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#set page(margin: 1cm, columns: 2)
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#show outline.entry.where(level: 1): it => {
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upper(it)
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}
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#set heading(numbering: "1.")
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#show outline.entry.where(level: 1): it => {
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v(12pt, weak: true)
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strong(it)
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}
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#set enum(numbering: "a1Ai)")
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#outline(indent: 1em)
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#set enum(numbering: "a)")
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= Logical axiom schemes <logical-axiom-schemes>
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// #show outline.entry.where(
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// level: 1
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// ): it => {
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// v(12pt, weak: true)
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// strong(it)
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// }
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$bold(L_1): B->(C->B)$
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#outline(indent: auto)
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$bold(L_2): (B->(C->D))->((B->C)->(B->D))$
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= Logical axiom schemes
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$bold(L_3): B and C->B$
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$bold(L_1): B→(C →B)$
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$bold(L_4): B and C->C$
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$bold(L_2): (B→(C →D))→((B→C)→( B→D))$
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$bold(L_5): B->(C->B and C)$
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$bold(L_3): B∧C→B$
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$bold(L_6): B->B or C$
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$bold(L_4): B∧C→C$
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$bold(L_7): C->B or C$
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$bold(L_5): B→(C →B∧C)$
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$bold(L_8): (B->D)->((C->D)->(B or C->D))$
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$bold(L_6): B→B∨C$
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$bold(L_9): (B->C)->((B->not C )-> not B)$
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$bold(L_7): C →B∨C$
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$bold(L_10): not B->(B->C)$
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$bold(L_8): (B→D)→((C →D)→(B∨C →D))$
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$bold(L_11): B or not B$
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$bold(L_9): (B→C)→((B→¬C )→¬B)$
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$bold(L_12): forall x F (x)->F (t) ("in particular," forall x F (x)->F(x)$)
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$bold(L_10): ¬B→( B→C)$
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$bold(L_13): F(t)->exists x F(x) ("in particular," F (x)->exists x F(x))$
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$bold(L_11): B∨¬B$
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$bold(L_14): forall x(G ->F (x))->(G->forall x F(x))$
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$bold(L_12): ∀x F (x)→F (t)$ (in particular, $∀x F (x)→F (x)$)
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$bold(L_15): forall x(F(x)->G)->(exists x F(x)->G)$
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$bold(L_13): F (t)→∃ x F( x)$ (in particular, $F (x)→∃ x F (x)$)
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$bold(L_14): ∀x(G →F (x))→(G→∀x F (x)) $
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$bold(L_15): ∀x(F (x) arrow G) arrow (exists x F (x)→G)$
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= Theorems
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= Theorems <theorems>
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== Prooving directly
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$[L_1, L_2, #[MP]]$:
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$[L_1, L_2, M P]$:
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+ $((A→B)→(A→C))→(A→(B→C))$. Be careful when assuming hypotheses: assume
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(A→B)→(A→C), A, B – in this order, no other possibilities!
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+ $(A→B)→((B→C)→(A→C))$. It's another version of the *Law of Syllogism* (by
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+ $((A->B)->(A->C))->(A->(B->C))$. Be careful when assuming hypotheses: assume
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$(A->B)->(A->C), A, B$ -- in this order, no other possibilities!
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+ $(A->B)->((B->C)->(A->C))$. It's another version of the *Law of Syllogism* (by
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Aristotle), or the transitivity property of implication.
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+ $(A→(B→C))→(B→(A→C))$. It's another version of the *Premise Permutation Law*.
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Explain the difference between this formula and Theorem 1.4.3(a): A→(B→C)├
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B→(A→C).
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+ $(A->(B->C))->(B->(A->C))$. It's another version of the *Premise Permutation
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Law*.
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== Deduction theorems
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== Deduction theorems <deduction>
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=== Theorem 1.5.1 (Deduction Theorem 1 AKA DT1)
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=== Theorem 1.5.1 (Deduction Theorem 1 AKA $D T 1$)
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If $T$ is a first order theory, and there is a proof of
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$[T, #[MP]]: A_1, A_2, dots, A_n, B├ C$, then there is a proof of
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$[L_1, L_2, T, #[MP]]: A_1, A_2, dots, A_n├ B→C$.
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$[T, M P]: A_1, A_2, ..., A_n, B tack.r C$, then there is a proof of
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$[L_1, L_2, T, M P]: A_1, A_2, ..., A_n tack.r B->C$.
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=== Theorem 1.5.2 (Deduction Theorem 2 AKA DT2)
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=== Theorem 1.5.2 (Deduction Theorem 2 AKA $D T 2$)
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If there is a proof $[T, #[MP], #[Gen]]: A_1, A_2, dots, A_n, B├ C$, where,
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If there is a proof $[T, M P, G e n]: A_1, A_2, ..., A_n, B tack.r C$, where,
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after B appears in the proof, Generalization is not applied to the variables
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that occur as free in $B$, then there is a proof of
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$[L_1, L_2, L_14, T, #[MP], #[Gen]]: A_1, A_2, dots, A_n├ B→C$.
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$[L_1, L_2, L_14, T, M P, G e n]: A_1, A_2, ..., A_n tack.r B->C$.
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== Conjunction
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== Conjunction <conjunction>
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=== Theorem 2.2.1.
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+ (C-introduction): $[L_5, #[MP]]: A, B├ A∧B$;
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+ (C-elimination): $[L_3, L_4, #[MP]]: A∧B ├ A, A∧B ├ B$.
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+ (C-introduction): $[L_5, M P]: A, B tack.r A and B$;
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+ (C-elimination): $[L_3, L_4, M P]: A and B tack.r A, A and B tack.r B$.
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=== Theorem 2.2.2.
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+ $[L_1, L_2, L_5, #[MP]]: (A→(B→C)) ↔ ((A→B)→(A→C))$ (extension of the axiom
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+ $[L_1, L_2, L_5, M P]: (A->(B->C)) <-> ((A->B)->(A->C))$ (extension of the axiom
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L_2).
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+ $[L_1-L_4, #[MP]]: (A→B)∧( B→C)→( A→C)$ (another form of the *Law of Syllogism*,
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or *transitivity property of implication*).
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+ $[L_1-L_4, M P]: (A->B) and (B->C)->(A->C)$ (another form of the *Law of
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Syllogism*, or *transitivity property of implication*).
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=== Theorem 2.2.3 (properties of the conjunction connective).
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$[L_1- L_5, #[MP]]$:
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$[L_1-L_5, M P]$:
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+ $A∧B↔B∧A$ . Conjunction is commutative.
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+ $ A∧(B∧C)↔( A∧B)∧C$. Conjunction is associative.
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+ $A∧A↔A$ . Conjunction is idempotent.
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+ $A and B<->B and A$ . Conjunction is commutative.
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+ $ A and (B and C)<->( A and B) and C$. Conjunction is associative.
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+ $A and A<->A$ . Conjunction is idempotent.
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=== Theorem 2.2.4 (properties of the equivalence connective).
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$[L_1- L_5, #[MP]]$:
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$[L_1- L_5, M P]$:
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+ $A↔A$ (reflexivity),
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+ $(A↔B)→(B↔A)$ (symmetry),
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+ $(A↔B)→((B↔C) →((A↔C))$ (transitivity).
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+ $A<->A$ (reflexivity),
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+ $(A<->B)->(B<->A)$ (symmetry),
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+ $(A<->B)->((B<->C) ->((A<->C))$ (transitivity).
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== Disjunction
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== Disjunction <disjunction>
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=== Theorem 2.3.1
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+ (D-introduction)$[L_6, L_7, #[MP]]: A├ A∨B; B├ A∨B$;
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+ (D-elimination) If there is a proof $[T, #[MP]]: A_1, A_2, ..., A_n, B├ D$, and
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a proof $[T, #[MP]]: A_1, A_2, ..., A_n, C├ D$, then there is a proof $[T,
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L_1, L_2, L_8, #[MP]]: A_1, A_2, dots, A_n, B∨C ├ D$.
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+ (D-introduction)$[L_6, L_7, M P]: A tack.r A or B; B tack.r A or B$;
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+ (D-elimination) If there is a proof $[T, M P]: A_1, A_2, ..., A_n, B tack.r D$,
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and a proof $[T, M P]: A_1, A_2, ..., A_n, C tack.r D$, then there is a proof $[T,
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L_1, L_2, L_8, M P]: A_1, A_2, ..., A_n, B or C tack.r D$.
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=== Theorem 2.3.2
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a) $[ L_5, L_6-L_8, #[MP]]: A∨B↔B∨A$ . Disjunction is commutative. b) $[L_1, L_2, L_5, L_6-L_8, #[MP]]: A∨A↔A$ .
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Disjunction is idempotent.
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+ $[L_5, L_6-L_8, M P]: A or B<->B or A$. Disjunction is commutative.
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+ $[L_1, L_2, L_5, L_6-L_8, M P]: A or A<->A$. Disjunction is idempotent.
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=== Theorem 2.3.3.
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=== Theorem 2.3.3
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Disjunction is associative: $[L_1, L_2, L_5, L_6-L_8, #[MP]]: A∨(B∨C)↔(
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A∨B)∨C$.
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Disjunction is associative: $[L_1, L_2, L_5, L_6-L_8, M P]: A or (B or C)<->(A or B) or C$.
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=== Theorem 2.3.4.
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=== Theorem 2.3.4
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Conjunction is distributive to disjunction, and disjunction is distributive to
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conjunction:
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+ $[L_1-L_8, #[MP]]: (A∧B)∨C ↔(A∨C)∧(B∨C)$ .
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+ $[L_1-L_8, #[MP]]: (A∨B)∧C ↔(A∧C)∨(B∧C)$ .
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+ $[L_1-L_8, M P]: (A and B) or C <->(A or C) and (B or C)$ .
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+ $[L_1-L_8, M P]: (A or B) and C <->(A and C) or (B and C)$ .
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=== Theorem 2.3.4.
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=== Theorem 2.3.4
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Conjunction is distributive to disjunction, and disjunction is distributive to
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conjunction:
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+ $[L_1-L_8, #[MP]]: (A∧B)∨C ↔(A∨C)∧(B∨C)$;
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+ $[L_1-L_8, #[MP]]: (A∨B)∧C ↔(A∧C)∨(B∧C)$ .
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+ $[L_1-L_8, M P]: (A and B) or C <->(A or C) and (B or C)$;
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+ $[L_1-L_8, M P]: (A or B) and C <->(A and C) or (B and C)$ .
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== Negation -- minimal logic
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=== Theorem 2.4.1.
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=== Theorem 2.4.1
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(N-elimination) If there is a proof
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$[T, #[MP]]: A_1, A_2, ..., A_n, B├ C$, and a proof $[T, #[MP]]: A_1, A_2, ..., A_n,
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B├ ¬C$, then there is a proof $[T, L_1, L_2, L_9, #[MP]]: A_1, A_2, ..., A_n├ ¬B$.
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$[T, M P]: A_1, A_2, ..., A_n, B tack.r C$, and a proof $[T, M P]: A_1, A_2, ..., A_n,
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B tack.r not C$, then there is a proof $[T, L_1, L_2, L_9, M P]: A_1, A_2, ..., A_n tack.r not B$.
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=== Theorem 2.4.2.
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=== Theorem 2.4.2
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a) $[L_1, L_2, L_9, #[MP]]: A, ¬B├ ¬(A→B)$. What does it mean? b) $[L_1-L_4, L_9, #[MP]]: A∧¬B→¬( A→B)$.
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+ $[L_1, L_2, L_9, M P]: A, not B tack.r not (A->B)$. What does it mean?
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+ $[L_1-L_4, L_9, M P]: A and not B->not (A->B)$.
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=== Theorem 2.4.3.
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=== Theorem 2.4.3
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$[L_1, L_2, L_9, #[MP]]: (A→B)→(¬B→¬A)$. What does it mean? It's the so-called
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*Contraposition Law*.
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$[L_1, L_2, L_9, M P]: (A->B)->( not B-> not A)$. What does it mean? It's the
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so-called *Contraposition Law*.
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Note. The following rule form of Contraposition Law is called *Modus Tollens*:
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$[L_1, L_2, L_9, #[MP]]: A→B, ¬B├ ¬A, or, frac(A→B \; ¬B, ¬A)$
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$[L_1, L_2, L_9, M P]: A->B, not B tack.r not A, or, ((A->B; not B)/(not A)$ // TODO: factcheck
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=== Theorem 2.4.4.
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=== Theorem 2.4.4
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$[L_1, L_2, L_9, #[MP]]: A→¬¬A$.
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$[L_1, L_2, L_9, M P]: A->not not A$.
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=== Theorem 2.4.5.
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=== Theorem 2.4.5
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a) $[L_1, L_2, L_9, #[MP]]: ¬¬¬A↔¬A$. b) $[L_1, L_2, L_6, L_7, L_9, #[MP]]: ¬¬( A∨¬A)$.
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What does it mean? This is a “weak form” of the *Law of Excluded Middle* that
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can be proved constructively. The formula $¬¬( A∨¬A)$ can be proved in the
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constructive logic, but $A∨¬A$ can't – as we will see in Section 2.8.
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+ $[L_1, L_2, L_9, M P]: not not not A<-> not A$.
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+ $[L_1, L_2, L_6, L_7, L_9, M P]: not not ( A or not A)$.
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What does it mean? This is a "weak form" of the *Law of Excluded Middle* that
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can be proved constructively. The formula $ not not ( A or not A)$ can be proved
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in the constructive logic, but $A or not A$ can't -- as we will see in
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@axiom-indempendence.
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=== Theorem 2.4.9.
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=== Theorem 2.4.9
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+ $[L_1, L_2, L_8, L_9, #[MP]]: ¬A∨¬B→¬( A∧B)$ . It's the constructive half of the
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so-called *First de Morgan Law*. What does it mean?
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+ $[L_1-L_9, #[MP]]: ¬(A∨B)↔¬A∧¬B$. It's the so-called *Second de Morgan Law*.
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+ $[L_1, L_2, L_8, L_9, M P]: not A or not B-> not ( A and B)$ . It's the
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constructive half of the so-called *First de Morgan Law*. What does it mean?
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+ $[L_1-L_9, M P]: not (A or B)<-> not A and not B$. It's the so-called *Second de
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Morgan Law*.
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== Negation -- constructive logic
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=== Theorem 2.5.1.
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=== Theorem 2.5.1
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+ $[L_1, L_8, L_10, #[MP]]: ¬A∨B→( A→B)$.
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+ $[L_1, L_2, L_6, #[MP]]: A∨B→(¬A→B) ├¬A→(A→B)$ . It means that the “natural”
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rule $A∨B ;¬ A ├ B$ implies $L_10$!
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+ $[L_1, L_8, L_10, M P]: not A or B->( A->B)$.
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+ $[L_1, L_2, L_6, M P]: A or B->( not A->B) tack.r not A->(A->B)$ . It means that
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the "natural" rule $A or B ; not A tack.r B$ implies $L_10$!
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=== Theorem 2.5.2.
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=== Theorem 2.5.2
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$[L_1-L_10, #[MP]]$:
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$[L_1-L_10, M P]$:
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+ $(¬¬A→¬¬B)→¬¬(A→B)$. It's the converse of Theorem 2.4.7(b). Hence, $[L_1-L_10,
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#[MP]]:├ ¬¬(A→B)↔(¬¬A→¬¬B)$.
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+ $¬¬A→(¬A→A)$. It's the converse of Theorem 2.4.6(a). Hence, [L1-L10,
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#[MP]]: $¬¬A↔(¬A→A)$.
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+ $A∨¬ A→(¬¬A→A)$ .
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+ $¬¬(¬¬A→A)$. What does it mean? It’s a “weak” form of the Double Negations Law –
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provable in constructive logic.
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+ $( not not A-> not not B)-> not not (A->B)$. It's the converse of Theorem
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2.4.7(b). Hence, $[L_1-L_10,
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M P]: tack.r not not (A->B)<->( not not A-> not not B)$.
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+ $ not not A->( not A->A)$. It's the converse of Theorem 2.4.6(a). Hence, $[L_1-L)10, M P]: not not A<->(not A->A)$.
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+ $A or not A->(not not A->A)$.
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+ $ not not (not not A->A)$. What does it mean? It’s a "weak" form of the Double
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Negations Law -- provable in constructive logic.
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== Negation -- classical logic
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=== Theorem 2.6.1. (Double Negation Law)
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$[L_1, L_2, L_8, L_10, L_11, #[MP]]: ¬¬A → A$. Hence, $[L_1-L_11, #[MP]]: ¬¬A ↔
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$[L_1, L_2, L_8, L_10, L_11, M P]: not not A -> A$. Hence, $[L_1-L_11, M P]: not not A <->
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A$.
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=== Theorem 2.6.2.
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=== Theorem 2.6.2
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$[L_8, L_11, #[MP]]: A→B, ¬A→B├ B$. Or, by Deduction Theorem 1, $[L_1, L_2, L_8,
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L_11, #[MP]]: (A→B)→((¬A→B)→B)$.
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$[L_8, L_11, M P]: A->B, not A->B tack.r B$. Or, by Deduction Theorem 1, $[L_1, L_2, L_8,
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L_11, M P]: (A->B)->(( not A->B)->B)$.
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=== Theorem 2.6.3.
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=== Theorem 2.6.3
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$[L_1-L_11, #[MP]]: (¬B→¬A)→(A→B)$. Hence, $[L_1-L_11, #[MP]]: (A→B) ↔ (¬B→¬A)$.
|
||||
$[L_1-L_11, M P]: ( not B-> not A)->(A->B)$. Hence, $[L_1-L_11, M P]: (A->B) <-> ( not B-> not A)$.
|
||||
|
||||
=== Theorem 2.6.3.
|
||||
=== Theorem 2.6.3
|
||||
|
||||
_(another one with the same number of because numbering error (it seems like it))_
|
||||
|
||||
$[L_1-L_9, L_11, #[MP]]: ˫ ¬(A∧B)→¬A∨¬B$ . Hence, $[L_1-L_9, L_11, #[MP]]: ˫
|
||||
¬(A∧B)↔¬A∨¬B$ .
|
||||
$[L_1-L_9, L_11, M P]: ˫ not (A and B)-> not A or not B$ . Hence, $[L_1-L_9, L_11, M P]: ˫
|
||||
not (A and B)<-> not A or not B$ .
|
||||
|
||||
=== Theorem 2.6.4.
|
||||
=== Theorem 2.6.4
|
||||
|
||||
$[L_1-L_8, L_11, #[MP]]: (A→B)→¬ A∨B $. Hence, (I-elimination) $[L_1-L_11, #[MP]]:
|
||||
(A→B)↔¬ A∨B$.
|
||||
$[L_1-L_8, L_11, M P]: (A->B)-> not A or B $. Hence, (I-elimination) $[L_1-L_11, M P]:
|
||||
(A->B)<-> not A or B$.
|
||||
|
||||
=== Theorem 2.6.5.
|
||||
=== Theorem 2.6.5
|
||||
|
||||
$[L_1-L_11, #[MP]]: ¬(A→B)→A∧¬B $.
|
||||
$[L_1-L_11, M P]: not (A->B)->A and not B $.
|
||||
|
||||
=== Theorem 2.7.1 (Glivenko's Theorem).
|
||||
=== Theorem 2.7.1 (Glivenko's Theorem)
|
||||
|
||||
$[L_1-L_11, #[MP]]:├ A$ if and only if $[L_1-L_10, #[MP]]:├ ¬¬A$.
|
||||
$[L_1-L_11, M P]: tack.r A$ if and only if $[L_1-L_10, M P]: tack.r not not A$.
|
||||
|
||||
== Axiom independence
|
||||
== Axiom independence <axiom-indempendence>
|
||||
|
||||
=== Theorem 2.8.1.
|
||||
=== Theorem 2.8.1
|
||||
|
||||
The axiom $L_9$: $(A→B)→((A→¬B)→¬A)$ can be proved in $[L_1, L_2, L_8, L_10,
|
||||
L_11, #[MP]]$.
|
||||
The axiom $L_9$: $(A->B)->((A-> not B)-> not A)$ can be proved in $[L_1, L_2, L_8, L_10,
|
||||
L_11, M P]$.
|
||||
|
||||
=== Theorem 2.8.2.
|
||||
=== Theorem 2.8.2
|
||||
|
||||
The axiom $L_9$ cannot be proved in $[L_1-L_8, L_10, #[MP]]$.
|
||||
The axiom $L_9$ cannot be proved in $[L_1-L_8, L_10, M P]$.
|
||||
|
||||
== Replacement Theorem 1
|
||||
|
||||
@ -256,7 +249,7 @@ $C$, and $o(B)$ is a propositional occurrence of $B$ in $C$. Let us denote by
|
||||
$C'$ the formula obtained from $C$ by replacing $o(B)$ by $B'$. Then, in the
|
||||
minimal logic,
|
||||
|
||||
$[L_1-L_9, #[MP]]: B↔B'├ C↔C'$.
|
||||
$[L_1-L_9, M P]: B<->B' tack.r C<->C'$.
|
||||
|
||||
== Replacement Theorem 2
|
||||
|
||||
@ -264,83 +257,80 @@ Let us consider three formulas: $B$, $B'$, $C$, where $B$ is a sub-formula of
|
||||
$C$, and $o(B)$ is any occurrence of $B$ in $C$. Let us denote by $C'$ the
|
||||
formula obtained from $C$ by replacing $o(B)$ by B'. Then, in the minimal logic,
|
||||
|
||||
$[L_1-L_9, L_12-L_15, #[MP], #[Gen]]: B↔B'├ C↔C'$.
|
||||
$[L_1-L_9, L_12-L_15, M P, G e n]: B<->B' tack.r C<->C'$.
|
||||
|
||||
== Theorem 3.1.1.
|
||||
== Theorem 3.1.1
|
||||
|
||||
$[L_1, L_2, L_12, L_13, #[MP]]: forall x B(x) arrow exists x B(x)$ . What does
|
||||
it mean? It prohibits "empty domains".
|
||||
$[L_1, L_2, L_12, L_13, M P]: forall x B(x) -> exists x B(x)$. What does it
|
||||
mean? It prohibits "empty domains".
|
||||
|
||||
Theorem 3.1.2.
|
||||
== Theorem 3.1.2
|
||||
|
||||
+ [L1, L2, L12, L14, MP, Gen]: ∀x(B→C)→(∀x B→∀xC).
|
||||
+ [L1, L2, L12-L15, MP, Gen]: ∀x(B→C)→(∃z(x+z+1=y).x B→∃z(x+z+1=y).xC).
|
||||
+ $[L_1, L_2, L_12, L_14, M P, G e n]: forall x(B->C)->(forall x B -> forall x C)$.
|
||||
+ $[L_1, L_2, L_12-L_15, M P, G e n]: forall x(B->C)->(exists x B->exists x C)$.
|
||||
|
||||
== Theorems 3.1.3.
|
||||
== Theorems 3.1.3
|
||||
|
||||
If F is any formula, then:
|
||||
If $F$ is any formula, then:
|
||||
|
||||
+ (U-introduction) [Gen]: F(x) ├∀x F(x) .
|
||||
+ (U-elimination) [L12, MP, Gen]: ∀x F(x) ├F(x) . What does it mean?
|
||||
+ (E-introduction) [L13, MP, Gen]: F(x) ├∃z(x+z+1=y).x F(x) . What does it mean?
|
||||
+ (U-introduction) $[G e n]: F(x) tack.r forall x F(x)$.
|
||||
+ (U-elimination) $[L_12, M P, G e n]: forall x F(x) tack.r F(x)$.
|
||||
+ (E-introduction) $[L_13, M P, G e n]: F(x) tack.r exists z(x+z+1=y).x F(x)$.
|
||||
|
||||
== Theorems 3.1.4.
|
||||
== Theorems 3.1.4
|
||||
|
||||
If F is any formula, and G is a formula that does not contain free occurrences
|
||||
of x, then:
|
||||
If $F$ is any formula, and $G$ is a formula that does not contain free
|
||||
occurrences of $x$, then:
|
||||
|
||||
+ (U2-introduction) [L14, MP, Gen] G →F (x) ├G →∀x F (x) . What does it mean?
|
||||
+ (E2-introduction) [L15, MP, Gen]: F (x)→G ├∃z(x+z+1=y).x F (x)→G . What does it
|
||||
mean?
|
||||
+ (U2-introduction) $[L_14, M P, G e n] G -> F (x) tack.r G -> forall x F (x)$.
|
||||
+ (E2-introduction) $[L_15, M P, G e n]: F(x)->G tack.r exists x F (x)->G$.
|
||||
|
||||
== Theorem 3.1.5.
|
||||
== Theorem 3.1.5
|
||||
|
||||
+ [L1, L2, L5, L12, L14, MP, Gen]: $forall x forall y B(x,y) ↔ forall y forall x B(x,y)$
|
||||
+ [L1, L2, L5, L13, L15, MP, Gen]: $exists x exists y B(x,y) ↔ exists y exists x B(x,y)$.
|
||||
+ [L1, L2, L12-L15, MP, Gen]: $exists x forall y B(x,y) ↔ forall y exists x B(x,y)$.
|
||||
+ $[L_1, L_2, L_5, L_12, L_14, M P, G e n]: forall x forall y B(x,y) <-> forall y forall x B(x,y)$
|
||||
+ $[L_1, L_2, L_5, L_13, L_15, M P, G e n]: exists x exists y B(x,y) <-> exists y exists x B(x,y)$.
|
||||
+ $[L_1, L_2, L_12-L_15, M P, G e n]: exists x forall y B(x,y) <-> forall y exists x B(x,y)$.
|
||||
|
||||
Theorem 3.1.6. If the formula B does not contain free occurrences of x, then
|
||||
[L1-L2, L12-L15, MP, Gen]: (∀x B)↔B;(∃z(x+z+1=y).x B)↔B , i.e., quantifiers ∀x
|
||||
;∃z(x+z+1=y). x can be dropped or introduced as needed.
|
||||
== Theorem 3.1.6
|
||||
If the formula $B$ does not contain free occurrences of $x$, then
|
||||
$[L_1-L_2, L_12-L_15, M P, G e n]: (forall x B)<->B;(exists x B)<->B$, i.e.,
|
||||
quantifiers $forall x; exists x$ can be dropped or introduced as needed.
|
||||
|
||||
Theorem 3.2.1. In the classical logic, [L1-L15, MP, Gen]: ¬ x¬B ∀ ↔ xB.
|
||||
== Theorem 3.2.1
|
||||
In the classical logic, $[L_1-L_15, M P, G e n]: not x not B forall <-> x B$.
|
||||
|
||||
== Theorem 3.3.1.
|
||||
== Theorem 3.3.1
|
||||
|
||||
+ [L1-L5, L12, L14, MP, Gen]: ∀x(B∧C)↔∀x B∧∀xC .
|
||||
+ [L1, L2, L6-L8, L12, L14, MP, Gen]: ├∀x B∨∀xC →∀x(B∨C) . The converse formula
|
||||
∀x(B∨C)→∀x B∨∀xC cannot be true. Explain, why.
|
||||
+ $[L_1-L-5, L_12, L_14, M P, G e n]: forall x(B and C)<-> forall x B and forall x C$.
|
||||
+ $[L_1, L_2, L_6-L_8, L_12, L_14, M P, G e n]: tack.r forall x B or forall x C -> forall x(B or C)$.
|
||||
The converse formula $forall x(B or C)-> forall x B or forall x C$ cannot be
|
||||
true.
|
||||
|
||||
== Theorem 3.3.2.
|
||||
== Theorem 3.3.2
|
||||
|
||||
a) [L1-L8, L12-L15, MP, Gen]: ∃z(x+z+1=y).x(B∨C)↔∃z(x+z+1=y). x B∨∃z(x+z+1=y).xC
|
||||
. b) [L1-L5, L13-L15, MP, Gen]: ∃z(x+z+1=y).x(B∧C)→∃z(x+z+1=y). x
|
||||
B∧∃z(x+z+1=y).xC . The converse implication ∃z(x+z+1=y).x B∧∃z(x+z+1=y). xC
|
||||
→∃z(x+z+1=y). x(B∧C) cannot be true. Explain, why. Exercise 3.3.3. a) Prove (a→)
|
||||
of Theorem 3.3.2. (Hint: start by assuming B∨C , apply D-elimination, etc., and
|
||||
finish by E2-introduction.)
|
||||
+ $[L_1-L_8, L_12-L_15, M P, G e n]: exists x(B or C)<-> exists x B or exists x C$
|
||||
+ $[L_1-L_5, L_13-L_15, M P, G e n]: exists x(B and C)-> exists x B and exists C$.
|
||||
The converse implication $exists x B and exists x C -> exists x(B and C)$ cannot
|
||||
be true.
|
||||
|
||||
= Three-valued logic
|
||||
|
||||
This is a general scheme (page 74) to define a three valued logic.
|
||||
|
||||
For example, let us consider a kind of "three-valued logic", where 0 means
|
||||
"false", 1 – "unknown" (or NULL – in terms of SQL), and 2 means "true". Then it
|
||||
would be natural to define “truth values” of conjunction and disjunction as
|
||||
"`false`", 1 -- "`unknown`" (or `NULL` -- in terms of SQL), and 2 means "true".
|
||||
Then it would be natural to define "truth values" of conjunction and disjunction
|
||||
as
|
||||
|
||||
$A∧B=min ( A, B)$ ;
|
||||
$A and B=min ( A, B)$ ;
|
||||
|
||||
$A∨B=max (A , B)$ .
|
||||
$A or B=max (A , B)$ .
|
||||
|
||||
But how should we define “truth values” of implication and negation?
|
||||
But how should we define "truth values" of implication and negation?
|
||||
|
||||
#table(
|
||||
columns: 5, [$A$], [$B$], [$A∧B$], [$A∨B$], [$A→B$], [$0$], [$0$], [$0$], [$0$], [$i_1$], [$0$], [$1$], [$0$], [$1$], [$i_2$], [$0$], [$2$], [$0$], [$2$], [$i_3$], [$1$], [$0$], [$0$], [$1$], [$i_4$], [$1$], [$1$], [$1$], [$1$], [$i_5$], [$1$], [$2$], [$1$], [$2$], [$i_6$], [$2$], [$0$], [$0$], [$2$], [$i_7$], [$2$], [$1$], [$1$], [$2$], [$i_8$], [$2$], [$2$], [$2$], [$2$], [$i_9$],
|
||||
columns: 5, $A$, $B$, $A and B$, $A or B$, $A->B$, $0$, $0$, $0$, $0$, $i_1$, $0$, $1$, $0$, $1$, $i_2$, $0$, $2$, $0$, $2$, $i_3$, $1$, $0$, $0$, $1$, $i_4$, $1$, $1$, $1$, $1$, $i_5$, $1$, $2$, $1$, $2$, $i_6$, $2$, $0$, $0$, $2$, $i_7$, $2$, $1$, $1$, $2$, $i_8$, $2$, $2$, $2$, $2$, $i_9$,
|
||||
)
|
||||
|
||||
#table(
|
||||
columns: 2, [$A$], [¬$A$], [$0$], [$i_10$], [$1$], [$i_11$], [$2$], [$i_12$],
|
||||
)
|
||||
#table(columns: 2, $A$, $not A$, $0$, $i_10$, $1$, $i_11$, $2$, $i_12$)
|
||||
|
||||
= Model interpreation
|
||||
|
||||
@ -350,68 +340,62 @@ But how should we define “truth values” of implication and negation?
|
||||
|
||||
Let L be a predicate language containing:
|
||||
|
||||
- (a possibly empty) set of object constants $c_1, dots, c_k, dots $;
|
||||
- (a possibly empty) set of function constants $f_1, dots, f_m, dots,$;
|
||||
- (a possibly empty) set of object constants $c_1, ..., c_k, ... $;
|
||||
- (a possibly empty) set of function constants $f_1, ..., f_m, ...,$;
|
||||
- (a non empty) set of predicate constants $p_1, ..., p_n, ...$.
|
||||
|
||||
An interpretation $J$ of the language $L$ consists of the following two entities
|
||||
(a set and a mapping):
|
||||
|
||||
+ A non-empty finite or infinite set DJ – the domain of interpretation (it will
|
||||
+ A non-empty finite or infinite set DJ -- the domain of interpretation (it will
|
||||
serve first of all as the range of object variables). (For infinite domains, set
|
||||
theory comes in here.)
|
||||
+ A mapping intJ that assigns:
|
||||
- to each object constant $c_i$ – a member $#[int]_J (c_i)$ of the domain
|
||||
$D_J$ [contstant corresponds to an object from domain];
|
||||
- to each function constant $f_i$ – a function $#[int]_J (f_i)$ from $D_J times dots
|
||||
times D_J$ into $D_J$ [],
|
||||
- to each predicate constant $p_i$ – a predicate $#[int]_J (p_i)$ on $D_J$.
|
||||
- to each object constant $c_i$ -- a member $"int"_J (c_i)$ of the domain $D_J$ [contstant
|
||||
corresponds to an object from domain];
|
||||
- to each function constant $f_i$ -- a function $"int"_J (f_i)$ from $D_J times ... times D_J$ into $D_J$ [],
|
||||
- to each predicate constant $p_i$ -- a predicate $"int"_J (p_i)$ on $D_J$.
|
||||
|
||||
Having an interpretation $J$ of the language $L$, we can define the notion of
|
||||
*true formulas* (more precisely − the notion of formulas that are true under the
|
||||
interpretation $J$).
|
||||
|
||||
*Example.* The above interpretation of the “language about people” put in the
|
||||
*Example.* The above interpretation of the "language about people" put in the
|
||||
terms of the general definition:
|
||||
|
||||
+ $D = {#[br], #[jo], #[pa], #[pe]}$.
|
||||
+ $#[int]_J (#[Britney])=#[br], #[int]_J (#[John])=#[jo], #[int]_J (#[Paris])=#[pa],
|
||||
#[int]_J (#[Peter])=#[pe]$.
|
||||
+ $#[int]_J (#[Male]) = {#[jo], #[pe]}; #[int]_J (#[Female]) = {#[br], #[pa]}$.
|
||||
+ $#[int]_J (#[Mother]) = {(#[pa], #[br]), (#[pa], #[jo])}; #[int]_J (#[Father]) =
|
||||
{(#[pe], #[jo]), (#[pe], #[br])}$.
|
||||
+ $#[int]_J (#[Married]) = {(#[pa], #[pe]), (#[pe], #[pa])}$.
|
||||
+ $#[int]_J (=) = {(#[br], #[br]), (#[jo], #[jo]), (#[pa], #[pa]), (#[pe],
|
||||
#[pe])}$.
|
||||
+ $D = {"br", "jo", "pa", "pe"}$.
|
||||
+ $"int"_J ("Britney")="br", "int"_J ("John")="jo", "int"_J ("Paris")="pa", "int"_J ("Peter")="pe"$.
|
||||
+ $"int"_J ("Male") = {"jo", "pe"}; "int"_J ("Female") = {"br", "pa"}$.
|
||||
+ $"int"_J ("Mother") = {("pa", "br"), ("pa", "jo")}; "int"_J ("Father") = {("pe", "jo"), ("pe", "br")}$.
|
||||
+ $"int"_J ("Married") = {("pa", "pe"), ("pe", "pa")}$.
|
||||
+ $"int"_J (=) = {("br", "br"), ("jo", "jo"), ("pa", "pa"), ("pe", "pe")}$.
|
||||
|
||||
=== Interpretations of languages − the standard common part
|
||||
|
||||
Finally, we define the notion of *true formulas* of the language $L$ under the
|
||||
interpretation $J$ (of course, for a fixed combination of values of their free
|
||||
variables – if any):
|
||||
variables -- if any):
|
||||
|
||||
+ Truth-values of the formulas: $¬B , B∧C , B∨C , B →C$ [those are not examples]
|
||||
must be computed. This is done with the truth-values of $B$ and $C$
|
||||
by using the well-known classical truth tables (see Section 4.2).
|
||||
+ Truth-values of the formulas: $ not B, B and C, B or C B->C$ (those are not
|
||||
examples) must be computed. This is done with the truth-values of $B$ and $C$
|
||||
by using the well-known classical truth tables (see @three-kinds-of-formulas).
|
||||
|
||||
+ The formula $∀x B$ is true under $J$ if and only if $B(c)$ is true under $J$
|
||||
+ The formula $ forall x B$ is true under $J$ if and only if $B(c)$ is true under $J$
|
||||
for all members $c$ of the domain $D_J$.
|
||||
|
||||
+ The formula $∃x B$ is true under $J$ if and only if there is a member c of the
|
||||
domain $D_J$ such that $B(c)$ is true under $J$.
|
||||
+ The formula $ exists x B$ is true under $J$ if and only if there is a member c
|
||||
of the domain $D_J$ such that $B(c)$ is true under $J$.
|
||||
|
||||
*Example.* In first order arithmetic, the formula
|
||||
|
||||
$
|
||||
y((x= y+ y)∨( x=y+ y+1))
|
||||
$
|
||||
$ y((x=y+y) or (x=y+y+1)) $
|
||||
|
||||
is intended to say that "x is even or odd". Under the standard interpretation S
|
||||
of arithmetic, this formula is true for all values of its free variable x.
|
||||
|
||||
Similarly, $∀x ∀y(x+ y=y+x)$ is a closed formula that is true under this
|
||||
interpretation. The notion “a closed formula F is true under the interpretation
|
||||
J” is now precisely defined.
|
||||
Similarly, $ forall x forall y(x+y=y+x)$ is a closed formula that is true under
|
||||
this interpretation. The notion "a closed formula F is true under the
|
||||
interpretation J" is now precisely defined.
|
||||
|
||||
*Important − non-constructivity!* It may seem that, under an interpretation, any
|
||||
closed formula is "either true or false". However, note that, for an infinite
|
||||
@ -419,22 +403,21 @@ domain DJ, the notion of "true formulas under J" is extremely non- constructive.
|
||||
|
||||
=== Example of building of an interpretation
|
||||
|
||||
In Section 1.2, in our "language about people" we used four names of people
|
||||
(Britney, John, Paris, Peter) as object constants and the following predicate
|
||||
constants:
|
||||
In our "language about people" we used four names of people (Britney, John,
|
||||
Paris, Peter) as object constants and the following predicate constants:
|
||||
|
||||
+ $#[Male] (x)$ − means "x is a male person";
|
||||
+ $#[Female] (x)$ − means "x is a female person";
|
||||
+ $#[Mother] (x, y)$ − means "x is mother of y";
|
||||
+ $#[Father] (x, y)$ − means "x is father of y";
|
||||
+ $#[Married] (x, y)$ − means "x and y are married";
|
||||
+ $"Male" (x)$ − means "x is a male person";
|
||||
+ $"Female" (x)$ − means "x is a female person";
|
||||
+ $"Mother" (x, y)$ − means "x is mother of y";
|
||||
+ $"Father" (x, y)$ − means "x is father of y";
|
||||
+ $"Married" (x, y)$ − means "x and y are married";
|
||||
+ $x=y$ − means "x and y are the same person".
|
||||
|
||||
Now, let us consider the following interpretation of the language – a specific
|
||||
“small four person world”:
|
||||
Now, let us consider the following interpretation of the language -- a specific
|
||||
"small four person world":
|
||||
|
||||
The domain of interpretation – and the range of variables – is: $D = {#[br],
|
||||
#[jo], #[pa], #[pe]}$ (no people, four character strings only!).
|
||||
The domain of interpretation -- and the range of variables -- is: $D = {b r,
|
||||
j o, p a, p e}$ (no people, four character strings only!).
|
||||
|
||||
Interpretations of predicate constants are defined by the following truth
|
||||
tables:
|
||||
@ -447,7 +430,7 @@ tables:
|
||||
columns: 6, [x], [y], [Father(x,y)], [Mother(x,y)], [Married(x,y)], [x=y], [br], [br], [false], [false], [false], [true], [br], [jo], [false], [false], [false], [false], [br], [pa], [false], [false], [false], [false], [br], [pe], [false], [false], [false], [false], [jo], [br], [false], [false], [false], [false], [jo], [jo], [false], [false], [false], [true], [jo], [pa], [false], [false], [false], [false], [jo], [pe], [false], [false], [false], [false], [pa], [br], [false], [true], [false], [false], [pa], [jo], [false], [true], [false], [false], [pa], [pa], [false], [false], [false], [true], [pa], [pe], [false], [false], [true], [false], [pe], [br], [true], [false], [false], [false], [pe], [jo], [true], [false], [false], [false], [pe], [pa], [false], [false], [true], [false], [pe], [pe], [false], [false], [false], [true],
|
||||
)
|
||||
|
||||
== Three kinds of formulas
|
||||
== Three kinds of formulas <three-kinds-of-formulas>
|
||||
|
||||
If one explores some formula F of the language L under various interpretations,
|
||||
then three situations are possible:
|
||||
@ -481,45 +464,43 @@ tests.
|
||||
|
||||
As an example, let us verify that the formula
|
||||
|
||||
$
|
||||
∀x( p( x)∨q( x))→∀x p(x)∨∀x q(x)
|
||||
$
|
||||
$ forall x(p(x) or q(x))-> forall x space p(x) or forall x space q(x) $
|
||||
|
||||
is not logically valid (p, q are predicate constants). Why it is not? Because
|
||||
the truth-values of p(x) and q(x) may behave in such a way that $p(x)∨q(x)$ is
|
||||
always true, but neither $forall x p(x)$, nor $forall x q(x)$ is true. Indeed,
|
||||
let us take the domain $D = {a, b}$, and set (in fact, we are using one of two
|
||||
possibilities):
|
||||
is not logically valid ($p$, $q$ are predicate constants). Why it is not?
|
||||
Because the truth-values of $p(x)$ and $q(x)$ may behave in such a way that $p(x) or q(x)$ is
|
||||
always true, but neither $forall x space p(x)$, nor $forall x q(x)$ is true.
|
||||
Indeed, let us take the domain $D = {a, b}$, and set (in fact, we are using one
|
||||
of two possibilities):
|
||||
|
||||
#table(
|
||||
columns: 3, [x], [p(x)], [q(x)], [b], [false], [true], [a], [true], [false],
|
||||
)
|
||||
|
||||
In this interpretation, $p(a)∨q(a) = #[true]$ , $p(b)∨q(b) = #[true]$, i.e., the
|
||||
premise $∀x( p( x)∨q(x))$ is true. But the formulas$forall p(x),
|
||||
forall q(x)$ both are false. Hence, in this interpretation, the conclusion $∀x
|
||||
p(x)∨∀x q(x)$ is false, and $∀x( p( x)∨q( x))→∀x p(x)∨∀x q(x)$ is false. We have
|
||||
built an interpretation, making the formula false. Hence, it is not logically
|
||||
valid.
|
||||
In this interpretation, $p(a) or q(a) = #[`true`]$ , $p(b) or q(b) = #[`true`]$,
|
||||
i.e., the premise $ forall x( p( x) or q(x))$ is true. But the formulas $forall p(x)$, $forall q(x)$ both
|
||||
are false. Hence, in this interpretation, the conclusion $ forall x
|
||||
p(x) or forall x q(x)$ is false, and $ forall x(p(x) or q(x))-> forall x space p(x) or forall x space q(x)$ is
|
||||
false. We have built an interpretation, making the formula false. Hence, it is
|
||||
not logically valid.
|
||||
|
||||
On the other hand, this formula is satisfiable – there is an interpretation
|
||||
On the other hand, this formula is satisfiable -- there is an interpretation
|
||||
under which it is true. Indeed, let us take $D={a}$ as the domain of
|
||||
interpretation, and let us set $p(a)=q(a)=#[true]$. Then all the formulas
|
||||
|
||||
$
|
||||
∀x( p( x)∨q( x)),∀x p(x),∀x q( x)
|
||||
$
|
||||
$ forall x(p(x) or q(x)), forall x space p(x), forall x space q( x) $
|
||||
|
||||
become true, and so becomes the entire formula.
|
||||
|
||||
== Gödel's Completeness Theorem
|
||||
|
||||
*Theorem 4.3.1.* In classical predicate logic $[L_1−L_15,#[MP],#[Gen]]$ all
|
||||
logically valid formulas can be derived.
|
||||
=== Theorem 4.3.1
|
||||
In classical predicate logic $[L_1−L_15,M P,G e n]$ all logically valid formulas
|
||||
can be derived.
|
||||
|
||||
*Theorem 4.3.3.* All formulas that can be derived in classical predicate logic
|
||||
$[L_1−L_15,#[MP],#[Gen]]$ are logically valid. In this logic it is not possible
|
||||
to derive contradictions, it is consistent.
|
||||
=== Theorem 4.3.3
|
||||
All formulas that can be derived in classical predicate logic
|
||||
$[L_1−L_15,M P,G e n]$ are logically valid. In this logic it is not possible to
|
||||
derive contradictions, it is consistent.
|
||||
|
||||
=== Gödel’s theorem usage for task solving
|
||||
|
||||
@ -529,23 +510,23 @@ rules of inference, deduction theorem, T 2.3.1 and other helping tools, we can
|
||||
just prove that $F$ is logically valid (by showing that none of interpretations
|
||||
can make it false). If we manage to do so, then we can announce: according to
|
||||
Gödel’s theorem, $F$ is derivable in classical predicate logic
|
||||
$[L_1−L_15,#[MP],#[Gen]]$.
|
||||
$[L_1−L_15,M P,G e n]$.
|
||||
|
||||
= Tableaux algorithm
|
||||
|
||||
== Step 1.
|
||||
|
||||
We will solve the task from the opposite: append to the hypotheses $F_1, dots
|
||||
F_n$ negation of formula $G$, and obtain the set $F_1, dots, F_n, ¬G$. When you
|
||||
will do homework or test, you shouldn’t forget this, because if you work with
|
||||
the set $F_1, ..., F_n, G$, then obtained result will not give an answer whether $G$ is
|
||||
derivable or not. You should keep this in mind also when the task has only one
|
||||
formula, e.g., verify, whether formula $(A→B)→((B→C)→(A→C))$
|
||||
is derivable. Then from the beginning you should append negation in front:
|
||||
¬((A→B)→((B→C)→(A→C))) and then work further. Instead of the set $F_1, dots,
|
||||
F_n, ¬G$ we can always check one formula $F_1∧...∧F_n∧¬G$. Therefore, our task
|
||||
(theoretically) is reducing to the task: given some predicate language formula
|
||||
F, verify, whether it is satisfiable or not.
|
||||
We will solve the task from the opposite: append to the hypotheses $F_1, ...
|
||||
F_n$ negation of formula $G$, and obtain the set $F_1, ..., F_n, not G$. When
|
||||
you will do homework or test, you shouldn’t forget this, because if you work
|
||||
with the set $F_1, ..., F_n, G$, then obtained result will not give an answer
|
||||
whether $G$ is derivable or not. You should keep this in mind also when the task
|
||||
has only one formula, e.g., verify, whether formula $(A->B)->((B->C)->(A->C))$
|
||||
is derivable. Then from the beginning you should append negation in front: not
|
||||
$((A->B)->((B->C)->(A->C)))$ and then work further. Instead of the set $F_1, ...,
|
||||
F_n, not G$ we can always check one formula $F_1 and ... and F_n and not G$.
|
||||
Therefore, our task (theoretically) is reducing to the task: given some
|
||||
predicate language formula F, verify, whether it is satisfiable or not.
|
||||
|
||||
== Step 2.
|
||||
|
||||
@ -554,33 +535,26 @@ so-called negation normal form. We can use the possibilities provided by
|
||||
Substitution theorem. First, implications are replaced with negations and
|
||||
disjunctions:
|
||||
|
||||
$
|
||||
(A→B)↔¬A∨B
|
||||
$
|
||||
$ (A->B)<-> not A or B $
|
||||
|
||||
Then we apply de Morgan laws to get negations close to the atoms:
|
||||
|
||||
$
|
||||
¬(A∨B)↔¬A∧¬B equiv \
|
||||
¬(A∧B)↔¬A∨¬B
|
||||
$
|
||||
$ not (A or B)<-> not A and not B eq.triple not (A and B)<-> not A or not B $
|
||||
|
||||
In such way all negations are carried exactly before atoms. After that we can
|
||||
remove double negations:
|
||||
|
||||
$
|
||||
¬¬A↔A
|
||||
$
|
||||
$ not not A<->A $
|
||||
|
||||
Example: $(p→q)→q$.
|
||||
Example: $(p->q)->q$.
|
||||
|
||||
First get rid of implications: $¬(¬p∨q)∨q$.
|
||||
First get rid of implications: $not (not p or q) or q$.
|
||||
|
||||
Then apply de Morgan law: $(¬¬p∧¬q)∨q$.
|
||||
Then apply de Morgan law: $(not not p and not q) or q$.
|
||||
|
||||
Then get rid of double negations: $(p∧¬q)∨q$.
|
||||
Then get rid of double negations: $(p and not q) or q$.
|
||||
|
||||
Now we have obtained equivalent formula in negation normal form – formula only
|
||||
Now we have obtained equivalent formula in negation normal form -- formula only
|
||||
has conjunctions and disjunctions, and all negations appear only in front of
|
||||
atoms.
|
||||
|
||||
@ -589,10 +563,10 @@ atoms.
|
||||
Next, we should build a tree, vertices of which are formulas. In the root of the
|
||||
tree we put our formula. Then we have two cases.
|
||||
|
||||
+ If vertex is formula A∧B, then each branch that goes through this vertex is
|
||||
+ If vertex is formula A and B, then each branch that goes through this vertex is
|
||||
extended with vertices A and B.
|
||||
+ If vertex is a formula A∨B, then in place of continuation we have branching into
|
||||
vertex A and vertex B.
|
||||
+ If vertex is a formula A or B, then in place of continuation we have branching
|
||||
into vertex A and vertex B.
|
||||
|
||||
In both cases, the initial vertex is marked as processed. Algorithm continues to
|
||||
process all cases 1 and 2 until all non-atomic vertices have been processed.
|
||||
@ -601,9 +575,9 @@ process all cases 1 and 2 until all non-atomic vertices have been processed.
|
||||
|
||||
When the construction of the tree is finished, we need to analyze and make
|
||||
conclusions. When one branch has some atom both with and without a negation
|
||||
(e.g., $A$ and $¬A$), then it is called closed branch. Other branches are called
|
||||
open branches.
|
||||
(e.g., $A$ and $ not A$), then it is called closed branch. Other branches are
|
||||
called open branches.
|
||||
|
||||
*Theorem.* If in constructed tree, there exists at least one open branch, then
|
||||
formula in the root is satisfiable. And vice versa – if all branches in the tree
|
||||
are closed, then formula in the root is unsatisfiable.
|
||||
formula in the root is satisfiable. And vice versa -- if all branches in the
|
||||
tree are closed, then formula in the root is unsatisfiable.
|
||||
|
||||
Loading…
Reference in New Issue
Block a user