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 2.4 Propositional Natural Deduction

Valentin Goranko Stockholm University November 2020

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Natural Deduction

◮ Natural Deduction (ND): System for structured logical derivation

from a set of assumptions, based on rules, specific to the logical connectives.

◮ For each logical connective: introduction rules and elimination rules. ◮ Introduction (opening) and cancelation ( closing, discharge) of

assumptions.

◮ Assumptions can be re-used many times before canceled. ◮ Cancelation of assumptions: only when the rules allow it, but not an

• bligation.

◮ All open assumptions at the end of the derivation must be declared.

NB: the fewer (or, weaker) are the assumptions, the stronger is the claim of the derivation.

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ND rules for the propositional connectives

Introduction rules: Elimination rules: (∧I) A, B A ∧ B (∨I) A A ∨ B B A ∨ B (∧E) A ∧ B A A ∧ B B (∨E) A ∨ B [A] . . . C [B] . . . C C

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Introduction rules: Elimination rules: (→ I) [A] . . . B A → B (¬I) [A] . . . ⊥ ¬A (→ E) A, A → B B (¬E) A, ¬A ⊥

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Two more ND rules

Ex falso quodlibet: Reductio ad absurdum: (⊥) ⊥ A (RA) [¬A] . . . ⊥ A

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Propositional Natural Deduction: Example 1

A ∧ B ⊢ND B ∧ A : (∧I) (∧E) A ∧ B B (∧E) A ∧ B A B ∧ A

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Propositional Natural Deduction: Examples 2

⊢ND A → ¬¬A : (→ I) (¬I)

(¬E) [A]2, [¬A]1

⊥ ¬¬A 1 A → ¬¬A 2

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Exercise:

⊢ND ¬¬A → A

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Propositional Natural Deduction: Examples 3

A → B ⊢ND ¬B → ¬A : [A]1, A → B B , [¬B]2 ⊥ ¬A ¬B → ¬A

2

1 ¬B → ¬A ⊢ND A → B : [¬B]1, ¬B → ¬A [¬A] , [A]2 ⊥

(→I)

B A → B

2

1

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Propositional Natural Deduction: Examples 4

A ∨ B ⊢ND ¬A → B : A ∨ B [¬A]1, [A]3 ⊥ B ¬A → B 1 [¬A]2, [B]3 ¬A → B 2 ¬A → B 3

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Propositional Natural Deduction: Examples 5

⊢ND (A → (B → C)) → ((A ∧ B) → C) : (→ E) (∧E) [A ∧ B]1 B (→ E)

(∧E) [A ∧ B]1

A

,

[A → (B → C)]2 B → C (→ I)

(→I)

C (A ∧ B) → C

1

(A → (B → C)) → ((A ∧ B) → C) 2

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Propositional Natural Deduction: a challenge

Derive in ND the formula: p ∨ ¬p

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Derivations in Natural Deduction: an intuitive definition

Intuitively, a derivation in ND is a finite tree-like object D, such that

◮ the leaves of D are labelled by assumptions (premises);

these may be open, or cancelled during the derivation.

◮ every internal node of D is labelled by a formula which is the

conclusion of an instance of some ND rule, applied to the formulae labelling its children nodes;

◮ the root of D is labelled by the derived formula (conclusion) of D.

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Derivations in Natural Deduction: inductive definition

Formally, a derivation in ND is a (tree-like) object of the type D

A where

the set DND of such derivations, as well as the set of open assumptions of each derivation D

A , denoted by O

D

A

• , are defined inductively as follows:

(D1) For every propositional formula A, the object A A belongs to DND and its set of open assumptions is O A

A

• = {A}.

(D2) If D

A is in DND and B is any propositional formula, then

D, B A is in DND and its set of open assumptions is O

• D, B

A

• = O

D

A

• ∪ {B}.

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Derivations in Natural Deduction: inductive definition continued

(∧I) If D

A and D′ B are in DND then D A D′ B

A ∧ B is in DND and its set of open assumptions is O D

A

• ∪ O
• D′

B

• .

(∧E) If

D A∧B is in DND then D A∧B

A and

D A∧B

B are in DND and the set of open assumptions of each is O

• D

A∧B

• .

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Derivations in Natural Deduction: inductive definition continued

(∨lI) If D

A is in DND then D A

A ∨ B is in DND and its set of open assumptions is O D

A

• .

(∨rI) If D

B is in DND then D B

A ∨ B is in DND and its set of open assumptions is O D

B

• .

(∨E) If

D A∨B , D, A C

and D, B

C

are in DND then

D A∨B D, [A] C D, [B] C

C is in DND and its set of open assumptions is O

• D

A∨B

• ∪
• O
• D, A

C

• \ {A}
• ∪
• O
• D, B

C

• \ {B}
• .

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Derivations in Natural Deduction: inductive definition continued

(→I) If D, A

B

is in DND then

D, [A] B

A → B is in DND and its set of open assumptions is O

• D, A

B

• \ {A}.

(→E) If D

A and D′ A→B are in DND then D A D′ A→B

B is in DND and its set of open assumptions is O D

A

• ∪ O
• D′

A→B

• .

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Derivations in Natural Deduction: inductive definition continued

(¬I) If D, A

⊥

is in DND then

D, [A] ⊥

¬A is in DND and its set of open assumptions is O

• D, A

⊥

• \ {A}.

(¬E) If D

A and D′ ¬A are in DND then D A D′ ¬A

⊥ is in DND and its set of open assumptions is O D

A

• ∪ O
• D′

A

• .

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Derivations in Natural Deduction: inductive definition completed

(⊥) If D

⊥ is in DND then D ⊥

A is in DND and its set of open assumptions is O D

⊥

• .

(RA) If D, ¬A

⊥

is in DND then

D, [¬A] ⊥

A is in DND and its set of open assumptions is O

• D, ¬A

⊥

• \ {¬A}.

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Derivability from set of assumptions in Natural Deduction: definition and soundness of ND

Definition

Given a finite set of propositional formulae Γ and a formula A, A is derivable from the set of assumptions Γ, denoted Γ ⊢ND A iff there is a derivation D

A such that its set of open assumptions O

D

A

• is Γ.

Now, for any set of propositional formulae Γ and a formula A, we define Γ ⊢ND A iff Γ′ ⊢ND A for some finite set Γ′ ⊆ Γ.

Theorem (Soundness of the system of Natural Deduction ND)

For every set of propositional formulae Γ and a formula A, if Γ ⊢ND A then Γ A.

• Proof. For finite Γ: by structural induction on derivations in ND.

Then extend for any Γ – straightforward. Exercise (for now). The system of Natural Deduction ND is also complete, and hence adequate, for the logical consequence (resp. validity) of the classical propositional logic (CPL). Sketch of the proof will be presented later.

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Constructive derivations and intuitionistic propositional logic

A Natural Deduction derivation in propositional logic is constructive if it does not use the rule Reductio ad absurdum. Respectively, the inductive definition of constructive derivations does not involve the clause (RA). The logical consequence (resp. validity) that correspond to constructive derivations defines the Intuitionistic Propositional Logic (IPL).