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

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Logic as a Tool Chapter 2: Deductive Reasoning in Propositional Logic 2.2 Axiomatic systems for propositional logics

Valentin Goranko Stockholm University November 2020

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Hilbert-style axiomatic systems

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Hilbert-style axiomatic systems

• Based on axioms (or, axiom schemes), and only one or two simple

rules of inference.

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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.

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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.

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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.

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The axiomatic system H for the classical propositional logic

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The axiomatic system H for the classical propositional logic

Axiom schemes for ¬ and →:

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The axiomatic system H for the classical propositional logic

Axiom schemes for ¬ and →: (→ 1) A → (B → A);

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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));

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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).

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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

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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.

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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.

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Adding axioms for ∧ and ∨ to H

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Adding axioms for ∧ and ∨ to H

Axiom schemes for ∧:

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Adding axioms for ∧ and ∨ to H

Axiom schemes for ∧: (∧1) (A ∧ B) → A;

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Adding axioms for ∧ and ∨ to H

Axiom schemes for ∧: (∧1) (A ∧ B) → A; (∧2) (A ∧ B) → B;

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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))).

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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 ∨:

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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;

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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;

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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)).

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Derivations and deductive consequence in H

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Derivations and deductive consequence in H

A formula A is derivable in H from a set of assumptions: Γ, denoted Γ ⊢H A,

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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 A1, ..., An, such that An = A and for each i ≤ n:

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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 A1, ..., An, such that An = A and for each i ≤ n: Ai is either an instance of an axiom of H,

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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 A1, ..., An, such that An = A and for each i ≤ n: Ai is either an instance of an axiom of H, or a formula from Γ,

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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 A1, ..., An, such that An = A and for each i ≤ n: Ai is either an instance of an axiom of H, or a formula from Γ,

• r is obtained from some Aj, Ak for j, k < i, by applying Modus Ponens.

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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 A1, ..., An, such that An = A and for each i ≤ n: Ai is either an instance of an axiom of H, or a formula from Γ,

• r is obtained from some Aj, Ak for j, k < i, by applying Modus Ponens.

A is a theorem of H if ∅ ⊢H A, also denoted ⊢H A.

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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 A1, ..., An, such that An = A and for each i ≤ n: Ai is either an instance of an axiom of H, or a formula from Γ,

• r is obtained from some Aj, Ak 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.

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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 A1, ..., An, such that An = A and for each i ≤ n: Ai is either an instance of an axiom of H, or a formula from Γ,

• r is obtained from some Aj, Ak 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.

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Inductive definition of derivations in H

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Inductive definition of derivations in H

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

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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.

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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.

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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.

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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.
• 3. If Γ ⊢H A is a derivation in H and Γ ⊢H A → B is a derivation in H

then Γ ⊢H B is a derivation in H.

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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.
• 3. If Γ ⊢H A is a derivation in H and Γ ⊢H A → B is a derivation in H

then Γ ⊢H B is a derivation in H.

• Exercise. Prove that for every Γ, A if Γ ⊢H A holds by the original

explicit definition of Γ ⊢H A, then Γ ⊢H A also belongs to the set of derivation in H inductively defined above.

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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.
• 3. If Γ ⊢H A is a derivation in H and Γ ⊢H A → B is a derivation in H

then Γ ⊢H B is a derivation in H.

• Exercise. Prove that for every Γ, A if Γ ⊢H A holds by the original

explicit definition of Γ ⊢H A, then Γ ⊢H A also belongs to the set of derivation in H inductively defined above. (Hint: use induction on the length n of the sequence A1, ..., An, claimed to exist in the explicit definition of Γ ⊢H A.)

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Proofs by structural induction on derivations in H

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Proofs by structural induction on derivations in H

Here is the corresponding scheme for proofs by induction on derivations. Given a property of derivations P, suppose that:

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Proofs by structural induction on derivations in H

Here is the corresponding scheme for proofs by induction on derivations. Given a property of derivations P, suppose that:

• 1. If A is an axiom then the derivation Γ ⊢H A has the property P.

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Proofs by structural induction on derivations in H

Here is the corresponding scheme for proofs by induction on derivations. Given a property of derivations P, suppose that:

• 1. If A is an axiom then the derivation Γ ⊢H A has the property P.
• 2. If A ∈ Γ then the derivation Γ ⊢H A has the property P.

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Proofs by structural induction on derivations in H

Here is the corresponding scheme for proofs by induction on derivations. Given a property of derivations P, suppose that:

• 1. If A is an axiom then the derivation Γ ⊢H A has the property P.
• 2. If A ∈ Γ then the derivation Γ ⊢H A has the property P.
• 3. If the derivation Γ ⊢H A has the property P

and the derivation Γ ⊢H A → B has the property P then the derivation Γ ⊢H B has the property P.

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Proofs by structural induction on derivations in H

Here is the corresponding scheme for proofs by induction on derivations. Given a property of derivations P, suppose that:

• 1. If A is an axiom then the derivation Γ ⊢H A has the property P.
• 2. If A ∈ Γ then the derivation Γ ⊢H A has the property P.
• 3. If the derivation Γ ⊢H A has the property P

and the derivation Γ ⊢H A → B has the property P then the derivation Γ ⊢H B has the property P. Then every derivation in H has the property P.

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Proofs by structural induction on derivations in H

Here is the corresponding scheme for proofs by induction on derivations. Given a property of derivations P, suppose that:

• 1. If A is an axiom then the derivation Γ ⊢H A has the property P.
• 2. If A ∈ Γ then the derivation Γ ⊢H A has the property P.
• 3. If the derivation Γ ⊢H A has the property P

and the derivation Γ ⊢H A → B has the property P then the derivation Γ ⊢H B has the property P. Then every derivation in H has the property P.

• Exercise. Prove by structural induction on the inductive definition of

derivations in H that every such derivation satisfies the original explicit definition of Γ ⊢H A.

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Proofs by structural induction on derivations in H

Here is the corresponding scheme for proofs by induction on derivations. Given a property of derivations P, suppose that:

• 1. If A is an axiom then the derivation Γ ⊢H A has the property P.
• 2. If A ∈ Γ then the derivation Γ ⊢H A has the property P.
• 3. If the derivation Γ ⊢H A has the property P

and the derivation Γ ⊢H A → B has the property P then the derivation Γ ⊢H B has the property P. Then every derivation in H has the property P.

• Exercise. Prove by structural induction on the inductive definition of

derivations in H that every such derivation satisfies the original explicit definition of Γ ⊢H A.

• Exercise. Prove by structural induction the soundness of H,

i.e. every derivation Γ ⊢H A has the property that Γ A.

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Proofs by structural induction on derivations in H

Here is the corresponding scheme for proofs by induction on derivations. Given a property of derivations P, suppose that:

• 1. If A is an axiom then the derivation Γ ⊢H A has the property P.
• 2. If A ∈ Γ then the derivation Γ ⊢H A has the property P.
• 3. If the derivation Γ ⊢H A has the property P

and the derivation Γ ⊢H A → B has the property P then the derivation Γ ⊢H B has the property P. Then every derivation in H has the property P.

• Exercise. Prove by structural induction on the inductive definition of

derivations in H that every such derivation satisfies the original explicit definition of Γ ⊢H A.

• Exercise. Prove by structural induction the soundness of H,

i.e. every derivation Γ ⊢H A has the property that Γ A. (Hint: for the first clause, show that every axiom is a tautology.)

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Derivations in H: example

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

• 1. ⊢H (p ∧ (p → q)) → p,

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

• 1. ⊢H (p ∧ (p → q)) → p,

by Axiom (∧1);

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

• 1. ⊢H (p ∧ (p → q)) → p,

by Axiom (∧1);

• 2. ⊢H (p ∧ (p → q)) → (p → q),

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

• 1. ⊢H (p ∧ (p → q)) → p,

by Axiom (∧1);

• 2. ⊢H (p ∧ (p → q)) → (p → q),

by Axiom (∧2);

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

• 1. ⊢H (p ∧ (p → q)) → p,

by Axiom (∧1);

• 2. ⊢H (p ∧ (p → q)) → (p → q),

by Axiom (∧2);

• 3. ⊢H ((p ∧ (p → q)) → (p → q)) →

(((p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q)),

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

• 1. ⊢H (p ∧ (p → q)) → p,

by Axiom (∧1);

• 2. ⊢H (p ∧ (p → q)) → (p → q),

by Axiom (∧2);

• 3. ⊢H ((p ∧ (p → q)) → (p → q)) →

(((p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q)), by Axiom (→ 2);

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

• 1. ⊢H (p ∧ (p → q)) → p,

by Axiom (∧1);

• 2. ⊢H (p ∧ (p → q)) → (p → q),

by Axiom (∧2);

• 3. ⊢H ((p ∧ (p → q)) → (p → q)) →

(((p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q)), by Axiom (→ 2);

• 4. ⊢H (p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q),

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

• 1. ⊢H (p ∧ (p → q)) → p,

by Axiom (∧1);

• 2. ⊢H (p ∧ (p → q)) → (p → q),

by Axiom (∧2);

• 3. ⊢H ((p ∧ (p → q)) → (p → q)) →

(((p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q)), by Axiom (→ 2);

• 4. ⊢H (p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q),

by 2,3 and Modus Ponens;

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

• 1. ⊢H (p ∧ (p → q)) → p,

by Axiom (∧1);

• 2. ⊢H (p ∧ (p → q)) → (p → q),

by Axiom (∧2);

• 3. ⊢H ((p ∧ (p → q)) → (p → q)) →

(((p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q)), by Axiom (→ 2);

• 4. ⊢H (p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q),

by 2,3 and Modus Ponens;

• 5. ⊢H (p ∧ (p → q)) → q,

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

• 1. ⊢H (p ∧ (p → q)) → p,

by Axiom (∧1);

• 2. ⊢H (p ∧ (p → q)) → (p → q),

by Axiom (∧2);

• 3. ⊢H ((p ∧ (p → q)) → (p → q)) →

(((p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q)), by Axiom (→ 2);

• 4. ⊢H (p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q),

by 2,3 and Modus Ponens;

• 5. ⊢H (p ∧ (p → q)) → q,

by 1,4 and Modus Ponens.

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Derivations in H: example

Example: ⊢H (p ∧ (p → q)) → q :

• 1. ⊢H (p ∧ (p → q)) → p,

by Axiom (∧1);

• 2. ⊢H (p ∧ (p → q)) → (p → q),

by Axiom (∧2);

• 3. ⊢H ((p ∧ (p → q)) → (p → q)) →

(((p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q)), by Axiom (→ 2);

• 4. ⊢H (p ∧ (p → q)) → p) → ((p ∧ (p → q)) → q),

by 2,3 and Modus Ponens;

• 5. ⊢H (p ∧ (p → q)) → q,

by 1,4 and Modus Ponens. Challenge: Derive ⊢H p → p.

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The Deduction Theorem

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The Deduction Theorem

Deduction Theorem: For any set of formulae Γ and formulae A and B: Γ ∪ {A} ⊢H B iff Γ ⊢H A → B.

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The Deduction Theorem

Deduction Theorem: For any set of formulae Γ and formulae A and B: Γ ∪ {A} ⊢H B iff Γ ⊢H A → B. Proof:

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The Deduction Theorem

Deduction Theorem: For any set of formulae Γ and formulae A and B: Γ ∪ {A} ⊢H B iff Γ ⊢H A → B. Proof: Right-to-left: Straightforward, by Modus Ponens, because if Γ ⊢H A → B then, moreover, Γ ∪ {A} ⊢H A → B.

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The Deduction Theorem

Deduction Theorem: For any set of formulae Γ and formulae A and B: Γ ∪ {A} ⊢H B iff Γ ⊢H A → B. Proof: Right-to-left: Straightforward, by Modus Ponens, because if Γ ⊢H A → B then, moreover, Γ ∪ {A} ⊢H A → B. Left-to-right: Induction on the derivation Γ ∪ {A} ⊢H B, using the axioms for →.

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The Deduction Theorem

Deduction Theorem: For any set of formulae Γ and formulae A and B: Γ ∪ {A} ⊢H B iff Γ ⊢H A → B. Proof: Right-to-left: Straightforward, by Modus Ponens, because if Γ ⊢H A → B then, moreover, Γ ∪ {A} ⊢H A → B. Left-to-right: Induction on the derivation Γ ∪ {A} ⊢H B, using the axioms for →. Exercise: complete the details.

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Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

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Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

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Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

instance of Axiom (∧1);

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Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

instance of Axiom (∧1);

• 2. p ∧ (p → q) ⊢H p,

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Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

instance of Axiom (∧1);

• 2. p ∧ (p → q) ⊢H p,

by (1) and the Deduction Theorem;

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Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

instance of Axiom (∧1);

• 2. p ∧ (p → q) ⊢H p,

by (1) and the Deduction Theorem;

• 3. ⊢H (p ∧ (p → q)) → (p → q)

Goranko

Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

instance of Axiom (∧1);

• 2. p ∧ (p → q) ⊢H p,

by (1) and the Deduction Theorem;

• 3. ⊢H (p ∧ (p → q)) → (p → q)

instance of Axiom (∧2);

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Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

instance of Axiom (∧1);

• 2. p ∧ (p → q) ⊢H p,

by (1) and the Deduction Theorem;

• 3. ⊢H (p ∧ (p → q)) → (p → q)

instance of Axiom (∧2);

• 4. p ∧ (p → q) ⊢H p → q

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Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

instance of Axiom (∧1);

• 2. p ∧ (p → q) ⊢H p,

by (1) and the Deduction Theorem;

• 3. ⊢H (p ∧ (p → q)) → (p → q)

instance of Axiom (∧2);

• 4. p ∧ (p → q) ⊢H p → q

by (3) and the Deduction Theorem;

Goranko

Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

instance of Axiom (∧1);

• 2. p ∧ (p → q) ⊢H p,

by (1) and the Deduction Theorem;

• 3. ⊢H (p ∧ (p → q)) → (p → q)

instance of Axiom (∧2);

• 4. p ∧ (p → q) ⊢H p → q

by (3) and the Deduction Theorem;

• 5. p ∧ (p → q) ⊢H q

Goranko

Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

instance of Axiom (∧1);

• 2. p ∧ (p → q) ⊢H p,

by (1) and the Deduction Theorem;

• 3. ⊢H (p ∧ (p → q)) → (p → q)

instance of Axiom (∧2);

• 4. p ∧ (p → q) ⊢H p → q

by (3) and the Deduction Theorem;

• 5. p ∧ (p → q) ⊢H q

by (2),(4), and Modus Ponens;

Goranko

Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

instance of Axiom (∧1);

• 2. p ∧ (p → q) ⊢H p,

by (1) and the Deduction Theorem;

• 3. ⊢H (p ∧ (p → q)) → (p → q)

instance of Axiom (∧2);

• 4. p ∧ (p → q) ⊢H p → q

by (3) and the Deduction Theorem;

• 5. p ∧ (p → q) ⊢H q

by (2),(4), and Modus Ponens;

• 6. ⊢H (p ∧ (p → q)) → q

Goranko

Using the Deduction Theorem: example 1

The derivation of ⊢H (p ∧ (p → q)) → q revisited:

• 1. ⊢H (p ∧ (p → q)) → p

instance of Axiom (∧1);

• 2. p ∧ (p → q) ⊢H p,

by (1) and the Deduction Theorem;

• 3. ⊢H (p ∧ (p → q)) → (p → q)

instance of Axiom (∧2);

• 4. p ∧ (p → q) ⊢H p → q

by (3) and the Deduction Theorem;

• 5. p ∧ (p → q) ⊢H q

by (2),(4), and Modus Ponens;

• 6. ⊢H (p ∧ (p → q)) → q

by (5) and the Deduction Theorem.

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

• 3. ⊢H (¬q → ¬p) → ((¬q → p) → q),

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

• 3. ⊢H (¬q → ¬p) → ((¬q → p) → q),

instance of Axiom (→ 3),

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

• 3. ⊢H (¬q → ¬p) → ((¬q → p) → q),

instance of Axiom (→ 3),

• 4. ¬p ⊢H (¬q → p) → q,

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

• 3. ⊢H (¬q → ¬p) → ((¬q → p) → q),

instance of Axiom (→ 3),

• 4. ¬p ⊢H (¬q → p) → q,

by (2),(3), and Modus Ponens;

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

• 3. ⊢H (¬q → ¬p) → ((¬q → p) → q),

instance of Axiom (→ 3),

• 4. ¬p ⊢H (¬q → p) → q,

by (2),(3), and Modus Ponens;

• 5. ⊢H p → (¬q → p),

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

• 3. ⊢H (¬q → ¬p) → ((¬q → p) → q),

instance of Axiom (→ 3),

• 4. ¬p ⊢H (¬q → p) → q,

by (2),(3), and Modus Ponens;

• 5. ⊢H p → (¬q → p),

instance of Axiom (→ 1);

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

• 3. ⊢H (¬q → ¬p) → ((¬q → p) → q),

instance of Axiom (→ 3),

• 4. ¬p ⊢H (¬q → p) → q,

by (2),(3), and Modus Ponens;

• 5. ⊢H p → (¬q → p),

instance of Axiom (→ 1);

• 6. p ⊢H ¬q → p,

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

• 3. ⊢H (¬q → ¬p) → ((¬q → p) → q),

instance of Axiom (→ 3),

• 4. ¬p ⊢H (¬q → p) → q,

by (2),(3), and Modus Ponens;

• 5. ⊢H p → (¬q → p),

instance of Axiom (→ 1);

• 6. p ⊢H ¬q → p,

by (5) and the Deduction Theorem;

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

• 3. ⊢H (¬q → ¬p) → ((¬q → p) → q),

instance of Axiom (→ 3),

• 4. ¬p ⊢H (¬q → p) → q,

by (2),(3), and Modus Ponens;

• 5. ⊢H p → (¬q → p),

instance of Axiom (→ 1);

• 6. p ⊢H ¬q → p,

by (5) and the Deduction Theorem;

• 7. p, ¬p ⊢H q,

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

• 3. ⊢H (¬q → ¬p) → ((¬q → p) → q),

instance of Axiom (→ 3),

• 4. ¬p ⊢H (¬q → p) → q,

by (2),(3), and Modus Ponens;

• 5. ⊢H p → (¬q → p),

instance of Axiom (→ 1);

• 6. p ⊢H ¬q → p,

by (5) and the Deduction Theorem;

• 7. p, ¬p ⊢H q,

by (4),(6), and Modus Ponens.

Goranko

Using the Deduction Theorem: example 2

Derivation of p, ¬p ⊢H q:

• 1. ⊢H ¬p → (¬q → ¬p),

instance of Axiom (→ 1);

• 2. ¬p ⊢H ¬q → ¬p,

by (1) and the Deduction Theorem;

• 3. ⊢H (¬q → ¬p) → ((¬q → p) → q),

instance of Axiom (→ 3),

• 4. ¬p ⊢H (¬q → p) → q,

by (2),(3), and Modus Ponens;

• 5. ⊢H p → (¬q → p),

instance of Axiom (→ 1);

• 6. p ⊢H ¬q → p,

by (5) and the Deduction Theorem;

• 7. p, ¬p ⊢H q,

by (4),(6), and Modus Ponens.

(In step (7) we use the Monotonicity Property of ⊢H, applied to (4) and (6).)