Natural Deduction Akim Demaille akim@lrde.epita.fr EPITA cole Pour - - PowerPoint PPT Presentation

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Natural Deduction Akim Demaille akim@lrde.epita.fr EPITA cole Pour - - PowerPoint PPT Presentation

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Natural Deduction Akim Demaille akim@lrde.epita.fr EPITA cole Pour lInformatique et les Techniques Avances June 10, 2016 Define logic [fuc, ] A. Demaille Natural Deduction 2 / 49 Natural Deduction 1 Logical Formalisms

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Natural Deduction

Akim Demaille akim@lrde.epita.fr

EPITA — École Pour l’Informatique et les Techniques Avancées

June 10, 2016

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Define “logic” [fuc, ]

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Natural Deduction

1

Logical Formalisms

2

Natural Deduction

3

Additional Material

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Natural Deduction 3 / 49

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Preamble

The following slides are implicitly dedicated to classical logic.

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Logical Formalisms

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Logical Formalisms Syntax Proof Types Proof Systems

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Natural Deduction

3

Additional Material

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Natural Deduction 5 / 49

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Syntax

1

Logical Formalisms Syntax Proof Types Proof Systems

2

Natural Deduction

3

Additional Material

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Natural Deduction 6 / 49

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Terminal Symbols

Propositional Calculus

Constants a, b, c, . . . Propositional Variables A, B, C, . . . Unary Connective ¬ Binary Connectives ∧, ∨, ⇒ Punctuation (, ), [, ].

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Terminal Symbols

Propositional Calculus

Constants a, b, c, . . . Propositional Variables A, B, C, . . . Unary Connective ¬ Binary Connectives ∧, ∨, ⇒ Punctuation (, ), [, ].

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Natural Deduction 7 / 49

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Terminal Symbols

Propositional Calculus

Constants a, b, c, . . . Propositional Variables A, B, C, . . . Unary Connective ¬ Binary Connectives ∧, ∨, ⇒ Punctuation (, ), [, ].

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Natural Deduction 7 / 49

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Terminal Symbols

Propositional Calculus

Constants a, b, c, . . . Propositional Variables A, B, C, . . . Unary Connective ¬ Binary Connectives ∧, ∨, ⇒ Punctuation (, ), [, ].

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Natural Deduction 7 / 49

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Terminal Symbols

Propositional Calculus

Constants a, b, c, . . . Propositional Variables A, B, C, . . . Unary Connective ¬ Binary Connectives ∧, ∨, ⇒ Punctuation (, ), [, ].

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Natural Deduction 7 / 49

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Terminal Symbols

Predicate calculus

Individual Variables x, y, z, . . . Functions f , g, h, . . ., with a fixed arity Predicates P, Q, R, . . ., with a fixed arity Quantifiers ∀, ∃ Punctuation ·.

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Natural Deduction 8 / 49

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Terminal Symbols

Predicate calculus

Individual Variables x, y, z, . . . Functions f , g, h, . . ., with a fixed arity Predicates P, Q, R, . . ., with a fixed arity Quantifiers ∀, ∃ Punctuation ·.

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Terminal Symbols

Predicate calculus

Individual Variables x, y, z, . . . Functions f , g, h, . . ., with a fixed arity Predicates P, Q, R, . . ., with a fixed arity Quantifiers ∀, ∃ Punctuation ·.

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Natural Deduction 8 / 49

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Terminal Symbols

Predicate calculus

Individual Variables x, y, z, . . . Functions f , g, h, . . ., with a fixed arity Predicates P, Q, R, . . ., with a fixed arity Quantifiers ∀, ∃ Punctuation ·.

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Natural Deduction 8 / 49

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Terminal Symbols

Predicate calculus

Individual Variables x, y, z, . . . Functions f , g, h, . . ., with a fixed arity Predicates P, Q, R, . . ., with a fixed arity Quantifiers ∀, ∃ Punctuation ·.

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Propositional Formulas

formula ::= propositional variable | ¬formula | formula ∧ formula | formula ∨ formula | formula ⇒ formula

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Terms

term ::= constant | function(term, . . .) With the proper arity.

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First Order Formulas

formula ::= propositional variable | ¬formula | formula ∧ formula | formula ∨ formula | formula ⇒ formula | predicate(term, . . .) | ∀individual variable · formula | ∃individual variable · formula With the proper arity.

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Syntactic Conventions

Associativity ∧, ∨ are left-associative (unimportant) ⇒ is right-associative (very important) Precedence (increasing)

1 ∀, ∃ 2 ⇒ 3 ∨ 4 ∧ 5 ¬

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Syntactic Conventions

Associativity ∧, ∨ are left-associative (unimportant) ⇒ is right-associative (very important) Precedence (increasing)

1 ∀, ∃ 2 ⇒ 3 ∨ 4 ∧ 5 ¬

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Free Variables

FV(X) = ∅ FV(P(x1, x2, · · · , xn)) = {x1, x2, · · · , xn} FV(¬A) = FV(A) FV(A ∨ B) = FV(A) ∪ FV(B) FV(A ∧ B) = FV(A) ∪ FV(B) FV(A ⇒ B) = FV(A) ∪ FV(B) FV(∀x · A) = FV(A) − {x} FV(∃x · A) = FV(A) − {x}

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Proof Types

1

Logical Formalisms Syntax Proof Types Proof Systems

2

Natural Deduction

3

Additional Material

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Different Proof Types

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Proof Systems

1

Logical Formalisms Syntax Proof Types Proof Systems

2

Natural Deduction

3

Additional Material

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Proof Systems

Hilbertian Systems Natural Deduction Sequent Calculus Natural Deduction in Sequent Calculus

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Proof Systems

Hilbertian Systems Natural Deduction Sequent Calculus Natural Deduction in Sequent Calculus

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Proof Systems

Hilbertian Systems Natural Deduction Sequent Calculus Natural Deduction in Sequent Calculus

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Proof Systems

Hilbertian Systems Natural Deduction Sequent Calculus Natural Deduction in Sequent Calculus

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Axioms

Axioms are formulas that are considered true a priori ∀x · x + 0 = x Axiom schemes use meta-variables (that range over a specific domain) X + Y = Y + X Axiom schemes are used when quantifiers are not welcome SXYZ → XZ(YZ) KXY → X Axiom schemes are used when quantifiers do not apply A ∨ ¬A

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Axioms

Axioms are formulas that are considered true a priori ∀x · x + 0 = x Axiom schemes use meta-variables (that range over a specific domain) X + Y = Y + X Axiom schemes are used when quantifiers are not welcome SXYZ → XZ(YZ) KXY → X Axiom schemes are used when quantifiers do not apply A ∨ ¬A

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Axioms

Axioms are formulas that are considered true a priori ∀x · x + 0 = x Axiom schemes use meta-variables (that range over a specific domain) X + Y = Y + X Axiom schemes are used when quantifiers are not welcome SXYZ → XZ(YZ) KXY → X Axiom schemes are used when quantifiers do not apply A ∨ ¬A

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Natural Deduction 19 / 49

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Axioms

Axioms are formulas that are considered true a priori ∀x · x + 0 = x Axiom schemes use meta-variables (that range over a specific domain) X + Y = Y + X Axiom schemes are used when quantifiers are not welcome SXYZ → XZ(YZ) KXY → X Axiom schemes are used when quantifiers do not apply A ∨ ¬A

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Inference Rules

H1 H2 · · · Hn Rule name C Axiom name A

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Logical Formalisms

David Hilbert (1862–1943)

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Hilbertian System

A single inference rule: the modus ponens A A ⇒ B modus ponens B Many axioms to define the connectives A ⇒ B ⇒ A ∧ B A ∧ B ⇒ A A ∧ B ⇒ B A ⇒ A ∨ B B ⇒ A ∨ B A ∨ B ⇒ (A ⇒ C) ⇒ (B ⇒ C) ⇒ C A ⇒ B ⇒ A (A ⇒ (B ⇒ C)) ⇒ (A ⇒ B) ⇒ A ⇒ C A ∨ ¬A A ⇒ ¬A ⇒ B

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Hilbertian System

A single inference rule: the modus ponens A A ⇒ B modus ponens B Many axioms to define the connectives A ⇒ B ⇒ A ∧ B A ∧ B ⇒ A A ∧ B ⇒ B A ⇒ A ∨ B B ⇒ A ∨ B A ∨ B ⇒ (A ⇒ C) ⇒ (B ⇒ C) ⇒ C A ⇒ B ⇒ A (A ⇒ (B ⇒ C)) ⇒ (A ⇒ B) ⇒ A ⇒ C A ∨ ¬A A ⇒ ¬A ⇒ B

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Hilbertian System

A single inference rule: the modus ponens A A ⇒ B modus ponens B Many axioms to define the connectives A ⇒ B ⇒ A ∧ B A ∧ B ⇒ A A ∧ B ⇒ B A ⇒ A ∨ B B ⇒ A ∨ B A ∨ B ⇒ (A ⇒ C) ⇒ (B ⇒ C) ⇒ C A ⇒ B ⇒ A (A ⇒ (B ⇒ C)) ⇒ (A ⇒ B) ⇒ A ⇒ C ⇒ A ∨ ¬A A ⇒ ¬A ⇒ B

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Hilbertian System: Prove A ⇒ A

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Hilbertian System: Prove A ⇒ A

(A ⇒ ((A ⇒ A) ⇒ A)) ⇒ (A ⇒ A ⇒ A) ⇒ A ⇒ A A ⇒ (A ⇒ A) ⇒ A (A ⇒ A ⇒ A) ⇒ A ⇒ A A ⇒ A ⇒ A A ⇒ A

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Natural Deduction

1

Logical Formalisms

2

Natural Deduction Syntax Normalization

3

Additional Material

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Syntax

1

Logical Formalisms

2

Natural Deduction Syntax Normalization

3

Additional Material

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Deduction

Deduction

A deduction is a tree whose root (A) is the conclusion and whose leafs (Γ) is the set of hypotheses. Γ · · · A Any formula A is a valid hypothesis.

Proof (Demonstration)

A proof is a deduction without hypotheses.

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Deduction

Deduction

A deduction is a tree whose root (A) is the conclusion and whose active leafs (Γ) is the set of hypotheses. Γ · · · A Any formula A is a valid hypothesis.

Proof (Demonstration)

A proof is a deduction without hypotheses.

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Deductions

What’s this? A

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Deductions

What’s this? A A deduction of A under the hypothesis A.

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Implication

[A] · · · B ⇒I A ⇒ B · · · A · · · A ⇒ B ⇒E B Deduction theorem, and Modus Ponens. Note the connection with (left) contraction: any number of A (including 0) is discharged.

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Implication

[A] · · · B ⇒I A ⇒ B · · · A · · · A ⇒ B ⇒E B Deduction theorem, and Modus Ponens. Note the connection with (left) contraction: any number of A (including 0) is discharged.

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Implication

[A] · · · B ⇒I A ⇒ B · · · A · · · A ⇒ B ⇒E B Deduction theorem, and Modus Ponens. Note the connection with (left) contraction: any number of A (including 0) is discharged.

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Implication

[A] · · · B ⇒I A ⇒ B · · · A · · · A ⇒ B ⇒E B Deduction theorem, and Modus Ponens. Note the connection with (left) contraction: any number of A (including 0) is discharged.

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Proving A ⇒ A in Natural Deduction

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Proving A ⇒ A in Natural Deduction

[A] ⇒I A ⇒ A

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Conjunction

· · · A · · · B ∧I A ∧ B · · · A ∧ B ∧lE A · · · A ∧ B ∧rE B

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Conjunction

· · · A · · · B ∧I A ∧ B · · · A ∧ B ∧lE A · · · A ∧ B ∧rE B

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Universal Quantification

· · · A[y/x] ∀I y ∈ FV(hyp(A)) ∀x · A · · · ∀x · A ∀E A[t/x] [A] ∀I ∀x · A ⇒I A ⇒ ∀x · A [A] ⇒I A ⇒ A ∀I ∀x · (A ⇒ A)

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Universal Quantification

· · · A[y/x] ∀I y ∈ FV(hyp(A)) ∀x · A · · · ∀x · A ∀E A[t/x] [A] ∀I ∀x · A ⇒I A ⇒ ∀x · A [A] ⇒I A ⇒ A ∀I ∀x · (A ⇒ A)

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Universal Quantification

· · · A[y/x] ∀I y ∈ FV(hyp(A)) ∀x · A · · · ∀x · A ∀E A[t/x] [A] ∀I ∀x · A ⇒I A ⇒ ∀x · A [A] ⇒I A ⇒ A ∀I ∀x · (A ⇒ A)

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Absurd

· · · ⊥ ⊥E A

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Disjunction

· · · A ∨lI A ∨ B · · · B ∨rI A ∨ B · · · A ∨ B [A] · · · C [B] · · · C ∨E C

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Existential Quantification

· · · A[t/x] ∃I ∃x · A · · · ∃x · A [A[y/x]] · · · B ∃E y ∈ FV(B, hyp(B)) B For elimination, y ∈ hyp(B), i.e., not in the hypotheses other than the discharged A.

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Negation

[A] · · · ⊥ ¬I ¬A · · · A · · · ¬A ¬E ⊥

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Negation

[A] · · · ⊥ ¬I ¬A · · · A · · · ¬A ¬E ⊥ Plus one of these equivalent formulations of the fact that classical negation is involutive. XM A ∨ ¬A · · · ¬¬A ¬¬ A [¬A] · · · B [¬A] · · · ¬B Contradiction A

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Normalization

1

Logical Formalisms

2

Natural Deduction Syntax Normalization

3

Additional Material

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Natural Deduction 36 / 49

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Cut

Cut: Introduction of a connective followed by its elimination. · · · A · · · B ∧I A ∧ B ∧lE A · · ·

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Normalization

The normalization process eliminates the cuts. · · · A · · · B ∧I A ∧ B ∧lE A ❀ · · · A · · ·

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Normalizing Conjunctions

· · · A · · · B ∧I A ∧ B ∧lE A · · · ❀ · · · A · · · · · · A · · · B ∧I A ∧ B ∧rE B · · · ❀ · · · B · · ·

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Normalizing Implications

· · · A [A] · · · B ⇒I A ⇒ B ⇒E B · · · ❀ · · · A · · · B · · ·

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Normalizing Universal Quantifiers

· · · A ∀I ∀x · A ∀E A[t/x] · · · ❀ · · · A[t/x] · · · x must not be free in the hypotheses, otherwise the reduction would change them.

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Normalizing Disjunction

· · · A ∨lI A ∨ B [A] · · · C [B] · · · C ∨E C · · · ❀ · · · A · · · C · · · · · · B ∨rI A ∨ B [A] · · · C [B] · · · C ∨E C · · · ❀ · · · B · · · C · · ·

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Additional Material

1

Logical Formalisms

2

Natural Deduction

3

Additional Material

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Logicians in a Bar

Three logicians walk into a bar, and the bartender asks “Would you all like a drink?” The first one says, “Maybe.” The second one says, “Maybe.” The third one says, “Yes.”

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Edukera

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Edukera

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Edukera

http://edukera.appspot.com 8245EEC

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Bibliography Notes

[Girard et al., 1989] A short (160p.) book addressing all the concerns of this course, and more (especially Linear Logic). Easy and pleasant to read. Now available for free. [Girard, 2004] A much more comprehensive book focusing on logic and its connections with computer science. In French.

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Bibliography I

Fuck theory — Experiments in visceral philosophy. http://fucktheory.tumblr.com/post/12492878134/ dictionary-of-philosophy-logic. Girard, J.-Y. (2004). Cours de Logique, Rome, Automne 2004. http://logica.uniroma3.it/uif/corso/. Girard, J.-Y., Lafont, Y., and Taylor, P. (1989). Proofs and Types. Cambridge University Press. http://www.cs.man.ac.uk/~pt/stable/Proofs+Types.html.

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