10. Propositional Logic Soundness

The Lecture

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Soundness

! Soundness of natural deduction means that deductions respect truth in the following sense: If A can be derived from the assumptions B1,…,Bn, and

v(B1)=…=v(Bn)=1, then also v(A)=1.

Jouko Väänänen: Propositional logic

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We show: If A has a natural deduction from B1,…,Bn, and v(B1)=v(Bn)=1, then v(A)=1.

Jouko Väänänen: Propositional logic

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We show: If A has a natural deduction from B1,…,Bn, and v(B1)=v(Bn)=1, then v(A)=1.

! The proof is “by induction” on the structure of a natural deduction.

Jouko Väänänen: Propositional logic

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We show: If A has a natural deduction from B1,…,Bn, and v(B1)=v(Bn)=1, then v(A)=1.

! The proof is “by induction” on the structure of a natural deduction.

! We proceed from simpler deductions to more complex ones.

Jouko Väänänen: Propositional logic

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Conjunction introduction rule

Jouko Väänänen: Propositional logic

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Conjunction introduction rule

! We assume v(A)=v(B)=1.

Jouko Väänänen: Propositional logic

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Conjunction introduction rule

! We assume v(A)=v(B)=1.

! We show v(A∧B)=1.

Jouko Väänänen: Propositional logic

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Conjunction introduction rule

! We assume v(A)=v(B)=1.

! We show v(A∧B)=1.! But this is trivial!

Jouko Väänänen: Propositional logic

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Conjunction elimination rule

Jouko Väänänen: Propositional logic

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Conjunction elimination rule

! We assume v(A∧B)=1.

Jouko Väänänen: Propositional logic

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Conjunction elimination rule

! We assume v(A∧B)=1.! We show v(A)=v(B)=1.

Jouko Väänänen: Propositional logic

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Conjunction elimination rule

! We assume v(A∧B)=1.! We show v(A)=v(B)=1.! But this is again trivial!

Jouko Väänänen: Propositional logic

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Disjunction introduction rule

Jouko Väänänen: Propositional logic

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Disjunction introduction rule

Jouko Väänänen: Propositional logic

! We assume v(A)=1.! We show v(AvB)=1.! But this is trivial!

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Disjunction introduction rule

Jouko Väänänen: Propositional logic

! We assume v(B)=1.! We show v(AvB)=1.! Again, this is trivial!

! We assume v(A)=1.! We show v(AvB)=1.! But this is trivial!

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Disjunction elimination rule

Jouko Väänänen: Propositional logic

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Disjunction elimination rule

! We assume v(AvB)=1.

Jouko Väänänen: Propositional logic

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Disjunction elimination rule

! We assume v(AvB)=1.! We also assume that the derivation of C from A, as

well as the derivation of C from B, are sound i.e. if

v(A)=1, then v(C)=1, and if v(B)=1, then v(C)=1.

Jouko Väänänen: Propositional logic

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Disjunction elimination rule

! We assume v(AvB)=1.! We also assume that the derivation of C from A, as

well as the derivation of C from B, are sound i.e. if

v(A)=1, then v(C)=1, and if v(B)=1, then v(C)=1.! We show v(C)=1.

Jouko Väänänen: Propositional logic

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Disjunction elimination rule

! We assume v(AvB)=1.! We also assume that the derivation of C from A, as

well as the derivation of C from B, are sound i.e. if

v(A)=1, then v(C)=1, and if v(B)=1, then v(C)=1.! We show v(C)=1.! But v(AvB)=1 implies v(A)=1 or v(B)=1.

Jouko Väänänen: Propositional logic

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Disjunction elimination rule

! We assume v(AvB)=1.! We also assume that the derivation of C from A, as

well as the derivation of C from B, are sound i.e. if

v(A)=1, then v(C)=1, and if v(B)=1, then v(C)=1.! We show v(C)=1.! But v(AvB)=1 implies v(A)=1 or v(B)=1. ! In either case we have v(C)=1.

Jouko Väänänen: Propositional logic

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Implication introduction rule

Jouko Väänänen: Propositional logic

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Implication introduction rule

! We assume that the derivation of B from A is sound, i.e. if v(A)=1, then v(B)=1.

Jouko Väänänen: Propositional logic

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Implication introduction rule

! We assume that the derivation of B from A is sound, i.e. if v(A)=1, then v(B)=1.

! We prove v(A→B)=1.

Jouko Väänänen: Propositional logic

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Implication introduction rule

! We assume that the derivation of B from A is sound, i.e. if v(A)=1, then v(B)=1.

! We prove v(A→B)=1.! Case 1: v(A)=0. Clear.

Jouko Väänänen: Propositional logic

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Implication introduction rule

! We assume that the derivation of B from A is sound, i.e. if v(A)=1, then v(B)=1.

! We prove v(A→B)=1.! Case 1: v(A)=0. Clear.! Case 2: v(A)=1. By

assumption, in this case v(B)=1, so v(A→B)=1.

Jouko Väänänen: Propositional logic

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Implication elimination rule

Jouko Väänänen: Propositional logic

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Implication elimination rule

! We assume v(A→B)=v(A)=1.

Jouko Väänänen: Propositional logic

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Implication elimination rule

! We assume v(A→B)=v(A)=1.

! We show v(B)=1.

Jouko Väänänen: Propositional logic

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Implication elimination rule

! We assume v(A→B)=v(A)=1.

! We show v(B)=1.! This is trivial!

Jouko Väänänen: Propositional logic

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Equivalence introduction rule

Jouko Väänänen: Propositional logic

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Equivalence introduction rule

! We leave both the formulation of the claim, and the details of the proof as an exercise.

Jouko Väänänen: Propositional logic

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Equivalence elimination rule

Jouko Väänänen: Propositional logic

! We leave both the formulation of the claim, and the details of the proof as an exercise.

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Negation introduction rule

Jouko Väänänen: Propositional logic

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Negation introduction rule

! We assume that the inference of B∧¬B from A is sound i.e. if v(A)=1, then v(B∧¬B)=1.

Jouko Väänänen: Propositional logic

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Negation introduction rule

! We assume that the inference of B∧¬B from A is sound i.e. if v(A)=1, then v(B∧¬B)=1.

! But v(B∧¬B)=0 always.

Jouko Väänänen: Propositional logic

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Negation introduction rule

! We assume that the inference of B∧¬B from A is sound i.e. if v(A)=1, then v(B∧¬B)=1.

! But v(B∧¬B)=0 always.

! So v(A)=0.

Jouko Väänänen: Propositional logic

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Negation elimination rule

Jouko Väänänen: Propositional logic

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Negation elimination rule

! We assume v(¬¬A)=1.

Jouko Väänänen: Propositional logic

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Negation elimination rule

! We assume v(¬¬A)=1.! We show v(A)=1.

Jouko Väänänen: Propositional logic

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Negation elimination rule

! We assume v(¬¬A)=1.! We show v(A)=1.! Clear!

Jouko Väänänen: Propositional logic

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Soundness Theorem

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Soundness Theorem

! If a propositional formula has a natural deduction, then it is a tautology.

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Soundness Theorem

! If a propositional formula has a natural deduction, then it is a tautology.

! If a propositional formula A has a natural deduction from assumptions which have truth value 1 in a valuation v, then also v(A)=1.

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Applications of Soundness

! We can show that a formula B is not derivable by natural deduction from a formula A by finding a valuation v such that v(A)=1 and v(B)=0.

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Applications of Soundness

! We can show that a formula B is not derivable by natural deduction from a formula A by finding a valuation v such that v(A)=1 and v(B)=0.

! Example: We show that p0∨(p1∧p2) is not derivable from (p0∨p2)→p1.

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Applications of Soundness

! We can show that a formula B is not derivable by natural deduction from a formula A by finding a valuation v such that v(A)=1 and v(B)=0.

! Example: We show that p0∨(p1∧p2) is not derivable from (p0∨p2)→p1.

! Solution: Let v(p0)=v(p1)=v(p2)=0.