Logic as a Tool Chapter 2: Deductive Reasoning in Propositional - PowerPoint PPT Presentation

Logic as a Tool Chapter 2: Deductive Reasoning in Propositional Logic 2.2 Axiomatic systems for propositional logics Valentin Goranko Stockholm University November 2020 Goranko

Hilbert-style axiomatic systems Goranko

Hilbert-style axiomatic systems • Based on axioms (or, axiom schemes), and only one or two simple rules of inference. Goranko

Hilbert-style axiomatic systems • Based on axioms (or, axiom schemes), and only one or two simple rules of inference. • Relatively easy to extract from the semantics and reason about. In particular, suitable to do induction on derivations. Goranko

Hilbert-style axiomatic systems • Based on axioms (or, axiom schemes), and only one or two simple rules of inference. • Relatively easy to extract from the semantics and reason about. In particular, suitable to do induction on derivations. • Practically not very convenient and useful, because the derivations are not well-structured. Goranko

Hilbert-style axiomatic systems • Based on axioms (or, axiom schemes), and only one or two simple rules of inference. • Relatively easy to extract from the semantics and reason about. In particular, suitable to do induction on derivations. • Practically not very convenient and useful, because the derivations are not well-structured. • In particular, not suitable for automated reasoning. Goranko

The axiomatic system H for the classical propositional logic Goranko

The axiomatic system H for the classical propositional logic Axiom schemes for ¬ and → : Goranko

The axiomatic system H for the classical propositional logic Axiom schemes for ¬ and → : ( → 1) A → ( B → A ); Goranko

The axiomatic system H for the classical propositional logic Axiom schemes for ¬ and → : ( → 1) A → ( B → A ); ( → 2) ( A → ( B → C )) → (( A → B ) → ( A → C )); Goranko

The axiomatic system H for the classical propositional logic Axiom schemes for ¬ and → : ( → 1) A → ( B → A ); ( → 2) ( A → ( B → C )) → (( A → B ) → ( A → C )); ( → 3) ( ¬ B → ¬ A ) → (( ¬ B → A ) → B ) . Goranko

The axiomatic system H for the classical propositional logic Axiom schemes for ¬ and → : ( → 1) A → ( B → A ); ( → 2) ( A → ( B → C )) → (( A → B ) → ( A → C )); ( → 3) ( ¬ B → ¬ A ) → (( ¬ B → A ) → B ) . The only rule of inference: Modus ponens: A , A → B B Goranko

The axiomatic system H for the classical propositional logic Axiom schemes for ¬ and → : ( → 1) A → ( B → A ); ( → 2) ( A → ( B → C )) → (( A → B ) → ( A → C )); ( → 3) ( ¬ B → ¬ A ) → (( ¬ B → A ) → B ) . The only rule of inference: Modus ponens: A , A → B B If we assume that the only propositional connectives in the language are ¬ and → while others are defined in terms of them, then the defining equivalences plus the axioms above provide a sound and complete axiomatization for classical propositional logic. Goranko

The axiomatic system H for the classical propositional logic Axiom schemes for ¬ and → : ( → 1) A → ( B → A ); ( → 2) ( A → ( B → C )) → (( A → B ) → ( A → C )); ( → 3) ( ¬ B → ¬ A ) → (( ¬ B → A ) → B ) . The only rule of inference: Modus ponens: A , A → B B If we assume that the only propositional connectives in the language are ¬ and → while others are defined in terms of them, then the defining equivalences plus the axioms above provide a sound and complete axiomatization for classical propositional logic. However, it is not convenient to treat ∧ and ∨ as definable connectives. Goranko

Adding axioms for ∧ and ∨ to H Goranko

Adding axioms for ∧ and ∨ to H Axiom schemes for ∧ : Goranko

Adding axioms for ∧ and ∨ to H Axiom schemes for ∧ : ( ∧ 1) ( A ∧ B ) → A ; Goranko

Adding axioms for ∧ and ∨ to H Axiom schemes for ∧ : ( ∧ 1) ( A ∧ B ) → A ; ( ∧ 2) ( A ∧ B ) → B ; Goranko

Adding axioms for ∧ and ∨ to H Axiom schemes for ∧ : ( ∧ 1) ( A ∧ B ) → A ; ( ∧ 2) ( A ∧ B ) → B ; ( ∧ 3) ( A → B ) → (( A → C ) → ( A → ( B ∧ C ))) . Goranko

Adding axioms for ∧ and ∨ to H Axiom schemes for ∧ : ( ∧ 1) ( A ∧ B ) → A ; ( ∧ 2) ( A ∧ B ) → B ; ( ∧ 3) ( A → B ) → (( A → C ) → ( A → ( B ∧ C ))) . Axioms schemes for ∨ : Goranko

Adding axioms for ∧ and ∨ to H Axiom schemes for ∧ : ( ∧ 1) ( A ∧ B ) → A ; ( ∧ 2) ( A ∧ B ) → B ; ( ∧ 3) ( A → B ) → (( A → C ) → ( A → ( B ∧ C ))) . Axioms schemes for ∨ : ( ∨ 1) A → A ∨ B ; Goranko

Adding axioms for ∧ and ∨ to H Axiom schemes for ∧ : ( ∧ 1) ( A ∧ B ) → A ; ( ∧ 2) ( A ∧ B ) → B ; ( ∧ 3) ( A → B ) → (( A → C ) → ( A → ( B ∧ C ))) . Axioms schemes for ∨ : ( ∨ 1) A → A ∨ B ; ( ∨ 2) B → A ∨ B ; Goranko

Adding axioms for ∧ and ∨ to H Axiom schemes for ∧ : ( ∧ 1) ( A ∧ B ) → A ; ( ∧ 2) ( A ∧ B ) → B ; ( ∧ 3) ( A → B ) → (( A → C ) → ( A → ( B ∧ C ))) . Axioms schemes for ∨ : ( ∨ 1) A → A ∨ B ; ( ∨ 2) B → A ∨ B ; ( ∨ 3) ( A → C ) → (( B → C ) → (( A ∨ B ) → C )) . Goranko

Derivations and deductive consequence in H Goranko

Derivations and deductive consequence in H A formula A is derivable in H from a set of assumptions: Γ, denoted Γ ⊢ H A , Goranko

Derivations and deductive consequence in H A formula A is derivable in H from a set of assumptions: Γ, denoted Γ ⊢ H A , if there is a finite sequence of formulae A 1 , ..., A n , such that A n = A and for each i ≤ n : Goranko

Derivations and deductive consequence in H A formula A is derivable in H from a set of assumptions: Γ, denoted Γ ⊢ H A , if there is a finite sequence of formulae A 1 , ..., A n , such that A n = A and for each i ≤ n : A i is either an instance of an axiom of H , Goranko

Derivations and deductive consequence in H A formula A is derivable in H from a set of assumptions: Γ, denoted Γ ⊢ H A , if there is a finite sequence of formulae A 1 , ..., A n , such that A n = A and for each i ≤ n : A i is either an instance of an axiom of H , or a formula from Γ, Goranko

Derivations and deductive consequence in H A formula A is derivable in H from a set of assumptions: Γ, denoted Γ ⊢ H A , if there is a finite sequence of formulae A 1 , ..., A n , such that A n = A and for each i ≤ n : A i is either an instance of an axiom of H , or a formula from Γ, or is obtained from some A j , A k for j , k < i , by applying Modus Ponens. Goranko

Derivations and deductive consequence in H A formula A is derivable in H from a set of assumptions: Γ, denoted Γ ⊢ H A , if there is a finite sequence of formulae A 1 , ..., A n , such that A n = A and for each i ≤ n : A i is either an instance of an axiom of H , or a formula from Γ, or is obtained from some A j , A k for j , k < i , by applying Modus Ponens. A is a theorem of H if ∅ ⊢ H A , also denoted ⊢ H A . Goranko

Derivations and deductive consequence in H A formula A is derivable in H from a set of assumptions: Γ, denoted Γ ⊢ H A , if there is a finite sequence of formulae A 1 , ..., A n , such that A n = A and for each i ≤ n : A i is either an instance of an axiom of H , or a formula from Γ, or is obtained from some A j , A k for j , k < i , by applying Modus Ponens. A is a theorem of H if ∅ ⊢ H A , also denoted ⊢ H A . An important observation: ⊢ H has the following Monotonicity Property: if Γ ⊢ H A and Γ ⊆ Γ ′ then Γ ′ ⊢ H A . Goranko

Derivations and deductive consequence in H A formula A is derivable in H from a set of assumptions: Γ, denoted Γ ⊢ H A , if there is a finite sequence of formulae A 1 , ..., A n , such that A n = A and for each i ≤ n : A i is either an instance of an axiom of H , or a formula from Γ, or is obtained from some A j , A k for j , k < i , by applying Modus Ponens. A is a theorem of H if ∅ ⊢ H A , also denoted ⊢ H A . An important observation: ⊢ H has the following Monotonicity Property: if Γ ⊢ H A and Γ ⊆ Γ ′ then Γ ′ ⊢ H A . Theorem Adequacy of H : The axiomatic system H is sound and complete for the classical propositional logic: Γ ⊢ H A iff Γ | = A . Goranko

Inductive definition of derivations in H Goranko

Inductive definition of derivations in H Consider derivations as objects of the type Γ ⊢ H A , not as derivability claims. Goranko

Inductive definition of derivations in H Consider derivations as objects of the type Γ ⊢ H A , not as derivability claims. Here is an inductive definition of (the set of) derivations in H . Goranko

Inductive definition of derivations in H Consider derivations as objects of the type Γ ⊢ H A , not as derivability claims. Here is an inductive definition of (the set of) derivations in H . 1. If A is an axiom then Γ ⊢ H A is a derivation in H . Goranko

Inductive definition of derivations in H Consider derivations as objects of the type Γ ⊢ H A , not as derivability claims. Here is an inductive definition of (the set of) derivations in H . 1. If A is an axiom then Γ ⊢ H A is a derivation in H . 2. If A ∈ Γ then Γ ⊢ H A is a derivation in H . Goranko