Introduction to Proof Theory Lecture 2 - Soundness and completeness for propositional logic Anupam Das & Thomas Powell University of Copenhagen & Technische Universität Darmstadt European Summer School on Logic, Language, and Information Sofia University 14 August 2018 These slides are available at http://www.anupamdas.com/esslli18. 1 / 29 Outline 1 Structural induction 2 Soundness 3 Completeness 4 Compactness 5 Questions and exercises 2 / 29 This gives an inductive definition of a tree-like derivation: Definition A tree-like derivation of A from Γ is defined as follows: • If A 2 Γ then A is a tree-like F derivation of A from Γ. • If A is an axiom then is a tree-like F derivation of A from Γ. A • If π0 and π1 are tree-like derivations of A and A ! B from Γ, then the following is a tree-like derivation of B from Γ: π0 π1 A A ! B (mp) B NB: While tree-like derivations prove the same theorems as usual ones, there can be a high cost in the size of the proof! Inductive definitions of proofs Recall that Tom mentioned that all proofs and derivations can be put into tree form. 3 / 29 NB: While tree-like derivations prove the same theorems as usual ones, there can be a high cost in the size of the proof! Inductive definitions of proofs Recall that Tom mentioned that all proofs and derivations can be put into tree form. This gives an inductive definition of a tree-like derivation: Definition A tree-like derivation of A from Γ is defined as follows: • If A 2 Γ then A is a tree-like F derivation of A from Γ. • If A is an axiom then is a tree-like F derivation of A from Γ. A • If π0 and π1 are tree-like derivations of A and A ! B from Γ, then the following is a tree-like derivation of B from Γ: π0 π1 A A ! B (mp) B 3 / 29 Inductive definitions of proofs Recall that Tom mentioned that all proofs and derivations can be put into tree form. This gives an inductive definition of a tree-like derivation: Definition A tree-like derivation of A from Γ is defined as follows: • If A 2 Γ then A is a tree-like F derivation of A from Γ. • If A is an axiom then is a tree-like F derivation of A from Γ. A • If π0 and π1 are tree-like derivations of A and A ! B from Γ, then the following is a tree-like derivation of B from Γ: π0 π1 A A ! B (mp) B NB: While tree-like derivations prove the same theorems as usual ones, there can be a high cost in the size of the proof! 3 / 29 Proposition (Structural induction) If P is some property of derivations such that: • P holds for every derivation A; • P holds for every derivation ; A • Whenever P holds for derivations π0 of A and π1 of B from Γ, P holds for: π0 π1 A A ! B (mp) B Then P holds for all proofs. Exercise: Reduce this proposition to strong induction on the length of a proof. Structural induction In logic (indeed, computer science), structural induction shows up all over the place. It can oten be seen as an instance of strong induction. 4 / 29 Exercise: Reduce this proposition to strong induction on the length of a proof. Structural induction In logic (indeed, computer science), structural induction shows up all over the place. It can oten be seen as an instance of strong induction. Proposition (Structural induction) If P is some property of derivations such that: • P holds for every derivation A; • P holds for every derivation ; A • Whenever P holds for derivations π0 of A and π1 of B from Γ, P holds for: π0 π1 A A ! B (mp) B Then P holds for all proofs. 4 / 29 Structural induction In logic (indeed, computer science), structural induction shows up all over the place. It can oten be seen as an instance of strong induction. Proposition (Structural induction) If P is some property of derivations such that: • P holds for every derivation A; • P holds for every derivation ; A • Whenever P holds for derivations π0 of A and π1 of B from Γ, P holds for: π0 π1 A A ! B (mp) B Then P holds for all proofs. Exercise: Reduce this proposition to strong induction on the length of a proof. 4 / 29 Outline 1 Structural induction 2 Soundness 3 Completeness 4 Compactness 5 Questions and exercises 5 / 29 According to its website, http://inutile.club/estatis/falso/, Falso has remarkable properties: What’s wrong with Falso? Let me introduce you to the Falso proof system, that extends F by a single axiom: ? 6 / 29 What’s wrong with Falso? Let me introduce you to the Falso proof system, that extends F by a single axiom: ? According to its website, http://inutile.club/estatis/falso/, Falso has remarkable properties: 6 / 29 A system S might also be called meaningful if it is consistent: • S does not prove ?. • S does not prove a contradiction. • S does not prove something! In structural proof theory it is oten the dynamics of a proof system which gives it meaning: • Proof search in S corresponds to some computational process. • There is a normalisation procedure for proofs in S that corresponds to a computational process. ‘proofs as programs’. Aside: On being meaningful Proofs should be meaningful in a concrete sense. In particular, here we will insist that they only prove true things (‘soundness’). 7 / 29 In structural proof theory it is oten the dynamics of a proof system which gives it meaning: • Proof search in S corresponds to some computational process. • There is a normalisation procedure for proofs in S that corresponds to a computational process. ‘proofs as programs’. Aside: On being meaningful Proofs should be meaningful in a concrete sense. In particular, here we will insist that they only prove true things (‘soundness’). A system S might also be called meaningful if it is consistent: • S does not prove ?. • S does not prove a contradiction. • S does not prove something! 7 / 29 Aside: On being meaningful Proofs should be meaningful in a concrete sense. In particular, here we will insist that they only prove true things (‘soundness’). A system S might also be called meaningful if it is consistent: • S does not prove ?. • S does not prove a contradiction. • S does not prove something! In structural proof theory it is oten the dynamics of a proof system which gives it meaning: • Proof search in S corresponds to some computational process. • There is a normalisation procedure for proofs in S that corresponds to a computational process. ‘proofs as programs’. 7 / 29 We will prove this by structural induction on a derivation of A from Γ. For the base cases, we must verify the axioms. Here are truth tables for (wk) and (neg): p q q ! p p ! (q ! p) 0 0 1 1 p :p ::p ::p ! p 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 Proof implies truth Recall that we write Γ A if, for every assignment α : Prop ! f0; 1g, if for every B 2 Γ we have α(B) = 1 then α(A) = 1. Theorem (Soundness) If Γ ` A then Γ A. 8 / 29 For the base cases, we must verify the axioms. Here are truth tables for (wk) and (neg): p q q ! p p ! (q ! p) 0 0 1 1 p :p ::p ::p ! p 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 Proof implies truth Recall that we write Γ A if, for every assignment α : Prop ! f0; 1g, if for every B 2 Γ we have α(B) = 1 then α(A) = 1. Theorem (Soundness) If Γ ` A then Γ A. We will prove this by structural induction on a derivation of A from Γ. 8 / 29 Proof implies truth Recall that we write Γ A if, for every assignment α : Prop ! f0; 1g, if for every B 2 Γ we have α(B) = 1 then α(A) = 1. Theorem (Soundness) If Γ ` A then Γ A. We will prove this by structural induction on a derivation of A from Γ. For the base cases, we must verify the axioms. Here are truth tables for (wk) and (neg): p q q ! p p ! (q ! p) 0 0 1 1 p :p ::p ::p ! p 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 8 / 29 p q r A B C D C ! D B ! (C ! D) 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 Verification of axioms (continued) For the axiom, (dist) : ((p ! (q ! r)) ! ((p ! q) ! (p ! r))) let us write: A : q ! r C : p ! q B : p ! A D : p ! r (so (dist) is B ! (C ! D)) 9 / 29 Verification of axioms (continued) For the axiom, (dist) : ((p ! (q ! r)) ! ((p ! q) ! (p ! r))) let us write: A : q ! r C : p ! q B : p ! A D : p ! r (so (dist) is B ! (C ! D)) p q r A B C D C ! D B ! (C ! D) 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 9 / 29 Exercise: Prove this for yourselves! Another way of structuring our inductive argument would be to have substitution as a bona fide rule: A (sub) σ(A) The proposition above would now just constitute one of our inductive cases.