature of the statement calculus, namely, the analysis of those logical relations among sentences which depend solely on their composition from constituent sentences using sentential connectives. The setting for such an analysis includes the presence of an initial set of sentences (the "prime sentences") and the following two assumptions. (i) Each prime sentence is a statement; that is, there may be assigned to a prime sentence a truth value. (ii) Each sentence under consideration is composed from prime sentences using sentential connectives and, for a given assignment of truth values to these prime sentences, receives a truth value in accordance with the truth tables given earlier for negation, conjunction, and so on. With this in mind, let us make a fresh start on the statement calculus. Suppose there is given a noncmpty set of distinct sentences and that we extend this set by adjoining precisely all of those sentences which can be formed by using, repeatedly and in all possible ways, the various sentential connectives. Then the extended set has the following property. If A and B are members, then so are each of -i A, A V B, A A B, A --* B, and A-+ B. We shall call the members of the extended class formulas. The members of the initial set are the prime formulas, and the others are composite formulas. The prime formulas which appear in a composite formula are said to be contained in that formula and are called its prime components. To display a composite formula unambiguously, parentheses are used. However, to avoid excessive use of parentheses, the conventions introduced earlier will be employed. The classical statement calculus, which is the only one we treat, assumes that with each prime formula there is associated exactly one member of IT, Fl. Further, it assumes that it is irrelevant if it is T or F that is associated with a prime formula. Thereby, maximum versatility in the applications |
is achieved-truth values may be assigned as the 170 Logic I CHAP. 4 occasion demands. The truth value of a composite formula is defined inductively in accordance with the following tables. AAB AvB A--*B A*-3B EXAMPLES A T F -,A F T 3.2. Let A be a formula having P,, Ps, 3.1. If the prime components in a formula A are P1, P2,, P,,, then the definition of the truth value of A in terms of truth values of P1, P2,, P" can be exhibited in a truth table, as described earlier. There are 2" rows in such a, P". table, each row exhibiting one possible assignment of T's and F's to P,, P2,., P as its prime components. Then A provides a rule for associating with any ordered n-tuple of T's and F's, whose ith coordinate is the assignment to Pi, for i = 1, 2,, n, one of T and F. If we set V = IT, F}, then we can rephrase our observation: A defines a function on V" into V. A function on V" into V we shall call a truth function (of n arguments). Truth functions will be designated by such symbols as.., P.), g(ql, q2,..., q"), and so on.,/ (P1) P2, Note that we depart from our practice of designating functions by single letters and use notation heretofore reserved for function values. Our excuse is that composition of functions can be described more simply. For example, the notation 1(pl,..., pi-1, g(gl)...' qm), pi-hl,..., pn) is self-explanatory as a function obtained by composition from the truth function f of n arguments and g of nl arguments. We shall refer to this function as that obtained by substitution of g for the ith variable in f. Clearly, such combinations of truth functions are again truth functions. An alternative approach to the statement calculus can be given in terms of truth functions: There are 22 different truth functions of n arguments. Of the four for n = 1, that whose value at T is F, and whose value at F is T, we shall denote by -Ip. Among the sixteen truth functions of two arguments appear the four listed below in tab |
ular form. The reason for the denotations chosen should be clear. A (p, q) v (p, q) -->(p, q) H(p, q) (T, T) (T, F) (F, T) (F, F.3 1 The Statement Calculus. Validity 171 Since the outfix notation for these functions seems unnatural, we shall put the reader at ease by employing the more familiar infix notation [for example, p A q instead of A(p, q)]. It is of interest that by using just those functions which have been mentioned so far and the operation of function composition in the form mentioned above, all truth functions of any number of arguments can be obtained. Indeed, the three functions -,p, p V q, and p A q suffice. To prove this, let f(p,, p2, be some truth function. If the value off is F for all values of is, f is the constant function F), then it is equal to, p (that (p, A A (P2 A -,p2) A.. A (pn A-,p,.). Otherwise, f assumes the value T at least once. For each element of the domain off such that f takes the value T, let its form the function q, A q2A... Aq,,, where qi is p (or -,p,) when p; has the value T (or F). Then we contend that f is equal to the function obtained "by disjunction" from all such functions. For example, iff(p, q) takes the value F when p = q = T and the value T otherwise, then f(1p,q)=(pA-1 q) V (-iPAq)V (-,pA-,q). The reader can verify this and supply a proof of the general statement. Actually, each of the pairs -,p, p A q and -,p, p V q is adequate to generate all truth functions, using the operation of function composition, since p V q = -, (-,p A q) and p A q = -1 (-,p V -, q). The same is true of the pair -,p, p --- q, as the reader can verify. Although no member of any of the three pairs mentioned can be discarded to obtain a single function which generates all truth functions, such functions do exist. For example, the function |
plq (as it is customarily written) of two arguments, whose value is T except at (T, T), where its value is F, suffices. To prove this it is sufficient to show (for example) that both -,p and p V q can be expressed in terms of it. As we have already observed, each formula of the statement calculus defines a truth function. It should be clear that it is only the structure of a composite formula A regarded as a truth function that one considers when making a truth value assignment to A for a given assignment of truth values to its prime components. When it is convenient, we shall feel free to regard a formula as a truth function. In such an event, the prime components (statement letters) will be considered as variables which can assume the values T and F. The statement calculus is concerned with the truth values of composite formulas in terms of truth-value assignments to the prime components and the interrelations of the truth values of composite formulas having some prime components in common. As we proceed in this study it will appear that those formulas whose truth value is T for every 172 Logic I CHAP. 4 assignment of truth values to its prime components occupy a central position. A formula whose value is T, for all possible assignments of truth values to its prime components, is a tautology or, alternatively, such a formula is valid (in the statement calculus). We shall often write GA for "A is valid" or "A is a tautology."f Whether or not a formula A is a tautology can be determined by an examination of its truth table., P,,, then A is a tautolIf the prime components in A are P1, P2, ogy if its value is T for each of the 2" assignments of T's and F's to, P,,. For example, P -- P and P A (1' -4 Q) -* Q are tauPt, P2, tologies, whereas P --' (Q --> R) is not. These conclusions are based on an examination of Tables I, II, and 111, below. Table I PJP--4P T F T T Table II P A (P -- Q) -- Table III Q R P -> (Q --> R The definition of validity provides us with a mechanical way to decide whether a given formula is valid--namely, the computation and examination of its truth table. Although it may be tedious, this method can |
always be used to test. a proposed formula for validity. But, clearly, it is an impractical way to discover tautologies. This state of affairs has prompted the derivation of rules for generating tautologies from tautologies. 'I he knowledge of a limited number of simple tautologies and several such rules make possible the detivation of a great variety of valid formulas. We develop next several such rules and then implement their with a list of useful tautologies. THEOREM 3.1. Let B be a formula and let B* be the formula resulting from B by the substitution of a formula A for all occurrences of a prime component P contained in B. If l- 13, then r- B*. t This symbol for validity (l=) appears to be due to Klecne. 4.3 1 Die Statement Calculus. Validity 173 Proof. For an assignment of values to the prime components of B* there results a value v(A) of A and a value v(B*) of 13*. Now v(B*) = v(B), the value of B for a particular assignment of values to its prime components, including the assignment of v(A) to P. If B is valid, then v(B) and hence v(B*) is always T. That is, if B is valid, then so is B*. EXAMPLES 3.3. From Table IV below it follows that G P V Q E-a Q V P. Hence, by Theorem 3.1,!_ (R --- S) V Q <-' Q V (R --I S). A direct verification of this result (Table V), using the reasoning employed in the proof of Theorem 3.1, should clarify matters, if need be. To explain the relationship of Table V to Table IV, we discuss the displayed line of Table V. Table IV 1' Q PV QE-+Q VP Table V R S Q (R - S) V Q <--' Q V (R --' S) T F T F T T F There was entered first (at two places) the value F of R -' S for the assignment of T to R and F to S. Then the value T assigned to Q was entered twice. The rest of the computation is then a repetition of that appearing in the third line of Table IV after the entries underlined there have been made. 3.4. The practical importance of Theorem 3.1 is |
that it provides a method to establish the validity of a formula without dissecting it all the way down to its prime components. An illustration will serve to describe the application we have in mind. Suppose the question arises as to whether the formula (RVS) A ((RVS)-4(PAQ))-'(PAQ) is a tautology. The answer is in the affirmative, with Theorem 3.1 supplying the justification, as soon as it is recognized that the formula in question has the "same form as" the tautology P A (P --+ Q) --' Q (Table II), in the sense that it results from P A (1' -' Q) -* Q upon the substitution of R V S for P and P A Q for Q. We introduce next a relation for formulas. For the definition it is convenient to interpret formulas as truth functions and observe that, P. may be regarded a formula whose prime components are P1, P2, as a function of an extended list P,,, P,,, of variables. Let us. now agree to call formula A equivalent to formula B, symbolized, 1',,,. A eq B, 174 Logic I CHAP. 4., Pm, if they are equal as truth functions of the list of variables P1, P2, where each P occurs as a prime component in at least one of A and B. In terms of truth tables, the definition amounts to this. Suppose that. -, P,4 is the union of the sets of prime components contained I P,, P2, in A and B, respectively, and that we compute the truth tables of A and of B as if both contained P,, P2,, P,,, as prime components. Then. A eq B if the resulting truth tables are the same. For example, from Tables VI and VII below we infer that (P--; Q) eq -,P V Q and Peg PA (QV -,Q). Table VI P Q P -> Q -1P Table VII It is left as an exercise to prove that eq is an equivalence relation on every set of formulas and, further, that it has the following substitutivity property: If CA is a formula containing a specific occurrence of the formula A and Ca is the result on replacing this occurrence of A by a formula B, then if B eq A, then egCA. Henceforth, equivalent formulas will be regarded as interchangeable, and the substitution property |
will often be employed without comment. Equivalence of formulas can be characterized in terms of the concept of a valid formula, according to the following theorem. TIIEOREM 3.2. KA+->BifAegB. :, P,,, be the totality of prime components apProof. Let P,, P2, pearing in A and B. For a given assignment of truth values to these components, the first part of the computation of the value of A -+ B consists of computing the values of A and B, after which the computation is concluded by applying the table for the biconditional. According to this table, the value of A 4-1 B is T if the values computed for A and B are the same. COROLLARY. Let CA be a formula containing a specified occurrence of the formula A and let CD be the result of replacing this occurrence of A by a formula: B. If t- A 4--> B, then t= CA 4-> CB. If t=CA,then t=Co. This proof is left as an exercise. 4.3 1 The Statement Calculus. Validity 175 If A and K A - - + B, then 1= B. THEOREM 3.3. Proof. Let P,, P2,, P,,, be the totality of prime components appearing in A and B. For a given assignment of truth values to these, the first part of the computation of the value of A --> B consists of computing the values of A and B, after which the computation is completed by applying the table for the conditional. The assumpA and 1= A --+ B imply that both the value obtained for A tions and that for A -- B are T. According to the table for A --> B, this implies that B must also have the value T. Since this is the case for all assignments of values to P,, P2,, Pm, B is valid. As the next theorem we list a collection of tautologies. It is not intended that these be memorized; rather, they should be used for reference. That many of the biconditionals listed are tautologies should be highly plausible on the basis of meaning, together with Theorem 3.1. That each is a tautology may be demonstrated by constructing a truth table for it, regarding the letters present as prime formulas. Then, once it is shown that the value is T for all assignments of values |
to the components, an appeal is made to the substitution rule of Theorem 3.1 to remove the restriction that the letters be prime formulas. In the exercises for this section the reader is asked to establish the validity of some of the later formulas by applying one or more of Theorems 3.1--3.3 to tautologies appearing earlier in the list. THEOREM 3.4. Tautological Conditionals 1.1=AA(A>B)-4 B. 2. 1= -,BA 3. k= -,AA (AV 13)>B. 4. tzA-a(B-) AA13). 5. k=AAB--->A. 6. 1= A --> A V B. 7. 1- (A -* B) A (B -> C) - > (.4 --a C). - (A A B-> C) -- (A > (B -- > C)). 8. 9. 1= (A --> (13 -+ C)) -> (A A B -- C). 10. 1= (A --> B A -, B) > -, A. 11. 12. (A -- B) --* (A V C --* B V C.'). (A -* B) --* ((B --> C) > (A -). C)). 14. 1= (A EI 13) A (B.-> C) > (A.--3 C). 176 Logic I CHAP. 4 Tautological Biconditionals A (A+-* B) +-4 (B A). (A -4 B) A (C ---> B) E--> (A V C ---* B). C) 4-4 (A -a B A C). 15. t= AAA. 16. - -, A 17. 18. 19. 20. K (A-4 B).+(-,B-- -, A). 21. KAVB4BvA. (A V B) V C H 22. t= (A -4 B) A (A AV(BVC). 23. 1= AV (BAC) - (A V B) A (A V C). 24. i=AvA<--'A. 25. t= -, (A V B). > -,A A -,B. 21'. tAABt-+BAA. 22'. 1= (AAB |
)ACH 23'. KAA(BVC)f--> A A (B A C). (AAB)V (A A C). 24'. IzAAA+-+A. 25'. t= -, (AAB)<--4 -,A V -,B. Tautologies for Elimination of Connectives 26. KA-'B4-,AVB. 27. I A -->BH 28. KAVB<-29. KAVB<-t7- AAB<-30. 31. l AAB 32. t= (A+-->B)*-+(A-). B) A (B--->A). We conclude this section with the description of a powerful method for obtaining tautologies from scratch. Initially we consider only for, 1' using -,, A, and mulas composed from prime formulas P,, P2, V. The denial, Ad, of such a formula A is the formula resulting from A by replacing each occurrence of A by V and vice versa and replacing each occurrence of P, by an occurrence of -, P; and vice versa. As illustrations of denials in the present context we note that the denial of P V Q is -1P A -, Q and the denial of -, (-, P A Q) is -1 (P V The theorem relating denials and tautologies follows. THEOREM 3.5. Let A be a formula composed from prime components using only -,, A, and V, Let Ad be the denial of A. Then K -,AHAd. 4.3 I The Statement Calculus. Validity 177 A proof of this assertion can be given by induction on the number of symbols appearing in a formula. We forego this, but do include in the first example below a derivation of an instance of the theorem. Another example describes the extension of the theorem to the case of a formula which involves --f or 4--p. EXAMPLES 3.5. An instance of Theorem 3.5 is the assertion that K-1((-,P V Q) V (Q A (R V -1 P))) 4 (P A -,Q) A (-i Q V (--iR A P)), or, in other words, that the left-hand side and the right-hand side of the biconditional are equivalent formulas. Using the properties of transitivity and substitutivity of equivalence, this is established below. Each step |
is justified by the indicated part of Theorem 3.4 (in view of Theorem 3.2). -, ((-,P V Q) V (Q A (R V -,P))) eq-,(-,PVQ)A-i(QA(RV-,P)) (25) eq (-,-,P A -,Q) A (-,Q V -,(R V -1 P)) (25, 25') eq (--i --i P A -,Q) A (-'Q V (-1R A -,-,P)) (25) eq (P A -,Q) A (-,Q V (-,RAP)) (16) 3.6. Using tautology 32 in Theorem 3.4 we can derive from a formula in appears an equivalent formula in which H is absent. For instance, which P 4-+ (Q A R) eq (P - Q A R) A (Q A R -4 P). That is, H can be eliminated from any formula. Similarly, using tautology 26 or 27, -a can be eliminated from any formula. Thus, any formula A is equivalent to a formula B composed from prime components using -,, A, and V. Then we may define the denial of A to be the denial of B. 3.7. According to the preceding example, H and -> can be eliminated from any formula. Using tautology 29 it is possible to eliminate V or (with tautology 31), equally well, A. That is, any formula is equivalent to one composed from prime components using -, and V or using -, and A. This conclusion should be recognized as merely another version of a result obtained in Example 3.2. 3.8. From tautology 22 follows the general associative law for V, which V A. to render asserts that however parentheses arc inserted in A, V A2 V it unambiguous, the resulting formulas are equivalent. From tautology 22' follows the corresponding result for A A. EXERCISES 3.1. Referring to Example 3.2, write each of the following formulas as a truth function in outfix notation. For example --,P --' (Q V (R A S)) becomes -+ (-,P, V (Q, A (R, S)). 178 Logic ( CHAP. 4 (a) P A (b) -,P - Q. (c) P V ( |
Q V R). (d) P A (Q-->B). 3.2. (a) Referring to Example 3.2, complete the proof that every truth func- (1) P V R -> (B A (S V -,P)). (g) (P - Q) --> (S A -,P -> Q). P -* (Q -R) H Q -- * (P - 4 R). tion can be generated from -,p, p V q, and p A q. (b) Referring to the same example, show that every truth function can be generated from p(q. 3.3. Suppose that P,, P2,, P. are prime components of A. Show that the, P," as prime components truth table of A, regarded as having P1, can be divided into 2'"`-" parts, each a duplicate of the truth table for A computed with P1, Pz,, P. as the prime components., P,,, 3.4. Prove that eq is an equivalence relation on every set of formulas and that it has the substitutivity property described in the text. 3.5. Prove the results stated in the Corollary to Theorem 3.2. 3.6. Derive each of tautologies 28-31 from earlier tautologies in Theorem 3.4, using properties of equivalence for formulas. As an illustration, we derive tautology 27 from earlier ones. From 26, A -a --I B eq --1 A V --I B, and, in turn, - A V -1 B eq -, (A A B) by 25'. Hence, A --> -i -1 B eq -I (A A --I B). Using 16 it follows that A --+ B eq -, (A A -1 B), which amounts to 27. 3.7. Instead of using truth tables to compute the value of a formula, an arithmetic procedure may be used. The basis for this approach is the representation of the basic composite formulas by arithmetic functions in the following way. Formula -'P P A Q PVQ 1' Q P f- Q Arithmetical representation 1 + P P -I- Q+ PQ PQ (1 -I- P) Q P -}- Q When the value T (respectively, F) is assigned to a prime component in a formula - for example, 1' |
-the value 0 (respectively, 1) is assigned to the variable P in the associated arithmetical representation. Further, values of the arithmetical functions are computed as in ordinary arithmetic, with one exception: namely, 1 + 1 = 0. In each case a simple calculation shows that when the formula takes the value T (respectively, F), then its arithmetical representation takes the value 0 (respectively, 1). In these terns, tautologies are represented by functions which are identically 0. For example, that K P V -j P is clear from the fact that P V --IP is represented by P(1 + P). To prove that the formula in 1 of Theorem 3.4 (regarding A and B as prime components) is a tautology, we form first [corresponding to A A (A -+ 13)J, 4.4 I The Statement Calculus. Consequence 179 A + (1 -1- A) B -1- A(1 + A) B, which reduces to A + (1 + A)B since A(1 + A) is identically 0. Then to the entire formula in I corresponds the function (I + A + (1 + A) B) B, which, as one sees immediately, is identically 0. In the algebra at hand, 2x = 0, x(x + 1) = 0, and x2 = x for all x. These facts make the simplification of long expressions an easy matter. Prove some of the tautologies in Theorem 3.4 by this method. 3.8. (a) With Exercise 3.7 in mind, show that the function (I + P)(1 + Q) is an arithmetical representation of the truth function PIQ defined in Example 3.2. (b) The result in (a), together with that in Exercise 3.2(b), may be reformulated as follows: Every mapping on (0, 1) ft into (0, 11 can be generated from the mapping f : {0, 1) 2 -} {0, 1 ) such that f (x, y) _ (1 + x) (1 +y). Show that the same is true ofg: {0, 1}a {0, 1}, where g(x, y, z) = 1 + x -1- y -1- xyz. 4 |
. The Statement Calculus. Consequence In the introduction to this chapter we said that it was a function of logic to provide principles of reasoning--that is, a theory of inference. In practical terms this amounts to supplying criteria for deciding in a mechanical way whether a chain of reasoning will be accepted as correct on the basis of its form. A chain of reasoning is simply a finite sequence of statements which are supplied to support the contention that the last statement in the sequence (the conclusion) may be inferred from certain initial statements (the premises). In everyday circumstances the premises of an inference are judged to be true (on the basis of experience, experiment, or belief). Acceptance of the premises of an inference as true and of the principles employed in a chain of reasoning from such premises as correct commits one to regard the conclusion at hand as true. In a mathematical theory the situation is different. There, one is concerned solely with the conclusions (the so-called "theorems" of the theory) which can be inferred from an assigned initial set of statements (the so-called "axioms" of the theory) according to rules which are specified by some system of logic. In particular, the notion of truth plays no part whatsoever in the theory proper. The contribution of the statement calculus to a theory of inference is just this: It provides a criterion, along with practical working forms thereof, for deciding when the concluding sentence (a statement) of an argu- 180 Logic I cnIAF. 4 ment is to be assigned the value T if each premise of the argument is assigned the truth value T. This criterion is in the form of a definition. The statement B is a consequence of statements At, A2, A,,, (by the statement calculus), symbolized, P,, occurring in one or more of A,, A2j A,, A2,..., Am 1= B, iff for every truth-value assignment to each of the prune formulas, A,,,, and B, the 1',, P2, formula B receives the value T whenever every A receives the value T., Am K B" means simply that if In terms of truth tables, "A,, A2, truth tables are constructed for A,, A2,, Am, and B, from the list, P. of prime formulas occurring in one or more of these P,, P2, formulas, then B receives the value T at least for each assignment to the F's which make all A's simultaneously T. |
EXAMPLE 4.1. From an inspection of Table VIII below we obtain the following three illustrations of our definition: P, R, Q A P --> -, R l= --, Q, P,P--*R,RK PV Q R, Q A P - --, R, --, Q, 1' - R l -, (P A Q). (line 3) (lines land 3) (lines 3, 7, 8) P Q R QAP--*-1R P P Q -,(PAQ) Table VIII THEOREM 4.1I) A 1= B iff l= A - - + B. (II) A1, A2,, Am 1= B iff A,AA2A KAlAA2A... AA,,--+ B(m> 2). AAm1=B or, if Proof. For (I), let A 1= B. By the table for --+, A > B receives the value F if A receives the value T, and, simultaneously, B receives the value F. From the hypoth-,sis, this combination of values does not 4.4 1 Die Statement Calculus. Consequence 181 occur. Hence A -+.13 always receives the value T, that is, t= A -> B. For the converse, let t= A --* B, and consider an assignment of values to the prime components such that A receives the value T. Since A -4 B receives the value T, it follows from the table for that B takes the value T, whence, A J-_ B. The first assertion in (II) follows from the table for A, and the second follows from the first by an application of (I). COROLLARY. A,,,Am_,,Am t- BiffA,, -, Am-, t-_ Am B., Am_t, Am l B iff t-- A, --, (A2 -.* ( More generally, At, (Am -.._, B)... ))., Am_,, A. f= B for m > 1. Then f= (At A Proof. For m = 1, the first assertion is (I) of the theorem. So, assume that A,,. A Am_,) A A,,,--* B, according to the theorem. From tautology 8 of Theorem 3.4 and Theorem 3.3, we deduce |
that A Am_,) (Am ---> B). According to (1) of the theorem, it follows that A, A A Am_, t= Am --+ B and hence, by (II), that A,,, A,,,_, I- Am -> B. The converse is established by reversing the foregoing steps. Finally, the second assertion follows by repeated application of the t= (A, A first. Thus, the problem of what statements arc consequences of others (by the statement calculus) is reduced to the problem of what statements arc valid (which accounts for the importance of tautologies). On the other hand, there is something to be said for approaching the concept of consequence directly. One reason is the possibility of converting the definition into a working form which resembles that used in mathematics to infer theorems from a set of axioms. Indeed, we can substantiate a working form as a sequence of formulas (the last formula being the desired consequence of the premises) such that the presence of each is justified by a rule, called a rule of inference (for the statement calculus). The basis for the rules of inference which we shall introduce is the following theorem. THEOREM 4.2. (I) At, A2,,A,,,1 Aifori= 1, 2, (II) If At, A2, -,At= B;for j = 1, 2, -, A. I= C. C, then At, A2, B,,,,n.,p,andifB,,B2, 182 Logic I CHAP. 4 - Proof. Part (I) is an immediate consequence of the definition of, A. 1= B." For (II) we construct a truth table from the "A,, A2,, P. of all prime components appearing in at least one list P1, P2j, A,,, of the A's, the B's, and C. Consider any row in which A,, A2, each receive the value T. Then, by the hypotheses, each B has the value T, and hence C has the value T. That is, for each assignment of values to the P's such that every A takes the value T, formula C receives the value T. This is the desired conclusion. With this result, a demonstration that a formula B (the conclusion), A. (the premises) may be preis a consequence of formulas A1, A2, |
sented in the form of a string (that is, a finite sequence) of formulas, the last of which is B and such that the presence of each formula E is justified by an application of one of the following rules. Rule p : The formula E is a premise. Rule t: There are formulas A, that t= A A A D -' E., D preceding E in the string such That is, we contend that A,, A2,, A. K B if we can concoct a string E,, E2,..., E,(= B) of formulas such that either each E is a premise (rule p) or there are preceding formulas in the string such that if C is their conjunction, C - E (rule t). Indeed, assuming that each entry in the disthen played sequence can be so justified, we shall prove that, A. i (any E in the sequence). A,, A2, - This is true of E_, by Theorem 4.2(I). Assume that each of E,, E2,., Ek_, is a consequence of the A's; we prove that the same is true of the next formula Ek. If Ek is a premise, then Theorem 4.2(I) applies. Otherwise, there are formulas preceding Ek such that if C is their conjunction, then i C-> Ek. Let us say E A E,, A... A E,. -f Ek. Then, by Theorem 4.1 (II),..., E`. (' Ek, E,,, E,,, and, by assumption, A1, A2,...,AmI=Er j = 1, 2,...,s. Hence, by Theorem 4.2(11), A1, A2,..., A. K Ek. 4.4 1 The Statement Calculus. Consequence 183 We note, finally, that by an application of rule t any tautology may be entered in a derivation. Indeed, if i D, then for any formula A we have K A -+ D. Thus, D may be included in a derivation by an application of rule l wherein we take any premise as the "A." The examples which follow illustrate the foregoing method for demonstrating that some formula is a consequence of given formulas. To make the method entirely definite, let us agree that when applying rule t, only the tautological conditionals which appear explicitly in Theorem 3.4 or are |
implicit in the biconditionals of that theorem (for exA H A yields the tautological conditional 1= A -a A and ample, -i A) may be used. K -, -, A H A yields 1= -i -1 A ---> A and i A --+ Admittedly, this is an arbitrary rule. Our excuse for making it is that it serves to make the game to be played a definite one. EXAMPLES 4.2. We demonstrate that AVB,A-*C,B--+DrCVD. An explanation of the numerals on the left is given below. {1} {1 } {3} {3) (1) A --+C (2) A V B --+ C V B (3) B -- 1) (4) C V B --' C V 1) {1, 3} (5) A V B --j C V I) (6) 11, 3, 6) (6) A V B (7) C V D Rulep Rule t; 1= (1) -a (2) by tautology 11. Rule p Rule i; G (3) - (4) by tautology 11. Rule t; l (2) A (4) - (5) by tautology 7. Rule p Rule t; 1= (6) A (5) --, (7) by tautology 1. The numbers in parentheses adjacent to each formula serve to designate that formula as well as the line of the derivation in which it appears. The set of numbers in braces for each line corresponds to the premises on which the formula in that line depends. That is, the formula in any line n is a consequence of the premises designated by the numbers in braces in that line. Thus, the formula in line 5 is a consequence of the premise in line 1 and the premise in line 3, and the formula in line 7 is a consequence of the premises in lines 1, 3, and 6--that is, of all the premises. In particular, for a line which displays a premise there appears in braces at the left just the number of that line, since such a formula depends on no other line. Using the brace notation in connection with the numerals on the left is deliberate in that it suggests that the for- 184 Logic I C. H A P. 4 mula in that line is a consequence of the set of premises designated |
by those numbers. We now rewrite the above derivation, incorporating some practical abbreviations. In this form the reader is called on to supply the tautologies employed. {1} {1) (3) {3} (1) A -> C (2) A V B--3CV B (3) B -- D (4) CV {6} {1,3,6) B (6) A V B (7) CV D D p It p 2,41 p 5,61 4.3. As a more elaborate illustration we prove that WVP->I,I-CVS,S-U,-,CA-,U1=-,W by the following string of thirteen formulas. {1) {1) (3) {1,3} {I} {1, 3} 11,3) {8} {9} {8,9} {1,3,8,9) {1,3,8,9) (1,3,8,9) p (1) -, C A -, U U 1 1 p 2,31 1 t 4, 5 t 6 t p p (2) (3) S -> U (4) -1S (5) -1C (6) -1C A --1S (7) -, (C V S) (8) W V P -+I (9) 1 -> C V S (10) WVP -). C VS 8,91 (11) -,(WVP) 7,101 (12) -,WA -,P 111 (13) -,W 121 We note that the foregoing takes the place of a truth table having 26 = 64 lines for the purpose of verifying that t= (WVP - I) A (I - C V S) A (S -3 U) A (-1 (: A U) -> -, W. 4.4. Many theorems in mathematics have the form of a conditional, the assumptions being the axioms of the theory under development. The symbolic form of such a theorem is A,, A2,, A. k= B. C, where the A's are the axioms and 1? -4 C is the consequence asserted. In order to prove such a theorem it is standard practice to adopt B as a further assumption and then infer that C is a consequence. Thereby it is implied that A1, A2,..., A |
,,, K B - C ifF A1, A2,..., A,,,, B k= C. This is correct according to the Corollary to Theorem 4.1. It is convenient to 4.4! The Statement Calculus. Consequence 185 formulate this as a third rule of inference, the rule of conditional proof, for the statement calculus. Rile rp: The formula B -. C is justified in a derivation having At, Az, A. as premises if it has been established that C is a consequence of At, Az, Am, and B., As an illustration of the use of this rule we prove that (1) {2} {3} {4) A-4(B-+C),-,D V A,B1=D--*C. (1) A -+ (B ---> C) p fi (2) -11) V A (3) B P (4) D p (introducing "D" as an additional premise) {2, 4} (1,2,4) {1, 2, 3, 41 {1, 2, 3} (5) A (6) B (7) C (8) D -* C C 2,41 1, 5 1 3, 6 t 4, 7 rp The usefulness of the braced numbers to show precisely what premises enter into the derivation of the formula in that line is clear. 4.5. Even if an alleged consequence of a set of premises does not have the form of a conditional, the application of the strategy as described in the preceding example may simplify a derivation. As an illustration we rework the first example, starting with the observation that the conclusion C V D is equivalent to -,C --> D. This suggests adding -,C as a premise and hoping that D can be derived as a consequence of this and the other premises. An advantage gained thereby is the addition of a simple assumption. The derivation follows. {1} {2} {3} {4} {2, 4} {1, 2, 4) {1,2,3,4) {1,2,3) {1,2,3} (1)AVB (2) A -- C I) (3) B (4) -,C (5) -, A (6) B (7) D (8) -,C -- D (9) CVD 4.6 |
. Each of the tautological implications in Theorem 3.4 generates a rule of inference, namely, the instance of rule t, which is justified by reference to that tautology alone. For example, tautology 1 in Theorem 3.4 determines the rule p p p p 2, 4 t 1, 5 t 3,61 4,7cli 81 from A and A --i B to infer B. This is called the rule of detachment or modus ponens. In a textbook devoted to logic, names for many rules of inference of this sort will be found. Probably modus ponens is the one used most frequently in derivations. 186 EXERCISES Logic I Char. 4 Note: It is intended that the restrictions described prior to Example 4.2 shall apply to applications of rule 1. 4.1. By an examination of Table VIII in Example 4.1, justify the conclusions drawn in that example. 4.2. Complete each of the following demonstrations of consequence by supplying the tautologies employed and the numbering scheme discussed in Example 4.2. (a) A --> B, --1(B V C) l= --,A (c) (A A B) V (C A D), A->11 -,(BVC) -,11A-,C -,11 A A C D->AVC D AVC A->B C->B B DrB A---> -,AA A A A A ---> (C --> B), -, D V A, CAD --'B -,DVA 1) A A-->(C-*B) C B B 4.3. Justify each of the following, using only rules p and 1. (a) -,AV B,C,->--, B A E ->F1 AF. (d) A->(BAC),-,BVD,(E->-,F)->-,D,B->(AA -,F.) K B-->E. -,A. (e) (A --> B) A (C -> D), (B --> E) A (D --> F), -, (E A F), A --> C 4.4. Try to shorten your proofs of Exercise 4.3(a), (b), (c), (d) using rule cp (along with rules p and t). 4.5. Can the rule of conditional proof be used to advantage in Exercise 4.3(e |
)? Justify your answer. 4.5! The Statement Calculus. Applications 187 5. The Statement Calculus. Applications We now turn to some household applications of the theory of inference which we have discussed. Usually the circumstances accompanying the presentation of an argument include the audience having the privilege of accepting or rejecting the contention that some statement, A,,,. In this event, the man B is a consequence of statements A,, A2, who thinks for himself will want to prove either that B is a consequence of the A's or that the argument is invalid, that is, that there can be made an assignment of truth values to the prime components at hand such that simultaneously each A receives value T, and B receives value F. The most expedient way to cope with the entire matter is this: Assume that B has value F and that each A has value T, and analyze the consequences so far as necessary assignments of truth values to prime components are concerned. Such an analysis will lead to either a contradiction, which proves that B is a consequence of the A's, or an assignment to each prime component such that all assumptions are satisfied, which proves that the argument is invalid. The foregoing method for proving that some formula is a consequence of others undercuts that promoted in the preceding section since it proceeds so quickly. However, the earlier method has (at least pedagogical) merits. For example, it leads to an acquaintance with the tautologies in Theorem 3.4. Instances of these are commonplace in proofs in mathematics, and the reader should learn to recognize them as such. As an illustration, tautology 20 justifies the familiar conclusion that if the contrapositive, - Q -a -i P, of P -+ Q is a consequence of A, then so is P--> Q. EXAMPLES 5.1. Consider the following argument. If 1 go to my first class tomorrow, then I must get up early, and if I go to the dance tonight, I will stay up late. If I stay up late and get up early, then I will be forced to exist on only five hours of sleep. I simply cannot exist on only five hours of sleep. Therefore, I must either miss my first class tomorrow or not go to the dance. To investigate the validity of this argument, we symbolize it using letters for prime statements. Let C be "I (will) go to my first class tomorrow," G be "I must get up early," D be "I (will) go |
to the dance tonight," S be "I will stay 188 Logic I CHAP. 4 up late," and E be "I can exist on five hours of sleep." Then the premises may be symbolized as (C - C) A (1) - S), SAC ->I, -, E, and the desired conclusion as ACV -iD. Following the method of analysis suggested above, we assume that -, C V -i D has value F and that each premise has value T. Then each of C and D must have value T. Further, according to the first premise, both C and S have value T. This and the second premise imply that E has value T. But this contradicts the assumption that the third premise has value T. Thus we have proved that -,C V -1D is a consequence of the premises. 5.2. Suppose it is asserted that A-'B,C-->D,AVCt= BAD. Assume that B A D has value F and each premise has value T. The first assumption is satisfied if T is assigned to B and F is assigned to D. Then C has value F, and A has value T. With these assignments, each premise receives value T, and B A D takes value F. Hence the argument is invalid. Related to the foregoing, but distinct from it, is the question of the satisfiability of a set of statements which is proposed as the set of prem- -, Am } of statements is satisfiable ises for an inference. A set { A,, A2, (within the statement calculus) iff there exists at least one assignment of truth values for the prime components such that the A's simultaneously receive value T. It is clear that {A,, A2,, Am} is satisfiable if - A A. is T for at least one combination of truth-value asAt A A2 A signments to the prime components and is not satisfiable if A, A A2 A A Am is F for all combinations of truth-value assignments to the prune components. The nonsatisfiability of a set of statements can be established within the framework of the methods described in the preceding section as soon as the following dclinition is made. A contradiction is a formula which always takes the value F (for example, A A --,A). THEOREM 5.1. A set { A,, A2, -, Am } of statements is not satisfiable if a contradiction can be derived |
as a consequence of the set. Proof. Assume that A,, A2,, A,,, 1= B A -, B for some formula B. A A. -, B n -, B, and the conclusion follows Then A, A A2 A from the truth table for the conditional. 4.5 I The Statement Calculus. Applications 189 Contradictions also play an important role in the method of indirect proof (also called proof by contradiction or reductio ad absurdum proof). The basis for this type of proof is the following result., THEOREM 5.2. A,, A2, K B if a contradiction can be, A,,, and -, B. derived as a consequence of A,, A2, Proof. Assume that A,, A2, A -, C for some formula C. Then A,, A2, A -, C. Consider now an assignment of values to the prim; components at hand such that every A receives value T. Then -, B ---3 C A -, C has value T. This and the fact that C A -, (, receives value F imply that -, B has value F and hence that B has value T.,,. EXAMPLES 5.3. We illustrate the usefulness of Theorem 5.1 in proving the nonsatisliiability of a set of statements. Such a proof follows the same pattern as one devised to establish the correctness of an argument in all but one respect: in a proof of the correctness of an argument the final line, which is the conclusion, is assigned in advance, whereas, in a proof of nonsatisliability the final line is any contradiction. For example, suppose that it is a question of the satisfiability of a set of statements which may be symbolized as AFiB, B --*C, - - 1 We adopt these as a set of premises and investigate what inferences can be made. {1} {2) {3} {4} {5} (4,5) {4, 5} (1,2) 11, 2, 4, 5} {3, 51 {1, 2,3, 4, 5} D (1) A <-> I. p (2) B -C p (3) -, C V 1) p (4) --,A --' D p (5) P (6) -, -, A 4, 5t ( |
7) A 6t (8) A --I C 1,21 7,81 (9) C (10) -,C 3, 5t (11) CA -,C 9,101 We conclude that the set is not satisfiable. 5.4. We could introduce a further rule of inference based on Theorem 5.2. Alternatively, we may employ the rule of conditional proof and the tautology r- (-, B --+ C A -,C) -b B to justify an indirect proof. As an illustration, we rework Example 5.1 in this section, starting with the negation of the desired conclusion as an additional premise. 190 Logic i CHAP. 4 p p p p (1) (C-'G) A (D --'S) (2) SAC-,E (3)-,E (4) -,(ACV -,D) (5) C A D (6) C (7) C--'C (8) G (9) D - S (10) D (11) S (12) S A G (13) E (14) Ii A -,E (15) -, (-,(; V -,D) - E A -1E (16) --j C V --,D It is left as an exercise to supply the missing details. EXERCISES Use the method discussed in this section to prove the validity or invalidity, whichever the case might be, of the arguments in Exercises 5.1-5.12 below. For those which are valid, construct a formal proof. In every case use the letters suggested for symbolizing the argument. 5.1. Either I shall go home or stay and have a drink. I shall not go home. Therefore I shall stay and have a drink. (II, S) 5.2. If John stays up late tonight, he will be dull tomorrow. If he doesn't stay up late tonight, then he will feel that life is not worth living. Therefore, either John will be dull tomorrow or he will feel that life is not worth living. (S, D, L) 5.3. Wages will increase only if there i3 inflation. If there is inflation, then the cost of living will increase. Wages will increase. 'Therefore, the cost of living will increase. (W, I, C) 5.4. If 2 is a prime, then it is |
the least prime. If 2 is the least prime, then I is not a prime. The number 1 is not a prime. Therefore, 2 is a prime. (P, L, N) 5.5. Either John is exhausted or he is sick. If he is exhausted, then he is con- trary. Ile is not contrary. Therefore, he is sick. (E, S, C) 5.6. If it is cold tomorrow, I'll wear my heavy coat if the sleeve is mended. It will be cold tomorrow, and that sleeve will not be mended. Therefore, I'll not wear my heavy coat. (C, H, S) 5.7. If the races are fixed or the gambling houses are crooked, then the tourist trade will decline, and the town will suffer. If the tourist trade decreases, then the police force will be happy. The police force is never happy. Therefore, the races are not fixed. (R, H, D, S, P) 4.5 1 The Statement Calculus. Applications 191 5.8. If the Dodgers win, then Los Angeles will celebrate, and if the White Sox win, Chicago will celebrate. Either the Dodgers will win or the White Sox will win. However, if the Dodgers win, then Chicago will not celebrate, and if the White Sox win, Los Angeles will not celebrate. So, Chicago will celebrate if and only if Los Angeles does not celebrate. (D, L, W, C) 5.9. Either Sally and Bob are the same age or Sally is older than Bob. If Sally and Bob are the same age, then Nancy and Bob are not the same age. If Sally is older than Bob, then Bob is older than Walter. Therefore, either -Nancy and Bob are not the same age or Bob is older than Walter. (S, 0, N, W) 5.10. If 6 is a composite number, then 12 is a composite number. If 12 is a composite number, then there exists a prime greater than 12. If there exists a t'rirne greater than 12, then there exists a composite number greater than 12. If 2 divides 6, then 6 is a composite number. The number 12 is composite. Therefore, 6 is a composite number. (S, W, P, G, D) 5.11. If I take the bus, and the bus is late, I'll miss my appointment. If I miss my appointment and start |
to feel downcast, then I should not go home. If I don't get that job, then I'll start to feel downcast and should go home. Therefore, if l. take the bus, and the bus is late, I will get that joh. (B, 1., M, D, FI, J) 5.12. If Smith wins the nomination, he will be happy, and if he is happy, he is not a good campaigner. But if lie loses the nomination, lie will lose the confidence of the party. lie is not a good campaigner if he loses the confidence of the party. If lie is not a good campaigner, then he should resign from the party. Either Smith wins the nomination or he loses it. Therefore, he should resign from the party. (N, H, C, P, R) 5.13. Investigate the following sets of premises for satisfiability. If you conclude that a set is not satisfiable by assigning truth values, then reaffirm this using Theorem 5.1 and vice versa. Substantiate each assertion of the satisfiability of a set of premises by suitable truth-value assignment,, (a) A -, (B A C) DV E- C C-.-> -(IIV I) -,CAEAI1 (b) AV B -'CAD DVE -'C AV --,G (c) (A --> B) A (C --), D) D) A (-1C--rA) (B (E -4G) A (G-'-, D) -,I- Z (d) (A-aBAC)A(D--), BAE) (G-a--,A)AH--i1 (1I--'I)-'GAD -,(-,C--GIs) (e) The contract is fulfilled if and only if the house is completed in February. If the house is completed in February, then we can move in March 1. If we can't move in March 1, then we must pay rent for March. If the contract is not fulfilled, then we must pay rent for March. We will not pay rent for March. (C, H, M, R) 1 92 Logic I CHAP. 4 5.14. Give an indirect proof of the validity of the argument in the following. (a) Example 4.3. (b) Example 4.4. (c) Example 5. |
1. 5.15. Prove that if A, -1B Ir- C (a contradiction), then A I-= B. (d) Exercise 5.7. (e) Exercise 5.11. (f) Exercise 5.12. 6. The Predicate Calculus. Symbolizing Everyday Language The theory of inference supplied by the statement calculus is quite inadequate for mathematics and, indeed, for everyday arguments. For example, from the premises every rational number is a real number, 3 is a rational number, certainly 3 is a real number is justified as a conclusion. Yet the validity of this argument cannot be established within the context of the statement calculus. The reason is that the statement calculus is limited to the structure of sentences in terms of component sentences, and the above inference requires an analysis of sentence structure along the sul)ject--predicate lines that grammarians describe. In other words, the statement calculus does not break down a sentence into sufficiently "fine" constituents for most purposes. On the other hand, with the addition of three additional logical notions, called terms, predicates, and quantifiers, it has been found that much of everyday and mathematical language can be symbolized in such a way as to make possible an analysis of an argument. We shall describe these three notions in turn. It is standard practice in mathematics to introduce letters such as "x" and "y" to reserve a place for names of individual objects. For example, in order to determine those real numbers such that the square of the number minus the number is equal to twelve, one will form the equation x2 - x = 12, thereby regarding "x" as a placcholder for the name of any such (initially unknown) number. Again, as it is normally understood, the "x" in such an equation as sine x + cost x = 1 reserves a place for the name of any real or, indeed, complex number. As it is employed in "x2 - x = 12," one is accustomed to calling "x" 4.6 I 77ie Predicate Calculus. Symbolizing Everyday Language 193 an unknown, and in "sing x + cos2 x = 1" one is likely to refer to "x" as a variable. The usage we shall make of letters from the latter part of the alphabet in symbolizing everyday language shall be like that just described- that is, as an unknown or a variable. In logic it is customary to employ the word "variable" for either usage; |
the decision as to whether "x" is intended to be a variable in the intuitive sense or an unknown is made on the basis of the form of the expression in which it appears. Since, ultimately, we intend to strip all symbols of any meaning whatsoever, it is simplest to do this at the outset for variables. This we do by defining an individual variable to be a letter or a letter with a subscript or superscript. Variables constitute one class of terms. We shall also find use for letters and symbols as names of specific, well-defined objects; that is, we shall use letters and symbols for proper names. Letters and symbols used for this purpose are called individual constants. For example, "3" is an individual constant, being a name of the numeral 3. Again, "Winston Churchill" is an individual constant. In order to achieve a compact notation we shall use a letter from the beginning of the alphabet to stand for a proper name if there is no accepted symbol for it. For example, we might let if we intend to translate the sentence a = Winston Churchill Winston Churchill was a great statesman into symbolic form. Proper names are often rendered by a "description," which we take to be a name that by its own structure unequivocally identifies the object of which it is a name. For example, the first president of the United States and the real number x such that for all real numbers y, xy = y are descriptions. If we let then we may write b = George Washington, b = the first president of the United States. Further, we have 1 = the real number x such that for all y, xy = y. Collectively, individual variables and individual constants (either in 194 Logic I CIIAP. 4 the form of proper names or descriptions) are classified as terms. The grammatical function of variables is similar to that of pronouns and common nouns in everyday language, and the function of individual constants is similar to the role of proper nouns. We now turn to the notion of predicates. In grammar a predicate is the word or words in a sentence which express what is said of the subject; for example, "is a real number," "is black," "is envious." In logic the word "predicate" has a broader role than it has in grammar. The basis for this is the observation that if a predicate is supplemented by including a variable as a placeholder for the intended subject (for example, "x is a real number"), the result behaves as |
a "statement function" in the sense that for each value of x (from an appropriate domain) a statement results. Although "John loves" is not a predicate in grammar, if "x" is introduced as a placeholder for the object (of John's affections), which yields John loves x, the result is a statement function in the sense just described. An obvious generalization is at hand, namely, the extension to statement functions of more than one variable. Examples are x is less than y, x divides y, z is the sum of x and y. The upshot is the notion of an n-place predicate P(x,, x2, as an expression having the quality that on an assignment of values to the -, x from appropriate domains, a statement results. variables x,, x2, For convenience we include 0 as a value of n, understanding by a 0-place predicate a statement., We now consider some examples of translations into symbolic form. EXAMPLES 6.1. The sentence (1) Every rational number is a real number may be translated as (2) For every x, if x is a rational number, then x is a real number. In ordinary grammar, "is a real number" is the predicate of (1). In the translation (2) the added predicate "x is a rational number" replaces the common noun "rational number." Using "Q(x)" for "x is a rational number" and "R(x)" for "x is a real number," we may symbolize (2) as 4.6 I The Predicate Calculus. Symbolizing Everyday Language 195 (3) Further, the statement "3 is a rational number" may be symbolized by For every x, Q(x) - R(x). (4) Q (3). In terms of symbolism available at the moment, (3) and (4) are the translations of the premises of the argument appearing at the beginning of this section. 6.2. The sentence we translate as Some real numbers are rational For some x, x is a real number and x is a rational number. Using the predicates introduced above, this may be symbolized as (5) For some x, R(x) A Q(x). 6.3. The sentence (6) For some x, R(x) --> Q(x) should have the same meaning as For some x, -, R(x) V Q(x), (7 |
) since we have merely replaced "R(x) -- Q(x)" by its equivalent "-y R(x) V Q(x)." Now (7) may be translated into words as There is something which is either not a real number or is a rational number. Certainly, this statement [which has the same meaning as (6)] does not have the same meaning as (5). Indeed, as soon as we exhibit an object which is riot a real number we must subscribe to (6). In summary, (6) and (5) have different meanings. By assumption, on suitable assignments of values to the variables in a predicate, a statement results. For example, if S(x) is "x is a sophomore," this predicate yields the statement "John is a sophomore." A statement may also be obtained from S(x) by prefixing it with the phrase "for every x": (8) For every x, x is a sophomore. No doubt, one would choose to rephrase this as (9) Everyone is a sophomore. The phrase "for every x" is called a universal quantifier. We regard "for every x," "for all x," and "for each x" as having the same meaning and symbolize each by (bx) or (x). 196 Logic I CrrAP. 4 Using this symbol we may symbolize (8) or (9) as (x)SW Similarly, prefixing S(x) with the phrase "there exists an x (such that)" yields a statement which has the same meaning as "There are sophomores." The phrase "there exists an x" is called an existential quantifier. We regard "there exists an x," "for some x," and "for at least one x" as having the same meaning, and symbolize each by (3x). Thus, "(3x)S(x)" is the symbolic form of "There are sophomores." In each of Examples 6.1 6.3 above a quantifier prefixes not merely a predicate but a "formula in x," by which we shall understand for the time being an expression compounded from one-place predicates using sentential connectives. Using the symbol introduced P(x), for the universal quantifier, we can now render "Every rational number is a real number" in its final form:. (x)(Q(x) -* R(x)). (10) Possibly it has already occurred to the reader that |
this means simply that Q C R. Indeed, if one recalls the definition of the inclusion relation for sets, it becomes clear that (10) is an instance of that definition. Further, we note that (10) is characteristic of statements of the form "Every so and so is a such and such." Similarly, the sentence "Some real numbers are rational" may be (3x)(R(x) A Q(x)). translated as (11) The meaning of this sentence is simply that R f1 Q is nonempty; that is, it is a symmetrical form of the original sentence. A mistake commonly made by beginners is to infer, since a statement of the form "Every so and so is a such and such" can be symbolized as in (10), that. the statement "Some so and so is a such and such" can be symbolized by (3x)(R(x) -* Q(x)). However, as is pointed out in Exaniple 6.3, this should have the same meaning as (3x)( R(x) V Q(x)). This should be accepted as true as soon as we exhibit an object which is not a real number. In particular, therefore, it has no relation to what it is intended to say, namely, that some real numbers are rational. 4.6 I The Predicate Calculus. Symbolizing Everyday Language 197 EXAMPLES 6.4. The notion of a formula in x, as (vaguely) described above, is the same as that given in Chapter 1. There it was stated that such an expression is often called a property (of x). Associated with a property is a set, according to the intuitive principle of abstraction. Extending in the obvious way the notion of a formula in x to that of a formula in x and y, one can associate with a formula A(x, y) those ordered pairs (a, b) such that A(a, b) is true. That is, a formula in x and y may be used to define a binary relation. This being so, formulas in two variables are often called binary relations, those in three variables are called ternary relations, and so on. 6.5. If A(x) is a formula in x, consider the following four statements. (a) (x)A(x) (b) (3x)A(x). (c) (x)(-1A(x)) |
(d) (3x)(-A(x)). We might translate these into words as follows. (a) Everything has property A. (b) Something has property A. (c) Nothing has property A. (d) Something does not have property A. Now (d) is the denial of (a), and (c) is the denial of (b), on the basis of everyday meaning. Thus, for example, the existential quantifier may be defined in terms of the universal quantifier by agreeing that "(3.x)A(x)" is an abbreviation for « --I (x) -, (A (x)). " 6.6. Traditional logic emphasized four basic types of statements involving quantifiers. Illustrations of these along with translations appear below. Two of these translations have been discussed. All rationals are reals. No rationals are reals. Some rationals are reals. Some rationals are not reals. (x)(Q(x) -' R(x)) (x)(Q(x) --+ n R(x)). (3x)(Q(x) A R(x)). (3x)(Q(x) A -R(x)). 6.7. If the symbols for negation and a quantifier modify a formula, the order in which they appear is relevant. For example, the translation of -I(x)(x is mortal) is "Not everyone is mortal" or "Someone is immortal," whereas the translation of (x)(--I (x is mortal)) is "Everyone is immortal." 6.8. By prefixing a formula in several variables with a quantifier (of either 198 Logic I CHAP. 4 kind) for each variable, a statement results. For example, if it is understood that all variables are restricted to the set of real numbers, then (x)(Y)(z)((x + y) + z = x + (y + Z)) is the statement to the effect that addition is an associative operation. Again, (x) (3y) (x2 - y = y2 - x) translates into "For every (real number) x there is a (real number) y such that x2 - y = y2 - x." This is a true statement. Notice, however, that (3y) (x) (x2 - y = y2 - x), obtained from the foregoing by interchanging the quantifiers, is a different |
indeed, a false- statement. 6.9. We supplement the first remark in the preceding example with the observation that a formula in several variables can also be reduced to a statement by substituting values for all occurrences of some variables and applying quantifiers which pertain to the remaining variables. For example, the (false) statement (x) (x < 3) results from the 2-place predicate "x < y" by substituting a value for y and quantifying x. We conclude this section with the remark that there are no mechanical rules for translating sentences from English into the logical notation which has been introduced. In every case one must first decide on the meaning of the English sentence and then attempt to convey that same meaning in terms of predicates, quantifiers, and, possibly, individual constants. Beginning with the exercises below we shall often omit parentheses when writing predicates. For example, in place of "A(x)" we shall write "Ax," and "A (x, y)" will be written simply as "Axy." EXERCISES 6.1. Let Px be "x is a prime," Ex be "x is even," Ox be "x is odd," and Dxy be "x divides y." Translate each of the following into English. (a) P7. (b) E2 A P2. (x) (D2x -' Ex). (c) (d) (3x) (Ex A Dx6). (i) (e) (x)(--,Ex -4 --i D2x). (x)(Ex -' (y)(Dxy -Ey)). (f) (g) (x)(Px -y (3y) (Ey A Dxy)). (h) (x)(Ox - (y)(Py --i -,Dxy)) (3x) (Ex A Px) A-,(3x)((Ex A Px) A (3y) (x 0 y A Ey A Py)). 6.2. Below are twenty sentences in English followed by the same number of 4.6 I The Predicate Calculus. Symbolizing Everyday Language 199 sentences in symbolic form. Try to pair the members of the two sets in such a way that each member of a pair is a translation of the other member of the pair. (a) (b) (c) (d) (e) (f) (g) (h) (i) |
(j) (k) (1) (m) (n) (0) (p) (q) (r) (s) (t) All judges are lawyers. (Jx, Lx) Some lawyers are shysters. (Sx) No judge is a shyster Some judges are old but vigorous. (Ox, Vx) Judge Jones is neither old nor vigorous. (j) Not all lawyers are judges. Some lawyers who are politicians are Congressmen. (Px, Cx) No Congressman is not vigorous. All Congressmen who are old are lawyers. Some women are both lawyers and Congressmen. (Wx) No woman is both a politician and a housewife. (lIx) There are some women lawyers who are housewives. All women who are lawyers admire some judge. (Axy) Some lawyers admire only judges. Some lawyers admire women. Some shysters admire no lawyer. Judge Jones does not admire any shyster. There are both lawyers and shysters who admire Judge Jones. Only judges admire judges. All judges admire only judges. (a)' (3x)(Wx A Cx A Lx). (b)' -, Oj A -, Vj. (c)' (x)(Jx -- -,Sx). (d)' (3x)(Wx A Lx A Hx). (e)' (x) (Ajx --r -m Sx). (f)' (x) (Jx -4 Lx). (g)' -, (x) (Lx -* Jx). (h)' (x)(Cx A Ox --+ Lx). (i)' Ox) (Lx A Sx). (j)' (3x)(Lx A Px A Cx). (k)' (x) (Wx --> -, (Px A IIx) ). (x) (Cx - Vx). (1)' (m)' (3x) (Jx A Ox A Vx). (n)' (x)(y)(Ayx A Jx ---'.Iy). (o)' (3x)(Sx A (y)(Axy -* -,Ly)) (p)' (3x) (3y) (Lx A Sy A AV A Ayj). (q)' (x) (Wx A Lx - (3y) (Jy A Axy)). ( |
r)' (3x) (Lx A (3y) (Wy A Axy) ) (x)(Jx --+ (y)(Axy - Jy)). (s)' (t)' *(3x) (Lx A (y) (Axy --4 Jy)). 200 Logic I afire['. 4 7. The Predicate Calculus. A Formulation The examples and exercises of the preceding section serve to substantiate the contention that if the sentential conncctives are supplemented with predicates and quantifiers, much of everyday language can be symbolized accurately. Predicate calculus is concerned with a theory of inference based on the structure of sentences in terms of connectives, predicates, and quantifiers. In particular, therefore, it is an extension of the statement calculus. The type we shall discuss admits of quantification only of individual variables. To distinguish this simple type from others, it is usually called restricted predicate calculus or predicate calculus of first order. Incidentally, it is not our intention to develop the restricted predicate calculus to the same degree of completeness as we did the statement calculus. Rather, we shall merely formulate it and sketch how it might be developed and applied. A formulation which is comparable to that of the statement calculus in Section 3 is our starting point. We assume that for each of n = 0, 1, 2, there is given an unspecified number of n-place predicates (or, statement functions of n variables). 'These we shall denote by such symbols as P(x, y) (to stand for some one 2-place predicate), P(x, y, z) (to stand for some one 3-place predicate which would necessarily represent a predicate different from that symbolized by P(x, y), being a function of a different number of variables), Q(x, y, z) (to stand for another 3-place predicate), R (to stand for some one 0-place predicate, that is, a statement), and so on. It is assumed that the set of all n-place predicates for n = 1, 2, is nonempty. Henceforth we shall call the given predicates predicate letters. From the given set of predicate letters we generate those expressions which we shall call "formulas (of the predicate calculus)." A prime formula is an expression resulting from a predicate letter by the substitution of any variables, not necessarily distinct, for those variables which appear in the predicate letter. For example, some |
of the prime formulas which the predicate letter P(x, y, z) yields are P(x, y, z), P(x, y, y), P(y, x, x), and I'(tt, u, u). We extend the set of all prime formulas by adjoining all those expressions which can be formed by using, repeatedly and in all possible ways, the sentential connectives and quantifiers. Precisely, we extend the set of all prime formulas to the smallest set such that each of the following holds. If A and B are members of the set, then so are -1(A), (A) A (B), (A) V (B), (A) -* (B), and (A) H (B). Also, 4.7 I The Predicate Calculus. A Formulation 201 if A is a member of the set and x is a variable, then (x)A and (3x)A are members of the set. The members of this extended set are called formulas. Those which are not prime formulas are called composite formulas. Parentheses are inserted automatically in a formula, but in some cases arc unnecessary. (Indeed, the sole purpose of such lavish use of parentheses is to make possible the formulation of a mechanical procedurc for demonstrating that some juxtaposition of symbols is a formula.) In other cases parentheses can be omitted by the same conventions established earlier. We extend those conventions by agreeing that quantifiers, along with -,, have the least possible scope. For example, (3x)A V B stands for ((3x)(A)) V (B). The foregoing description is vague only with respect to the nature of a predicate letter. From the standpoint of the theory of the firstorder predicate calculus, the nature of predicate letters is irrelevant, for there they are treated in a purely formal way, that is, simply as certain strings of letters, parentheses, and commas. From the standpoint of the applications, the vagueness is deliberate, for thereby versatility is achieved. The examples which follow may serve to substantiate this assertion. Each example describes the initial steps which one might take in axiomatizing a mathematical theory. EXAMPLES 7.1. Suppose that a practitioner of the axiomatic method were to set out to reconstruct the set theory of Chapter 1 as an axiomatic theory. After analyzing how that subject matter was developed, he might conclude that all concepts stemmed from the membership relation --that is |
, the 2-place predicate "is a member of." This might motivate the practitioner to set up a system of the type introduced above, one having a single predicate letter C(x, y) intended to denote the membership relation. Of course, the intended interpretation of individual variables would be as sets. The prime formulas of the system would consist of all expressions of the form C(x, y) or, using more suggestive notation, x C y. Then, for convenience, further predicates could be introduced by definition. Following are some instances: xtZyfor -,(xCy), xcyfor (a)(aCx-aCy), x= y for (x _C y) A (y c x), x0yfor -i(x=y), xCyfor(x9y)A(xg-y). The next step would be the adoption of certain formulas as axioms. 202 Logic I CHAP. 4 geometry are "points," "lines," and the relation of incidence, " 7.2. As every high school student knows, the basic ingredients of elementary lies on _." In formulating an axiomatic theory intended to have intuitive geometry as an interpretation, one might choose as primitive terms a list of individual variables (intended to range over points and lines), two 1-place predicate letters, P(x) and L(x), and one 2-place predicate letter, I(x, y). These might be read, in turn, "x is a point," "x is a line," and "x is on y." Among the axioms might appear the following: (3x)P(x), (3x)L (x), (x) (y) (AX, y) <- -' I(.y, X)), (x)(P(x) -, (3y)(L(y) A I(x, y))) 7.3. As the first step in axiomatizing the theory of partially ordered sets as described in Chapter 1, one might introduce as the primitive terms a list of individual variables and two 2-place predicate letters, = (x, y) and < (x, y). Then the prime formulas would consist of all expressions of the form x = y and x < y, using more familiar notation. As nonlogical axioms for the theory (that is, those axioms which serve as a basis for the intended mathematical structure |
), we might then take (x) (x = x), (x) (y) (x = y - y = x), (x) (y) (z) () (which mean that = is an equivalence relation), (x) (y) (z) (x = y A x < z --b y < z), (x) (y) (z) (x = y A z < x -- z < y) (which assert that "equals may be substituted for equals"), and, finally, (x) -, (x < x), (x) (y) (z) () (which establishes < as an ordering relation). As part of the formulation of the predicate calculus there must be introduced definitions for distinguishing between the circumstances in which a variable is intended to play the role of a variable or an unknown in the intuitive sense. As a preliminary to this we define the scope of a quantifier occurring in a formula as the formula to which the quantifier applies. A possible ambiguity is removed by use of parentheses. Below are several examples illustrating the scope of the quantifier "(x)," in which the scope is indicated by the line underneath: (x) P(x) A Q (x), (z) Q (z) ), (3y) (x) (P(x, y) (x) (y) (P(x, y) A Q(y, z)) A (3x)P(x, y), (x)(P(x) A (3x)Q(x,z) -* (3y)R(x,y)) v Q(x,y) 4.7 I The Predicate Calculus. A Formulation 203 It is now possible to give the key definitions in connection with the matter at hand. An occurrence of a variable in a formula is bound if this occurrence is within the scope of a quantifier employing that variable or is the explicit occurrence in that quantifier. An occurrence of a variable is free ill this occurrence of the variable is not bound. For example, in (x)P(x, y) both occurrences of x are bound, and the single occurrence of y is free. Again, in the formula (3y) (x) (P(x, y) -* (z) Q W) each occurrence of every variable is bound. A variable is free in a formula if at least one occurrence of it is free, and a variable is |
bound in a formula if at least one occurrence of it is bound. A variable may be both free and bound in a formula. This is true of z in the formula (z)(P(z) A (3x) Q(x, z) -a (3y) R(z, y)) V Q(z, x). If a variable is free in a formula, then, on an assignment of meaning to the predicates involved, that variable behaves as an unknown in the familiar sense, since the formula becomes a statement about that variable. The formulas x < 7 and (3y)(y < x), in each of which x is free, serve to illustrate this point. The formula (3y) (y < x) A (x) (x > 0), wherein the first occurrence of x is free and the other two are bound, illustrates the remark that insofar as meaning is concerned, the free and bound occurrences of the same variable in the same formula have nothing to do with each other. Indeed, the formula (x) (x > 0) is simply a statement and has the same meaning as (u)(u > 0) and (W) (W > 0). In bound occurrences in a formula a variable behaves like a variable in the intuitive sense. For example, in (x) (x 2 - 1 = (x - 1)(X + 1) ) all occurrences of x are bound and, clearly, x serves as a variable. That x in the formula (3x) (y ; x) 204 Logic I CHAP. 4 serves as a variable is made more plausible on recalling that this formula has the same meaning as (x) --I (Y 7,!5 X) In conclusion, we note that it is now possible to give a precise definition of the word "statement." A statement is a formula which has no free variables. EXERCISES 7.1. List the bound and the free occurrences of each variable in each of the following formulas. (a) (x)P(x). (b) (x)P(x) --' P(J') (c) P(x) - (3x)Q(x) 7.2. Using the letters indicated for predicates, and whatever symbols of arithmetic (for example, "+" and "<") may be needed, translate the following. (3x)(y)(P(x) A Q(y)) --> (x)R(x). |
(3x)(3y)(1'(x, y) A Q(z)). (d) (3x)A(x) A 11(x). (e) (f) (a) If the product of a finite number of factors is equal to zero, then at least one of the factors is equal to zero. (Px for "x is a product of a finite number of factors," and Fxy for "x is a factor of y.") (b) Every common divisor of a and b divides their greatest common divisor. (Fxy for "x is a factor of y," and Gxyz for "z is the greatest common divisor of x and y.") (c) For each real number x there is a larger real number y. (Rx) (d) There exist real numbers x, y, and z such that the sum of x and y is greater than the product of x and z. (c) For every real number x there exists a y such that for every z, if the sum of z and 1 is less than y, then the sum of x and 2 is less than 4. 7.3. An abelian group may be defined as a (noncmpty) set A together with a binary operation -l- in A which is associative, commutative, and such that for given x and y in A the equation x -l- z = y always possesses a solution z in A. A familiar example is that of L_ with ordinary addition as the operation. A formulation within the predicate calculus can he given by taking as primi(x, y), and a 3-place tive terms a list of variables, a 2-place predicate letter predicate letter S(x, y, z). The prime formula x = y is read "x equals y," and the prime formula S(x, y, z) is read "z is the sum of x and y." As axioms we take th.e following formulas. (x)(x = x)(x) (y) (x = y --> y = X). (x) ()) (z) ().. 4.8 I The Predicate Calculus. Validity 205 (u) (v) (w) (x) (y) (z) (S(u, v, w -+ S(x, y, z) ). (x) (y) (3z) |
S(x, y, z) (x) (y) (z) (w) (S(x, y, z) A S(x, y, w) - + z = w). (u) (v) (w) (x) (y) (z) (S(u, v, w) A S(w, x, y) A S(v, x, z) --3 S(u, z, y)). (x) (y) (z) (S(x, y, z) --, S(.Y, x, z)) (x) (y) (3z)S(x, z, y) Write a paragraph in support of the contention that, collectively, these axioms do serve to define abelian groups. 8. The Predicate Calculus. Validity The system described in the preceding section is essentially the common starting point in the formulation of various predicate calculi. Distinguishing features of the classical predicate calculus (which is our concern) include further assumptions which extend the one assumption made in Section 3 for the statement calculus, namely, that with each prime formula there is associated exactly one of T and F. The corresponding assumption about a prime formula in the sense of the predicate calculus is much more complicated. We shall introduce it in several steps. First, it is assumed that with the system described in the preceding section there is associated a nonempty set D, called the domain, such that each individual variable ranges over D. Further, it is assumed that with each n-place predicate letter there is associated a logical function, that is, a function on D" into IT, F}. (For 0-place predicates the associated function is assumed to be a constant, one of T or F.) Finally, it is assumed that a truth-value assignment to a prime formula, y") can be made, relative to an assignment of an element P(yt, y23, y", in the following way. in D to each distinct variable among yl, y2, If toy; is assigned d; in D and if to the predicate letter P(xt, x2, is assigned X: D" --*- IT, F }, then the truth value of 1'(yt, y2, is, d"). For example, if P(x, y, x) is the prime formula and X A(dt, r12, is assigned to P(x |
, y, z), then the truth value of P(x, y, x), relative to the assignment of a to x and b toy, is A(a, b, a).,, For the theory of the statement calculus, that one of T and F which is assigned to a prime formula is assumed to be irrelevant. In the predicate calculus this is extended to the assumption that the theory is independent of the domain D and the assignment of functions to predicate letters. 206 Logic I CHAP P. 4 The foregoing is the basis of the valuation procedure for a formula C of the predicate calculus. For this it is assumed that (i) a domain D is given, (ii) a function is assigned to each predicate letter appearing in C, and (iii) to each distinct free variable in C is assigned a value in D. Collectively, these constitute an assignment to C. A truth value is assigned to C by a procedure which parallels the formation of C. (I) If P(y1j Y2, P(x1, x2, P(Y1,Y2,...,yn) is X(d1, d2,..., dn)., yn) is a prime formula in C and A is assigned to, xn) and d= is assigned to ys, then the truth value of (II) For a given assignment of values to the predicate letters and free variables of --,A, the value of --,A is F if the value of A is T, and the value of -, A is T if the value of A is F. Similarly, for a given assignment of values to the predicate letters and free variables of A V B, A A B, A --+ B, and A <--, B, the truth tables from the statement calculus apply. (III) For a given assignment of values to the predicate letters and free variables of (x)A, the value of (x)A is T if the value of A is T for every assignment to x, and the value of (x)A is F if the value of A is F for at least one assignment to x. For'a given assignment of values to the predicate letters and free variables of (3x)A, the value of (3x)A is T if the value of A is T for at least one value of x, and otherwise it is F. As an illustration, we consider the problem of the assignment of truth values to |
the formula (x) (P(x) --+ Q) V (Q A P(Y)) Although the domain D is fixed, it is unknown. Suppose D = {a, b}. By assumption there is associated with P(x) a logical function on D into IT, F} and with Q a truth value. Further, the free variable y may assume any value in D. The possible logical functions which may be associated with P(x) are tabulated( here: x I X1(x) X2(x) A3(x) X4(x The possible values which may be associated with Q are T and F, and to y may be assigned the value a or b. Thus, we may fill out a table 2) entries exhibiting the truth-value assignment in all with 16(= 4 2 possible cases: 81 ) Qy ( TA T F T T T T T 'T a - sa sT fT 208 Logic I CHAP. 4 predicate calculus, we have introduced the symbols to be employed, given the definition of a formula, and described a valuation procedure. We imitate the next step in the earlier theory by defining validity in the predicate calculus. A formula is valid in a given domain iff it takes the value T for every assignment to the predicate letters and free variables in it. A formula is valid if it is valid in every domain. For "A is valid" we shall write K A. It is appropriate to use the same terminology and symbolism as before, since this definition of validity is an extension of the earlier one. It is obvious that to establish the validity of a formula, truth tables must give way to reasoning processes. On the other hand, to establish nonvalidity, just one D and one assignment based on this domain will suffice. For example, the fourth line of the above table demonstrates that the formula considered there is not valid. The case with which the validity of some formulas can be established may come as a surprise. EXAMPLES 8.1. Let us illustrate the assignment of functions to predicate letters in an application of the predicate calculus. Suppose that L is the domain and that we are told that 1'(x, y, z) is to be interpreted as "z is the sum of x and y." Then to this predicate letter we would assign the function X: Z' ->- IT, F} such that X (a, b, c) = T if a + b = c, and X ( |
a, b, c) = F otherwise. If, on the other hand, we are told that P(x, y, z) is to be interpreted as "z is the product of x and y," then we would define X(a, b, c) to be T if ab = c, and to be F otherwise. 8.2. We prove that l= (x)P(x) --> P(y). A prerequisite for the formula to take the value F is that. P(y) receive the value F for some assignment in some domain. But in that event, (x)1'(x) receives the value F. Hence, (x)P(x) -+ P(y) always receives the value T. 8.3. Let us prove that K P(y) - (3x)P(x). As in the preceding example, we need concern ourselves only with assignments in some domain D such that (3x)P(x) takes value F. This is the case if P(x) receives value F as x ranges over D. But then P(y) must receive the value F. 8.4. Let us establish the nonvaliclity of the formula (3x)P(x) -* (x)P(x). Let D contain at least two individuals, a and b. Assign to P(x) a logical function A such that X(a) = T and,\(b) = F. Then (3x)P(x) receives the value T and (x)P(x) receives the value F. Hence, the entire formula receives the value F. 8.5. A proof that (x)P(x) V (x)Q(x) -' (x)(P(x) V Q(x)) 4.8 I The Predicate Calculus. Validity 209 may be given as follows. Suppose that the consequent takes the value F for an assignment A1, A2, and a to P(x), Q(x), and x, respectively. Then, for this assignment, P(x) V Q(x) takes the value F. Hence, Ai(a) = F and A2(a) = F, from which it follows that (x)P(x) and (x)Q(x), and hence their disjunction, each take the value F. We turn |
now to the question of general methods for proving validity, looking first at what we can take over from the statement calculus. Theorem 3.2 (with "A eq B" now assigned a meaning in terms of our present valuation procedure) and Theorem 3.3 carry over unchanged. The proofs employ essentially the earlier reasoning. The substance of Theorem 3.1 is the possibility of proving validity of a formula without dissecting it into prime components. This same technique has applications in the predicate calculus. To proceed with our first illustration, let us call a formula of the predicate calculus prime for the statement calculus if no sentential connectives appear in it. In terms of the composition of a formula from such prime formulas we can introduce the notion of tautology into the predicate calculus. For example, P(x) -4 P(x) is a tautology, and we may recognize tautologies (for example, A --+ A) even when the prime formulas are not displayed. Clearly, a tautology is a valid formula. In particular, Theorem 3.4 holds for the predicate calculus. In order to illustrate further the technique under discussion some definitions are required. To substitute a variable y for a variable x in a formula A means to replace each free occurrence of x in A by an occurrence of y. If y is to be substituted for x in A, it is convenient to introduce a composite notation such as "A(x)" for the substituend and then write "A(y)" for the result of the substitution. Such notation as "A(x)" for the formula A is used solely to show the dependence of A on x and is not to be confused with the notation for predicate letters; indeed, we do not require that x actually occur free in A and do not exclude the possibility that A(x) may contain free variables other than x. In the future we shall often use such notations as "A(x)" or "A(x, y)" instead of "A" when we are interested in the dependence of A on a variable x or variables x and y, whether or not we plan to make a substitution. Let us consider an example. If A(x) is (x = 1) A (3y) (y $ x), (1) then, clearly, A(y) says something different about y than A(x) says about x. The reason is that the occurrence of x in (3y) (y x |
) is free, whereas an occurrence of y in the same position is bound. In everyday 210 Logic I c1AP. 4 mathematics we are not likely to make a substitution which changes the meaning of a formula. A safeguard against inappropriate substitutions in purely formal situations can be given. A formula A(x) is free for y if no free occurrence of x in A(x) is in the scope of a quantifier (y) or (3y). For example, if A(x) is P(x, Y) A (y)Q(y), then it is free for y, whereas if A(x) is (1), above, then it is not free for y. If substitutions for x in A(x) are restricted to variables y such that A(x) is free for y, difficulties of the sort mentioned are avoided. We turn now to Example 8.3, where we proved that K P(y) -, (3x)P(x) for a predicate letter P(x). Using the same reasoning we can prove that 1= A(y) --> (3x)A(x), where A(x) is any formula which is free for y. The computation of the value of the formula at hand for a given assignment consists of (i) the computation of a value of the logical function assigned to A, and (ii) the computation of the value of the formula. The second step will coincide with that by which the value of P(y) -, (3x)P(x) is computed for some assignment; this, as we have seen, is always T. In general, although a formula A may contain several prime formulas, we may consider A as a prime formula and speak of "the logical function assigned to A." We state the result just derived along with a companion valid formula as our next theorem. THEOREM 8.1. Let A (x) be a formula which is free for y. Then (I) 1= (x)A(x) --+ A(y). (II) A(y) -> (3x)A(x). COROLLARY. If (x)A(x), then A(x). Proof. We apply (I) of the theorem, taking x as the y to obtain (x)A(x). Then we may conclude that K A(x) by the extension of Theorem 3.3 mentioned above. (x |
)A(x) - A(x). Now assume that 'r I-1 E O R E M 8.2. Let x be any variable, B be any formula not containing any free occurrence of x, and A(x) be any formula. Then (1) If I-- B -* A(x), then 1= B -> (x)A(x). (II) if r- A(x) - B, then l (3x)A(x) -> B. Proof. To prove (I), we assume that K f3 -. A(x). Let. D be any domain and for this domain consider any assignment a to the formula B-+ (x)A(x). Note that since x does not occur free in either B or (x)A(x), a does not include an assignment of a value in D to x. For a, 4.8 I The Predicate Calculus. Validity 211 B takes either the value F or T. If B takes the value F, then B -* (x)A(x) takes the value T. If B takes the value T, then this is still the case when a is extended to include any assignment of a value in D to x. Hence, for a so extended, A(x) receives the value T, since, by assumption, B --+ A(x) has value T. That is, for each assignment to x along with the given assignment a, A(x) receives the value T. It follows that t-- B --* (x)A(x). The proof of (11) is similar and is left as an exercise. COROLLARY. If t= A(x), then 1= (x)A(x). l= A(x). Since J= B ---, (C -* B), if P is any Proof. Assume that 0-place predicate letter, then 1= A(x) --+ (P V -, P --* A(x)). hence, K P V -,P --- A(x) by Theorem 3.3. By (1) of the above theorem, it follows that r. P V --1 P -+ (x)A(x). Finally, since P V --j P, another application of Theorem 3.3 gives K (x)A(x). An illustration in familiar terms of the above corollary is this. |
A proof of "For all real numbers x, sine x + cos" x = 1" begins by regarding x as some unknown (but fixed) real number. After proving that, for this x, Sine X + COS2 X = 1, it is argued that since x is any real number, the assertion follows. Note that this involves the transition from the consideration of x as a free variable to that of a bound variable. When we initially raised the questions of what methods for proving validity in the statement calculus carry over to the predicate calculus, we ignored the possibility of a direct generalization of Theorem 3.1. It has generalizations to the predicate calculus, but they are complicated because of the necessity of the avoidance of binding, in a way which is not intended, of free variables by quantifiers which may be present. In order to present one theorem of this type, we must describe the mechanics of substituting in a formula for all occurrences of prime formulas resulting front a particular predicate letter. We begin with an illustration. In the formula (x)P(.r) -. P (y) there are two occurrences of prime formulas resulting from the predicate letter P(w). By the result of substituting a formula A(zu) for the predicate letter P(w) in (x)P(x) --* P(y) we shall mean the result of replacing P(x) by A(x) and P(y) by A(y). For instance, if we take A(w) to be (3z)Q(w, z), then the result of the substitution is (2) (x)(3z)Q(x, z) --- (3z)Q(y, z), 212 Logic I CHAP. 4 and if we take A(w) to be (3y)Q(w, y), then the result of the substitution is (3) (x) (3y) Q (x, y) -> (3y) Q (y, y) There is a basic difference between results (2) and (3). Namely, the free y in P(y) remains free in (2) but in (3) it becomes bound by the quantifier (3y) of our second choice for A(w). The effect of binding y in (3) is disastrous as may be seen by considering, for instance, the interpretation of (3) which results on choosing Z as the domain |
and Q(x, y) as x < Y. Such mixups in the way the variables are bound after a substitution can be avoided by observing two restrictions. To formulate the substitution process and these restrictions in general, let us suppose the -, wk) for the predicate letter substitution is of the formula A(wi, W2,, ZOO in a formula B not containing any one of wI, w2i P(wI, w2,, wk. The substitution with result B* is effected by replacing each part of B of the form P(rI, r2, - -, rk) by A(r1, r2,, rk), where., rk) is the result of substituting rI, r2j, wk A(r,, r2,, Wk). The substitution is called admissible iff' none of in A(wj, w2,, wk) and none of the the variables in B occur bound in A(w1, w2, -, wk) occur bound in B. The generalizafree variables in A(wI, w2, tion of Theorem 3.1 which we have in mind can then be stated as follows (the proof is omitted)., rk for wI, w2, THEOREM 8.3. Let B be a formula containing a prime formula resulting from the predicate letter P(wI, IV2, -, Wk) and let B* be the formula resulting from B by an admissible substitution of the., wk). If 1= B, then K B*. formula A(w,, w2, - -, rvk) for P(wI, w2, Although this theorem is not the most general of its kind, it serves to reduce the proof of the validity of each of the formulas in the next theorem to the case of prime formulas in place of arbitrary formulas. Since the formulas of Theorem 3.4 extend to the predicate calculus, we continue the numbering used there to emphasize that we arc introducing additional valid formulas for the predicate calculus. THEOREM 8.4. Let x and y be distinct variables, A(x), B(x), and A(x, y) be any formulas, and A be any formula not containing any free occurrences of x. Then 33. t (3x) (3y)A(x, y) E-> ( |
3y) (3x)A(x, y) 33'. i (x)(y)A(x,y) -- (y)(x)A(x,y'). 4.8 1 The Predicate Calculus. Validity 213 34. 1= (3x)A(x) E-> -, (x) -, A(x). 34'. 1 (x)A(x) F-4 -, (3x) -,A(x). 35. -, (3x)A(x) H (x) -,A(x). 35'. 1= -i (x)A(x) - (3x) -, A(x). 36. K (3x)(y)A(x,y)--. (y)(3x)A(x,y) 37. (3x)(A(x) V B(x)) H (3x)A(x) V (3x)B(x). 37'. G (x)(A(x) A B(x)) <-+ (x)A(x) A (x)B(x). 38. K (x)A(x) V (x)B(x) -' (x)(A(x) V B(x)). 38'. (3x)(A(x) A B(x)) - (3x)A(x) A (3x)13(x). 39. K (3x) (A V B(x)) E-> A V (3x) B(x). 39'. K (x)(A A B(x)) A A (x)B(x). 40. K (x)(A V B(x)) --' A V (x)B(x). 40'. I_ (3x)(A A B(x)) H A A (3x)B(x). The proofs of the validity of these formulas are left as exercises. That some of the formulas are valid should be highly plausible on the basis of meaning; formulas 33 and 33', which mean that existential (or universal) quantifiers can be interchanged at will, are in this category. Again, formulas 34 and 34', which describe how an existential quantifier can be expressed in terms of a universal quantifier and vice versa, were discussed in the preceding section. Formulas 35 and 35' provide rules for transferring -, across quantifiers. Formulas |
37, 37', 38, and 38' are concerned with transferring quantifiers across V and A in general, and formulas 39, 39', 40, and 40' treat special cases of such transfers. EXAMPLES 8.6. We consider some practical illustrations of the use of formulas 35 and 35' in arithmetic. That is, we take as the domain D the set of natural numbers. Further, let < and + have their familiar meanings; thus <(x, y) is a 2-place predicate letter, and + (x, y, z) is a 3-place one. The (true) statement "There does not exist a greatest natural number" may be symbolized by Its negation, (x)(3y)(x < y)- -, (x) (3y) (x < y), may be rewritten, using 35', as In turn, using 35, this may be rewritten as (3x) -i ((3y) (x < y)) (3x) (y) -, (x < y) or (ax)(Y)(x? Y) - 214 Logic I CHAP. 4 In English this last formula reads "There exists a greatest natural number." The (false) statement "For every pair m, n of natural numbers there is a natural number p such that m + p = n" may be symbolized by Its negation may be transformed into (m)(n)(3p)(m + p = n). (3m) (3n) (p) (m +p n). The reader can translate this into acceptable English. 8.7. Take for D the set R of real numbers. The definition of continuity of a function J at a, namely, "J is continuous at a iff for every e > 0 there exists a 5 > 0 such that for all x, if Ix - al < 5, then If (x) - f(a)l < e" can be translated into the symbolic form (E) (E > 0 -. ((36) (6 > 0 A (x)(Ix - al < S If(x) - 1(a)I < e)))) This can be shortened considerably using the notion of restricted quantification, which in practical terms amounts to restricting the range of a and S to the set R 1. Then the above may be contracted to (E)(35)(x)(Ix - aI < S --: 11( |
x) - J(a)I < e). With mild restrictions, the valid formulas of Theorem 8.4 remain valid when some quantifiers are restricted. This makes it possible, for example, to obtain the negation of complicated formulas quickly and in greatly abbreviated form. As an illustration, the reader is asked to form the negation of the original formula above and show that, in terms of restricted quantifiers', it reduces to the negation of the abbreviation of the original formula, which is (3E)(5)(3x)(Ix - al < S A 11(x) - 1(a)I > e). EXERCISES 8.1. For a domain of two elements, construct a truth table for the formula (x)(P V Q(x)) *-> P V (x)Q(x). 8.2. Prove that the formula in Example 8.4 is valid in a domain consisting of one element. 8.3. Establish the validity of formulas 34, 35, and 36 in Theorem 8.4, regard- ing all constituent formulas as primes. 8.4. Establish the validity of formulas 37, 38, and 39 in Theorem 8.4, regard- ing all constituent formulas as prunes. 8.5. Supply an example to show that the converse of formula 36 in Theorem 8.4 is nonvalid. 8.6. Prove Theorem 8.2 (II). 8.7. As in Example 8.6, let us take for 19 the set of natural numbers. Using Theorem 8.4, justify the equivalence of the left-hand and right-hand members of each of the following pairs of formulas. 4.9 I The Predicate Calculus. Consequence 215 (a) (3x)(y) -, (y > x), (3x) -i (3y)(y > x). (b) (3x)(y)(y > x V -, (9 > 0)), (3x)(y)(y > 0 ->y > x). (c) (x) (3y) (3z) (x < y A z2 > y), (x) (3y) (x < y A (3z) (z2 >y)). 8.8. Let a0, a,,, a,,, quantification, translate into symbolic form be a sequence of real numbers. Using restricted (a |
) the assertion that a is the limit of the sequence, (b) the assertion that the sequence has a limit, (c) the assertion that the sequence is a Cauchy sequence (that is, given e > 0 there exists a positive integer k such that if n, m > k, then Ia. - a,,,j < e). 8.9. Write the negation of each of the formulas obtained in the preceding exercise. 8.10. With R as domain, translate each of the following statements into symbolic form, write the negation of each (transferring -I past the quantifiers), and translate each resulting formula into English. (a) For x, y C R and z E R+, xz = yz implies x = Y. (b) The number a is the least upper bound of A R. (c) The set A has a greatest element. 9. The Predicate Calculus. Consequence The concept of consequence for the predicate calculus is an extension of that for the statement calculus as given in Section 4. In this extension, statement letters give way to predicate letters, and assignments of truth values give way to the more elaborate assignments of the predicate calculus. In addition, a further ingredient appears for the first time: the possibility that an assumption forrmrla contains a free occurrence of a variable. For example, in a theorem in an assurnption may have the form "Let x be an integer greater than 0" or "Suppose that x is divisible by 3." An examination of how such an x is "treated" in a proof' reveals that it is regarded as a constant; that is, it is regarded as a name of one and the same object throughout the proof. Outside of the context of the proof, however, it is it variable. (For exam ple, having proved some result concerning an x which is divisible by 3, one feels free to apply it to all such numbers.) The reader is familiar with. such names as "parameter" and "arbitrary constant" for symbols employed in this way. This brings its to our basic definition. The formula 13 is a consequence, A. (in the predicate calculus), symbolized by of formulas A,, A2, A,, A2, ---,A,,,K B, 216 Logic ( CCHHAP. 4 if for each domain D and for each assignment to the A's in D the formula 13 receives the value T whenever each A receives the value |
T. Further, if a variable x occurs free in any A, then in each assignment to the A's one chooses for all free occurrences of x one and the same value in D; that is, in making an assignment to the A's, such an x is regarded as a constant., A. The statement and proof of Theorem 4.1 and its Corollary carry over unchanged to the present case. Thus, these results are available. In particular, to conclude that A1, A2, B, it is sufficient to prove that K A, A A,. A.. A A. -p B. Since Theorem 4.2 likewise extends to the predicate calculus, it is possible to give a demonstration, A. in the form of a that a formula B is a consequence of A1, A2, finite sequence of steps, the last of which is B. In addition to the two basic rules p and 1, which in the statement calculus serve to justify the appearance of a formula E in a demonstration, we may introduce others for the predicate calculus. The most fundamental of these are the following two. Rule (of universal specification) us: There is a formula (x)A(x) preceding E such that E is A(y), the result of substituting y for x in A(x), such substitutions being restricted by the requirement that none of the resulting occurrences of y is bound. Rule (of universal generalization) ug: E is of the form (x)A(x) where A(x) is a preceding formula such that x is not a variable having a free occurrence in any premise. The state of affairs regarding a demonstration of consequence in the, Am K B if predicate calculus is then this. We contend that A,, A2, we can devise a string E1,E2j...,Er(=13) of formulas such that the presence of each E can be accounted for on the basis of one of the rules p, 1, us, or ug. Indeed, as in Section 4, it is possible to prove that if the presence of each E can be so justified, then A1, A2, (any E in the sequence)., The earlier proof carries over (using the extended form of Theorem 4.2) to dispose of the case where the presence of an E is justified by either rule p or rule 1. The cases which involve rule us or Kg are dispatched using Theorem 8.1( |
1) and Theorem 8.2(I). The details are left as an exercise., We are now in a position to construct formal derivations of simple arguments in the style developed in Section 4. 4.9 I The Predicate Calculus. Consequence 217 EXAMPLES 9.1. Consider the following argument. No human beings are quadrupeds. All women are human beings. Therefore, no women are quadrupeds. Using the methods of translation' of Section 6, we symbolize this as follows. (x) (Hx -+ --i Qx) (x) (Wx --> Hx) (x) (Wx -* -, Qx) The derivation proceeds as follows. {1} {2} {2} {1} {l, 2} {1, 2} (1) (x) (Hx --> -, Qx) (2) (x) (Wx - a Hx) (3) Wy --> Hy (4) Hy -' -' Qy (5) Wy -' Qy (6) (x) (Wx -a Qx) p p 2 us I us 3,4t 5 ug 9.2. The following argument is more involved. Everyone who buys a ticket receives a prize. Therefore, if there are no prizes, then nobody buys a ticket. if Bxy is "x buys y," Tx is "x is a ticket," Px is "x is a prize," and Rxy is "x receives y," then the hypothesis and conclusion may be symbolized as follows. (x)((3y)(Bxy A Ty) - (y)(Py A Rxy)) -i (3x)Px -+ (x) (y) (Bxy --> -, Ty) Since the conclusion is a conditional, we employ the rule cp in the derivation below. The deduction of line 3 from line 2, that of line 7 from line 6, and that of line 11 from 10 should be studied and justified by the reader. {1} {2} {2} {2} {2} {2} {2} {l} 11,2) 11,21 11,2) {l, 2} {1} (1) (x)((3y)(Bxy A Ty) --' (3y)(Py A Rxy)) p (2) -, (3x)P |
x p (3) (x) -1 Px 2 t -, Py (4) 3 us -, Py V Rxy 4 t (5) (6) (y) (-, Py V -,Rxy) 5 ug 6t (7) -,Gy)(Py A Rxy) (8) (3y) (Bxy A Ty) --> (yy)(Py A Rxy) I us (9) -, (3y) (Bxy A Ty) 7, 8 t (10) (y)(-,Bxy V -,Ty) 9 t lot (11) (y)(Bxy --> -, Ty) hug (12) (x)(y)(Bxy -> (13) -, (3x)Px -4 (x) (y) (Bxy --, 2, 12 cp Ty) Ty) 218 Logic I CHAP. 4 9.3. Once the reader has subscribed to the soundness of the derivation in the preceding example, he has, in effect, endorsed further rules of inference which serve to expedite derivations. We introduce two further derived rules of inference which render the same service. These are formal analogues of two familiar everyday occurrences in mathematics. If one is assured that "(3x)A(x)" is true, one feels at liberty to "choose" ay such that A(y). Then y is an unknown fixed quantity such that A(y). Conversely, given that there is some y such that A(y), one does not hesitate to infer that "(3x)A(x)" is true. In the predicate calculus the rule which permits the passage from (3x)A(x) to A(y) is called the rule (of existential specification) es. The rule which permits the passage from A(y) to (3x)A(x) is called the rule (of existential generalization) eg. These are the analogues for existential quantifiers of the rules us and ug for universal quantifiers. We shall not validate these rules nor even discuss the restrictions which must be observed in using them. In the following simple example illustrating them we employ a lower-case Greek letter to designate an object which is involved in the "act of choice" accompanying an instance of the rule es. Every member of the committee is wealthy and a Republican. Some committee members are old. Therefore, there are some old Republicans. {1} {2} |
{2} {1} {2} {1, 2} {2} {1, 2} {1, 2) (1, 2} (x) (Cx -, Wx A Rx) p p 2 es (1) (2) (3x) (Cx A Ox) (3) Ca A Oa (4) Ca - Wa A Ra (5) Ca (6) Wa A Ra (7) Oa (8) Rot (9) Oa A Ra (10) (3x)(Ox A Rx) 1 us 3t 4, 5t 3t 61 7, 8 t 9 eg 9.4. The derivation corresponding to the following argument employs all of the rules which we have described. Some Republicans like all Democrats. No Republican likes any Socialist. Therefore, no Democrat is a Socialist. The reason for the introduction of "x" in line 3 below is this. By virtue of the form of the conclusion, (x)(Dx -> -,Sx), a conditional proof is given. Thus, Dx is introduced as a premise in line 3. Since x occurs free here, we note its presence (as well as in subsequent lines which depend on this premise) to assist in avoiding any abuse of rule ug. 4.9 The Predicate Calculus. Consequence 219 {1} {2} {3} (1) {1} {1} {1, 31 {2} {l} {1, 21 {1,2} {l, 2, 31 {1,2} (1,21 (1) (3x)(Rx A (y)(Dy - Lxy)) (2) (x) (Rx -+ (y) (Sy -> -, Lxy)) (3) Dx (4) Ra A (y)(Dy - Lay) (5) (y)(Dy -> Lay) (6) Dx -, Lax (7) Lax (8) Ra --> (y) (Sy --> -i Lay) (9) Ra (10) (y)(Sy - -,Lay) (I1) Sx --i --,Lax (12) -,sx (13) Dx -> --,Sx (14) (x)(Dx -> --1Sx) p p x, p 1 es 4 t 5 us x, 3, 6 t 2 its 4 t 8, 9 |
t 10 us x, 7, 11 t 3,12cp 13 ug The foregoing examples lend plausibility to the contention that the predicate calculus is adequate for formalizing a wide variety of arguments. Lest there be concern over the lengths of derivations of such simple arguments as those considered, we assure the reader that an extended treatment would include the introduction of further derived rules of inference to streamline derivations. The outcome is the concept of an "informal proof." In mathematics this amounts to a derivation in the conversational style to which one is accustomed: mention of rules of inference and tautologies used is suppressed, and attention is drawn only to the mathematical (that is, nonlogical) axioms and earlier theorems employed. (Further details of this are supplied in the next chapter.) The principal advantage accrues in having informal proofs as the evolution of formal derivations is this: One has a framework within which it can be decided in an objective and mechanical way, in case of disagreement, whether a purported proof is truly a proof. EXERCISES Construct a derivation corresponding to each of the following arguments. 9.1. No freshman likes any sophomore. All residents of Dascornh are sophomores. Therefore, no freshman likes any resident of Dascomb. (Fx, LV, Sv, Dx) 9.2. Art is a boy who does not own a car.,Jane likes only boys who own cars. Therefore,,Jane does not like Art. (Bx, Ox, Lxy, a, j) 9.3. No Republican or Democrat is a Socialist. Norman Thomas is a Socialist. Therefore, he is not a Republican. (Rx, Dx, Sx, t) 9.4. Every rational number is a real number. There is a rational number. Therefore, there is a real number. 220 Logic I CHHAP. 4 9.5. All rational numbers are real numbers. Some rationals are integers. Therefore, some real numbers are integers. (Qx, Rx, Zx) 9.6. All freshmen date all sophomores. No freshman dates any junior. There are freshmen. Therefore, no sophomore is a junior. 9.7. No pusher is an addict. Some addicts are people with a record. There- fore, some people with a record are not pushers. 9.8. Sonle freshmen like all sophomores. No freshmen likes any junior. |
There- fore, no sophomore is a junior. (Fx, Lxy, Sx, Jx) 9.9. Some persons admire Elvis. Some persons like no one who admires Elvis. Therefore, some persons are not liked by all persons. (Px, Ex, Lxy) BIBLIOGRAPHICAL NOTE Extended treatments of symbolic logic, pitched at approximately the same level as that of this chapter, appear in Copi (1954), Exncr and Rosskopf (1959), Suppes (1957), and Tarski (1941). Formulations of both the statement calculus and the first-order predicate calculus as axiomatic theories are given in Chapter 9 of this book. The bibliographical notes for that chapter include references to more comprehensive accounts of this subject matter. CHAPTER 5 Informal Axiomatic Mathematics ONE OF THE striking aspects of twentieth century mathematical research is the enormously increased role which the axiomatic approach plays. The axiomatic method is certainly not new in mathematics, having been employed by Euclid in his Elements. However, only in relatively recent years has it been adopted in parts of mathematics other than geometry. This has become possible because of a fuller understanding of the nature of axioms and the axiomatization of logic. The axiomatization (in the way we shall discuss it presently) of various fragments of mathematics was the main subject of studies of the foundations of mathematics, from the late 1880's until the 1920's. At that time the present-day approach began to flourish. Distinctive features of this modern approach include the explicit incorporation into the set of axioms of a theory, those which provide for a "built-in" theory of inference, and the concentration on the theory of models for structures characterized by sets of axioms. Chapter 9 is devoted to an introduction to this modern approach. The present chapter, when judged relative to standards imposed by the present stage of investigations of the foundations of mathematics, belongs to the past. But, we repeat, it expounds the axiomatic method as it is used currently in everyday mathematics. 1. The Concept of an Axiomatic Theory The concept to be described is an outgrowth of the method used by Euclid in his Elements to organize ancient Greek geometry. The plan of this work is as follows. It begins with a list of definitions of such notions as point and line; for example, a line is defined as length without breadth. Next appear various statements, |
some of which are labeled axioms and the others postulates. It appears that the axioms are intended to be principles of reasoning which are valid in any science (for example, 221 222 Informal Axiomatic Mathematics I CtfAP. 5 one axiom asserts that things equal to the same thing are equal to each other) while the postulates are intended to be assertions about the subject matter to be discussed-geometry (for example, one postulate asserts that it shall be possible to draw a line joining any two distinct points). From this starting point of definitions, axioms, and postulates, Euclid proceeds to derive propositions (theorems) and at appropriate places to introduce further definitions (for example, an obtuse angle is defined as an angle which is greater than a right angle). Several comments on Euclid's work are in order. It is clear that his goal was to deduce all of the geometry known in his day as logical consequences of certain unproved propositions. On the other hand, we can only conjecture as to his attitude toward other facets of his point of departure. From a modern viewpoint it may be said that he treated point and line essentially as primitive or undefined notions, subject only to the restrictions stated in the postulates, and that his definitions of these notions offer merely an intuitive description which assists one in thinking about formal properties of points and lines. I lowevcr, since the geometry of that era was intended to have physical space as an interpretation, it is highly plausible that Euclid assigned physical meaning to these notions. Further evidence to support this conclusion is to be found in some proofs where Euclid made assumptions that cannot be justified on the basis of his primitive notions and postulates, yet which, on the basis of the intended interpretation of his primitive notions, appear to be evident. If, indeed, Euclid was confused between formal or axiomatic questions and problems concerning applications of geometry, then herein lies the source of the only flaws in his work as judged by modern standards. Concerning the postulates, he probably believed them to be true statements on the basis of the meaning suggested by his definitions of the terms involved. Since proofs were not provided for the postulates, they acquired the status of "self-evident truths." This attitude with respect to the nature of postulates or axioms (now, incidentally, no distinction is drawn between these two words) still persists in the minds of many. Indeed, in current nonmathematical writings it is |
not uncommon to see such phrases as "It is axiomatic that" and "It is a fundamental postulate of" used to mean that some statement is beyond all logical opposition. Within mathematics this point of view with respect to the nature of axioms has altered radically. The change was gradual and it accompanied the full understanding of the discovery by J. Bolyai and (independently) N. Lobachevsky of a non-Euclidean geometry. Let us elaborate on this matter. 5.1 1 The Concept of an Axiomatic Theory 223 In the traditional sense a non-Euclidean geometry is a geometry whose formulation coincides with that of Euclidean geometry with the one exception that Euclid's fifth postulate (the "parallel postulate") is denied. The fifth postulate is "If two lines are cut by a third so as to make the surn of the two interior angles on one side less than two right angles, then the two lines, if produced, meet on that side on which the interior angle sum is less than two right angles." An equivalent formulation, in the sense that either, together with the remaining postulates, implies the other, and one which is better suited for comparison purposes, is "In a plane, if point A is not on the line 1, then there is exactly one line on A parallel to 1." This is one of many axioms equivalent to the parallel postulate which were obtained as by-products of unsuccessful attempts to substantiate the belief that the parallel postulate could be derived from Euclid's remaining axioms. Bolyai and Lobachevsky dispelled this belief by developing a geometry in which the parallel postulate was replaced by the statement "In a plane, if the point A is not on line 1, then there exists more than one line on A parallel to 1." Apparently, the "truth" of this new geometry was initially in doubt. But on the basis of measurements that Could be made in the portion of physical space available, there appeared to be no measurable differences between the predictions of the Bolyai-Lobachevsky geometry and those of Euclidean geometry. Also, each geometry, when studied as a deductive system, appeared to be consistent so far as riot yielding contradictory statements. The ability to examine these geometries from the latter point of view represented a great advance, for, in essence, it amounted to the detachment of physical meaning from the primitive notions of point, |
line, and so on. A second advance in the attitude toward the axiomatic method accompanied the creation of various models in Euclidean geometry of the Bolyai-Lobachevsky geometry. A typical example is the model proposed in 1871 by helix Klein, for which he interpreted the primitive notions of plane, point, and line, respectively, as the interior of a fixed circle in the Euclidean plane, a Euclidean point inside this circle, and an openended chord of this circle. If, in addition, distances and angles are computed[ by formulas developed by A. Cayley, in 1859, then all axioms of plane Bolyai-Lobachevsky geometry become true statements. The immediate value of such an interpretation was to establish the relative consistency (a concept which will be described in detail later) of the Bolyai-Lobachevsky geometry. That is, if Euclidean geometry is a consistent logical structure, then so is the Bolyai-Lobachevsky geometry. 224 Informal Axiomatic Mathematics i CH A P. 5 Of greater significance, so far as understanding the nature of axiomatic theories, was the entertainment of the possibility of varying the meaning of the primitive notions of an axiomatic theory while holding fixed its deductive structure. This evolution in the understanding of the nature of the axiomatic method set the stage for the present-day concept of an axiomatic theory. In its technical sense the word "theory" is applied to two sets of statements, of which one is a distinguished subset of the other. The entire set of statements defines the subject matter of the theory. In the sciences, apart from mathematics, the members of the distinguished subset arc those statements which are classified as true statements about the real world, with experiment the ultimate basis for the classification. In sharp contrast, it is a characteristic feature of an axiomatic theory that the notion of truth plays no role whatsoever in the determination of the distinguished subset. Instead, its members, which arc called theorems or provable statements, are defined to be those statements of the theory that can be deduced by logic alone from certain initially chosen statements called axioms (or postulates). A precise definition of theorem can. be given in terms of the notion of proof. A (formal) proof -, Sk) of statements of the theory such that is a finite column (Si, S2, each S either is an |
axiom or comes from one or more preceding S's by the rules of inference of the system of logic employed. A theorem or provable statement is a statement which is the last line of some proof. Note that, in particular, an axiom is a theorem with a one-line proof. In the consideration of an axiomatic theory the notion of truth is relegated to possible applications of the theory. In any circumstance in which the axioms are accepted as true statements and the system of logic is accepted, then the theorems must be accepted as true statements since the theorems follow from the axioms by logic alone. That is, it is the potential user of an axiomatic theory who is concerned with the question of the truth of the axioms of the theory. Today, axiomatic theories are usually presented in essentially the same way that Euclid began his development of geometry--by listing the primitive notions and the axioms of the theory. However, in order to meet one of the present-day requirements of an axiomatic theorythat truth play no role--the primitive potions are taken to be undefined and the axioms are taken as simply an initial stock of theorems. We, shall elaborate on these matters in connection with a discussion of the evolu- 5.1 I The Concept of an Axiomalii: Theory 225 tion of axiomatic theories from intuitive theories (which constitute a primary source of axiomatic theories). Usually one's first exposure to some branch of science is by way of an intuitive approach; subjects such as arithmetic, geometry, mechanics, and set theory, to cite just a few, are approached in this way. An axiomatization of such an intuitive theory can be attempted when the fundamental notions and properties are believed known and the theory appears to be sound to the extent that reliable predictions can be made with it. The first step in such an attempt is to list what are judged to be the basic notions discussed by the theory together with what are judged to be a basic set of true statements about these notions. In order to carry out this step efficiently, one often elects to presuppose certain theories previously constructed. In most axiomatic work in mathematics it is customary to assume a theory of logic along with a theory of sets. t In axiomatic work in an empirical science such as economics or physics it is standard procedure to assume, in addition to logic and general set theory, parts of classical mathematics. Once it has been |
decided what theories will be assumed, the key steps in the axiomatization can be carried out. The first of these is the introduction of symbols (including, possibly, words) as names for those notions which have been judged to be basic for the intuitive theory. These are called the primitive symbols (or, terms) of the axiomatic theory. The only further symbols which are admitted (aside from symbols of the presupposed theories) are defined symbols, that is, expressions whose meaning is explicitly stated in terms of the primitive symbols. (The intuitive theory in mind often suggests the introduction of some such symbols.) The next step is the translation of those statements that were singled out as expressing fundamental properties of the basic notions of the intuitive theory into the language which can be constructed from just the primitive. and defined terms (and those of any theory which is presupposed). To obtain an example of a language of the sort mentioned above, let us consider an axiomiatization of intuitive set theory with the first-order predicate calculus as the only presupposed theory. In addition to logical symbols, only one further (primitive) symbol, the familiar one for the membership relation, shall be employed. Then the language which is available is that described in Section 4.7, with expressions of the form xCy t By a theory of sets we mean some development which includes roughly the content of Chapters 1, 2, and 3. Often a theory of sets which encompasses this material is referred to as "general set theory." 226 Informal Axiomatic Mathematics I CHAP. 5 constituting the totality of prime sentences (formulas). A list of useful defined symbols for the theory appears in Example 4.7.1. If some theory of logic is not assumed for an axiomatization, then one must include in the presentation of the theory an axiomatized version of a theory of inference. A detailed discussion of this begins in Section 9.3. In a program of the sort we have described for axiomatizing an intuitive theory, there is often considerable leeway in the choice of primitive notions. Different. sets may be suggested by various combinations of notions which occur in the intuitive theory. In the modern axiomatization of Euclidean geometry devised by D. Hilbert there are six primitive notions: point, line, plane, incidence, betweenness, and congruence. On the other hand, in that created by M. Pieri there are but two primitive notions: point and motion. Obviously the choice |
of primitive notions for an axiomatic theory influences the choice of axioms. A great variety of more subtle remarks can be made concerning the selection of axioms for a particular theory. Some are presented in Section 4. While we are dealing in generalities we will mention another stimulus for the creation of axiomatic theories-the observation of basic likenesses in the central features of several different theories. This may prompt an investigator to distill out these common features and use them as a guide for defining an axiomatic theory in the manner described above. Any one of the theories which an axiomatic theory is intended to formalize serves as a potential source of definitions and possible theorems of this axiomatic theory. An axiomatic theory which successfully formalizes an intuitive theory is a source of insight into the nature of that theory, since the axiomatic theory is developed without reference to meaning. One which formalizes each of several theories to some degree has the additional merit that it effects simplicity and efficiency. Since such an axiomatic theory has an interpretation in each of its parent theories (on a suitable assignment of meaning to its primitive terms), it produces simplicity because it tends to reduce the number of assumptions which have to be taken into account for particular theorems in any one of the parent theories. Efficiency is effected, because a theorem of the axiomatic theory yields a theorem of each of the parent theories. Herein lies one of the principal virtues of taking the primitive terms of an axiomatic theory as undefined. A by-product of the creation of an. axiomatic theory which is the common denominator of several theories is the possibility of enriching and extending given theories in an inexpensive way. For example, a 5.2 I Informal Theories 227 theorem in one theory may be the origin of a theorem in the derived theory and it, in turn, may yield a new result in another parent theory. In addition to the possible enrichment in content of one theory by another, by way of an axiomatic theory derived from both, there is also the possibility of "cross-fertilization" insofar as methods of attack on problems are concerned. That is, a method of proof'whic h is standard for one theory may provide a new method in another theory with a derived theory serving as the linkage. A full understanding of such remarks as the foregoing cannot possibly be achieved until one has acquired some familiarity with it variety of specific theories and analyzed some successful attempts to bring diverse |
theories under a single heading. The field of algebra abounds in such successful undertakings. Indeed, it is perhaps in algebra that this type of genesis and exploitation of theories has scored its greatest successes. Several important examples of algebraic (axiomatic) theories are discussed later. 2. Informal Theories In this section we shall discuss the formulation of axiomatic theories when a theory of inference and general set theory are presupposed as already known. Such axiomatic theories will be called informal theories. As has already been mentioned, it is common practice in mathematics to present axiomatic theories as informal theories. that is, The first [natter to be thoroughly understood about informal theories is the working forms which are adopted for the assumed theories of inference and of sets the actual settings in which informal theories are presented. Concerning the theory of inference, it is simply the intuitive theory which one. absorbs by studying That this theory is clearly defined is suggested by the fact that what is judged to be it proof by one competent is usually acceptable to other mathematicians. '['his is not the end of the matter, however. The contents of Chapter 4 indicate that there is it systctu of logic (the firstorder predicate calculus) which is adequate for much of mathematics and which can be described in precise terms. Both the preciseness and adequacy of the first-order predicate calculus take on sharp forms later version of this theory (Section 9.3) and when we give an in the sense that every valid formula is a then prove its theorem. Further, there is considerable evidence to support the contention that the definition of logical correctness which is supplied by this 228 Informal Axiomatic Mathematics I c tt A P. 5 symbolic logic is closely attuned to the corresponding intuitive notion which mathematicians acquire. Such a book as Logic for Mathematicians, by J. B. Rosser (1953), is rich in examples which illustrate his thesis that logical principles which are judged correct by most mathematicians are classified as correct by symbolic logic and vice versa. That is, there is considerable evidence in support of the thesis that the system of logic which is presupposed for an informal theory is a clearly defined theory which can be spelled out if necessary. This empirical conclusion does not evidence itself in mathematicians giving formal proofs and then using the mechanical procedures provided by the predicate calculus for testing their correctness. However, it is usually not difficult to convince oneself that an accepted, informal proof could be formalized if demanded. The set- |
theoretical framework which is assumed for an informal theory is the general set theory developed in Chapters 1-3. Although contradictions can be devised within this intuitive theory, that part which is employed in developing informal theories does not lead to such difficulties so far as is known. For the moment we shall support this latter statement with only the following remark. The intuitive set theory we have discussed can be axiomatizcd in such a way that (i) so far as is known, all undesirable features (that is, the known paradoxes) are avoided, and (ii) all desirable features consonant with (i) are retained. An outline of such a development is given in Chapter 7. We turn now to some examples of informal theories. These will serve to illustrate the two circumstances described at the end of the preceding section under which axiomatic theories are devised (namely, to axiomatize some one intuitive theory and to formalize simultaneously several theories). Further, they will serve to illuminate our later discussion of informal theories. EXAMPLES 2.1. In Example 2.1.2 appears what is essentially Peano's axiomatization of the natural number system. The primitive notions are natural number, zero (0), and successor ('), and the axioms are the statements Pi--P6 appearing there. 2.2. Immediately following Theorem 3.4.1 we called attention to certain likenesses in the properties of the rational numbers and integers. Specifically, we noted that the system consisting of Q, the operation of addition, and 0,, as well as the system consisting of 0 - {0,}, multiplication, and 1, share, with the system made up of Z, addition, and 0;, properties (1)-(4) of Theorem 3.3.1. Thus, we argued, any further properties of the integers which can be derived from (1)-(4) (for example, those mentioned in Exercise 3.3.5) also hold for the other 5.2 I Informal Theories 229 two systems. In terms of our current discussion we may classify that argument as a bit of axiomatic mathematics. Before formulating explicitly the axiomatic theory involved we remark that for the derivation of the results stated in Exercise 3.3.5 the property of commutativity of addition is not required (we "allowed" the reader to use this property because simpler proofs can be given with it). Essentially the same simplifications in the proofs can |
be achieved if commutativity is assumed only in part, as in the axioms below. The axiomatic theory to be described is called group theory. The primitive notions are an unspecified set C, a binary operation in G, for which we use multiplicative notation (that is, the operation will be symbolized by and the value at (a, b) of this function on C X G into G will be designated by a b), and an element e of G. The axioms are the following. G1. For all a, b, and c in C, a G2. For all a in (b c) = (a b) c. G there exists an a' in G such that a a' = a' a = e. The above is a formulation of group theory as one might find it in an algebra text. In harmony with the agreement to write the value of b, we call this element the product of a and b. Henceforth we shall use the simpler notation ab for it. An element which has the property assumed for e in G2 is called an identity element and an element which satisfies Ga for a given a is called an inverse of a (relative to e). at (a, b) as a A few theorems of group theory, including those to which reference has been made in connection with number systems, are proved next. G4. G contains exactly one identity element. In view of G2, only a proof of the uniqueness is required. Assume that Proof. each of el and e2 is an identity element of G. Then ela = a for every a, and ae2 = a for every a. In particular, ele2 = e2 and eie2 = el. Hence, el = e2 by properties of equality. G6. Each element in G has exactly one inverse. Proof. Since G$ asserts the existence of an inverse for each element a, only the proof of its uniqueness remains. Assume that both a' and a" are inverses of a. Then a"a = e and aa' = e. By G1, (a"a)a' = a"(aa'), and, hence, ea' = a"e. Using G2 it follows that a' = a". In multiplicative notation the inverse of a is designated by "a'1"; thus a -'a = as 1 = e (the unique identity element of C). |
Ge. For every a, b, and c in G, if ab = ac, then b = c, and, if ba = ca, then b = c. 230 Informal Axiomatic Mathematics CHAP P. 5 I Proof. Assume that ab = ac. Now a-'(ab) = (a-'a)b = eb = b. On the other hand, a-'(ab) = a-'(ac) _ (a-'a)c = ec = c. Hence, b = c. The proof of the remaining assertion is similar. Proofs of the next two theorems are left as exercises. G7. For all a and b in G, each of the equations ax = b and ya = b has a unique solution in G. G8. For all a and b in G, (ab)-' = b-'a 2.3. The theory to be described has its origin in Euclidean plane geometry. It is that generalization of Euclidean geometry known as afline geometry. The primitive notions arc a set (1' (whose members are called points and will be denoted by capital letters), a set 2 (whose members are called lines and will be denoted by lower-case letters), and a set q called the incidence relation. The axioms are as follows. AG,. J C 6' X 2. ((P, 1) C 9 is read "P lies on l," or "l contains P," or "1 passes through P.") AGs. For any two distinct points P and Q there is exactly one line passing through P and Q. (This line will be denoted by P + Q.) Before stating the next axiom we make a definition. If I and in are two lines such that either I = m or there exists no point which lies on both l and in, then I and in are called parallel. AG3. AG,. For any point P and any line I there exists exactly one line in passing through P and parallel to I. If A, B, C, D, E, and F are six distinct points such that A + 13 is parallel to C + 1), C + I) is parallel to E -F F,, A 4- C is parallel to B + D, and C + E is parallel to 1) + F, then A -F- E is parallel to B + 1". AG6. There exist three distinct points not on one line. Proofs |
of a few simple theorems are called for in the following exercises. Since axiomatic theories are often elaborate structures, they deserve elaborate symbols as n;lnes. To our mind, capital (;ernrut letters suffice. Consielcr now an iniorinal theory T. Associated with it is a language which can be constructed from the primitive and defined terms of I and the terminology of set theory and logic. We shall call this language the T -language and its nlcniber sentences `;-sentences. "Those T-sentences which involve no free variables shall be called a-statements. (Parenthetically we remark that Z--sentences, are usually 5.2 I Informal Theories 231 written using a combination of words and symbols, as in the foregoing examples, instead of the purely symbolic style of Examples 4.7.1 -4.7.3.) An interpretation of `;" consists of selecting a particular nonempty set D (called the domain of the interpretation) as the range for the individual variables of ` and assigning to each primitive term an object of the same "character" constructed from D; that is, to a binary relation symbol we assign a binary relation in D, to a binary operation symbol we assign a binary operation in D, to an individual constant we assign an element of D, and so on. This can scarcely be regarded as a definition of an interpretation in view of its vagueness. Until such time as we correct this deficiency we shall rely on the reader's intuition and the examples below. If I is an interpretation of `3: in a system X9J1 and if S is a `3"-sentence, then we shall call the sentence, which results on the assignment of meaning (as specified by I) to the primitive terms of `;" that occur in S, an interpretation of Sin 931. If an interpretation of S in 931 is it true statement of 931, we shall say that S is true in 931, or that 931 is a model of S. If 2; is a set of a-sentences, then 9X is called a model of I iff it is a model of each member of 2;. If T1 is a model of the set of axioms of T, then 931 is called a model of `i;". Notice that such definitions are relative to some one interpretation of `; in P. As our first illustration of the notion of it model we note |
that each of the progressions described in Example 2.1.1 is a model of the Peano axioms under the obvious interpretations of natural number, zero, and successor. Next, the set 0(X) of all one-to-one mappings on a nonempty set X onto itself together with function composition and ix is a model of the theory of groups or, more simply, is a group. Again, the power set of any set together with the symmetric difference operation and the empty set is a group. As for models of atline geometry, one who is familiar, to some degree, with intuitive Euclidean geometry will undoubtedly accept it as an afline geometry. A radically different model results on setting (f' = 11, 2, 3, 4},,c _ {{1, 21, {1, 3}, 11, 4}, {2, 3}, {2, 4}, 13, 4}} and defining P to be on I iff P C 1. The verification that all axioms are satisfied is left as an exercise. It is an accepted property of a model T1 of an informal theory `;` that is true in 91.f The supporting argument is simply that each theorem of (by definition of a model of Z) each axiom is true in 931 and each theorem of T_ is derived from the axioms by logic alone. An illustration may be given in terms of Theorem G8 of Example 2.2. 'The interpretat A 2-sentence in which an individual variable x has a free occurrence is interpreted as if the quantifier "For all x" were prefixed to the sentence. 232 Informal Axiotnatic Mathematics I e n n t'. 5 Lion of Gg in the group G(X) of mappings is the statement that if a, b C G(X) then (a o b)-' = b'-' o a-', which is an important property of functional inversion. The interpretation of Gg in the group consisting of Z, addition, and 0 is the statement that -(a + b) = (-b) + (-a). Thus, these two results, diverse in appearance, are interpretations of a single statement of group theory. EXERCISES 2.1. Prove Theorems G7 and Cg in Example 2.2. 2.2. The theory of commutative groups differs from the theory of groups in that it includes one |
further axiom: G9. For all a and b in G, ab = ba. It is common practice to use additive notation for the operation in a commutative group (that is, to write a -I- h instead of ab), to write 0 instead of e, and to write -a instead of a`. Suppose that G together with A- and 0 is a commutative group. Prove each of the following theorems. (a) -(a -l- b) = (-a) + (-b). (b) If "a - b" is an abbreviation for "a + (-b)," then a + b = c iff b = c - a. (c) a-(-b)=a-I-batic] -(a-b)=b-a. (d) If f: G-+- G where f(a) _ -a, then f is a one-to-one and onto mapping. 2.3. Let Z be the set of residue classes [a] of Z modulo n (see Section 1.7). Show that the relation {(([a], [b]), [a -I- bbl)I Jal, [b] F_ Z,,} is a binary operation in 7.,,. Show that Z. together with this operation and [0] is a commutative group. 2.4. Show that an operation + can be introduced in the set I of equivalence classes defined in Exercise 1.7.11, by the definition [a, bJ A- [c, d] = [a + c, b + d], where [a, b] is the equivalence class determined by (a, b), and so on. Prove that I together with this operation and [1, 1] is a commutative group. 2.5. Show that R together with the operation * such that x * y = (x3 +ya) 13 and 0 is a group. 2.6. Write out the elements of G(X) for X = 11, 2} and for X = 11, 2, 3}. Show that the group associated with the latter set of mappings is not commutative. 2.7. Let G be a nonempty set and be a binary operation in G such that G, and G7 hold. Prove that C,, and a suitable clement of C is a group. 2.8 |
. Let G be a nonempty finite set and be a binary operation in C such that G, and G6 hold. Prove that C,, and a suitable clement of C is a group. 5.3 I Definitions of Axiomatic Theories 233 2.9. This exercise is concerned with afine geometry as formulated in Ex- ample 2.3. (a) Prove that "is parallel to" is an equivalence relation on C. An equivalence class is called a pencil of lines. (b) Let a, and 7r2 be two distinct pencils of lines. Using only AG2 and AG3, prove that the number of points on any line 1 of 7r, is the same as the number of lines of 7r2. (c) Using (b), prove that if there exist three distinct pencils of lines, then all lines have the same number of points, all pencils have the same number of lines, and every pencil has the same number of lines as the number of points on every line. (d) From AG6 infer that there exist at least three distinct pencils of lines. (c) Show that the set of four points and six lines given in the text is a model of the theory. (f) Show that any affine geometry contains at least four points and six lines. 2.10. Let S be the axiomatic theory having as its primitive notions two sets P and L and as its axioms the following. A,. As. If l C L, then I -C P. If a and b are distinct elements of P, then there exists exactly one member I of L such that a, b C 1. A3. For every I in L there is exactly one I' in L such that I and I' are disjoint. A4. L is nonempty. A6. Every member of L is finite and nonempty. Establish the following theorems for e. (a) Each member of L contains at least two elements. (b) P contains at least four elements. (c) L contains at least six elements. (d) Each member of L contains exactly two elements. 3. Definitions of Axiomatic Theories by Set-theoretical Predicates We continue our discussion of the axiomatization of intuitive theories with a description of a uniform approach which takes fuller advantage of the expressive powers of general set theory. The point of departure is the observation ( |
which is substantiated, in part, by those theories discussed in Examples 2.1-2.3) that the primitive notions of a great variety of mathematical theories consist of a set X and certain constants 234 Informal Axiomatic Mathematics I CHAP. 5 associated with X. These constants may be of various types: elements of X (such as the identity element of a group), subsets of X, collections of subsets of X (such as the lines of an affine geometry), subsets of X" for some n (which include relations in X and operations in X), and so on. Collectively, the constants serve as the basis for imposing a certain structure on X (which is the object of study of the theory). The structure itself is given in the axioms, which are the properties assigned to X and the constants (including, possibly, the existence of inner relations among them). The approach to the axiomatization of theories which stems from the foregoing observations calls for definitions of axiomatic theories by way of set-theoretical predicates. A consideration of several examples will serve to bring the procedure into focus. In our first example we consider the theory of partially ordered sets. The purely set-theoretical character of the predicate "is a partially ordered set," which is defined should be apparent. DEFINITION A. W is a partially ordered set if there is a set X and a binary relation p such that 4C = (X, p) and p is reflexive in X, p is antisymmetric in X, 01. 02. 03. p is transitive in X. This definition illustrates a convention which we shall follow in this discussion, namely, to exhibit the basic set as the first coordinate of an ordered n-tuple, and the associated constants, in some order, as the remaining coordinates. The sentence in Definition A may be regarded as being in need of recasting if it is to appear in the running text since it begins with a symbol. The following version meets this objection. A partially ordered set is an ordered pair (X, p) where X is a set, p is a binary relation, and the following conditions are satisfied. 01. p is reflexive in X. 02. O. p is antisymmetric in X. p is transitive in X. 5.3 1 Definitions of Axiomatic Theories 235 An alternative to Definition A, which is closer to standard mathe- matical practice, is a conditional |
definition. DEFINITION B. Let X be a set and p be a binary relation. Then (X, p) is a partially ordered set if p is reflexive in X, p is antisyrnmetric. in X, 0t. 02. 03. p is transitive in X. This definition is conditional in the sense that the proper definition is prefaced by a hypothesis. When it definition is so formulated it is common practice to omit the hypothesis in stating theorems of the theory. Our second example is a definition of group theory along the lines suggested by the axiomatization appearing in Example 2.2. DEFINITION C. a binary operation that V = (X,, e) and (i is a group if there is a set X, in X, and an element e of X such G1. Cs. for all a, b, and c in X, a for all a (b c) _ (a b) c, in X there exists an a' in X such that a- a' = a theory Z is axiomatized by defining a set-theoretical predicate, what we have called up to this point the primitive symbols (or terms) of the theory appear in the running text immediately preceding the axioms. Also in this circumstance models of `. are simply those entities which satisfy the predicate. For the theory of groups, for example, the point can be put quite trivially as follows: If (X,, e) is a group, then (X,, e) is a model for the theory of groups. EXERCISES 't'hese exercises are concerned with the theory of simply ordered commutative groups, which may be defined as follows: (SS is it simply ordered continut,ttive group (.r.o.c.g.) iff (SS = (C, 1-, 0, <), where SG,. SG2. SG3. (G, -+, 0) is a commutative group, ((;, <) is a simply ordered set, for all a, b, and e in G, if a < b, then a -- c < b -1- c. (I Iere, "a < b" is an abbreviation for "a < b and a X. b.") 236 Informal Axiomatic Mathematics I C H A P. 5 All results obtained earlier for groups, in particular, commutative groups, |
may be used when needed. Also, properties of simply ordered sets may be used. 3.1. Find two s.o.c.g. within the real number system. 3.2. If (G, +, 0, <) is a.s.o.c.g., define. GE to be (a C GI0 < a). Prove the following properties of G+. (a) If a E G+, then -a (Z GE. (b) If a ; 0, then either a E G+ or -a C G1. (c) If,a,bCG',then a+bCG+. 3.3. Prove the following theorems for a s.o.c.g. (a) Ifa <b,thena-c<b-c. (b) If a + c < b + c, then a < b. (c) If a<bandc<d,then a-i-c<b+d. (d) If a < b, then -b < -a. 3.4. Prove the following theorem. If G has more than one clement and (G, +, 0, <) is a s.o.c.g., then G has infinitely many elements. 4. Further Features of Informal Theories In this section we introduce a variety of notions which have relevance to informal theories. Most of these serve to provide a classification scheme for a given theory. Thereby its status and its merits can be summarized concisely. Suppose that A is a formula of some theory Z and that both A and -1A are theorems. Then, if the system of logic employed includes the statement calculus with modus ponens as a rule of inference, any formula B of the theory is a theorem. Indeed, A (-i A---> B) is a theorem since it is a tautology, and two uses of modus ponens establish B as a theorem. A theory Z is called inconsistent if it contains a formula A such that both A and -i A are theorems. A theory is called consistent if it is not inconsistent-that is, if it contains no formula A such that both A and --,A are theorems. Since in any theory which we shall consider the logical apparatus will include what was used above, we regard an inconsistent theory as worthless, since every formula is a theorem. Thus, the |
question of establishing the consistency of a theory becomes of primary importance. A moment's reflection will point out the high degree of improbability of reaching an answer by direct application of the definition and, consequently, of the need for a "working form" of the definition of consistency. That which is usually adopted in mathematics is: the existence 5.4 I Further Features of Informal Theories 237 of a model of a theory implies the consistency of the theory. The supporting argument is based on (i) the property of a model mentioned at the end of Section 2, namely, if 9x1 is a model of the theory Z, then each theorem of X is true in T Z, and (ii) the assumption that if S is a Z-statement then not both of S and -,S are true in 92. Indeed, assuming (i) has a model TZ. If both of the T-statements S and (ii), suppose that and -,S are theorems, then both S and --,S are true in X71 by (i) and this is a contradiction by (ii). Hence, if Z has a model, then Z is consistent. In essence, the foregoing working form of consistency merely substitutes an inspection of true statements about a model of a theory for an inspection of theorems of the theory. If a model of a theory (X, ) can be found such that the interpretation of X is a finite set, one may expect that the question of whether it is free from contradiction can be settled by direct observation. For example, the fact that ((e},, e), e = e, is a model of group theory establishes the consistency where e of group theory beyond all doubt. If, on the other hand, a theory has only infinite models (that is, models where the interpretations of the basic set are infinite), then no net gain results upon substituting an inspection of true statements about a model for that of theorems of the theory. Such models of a given theory T really amount to interpretations of T in another theory such that the interpretation of each axiom of Z is a theorem of the other theory. If this other theory is consistent, then T must be. For suppose that a contradiction were deducible from the axioms of Z. Then, in the other theory, by corresponding inferences about the objects constituting the model, a contradiction would be deducible from the corresponding theorems. Such demonstrations of consistency are merely relative: The |
theory for which a model is devised is consistent if that from which the model is taken is consistent. Let us consider some examples. As described in Section 1, the plane geometry of Bolyai-Lobachevsky has a model in Euclidean plane geometry. Thereby the relative consistency of this nonEuclidean geometry is established in the form : If Euclidean geometry is consistent, then so is the Bolyai-Lobachevsky geometry. A proof of the consistency of Euclidean geometry, as precisely formulated in Hilbert (1899), can be given by interpreting a point as an ordered pair of real numbers and a line as a linear equation; in more familiar guise this is simply the standard coordinatization of the Euclidean plane. However, since the theory of real numbers has never been proved 238 Informal Axiomatic Mathematics I C H A P. 5 consistent, one may conclude merely that if the theory of real numbers is consistent, then so is Euclidean geometry. In other words, we obtain a relative consistency proof. In turn, since we have seen that a construction of the real numbers can be given, starting from Peano's axioms, within a sufficiently rich theory of sets, a consistency proof of the theory of real numbers can be given relative to a theory which embraces both Peano's theory and this theory of sets. Assuming that the consistency of a theory has been settled in the affirmative by proof or by faith, the question of its completeness may be raised. In rough terms, a theory is called complete if it has enough theorems for some purpose. The variety of purposes which may enter in this connection are responsible for a variety of technical meanings being assigned to this notion. However, most definitions of completeness fit into either the category which corresponds to a positive approach or that which corresponds to a negative approach to the question of a sufficiency of theorems. We shall give one definition in the first category and two in the second. The setting for the first of these, which is in the positive vein, is as follows. We know that if Dl is a model of a theory Z and T is a theorem of Z, then T is true in V. We might regard T as being complete with respect to T1 if, conversely, whenever a Z-statement has a true statement of 1J as its interpretation, then that a-statement is a theorem. This suggests calling Z complete if it is complete with respect to every |
model. If we understand by a (universally) valid statement of a theory one which is true in every model, then the notion of completeness which we have in mind may be formulated as: A theory Z is deductively complete if every valid statement is provable. The statement calculus can be formulated as an of axiomatic theory which is complete in this sense (see Section 9.2); that is, every tautology is a theorem. If we approach the question of a sufficiency of theorems in a negative fashion, we are led to a second category of formulations of completeness. For example, we might say that a theory is complete if the axioms provide all theorems we can afford to have without some dire consequence (such as inconsistency) ensuing. A circumstance which might suggest this interpretation of completeness is an attempt to devise an axiomatic theory intended to formalize some intuitive theory. For then one strives to include sufficient axioms that as many as possible true propositions of the intended model aan be obtained as interpretations of theorems of the theory. Hence, one keeps adding, as axioms, formulas 5.4 I Further Features of Informal Theories 239 which express true propositions of the model up to the point that an inconsistent theory results. This approach to completeness may be crystallized in the following definition. An axiomatic theory Z is formally complete provided that any theory ', which results from Z of a statement of which is not by the adjunction to the axioms of already a theorem of ';', is inconsistent. A theory which is formally complete may be said to have maximum consistency. An axiomatic theory is said to be negation complete if, for any statement A of the theory, either A or --,A is a theorem. It is clear that negation completeness implies formal completeness. Conversely, if the theory of inference employed in developing an axiomatic theory includes a deduction theorem-that is, a theorem which asserts that if a formula., A,,,, then A. B is deducible from formulas A,, A2, B is deducible -then formal completeness implies negation corn, A,,,_, from A,, A2, is formally complete pleteness. To show this, suppose that a theory T. and that the T--statement A is not a theorem. Th.'n the theory which results on the adjunction of A as an axiom is inconsistent. That is, if |
I' is the set of axioms of :?", then a contradiction C can be derived from IF U I A y, whence A - C can be derived from 1'. In turn, since (A --> C) --+ --,A is a theorerrt (being a tautology), A can be derived from A > C. I fence, -, A can be derived from I'; that is Z is negation complete. We may loosely relate consistency and completeness in the following way. An axiomatic theory is consistent if it does not have too many theorems and it is complete if it does not have too few. If an axiomatic theory is both consistent and negation complete, then all questions which arise within the framework of the theory are theoretically decidable in exactly one way. For any statement of the theory is either provable or refutable (that is, its negation is provable) because of completeness, and cannot be both proved and refuted because of consistency. Such a state of affairs for a theory does not always imply that proofs or refutations of specific statements of the theory are automatically made available, but in some interesting cases it does. 't'hat is, for some consistent and complete theories there exists a method which can be described in advance for deciding in a finite number of steps whether a given formula of the theory is a theorem. Such theories are called decidable (sere Section 9.5). Notions of the sort which we have introduced so far in this section as well as that of categoricity (which is described next) cannot, in general, 240 Informal Axiomatic Mathematics ( CFI A P. 5 be discussed in a precise and definitive way at our present intuitive level of discourse. A precise account is possible only when the theory of inference is explicitly incorporated into an axiomatic theory. In Chapter 9 we shall show how this can be done for an important class of theories. Then we shall re-examine for this class the concepts of consistency, completeness, categoricity, and decidability, including interrelations which exist among them. The remaining notion which we shall introduce as an ingredient of a classification scheme for informal theories arises in connection with the purpose for which a theory is devised. If it is intended that an axiomatic theory formalize some one intuitive theory, a natural requirement for the successfulness of the axiomatization is the presence of a theorem to the effect that any two models of the |
theory arc indistinguishable apart from the terminology they employ. In other words, the theory has essentially only one model. For example, one would certainly hope to have such a theorem for any theory designed to formalize Euclidean geometry or the real number system, since we think of each of these as a single clearly delimited theory. A theory is called categorical if it has essentially only one model. This will qualify as a definition as soon as the vague notion that models of a theory are indistinguishable is made precise. The sort of indiscernibility of models which is involved is known as isomorphism. A definition which could cover all conceivable situations would be too unwieldy to attempt. This is the reason for the repeated occurrence of definitions bearing this name. Each is tailored to fit the distinguishing features of the theory under consideration. Already we. have given three such definitions: one for partially ordered sets, one for integral systems, and another which is applicable to systems consisting of a set with two binary operations and an ordering relation. In order to further strengthen the reader's comprehension of the concept and to serve as a vehicle for several general comments, we offer definitions in three specific cases (labeled I,, Ia, and 13). These together with those definitions given earlier should serve to clarify the essence of isomorphism. I,. Let (X,, pi) and (X2, p2) be two models of a theory having a set and a pertinent relation as primitive notions. Then (X,, p,) is isomorphic to (X2, p2) if there exists a function f such that- (i) f is a one-to-one correspondepce between X, and X2, (ii) if x, y C X, and x p, y, then f(x) P2 f(y), (iii) if x, y C X2 and x ply, then f-'(x) pl f-'(y) 5.4 I Further Features of Informal Theories 241 This definition is patterned after that of isomorphism for partially ordered sets (Section 1.11). It is applicable to the case where pi is a function on Xi into Xi, i = 1, 2. In this event the definition of isomorphism can be simplified to the following, as the reader can verify. Let (X1, f,) and (X2, fz) be models of a theory whose primitive notions are a set and a function on that set into itself. Then |
(XI, fl) is isomorphic to (Xz, fz) if there exists a function f such that (i) f is a one-to-one correspondence between X, and Xz, (ii) if x C Xi, then f(f,(x)) = fz(f(x)). Thus, in this case only one of the two requirements for isomorphism must be proved; the other, which completes the symmetry inherent in the concept of isomorphism, necessarily follows. 12. Let (XI, -i) and (Xz, 02) be two models of a theory having a set and a binary operation in that set as its primitive notions. Then (Xi, ^i) is isomorphic to (X2j -z) if there exists a function f such that (i) f is a one-to-one correspondence between X, and Xz, (ii) if x, y C XX, then f(x -,y) = f(x) -z f(y). It is left as an exercise to show that this formulation of isomorphism is an equivalence relation in any collection of models of the theory described. In particular, therefore, as in the specialized version of I, given above, the symmetric nature of the concept follows automatically. Is. Let (Xi, Y,, p,) and (Xz, Y2, P2) be two models of a theory having as its primitive notions two sets and a relation whose domain is the first set and whose range is the second set. Then (X,, Y,, pi) is isomorphic to (X2j Yz, p2) iff there exists a function f such that (i) f is a one-to-one correspondence between X, U Y, and Xz U Yz such that f(XI) = Xz and f(Y,) = Y2, (ii) f preserves the relations p, and P2 in the sense of definition I,. This is not the only definition of isomorphism which might be made under the circumstances. The one given takes into account the preservation of set-theoretical interconnections between Xi and Yi, i = 1, 2. We now define an informal theory to be categorical if any two models of it are isomorphic. In view of Theorem 2.1.8, the theory of integral systems, which was devised to axiomatize the natural |
number 242 Informal Axiomatic Mathematics I CHAP. 5 sequence, is categorical. t This result is one which might be hoped for since the theory is intended to formalize just one intuitive theory. An elementary example of a categorical theory is obtained by adding to the five axioms for affinc geometry (Example 2.3), the following. AG6. The set (P has exactly four members. The resulting theory is consistent by virtue of the model given in Section 2. The proof that it is categorical is left as an exercise. Analogous to the acceptance of the existence of a model as a criterion for consistency, the existence of essentially only one model (that is, categoricity) is often accepted as a criterion for negation completeness. To state the pertinent result we make a definition. A statement of a if it is true in consistent theory T will be called a consequence of every model of T. Then, if Z is a consistent and categorical theory, for each T-statement S, either S is a consequence of Z or -1 S is a consequence of Z. This, it will be noted, amounts to negation completeness with provability replaced by a weaker notion. The proof makes use of the following property of models. If Y, and 9Jtz are isomorphic models of a theory Z, then for every a-statement S, either S is true in both f l1 and 9)22 or S is false in both. Assuming this as proved, the main result can be derived as follows. Suppose that the T-statement S is not a consequence of the consistent theory Z. Then, by the definition of consequence, there exists a model 9N, of Z which does not satisfy S. Let 9N be any model of Z. Then, since 99J1 is isomorphic to 9 lt, S is not true in T2, and, hence -,S is true in P. Since 9t is any model of T, this means that --is is a consequence of T. A theory which is consistent and noncategorical has essentially different (that is, nonisomorphic) models. This is precisely what should be anticipated for a theory intended to axiomatize the common part of several different theories. The theory of groups is an excellent example. Because it has such a general character it has a wide variety of models, which means that it has a wide range of application. We conclude this section with several miscellaneous remarks. |
The first involves assigning a precise meaning to the word "formulation" which we have used frequently. As we described it, an informal theory Z includes a list TO of undefined terms, a list T1 of defined terms, a list P of axioms, and a list Pl of all thqse other statements which can be inferred from Po in accordance with some system of logic. The set TO t Later we shall find it necessary to modify this assertion. 5.4 I Further Features of Informal Theories 243 serves to generate TO U Ti, the set of all technical terms of ;'; the set P0 serves to generate Pu U P1, the set of all theorems of Z. For the ordered pair (To, Po) we propose the name of a "formulation" for T. A study of T may very well culminate in the discovery of other useful formulations. To obtain one amounts to the determination of: (i) a set To' which is a subset of To U T, (which may or may not differ from 7o), and (ii) a subset P of Po U P, whose member statements are expressed in terms of the members of To' and from which the remaining theorems of the theory can be derived. For a pair of the form (To', PP) to be a formulation of T, it is clearly sufficient that the members of To can be defined by means of those in To and that the statements of Po can be derived from those of P. For many of the well-known axiomatic theories there exists a variety of formulations. This is true, for example, of the theory of Boolean algebras discussed in Chapter 6. A rather trivial example appears in Section 1.11, and we may rephrase it to suit our present purposes: As a different formulation of the theory of partially ordered sets we may take that consisting of a set X together with a relation that is it-reflexive and transitive on X (see Exercise 1.11.3). Another example is implicit in a remark made in Section 1; rephrased, it amounts to the assertion that Hilbert and Pieri gave different formulations of a theory which axiomatizes intuitive plane geometry. Different formulations of a theory amount to one variety of possible approaches which can be made to one and the same mathematical structure. Depending on the criteria adopted, one may show a marked preference for one formulation over others. Aesthetic considerations may influence one's judgment, and |
the simplicity of the set of axioms in conjunction with the elegance of the proofs may also play an important role. One may prefer a particular formulation because he feels it has a "naturalness" that others lack. He may favor a formulation which involves the fewest number of primitive notions or axioms. A notion which is pertinent to a formulation of an informal theory is that of the independence of the set of axiomns. A set of axioms is independent if the omission of any one of them causes the loss of a theorem; otherwise it is dependent. A particular axiom (considered as a member of the set of axioms of some formulation) is independent if its omission causes the loss of a theorem; otherwise it is dependent. Clearly, an independent axiom cannot be proved from the others of a set of which it is a member, and conversely. Further, the set of axioms of a formulation is independent iff each of its members is independent. Models may be used to establish the independence of axioms. For 244 Informal Axiomatic Mathematics I c t r A P. 5 example, the independence of the axioms 0r, 02, 03 for the theory of partially ordered sets (see Section 3) may be shown by constructing a model of each of the three theories having exactly two of 0r, 02, and 03 as axioms and in which the interpretation of the missing axiom is false. Otherwise expressed, the independence of 03, for example, is equivalent to the consistency of the theory having 01, 02, and the negation of 03 as axioms. The independence of a set of axioms is a matter of elegance. A dependent set simply contains one or more redundancies; this has no effect on the theory involved. The foregoing concepts of independence for both individual axioms and sets of axioms have analogues for primitive terms. A given primitive term (considered as a member of the set of primitive terms in a formulation of a theory) is independent if it cannot be defined by relation to the remaining primitive terms and a set of primitive terms is indepcndent if each of its members is independent. Models are also used to show such independence in the following way. To prove that a particular primitive symbol Q of some formulation of a theory Z is independent of the remaining primitives, we exhibit two models 9121 and J 2 of $ which have the same domain and in which the interpretation of each primitive term |
except Q is the same but which give different interpretations to the symbol, Q. This is known as Padoa's method for demonstrating definitional independence. A complete account of this method, which is due to the Italian logician, A. Padoa, is given in J. C. C. McKinsey (1935); we shall be content. to consider an example. In the exercises for Section 3 is a formulation of the theory of simply ordered commutative groups. We will show that the binary relation < is an independent primitive. For this we introduce the interpretations I2r and 9)22 in both of which we take G as Z, + as ordinary addition, 0 as zero and, in Mgr we take < to be the familiar relation of less than or equal to, while in 122 we take < to be the familiar relation of greater than or equal to. Then, clearly, the interpretations of < are different (for example, 2 < 3 is true in 912r but false in T22)- We conclude that < cannot be defined in terms of the remaining primitives, for otherwise its interpretation would have to be the same in both models since the other primitives are the same. In order to motivate the final remark we recall'i'heorern 1.11.1, which asserts that every partially ordered set is isomorphic to a collection of sets partially ordered by inclusion. That is, to within isornorphisrn, all models of the theory of partially ordered sets are furnished by collections of sets. In general, a theorem to the effect that for a given 5.4 I Further Features of Informal Theories 245 axiomatic theory Z a distinguished subset of the set of all models has the property that every model is isomorphic to some member of this subset is a representation theorem for Z. Analagous to the case of the theory of partially ordered sets where, from the outset, collections of sets constitute distinguished models, in the case of an arbitrary theory Z, even though it is noncategorical, one particular type of model may seem more natural. In this event a representation problem arisesthe question whether there can be proved a representation theorem for X which assi its that this type of model yields all models to within isomorphism. When such a problem is answered in the affirmative, new theorems may follow for Z by imitating proof techniques that have proved useful in those theories which, in effect, supply all models. EXERCISES |
4.1. (a) Establish the consistency of the theory of partially ordered sets by way of a model. (b) Show that this is a noncategorical theory. (c) Show that the set of axioms {O,, 02, 03) for partially ordered sets is independent. 4.2. (a) Show that the theory of groups is noncategorical. (b) Defining a group as an ordered triple (C,, e) such that G1, G2, and G3 of Example 2.2 hold, establish the independence of {G,, G2, G3}. (Suggestion: Use a multiplication table for displaying the operation which you introduce into any set.) 4.3. Consider the axiomatic theory having as its primitive notions two sets A and (Id and having as axioms the following. (i) Each element of B is a two-element subset of A. (ii) If a, a' is a pair of distinct elements of A, then {a, a') C B. (iii) A V B. (iv) If B, B' is a pair of distinct elements of B, then B (1 B' C A. Show that this theory is consistent. Is it categorical? 4.4. Consider the axiomatic theory whose primitive notions are a nonempty set A and a binary operation (x, y) -9- x - y (that is, we write the image of (x, y) as x - y) in A, which satisfies the identity y=x-[(x-z)-(y-z)]. Show that this theory is consistent. 4.5. Consider the axiomatic theory whose primitive notions are a nonempty x X y in A, and a unary operation x-4- x' set A, a binary operation (x, y) in A. The axioms are the following. (i) X is an associative operation. (ii) (x X y)' = y' X x'. 246 Informal Axiomatic Mathematics CHAP. 5 (iii) IIxXv= zXi for some z, then x=y. (iv) If x = y', then x X y = z Xi for all z. (a) Show that the theory is consistent. (b) Show that this set of axioms is dependent. 4 6. Prove the assertion made in the |
text to the effect that if p; is a function or. A, into X;, i = 1, 2, then (X,, pl) is isomorphic to (X2, p2), provided there exists a one-to-one correspondence f : X, -+- X2 such that f (pi(x)) = ps(f (x)) for all x in X. 4.7. Prove that the type of isomorphism labeled 12 is an equivalence relation in any set whose members are systems consisting of a set together with an operation in that set. 4.8. Assume that of two isomorphic models of the theory considered in Ex- ercise 4.4, one is a group. Prove that the other is a group. 4.9. The set {e, a, b, c} together with the operation defined by the following multiplication table is a group. Determine six isomorphisms of this group with itself.10. Devise a definition of isomorphism for systems consisting of a set to- gether with two operations. 4.11. Consider an axiomatic theory Z formulated in terms of two sets, whose members are called points and lines, respectively, and whose axioms are as follows. (i) Each line is a nonempty set of points. (ii) The intersection of two lines is a point. (iii) Each point is a member of exactly two lines. (iv) There are exactly four lines. (a) Show that Z is a consistent theory. (b) Show that there are exactly six points in a model of Z. (c) Show that each line consists of exactly three points. (d) Find two models of T. (e) Is T categorical? Give reasons for your answer. 4.12. Show that the axiomatic theory defined in Exercise 4.4 is a formulation of the theory of commutative groups. 4.13. Show that the axiomatic theory defined in Exercise 4.5 is a formulation of the theory of groups. 4.14. Show that the following is another formulation of the theory of groups. Bibliographical tote 247 A group is an ordered triple (C,, ') such that G is a set, in C,'is a unary operation in C, and is a binary operation (i) C is nonempty, is associative, (ii) (iii) a' (a b) |
= b = (b a) a' for all a and b. 4.15. Show that the following is another formulation of the theory of groups. is a binary operation A group is an ordered triple (C,, e) such that C is a set, in C, e is a member of C, and is an associative operation, (i) (ii) for each a in C, e a = a, and there exists a' in G such that a' a = e. 4.16. Consider the theory whose primitive notions are a set X, a binary opera- tion in X, and whose axioms are the following. is an associative operation. (i) X is nonempty. (ii) (iii) To each element a in X there corresponds an element e of X such that e = a, and a possesses an inverse a' relative to e in X (that is, e a = a Show that if (S, ) is a model of the theory, then there exists a partition of S such that each member set determines a group. 4.17. Consider the theory T whose primitive notions are the power set of a set S and a mapping f on (P(S) into itself, and whose axioms are as follows. (i) For all.Y in (P(S), Xf 13 X. (ii) For all X in P(S), (Xf)f = X'. (iii) For all X and Y in PPS), X D Y implies Xf 1) Y'. Show that another formulation of T. results on adopting as the sole axiom: (X U Y)f D (Xf)f U Yf U Y, for all X and Y in 6'(S). BIBLIOGRAPHICAL NOTE Discussions of axiomatic theories and the axiomatic method, pitched at about the same level as ours, appear in R. L. Wilder (1952), E. It. Stabler (1953), and A. Tarski (1941). CHAPTER 6 Boolean Algebras T; r. THEORY or Boolean algebras has historical as well as present-day practical importance. For the beginner its exposition should prove a serviceable vehicle for assimilating many of the concepts discussed in relation to informal theories in Chapter 5. Moreover, it illustrates the important type of axiomatic theory known as an "algebraic theory |
." The theory of Boolean algebras is, on one hand, relatively simple and, on the other hand, exceedingly rich in structure. Thus, its detailed study serves in some respects as an excellent introduction to techniques which one may employ in the development of a specific axiomatic theory. The only possible shortcoming is that the ease with which it may be put into a relatively completed form is somewhat misleading, so far as axiomatic theories in general are concerned. This chapter presents first a natural formulation of the theory. Then a formulation which is commonly regarded as being more elegant is given. This second formulation is used in the development of the next topic, the representation of Boolean algebras as algebras of sets. Next, it is shown that a statement calculus determines a Boolean algebra in a natural way. It is by way of this Boolean algebra associated with a statement calculus that statement calculi can be analyzed by so-called Boolean methods and interconnections be established between the theory of Boolean algebra-, and that of statement calculi. This is developed in the last three sections of the chapter. 1. A Definition of a Boolean Algebra By an algebra of sets based on U we shall mean a nonempty collection a of subsets of the rronempt.y set U such that if A, B E a, then A U B, A fl B C a, and if A C a, then fl C a. For example, the power set of U, (P(U), is an algebra of sets. However, certain proper subsets of (p(U) may be an algebra of sets (see Exercise 2.6). If a is an algebra of sets based on U, then U C a (since if A C a, then U = A U A C a) and 0 C a (since if A C a, then 0 = A n A E a). Further, Theorem 248 6.1 I A Definition of Boolean Algebra 249 1.5.1 may be interpreted as a list of properties of an algebra of sets. That this is a fundamental list of properties is suggested by the variety of other properties (for example, those in Theorem 1.5.2) which may be deduced solely from them. As formulated below, the theory of Boolean algebras may be regarded as the axiomatized version of algebras of sets when viewed as systems having the properties appearing in Theorem 1.5.1. A Boolean |
algebra is a 6-tuplc (B, U, n, ', 0, 1), where B is a set, U is a binary operation (called union or join) in B, n is a binary operation (called intersection or meet) in B,'is a binary relation in B having B as its domain, 0 and 1 are distinct elements of B, and the following axioms are satisfied. (i) Each operation is associative: for all a, b, c C B, aU (bUc) _ (aUh)Uc and an (bnc) _ (anb)nc. (ii) Each operation is commutative: for all a, b E B, aUb=bUa and anb=bna (iii) Each operation distributes over the other: for all a, b, c C B, and (iv) For all a in B, aU (bnc) = (aUb) n (aUc) an (bUc) = (anb) U (anc). aUO=a and and =a. (v) For each a in B there exists a '-related element a' such that a U a' = I and ana'=0. The consistency of the theory that we have just formulated can be established by choosing for B the power set of a nonempty, finite set U,'as taking U and n as set-union and set-intersection, respectively, complementation relative to If, and, finally, choosing 0 and 1 as 0 and U, respectively. The uniqueness of the elements 0 and I is established in Theorem 2.1. These uniquely determined elements are called the zero element and unit element, respectively, of a Boolean algebra. It was in anticipation of this uniqueness and terminology that the symbols "0" and "1" were used in the axioms. We might have postulated their uniqueness; however, we would then be obligated to prove uniqueness as part of any verification that an alleged Boolean algebra is truly just that. An element which is '-related to an element a is called a complement of a; that each element has a unique complement (and, 250 Boolean Algebras I CHAP P. 6 hence, that'is a function having B as its domain) is proved below. The set of axioms is not independent, since the two associative laws can be derived from the remaining axi |
oms. A hint as to how this can be done is given in an exercise accompanying the next section. If the set of remaining axioms is regarded as having seven members, which is the case when each of (ii)-(iv) is divided into two parts, then it is an independent set of axioms. This fact, which is interesting but unimportant, was established by E. V. Huntington (1904) with appropriate models. EXERCISES 1.1. Accepting for the moment the fact that the associative laws (i) in the formulation of the theory of Boolean algebras are redundant, the independence of the remaining set of seven axioms can be demonstrated by a collection of seven systems of the form (B, U, n,', 0, 1), one of which satisfies (ii)-(v) except the commutativity of U, another of which satisfies (ii)-(v) except the commutativity of n, and so on. For a B having just a few elements, an operation in B can be defined by means of a "multiplication table," that is, a square array whose rows and columns are numbered with the elements of B and such that at the intersection of the ath row and the bth column the composite of a and b appears. For example, the following two tables define two operations in the set B = {a, b}. Show that (B, y, n,', 0, 1)-where B = {a, b}, U and n are defined as above,'is the relation {(a, b), (b, a)) (that is, a' = b and b' = a), 0 is a and 1 is b-satisfies all of (ii)-(v) except the first half of (iii), thereby demonstrating the independence of this axiom. Next, show that the system which results from the foregoing upon substituting the multiplication tables for U and n establishes the independence of the second half of (iii). 1.2. Construct five other systems which demonstrate the independence of the other axioms. 2. Some Basic Properties of a Boolean Algebra The properties of a Boolean algebra which are derived in this section are the abstract versions of the results obtained in Section 1.5 for an, 6.2 1 Some Basic Properlies of a Boolean Algebra 251 algebra of sets. The only essential difference is that now the set of axioms of a Boolean algebra is used |
in place of the first theorem of that earlier section. We begin by describing the principle of duality for Boolean algebras. By the dual of a statement formulated within the framework of a Boolean algebra is meant the statement that results from the original upon the replacement of U by n and n by U, I by 0 and 0 by 1. We observe that each axiom is a dual pair of statements, with (v) regarded as self-dual. Hence, if 1' is any theorem of Boolean algebras, then the dual of T is a theorem, the duals of the steps appearing in the proof of 7' providing a proof of the dual. This is the principle of duality for the theory at hand; it yields a free theorem for each theorem which has been obtained, unless that theorem happens to be its own dual. Turning to theorems of the theory of Boolean algebras, we mention first the validity of the general associative law and the general commutative law for each operation, as well as the general distributive law for each operation with respect to the other. Theorem 2.2.2, Exercise 2.2.4, and Exercise 2.2.5 dispose of these matters. The next group of results, which make up our next theorem, is the Boolean algebra version of Theorem 1.5.2. THEOREM 2.1. following hold. In each Boolean algebra (B, U, 0, 1) the (vi) The elements 0 and 1 are unique. (vii) Each element has a unique complement. (viii) For each element a, (a')' = a. (ix) 0' = 1 and 1' = 0. (x) For each element a, a U a = a and a n a = a. (xi) For each element a, a U 1 = 1 and a n 0 = 0. (xii) For all a and b, a U (a n b) = a and a n (a u b) = a. (xiii) For all a and b, (a U b)' = a' n b' and (a n b)' = a' U Y. Proof. For (vi) assume that Ol and 02 are elements of B such that a U OL =a and a U 02 = a for all a. Then 02 U Oi = 02 and OL U 02 = 01. By axiom (ii |
), 02 U 01 = 01 U 02, and, hence, 02 = 01. Thus there is a single element in B satisfying the first property in (iv). (The uniqueness of 1 follows by the principle of duality.) For (vii) assume that a,' and a. are both complements of a. Then a. = a. U 0, = al U (a 0 a6), by (iv); since anal = 0; 252 Boolean Algebras CHAP. 6 = (a'. U a) n (al' U as), = (a U ai) n (a,' U as), = 1 n (ai U ai), _ (al' U ae) n 1, a, = By a similar proof we get U al 2) az=a'Uat. by (iii) ; by (ii); since a U a,' = 1; by (ii) ; by (iv). Hence, by (ii), a, = aa. For (viii), by definition of the complement of a, a U a' = 1 and a n a' = 0. Hence, by (ii), a' U a = 1 and a' n a = 0. That is, (a')' = a, by (vii). The proof of (ix) is left as an exercise. The proof of (x) is the following computation. aUa=(aUa)n1, _ (a U a) n (a U a'), =aU(ana'), =aUO, = a, by (iv) ; by (v); by (iii) ; by (v); by (iv). The proofs of the remaining parts of the theorem are left as exercises. The property of cornpleuientation stated as (vii) means that { (a, a')Ia C B) is a function on B into B (that is, complementation is a unary operation in B). According to (viii) this function is of period 2 and, consequently, one-to-one and onto. It is possible to introduce into the set B of an arbitrary Boolean algebra (B, U, n,', 0, 1) a partial ordering relation which resembles that of set inclusion. The characterization of inclusion in Theorem 1.5.3 in terms of set intersection is the origin of the following definition. If (B, U, n, ', O, |
1) is a Boolean algebra, then for a, b E B a <b if anb =a. There is no need to give preference to the meet operation, since, just as for the algebra of sets, anb=a if aUb=b. The proof of this as well as the proofs of such related facts as a < b iff a nb' = 0 and a < b if b' < a' are left as exercises. Important features of the new relation are stated in the next theorem. 6.2 I Some Basic Properties of a Boolean Algebra 253 THEOREM 2.2. If (B, U, n, ', 0, 1) is a Boolean algebra, then (B, <) is a partially ordered set with greatest element (namely, 1) and least element (namely, 0). Moreover, each pair { a, b } of elements has a least upper bound (namely, a U b) and a greatest lower bound (namely, a n b). The proof is straightforward and is left as an exercise. EXERCISES 2.1. Referring to Theorems 1.5.2 and 2.1, it is obvious that (viii)-(xiii) of Theorem 2.1 are the abstract versions of 8, 8'-13, 13' of Theorem 1.5.2. Show that (vi) and (vii) of Theorem 2.1 are the abstractions of 6, 6' and 7, 7', respectively, of Theorem 1.5.2. 2.2. Supply proofs for parts (ix), (xi), (xii), and (xiii) of Theorem 2.1. 2.3. In regard to a proof of the assertion that the associative laws for U and n can be derived from the remaining axioms for a Boolean algebra, we observe first that the given proofs of (vi)-(viii) and (x) do not employ (i). Further, the proofs of (ix), (xi), and (xii) called for in the preceding exercise need not use (i). Hence, (ii)-(xii) are available to prove (i). Supply such a proof. Hint: Given a, b, and c, define x=aU(bUc) and y=(aUb)Uc, and then deduce, in turn, that all x = any |
,a,nx=a'ny,x=y. 2.4. Establish each of the following as a theorem for Boolean algebras. (a) a<biffaUb=b. (b) a <biffanb' = 0iffa'Ub = 1. (c) a<bifb'<a'. (d) For given x and y, x = y if 0 = (x n y') U (y n x'). 2.5. Prove Theorem 2.2. 2.6. Let d be the collection of all subsets A of Z+ such that either A or A is finite. Show that (a, u, n, ^, 0, z+), where the operations are the familiar set-theoretical union and intersection, is a Boolean algebra. Remark. The remaining problems in this section are concerned with a type of generalization of a Boolean algebra called a lattice. A lattice is a triple (X, U, n), where X is a nonempty set, U and n are binary operations in X (read "union" and "intersection," respectively), and the following axioms are satisfied. For all a, b, c C X, L,. aU(bUc)_(aUb)Uc, U. an(bnc)_(a() b)flc, L2. aUb=bUa, L3. U. anb=bna, U. (a fl b) U a = a. (a U b) n a = a, 254 Boolean Algebras I C11 A P. 6 2.7. State and prove a principle of duality for a lattice. 2.8. Derive the following properties of a lattice. (a) For all a, a U a = a andafla = a. (b) For all a, b, the relations a U b = a and a () b = b are equivalent. (c) For all a, b, the relations a (1 b = a and a U b = b are equivalent. 2.9. Let (X, <) be a partially ordered set such that each pair of elements has a least upper bound and a greatest lower bound in X. Thus, if we set a U b = lub {a, b} and a (1 b = gib {a, b |
}, then U and (1 are operations in X. Prove that (X, U, fl) is a lattice. Next, prove that, conversely, if in a lattice (X, U, (l) we define the relation < by a < b if a (1 b = a, then (X, <) is a partially ordered set such that each pair of elements has a least upper bound (namely, a U b) and a greatest lower bound (namely, a fl b). Remark. This result gives, in effect, a second formulation of the axiomatic theory called lattice theory. Thus, one may think of a lattice in either way. If the formulation is in terms of <, then, by U and (l, one understands the operations in Exercise 2.9. If the formulation is in terms of U and fl, then, by <, one understands the ordering relation defined, again, in Exercise 2.9. 2.10. Let (X, U, fl) and (X', y', fl') be lattices. Show that they are isomorphic (using the definition of isomorphism suggested by 12 in Section 5.4) iff the associated partially ordered sets (X, <) and (X', <') are isomorphic (using the definition in Section 1.11). 2.11. Show that there are exactly five nonisomorphic lattices of fewer than five elements and that there are exactly five nonisomorphic lattices of five elements. (Hint: For this problem it is more convenient to think of a lattice as a partially ordered set.) 3. Another Formulation of the Theory The formulation which we have given of the theory of Boolean algebras has much to recommend it. The primitive notions are few, and the simplicity and symmetry of the axioms lend aesthetic appeal. Moreover, if the associative laws are omitted, the resulting set is independent. Finally, the formulation clearly reflects the type of system that motivated it. However, it is always a challenge to see if a formulation can be pared down in one or more respects. In the case of Boolean algebras this challenge has been successfully met by a great variety of formulations. We shall describe one that has become quite popular. It achieves for arbitrary Boolean algebras the analogue of the familiar fact for an algebra of sets that either of the operations of union and intersection can be eliminated in terms of the other together with |
complementation [for example, A U B = (A n B)). 6.3 1 Another Formulation of the Theory 255 If (B, V, n,', 0, 1) is a Boolean algebra, then B is a set with at least two distinct members. Moreover, the binary operation fl and the unary operation'have the following properties. fl is commutative. fl is associative. For a,binB,ifaflb'=cflc'forsome cinB,then aflb=a. For a,binB,ifaflb=a,then aflb'=cflc'forallcinB. The first two properties are axioms, and the last two follow from the facts that for all c in B,cflc' = 0, andaflb' = 0iffaflb = a. We shall prove next that a triple (B, fl, ') having the properties mentioned above (a precise description appears in the next theorem) may be taken as a formulation of the theory of Boolean algebras. That is, the primitive notions of the initial formulation of the theory can be defined and the axioms (i)-(v) can be derived as theorems. THEOREM 3.1. The following is a formulation of the theory of Boolean algebras. The primitive notions are an unspecified set B of at least two elements, a binary operation fl in B, and a unary operation'in B. The axioms are as follows. B1. fl is a commutative operation. fl is an associative operation. B2. B3. For all a, b in B, if a fl b' = c fl c' for some c in B, then aflb=a. B4. For all a,binB,ifaflb=a,then aflb' =cflc'forallc in B. It remains to prove that the primitive notions of the original Proof. formulation can be defined and the axioms derived from a triple (B, fl, ') satisfying B1--B4. As the undefined set, the meet operation, and the binary relation'of the original formulation we take B, fl, and ', respectively. A Join operation and the distinguished elements 0 and 1 are defined below. The first ten results (TI-T 10) which we prove, al.)uut |
(13, fl, '), together with 13, and 133, establish the validity of all axioms of the original formulation except the distributive laws. The remainder of' the proof is concerned with them. A,telegraphic style of presentation is used for ease in reading. Pr. anal = anal. Now apply 113. Ti. T2. afla=a. afla'=bflb'. Pr. T1 and 134. 256 Boolean Algebras ( CHAP. 6 This result justifies the following definition. 0 = a fl a' and 1 = 0'. an0=0. Dl. T3. Pr. afl0=afl(afla'), = (ana)na', T4. Pr. 3. 4. 2. 0, all = 'a. 1. a" l a' = 0, a" fla=a", a"" n a" = a.". a"" al", n a a"', na'=0, a' n a". = a' a," n a' = a = a, ana"'=0, ana"a, 7. 6. 9. 5. 8., 10. by DI; by B2; by TI and DI. from D1 and B2. from I by B3. from 2. from 2 and 3, by B2. from 4, by B4 and Dl. from 5, by B, and B3. from 2. from 6 and 7. from 8 and D1. from 9 by B3. from 2 and 10, by B1. 11. T5. Pr. 5. D2. T7. Pr. T8. Pr. T9. Pr. TI0. Pr. afl(ana')"=0, by T4, Ti, and DI, from the above, by B3. by Dl. a" = a, afll =a. afl(afla')'=a, afll =a, 001. Assume 0 = 1. a fl 0 = a, an0 = 0, a = 0, This contradicts the assumption that there exist at least two distinct elements in B. a U b = (a'nb')'. (a U b)' = a' fl b' and (a fl b)' = a'Ub'. from 1 and T5. by T3. from 2 and 3. |
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