| { | |
| "domains": ["modal_logic", "order_theory", "group_theory", "set_theory"], | |
| "lexicon": { | |
| "assumptions": { | |
| "assume:transitive": { | |
| "positive": ["transitive", "推移的", "推移性", "推移律", "連鎖的"], | |
| "negative": ["not transitive", "non-transitive", "反推移", "推移的ではない"], | |
| "description": "R is transitive: (wRv and vRu) => wRu" | |
| }, | |
| "assume:reflexive": { | |
| "positive": ["reflexive", "反射的", "反射性", "自己到達", "wRw"], | |
| "negative": ["not reflexive", "non-reflexive", "反射的ではない"], | |
| "description": "R is reflexive: for all w, wRw" | |
| }, | |
| "assume:symmetric": { | |
| "positive": ["symmetric", "対称的", "対称性", "双方向"], | |
| "negative": ["not symmetric", "non-symmetric", "非対称"], | |
| "description": "R is symmetric: wRv => vRw" | |
| }, | |
| "assume:euclidean": { | |
| "positive": ["euclidean", "ユークリッド的", "ユークリッド性"], | |
| "negative": ["not euclidean", "non-euclidean"], | |
| "description": "R is euclidean: wRv and wRu => vRu" | |
| }, | |
| "assume:antisymmetric": { | |
| "positive": ["antisymmetric", "反対称的", "反対称性"], | |
| "negative": ["not antisymmetric"], | |
| "description": "xRy and yRx => x=y" | |
| }, | |
| "assume:associative": { | |
| "positive": ["associative", "結合的", "結合律"], | |
| "negative": ["not associative", "非結合的"], | |
| "description": "(a * b) * c = a * (b * c)" | |
| }, | |
| "assume:commutative": { | |
| "positive": ["commutative", "可換", "交換法則"], | |
| "negative": ["not commutative", "非可換"], | |
| "description": "a * b = b * a" | |
| } | |
| } | |
| }, | |
| "correspondence": { | |
| "axioms": { | |
| "axiom:T": { | |
| "requires": ["assume:reflexive"], | |
| "schema": "[]P -> P", | |
| "description": "Reflexivity corresponds to Axiom T." | |
| }, | |
| "axiom:4": { | |
| "requires": ["assume:transitive"], | |
| "schema": "[]P -> [][]P", | |
| "description": "Transitivity corresponds to Axiom 4." | |
| }, | |
| "axiom:5": { | |
| "requires": ["assume:euclidean"], | |
| "schema": "<>P -> []<>P", | |
| "description": "Euclidean property corresponds to Axiom 5." | |
| }, | |
| "axiom:B": { | |
| "requires": ["assume:symmetric"], | |
| "schema": "P -> []<>P", | |
| "description": "Symmetry corresponds to Axiom B." | |
| }, | |
| "axiom:group_identity": { | |
| "requires": ["assume:group_structure"], | |
| "schema": "e * x = x", | |
| "description": "Identity element existence." | |
| } | |
| }, | |
| "schemas": { | |
| "[]p->p": { "explain": "T-axiom (Reflexivity)", "supports": ["axiom:T"] }, | |
| "[]p->[][]p": { "explain": "4-axiom (Transitivity)", "supports": ["axiom:4"] }, | |
| "<>p->[]<>p": { "explain": "5-axiom (Euclidean)", "supports": ["axiom:5"] }, | |
| "p->[]<>p": { "explain": "B-axiom (Symmetry)", "supports": ["axiom:B"] } | |
| }, | |
| "missing_assumption_hints": { | |
| "[]p->p": ["assume:reflexive"], | |
| "[]p->[][]p": ["assume:transitive"], | |
| "<>p->[]<>p": ["assume:euclidean"], | |
| "p->[]<>p": ["assume:symmetric"] | |
| } | |
| }, | |
| "structures": { | |
| "order:partial_order": { | |
| "name": "Partial Order", | |
| "assumptions": ["assume:reflexive", "assume:antisymmetric", "assume:transitive"], | |
| "intuition": "A set with a hierarchy where not all elements need to be comparable." | |
| }, | |
| "algebra:group": { | |
| "name": "Group", | |
| "assumptions": ["assume:associative", "assume:has_identity", "assume:has_inverse"], | |
| "intuition": "A set with an operation that allows symmetry and reversibility." | |
| } | |
| }, | |
| "counterexample_notes": { | |
| "missing_reflexive_breaks_T": { | |
| "explain": "反射性が無いと wRw が保証されないため、□P が真（到達先がない）でも P が偽になる世界が存在しうる。" | |
| }, | |
| "missing_transitive_breaks_4": { | |
| "explain": "推移性が無いと、2歩先の到達点が見えないため、□P から □□P が導けない。" | |
| } | |
| } | |
| } |