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

Natural Deduction Akim Demaille akim@lrde.epita.fr EPITA — École Pour l’Informatique et les Techniques Avancées June 10, 2016

Define “logic” [fuc, ] A. Demaille Natural Deduction 2 / 49

Natural Deduction 1 Logical Formalisms Natural Deduction 2 Additional Material 3 A. Demaille Natural Deduction 3 / 49

Preamble The following slides are implicitly dedicated to classical logic. A. Demaille Natural Deduction 4 / 49

Logical Formalisms Logical Formalisms 1 Syntax Proof Types Proof Systems Natural Deduction 2 Additional Material 3 A. Demaille Natural Deduction 5 / 49

Syntax Logical Formalisms 1 Syntax Proof Types Proof Systems Natural Deduction 2 Additional Material 3 A. Demaille Natural Deduction 6 / 49

Terminal Symbols Propositional Calculus Constants a , b , c , . . . Propositional Variables A , B , C , . . . Unary Connective ¬ Binary Connectives ∧ , ∨ , ⇒ Punctuation ( , ) , [ , ] . A. Demaille Natural Deduction 7 / 49

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 · . A. Demaille Natural Deduction 8 / 49

Propositional Formulas � formula � � propositional variable � ::= | ¬� formula � | � formula � ∧ � formula � | � formula � ∨ � formula � | � formula � ⇒ � formula � A. Demaille Natural Deduction 9 / 49

Terms � term � ::= � constant � | � function � ( � term � , . . . ) With the proper arity. A. Demaille Natural Deduction 10 / 49

Syntactic Conventions Associativity ∧ , ∨ are left-associative (unimportant) ⇒ is right-associative (very important) Precedence (increasing) 1 ∀ , ∃ 2 ⇒ 3 ∨ 4 ∧ 5 ¬ A. Demaille Natural Deduction 12 / 49

Free Variables FV ( X ) = ∅ FV ( P ( x 1 , x 2 , · · · , x n )) = { x 1 , x 2 , · · · , x n } 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 } = A. Demaille Natural Deduction 13 / 49

Proof Types Logical Formalisms 1 Syntax Proof Types Proof Systems Natural Deduction 2 Additional Material 3 A. Demaille Natural Deduction 14 / 49

Different Proof Types A. Demaille Natural Deduction 15 / 49

Proof Systems Logical Formalisms 1 Syntax Proof Types Proof Systems Natural Deduction 2 Additional Material 3 A. Demaille Natural Deduction 17 / 49

Proof Systems Hilbertian Systems Natural Deduction Sequent Calculus Natural Deduction in Sequent Calculus A. Demaille Natural Deduction 18 / 49

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 → XZ ( YZ ) SXYZ KXY → X Axiom schemes are used when quantifiers do not apply A ∨ ¬ A A. Demaille Natural Deduction 19 / 49

Inference Rules · · · H 1 H 2 H n Rule name C Axiom name A A. Demaille Natural Deduction 20 / 49

Logical Formalisms David Hilbert (1862–1943) A. Demaille Natural Deduction 21 / 49

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 A. Demaille Natural Deduction 22 / 49

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 A. Demaille Natural Deduction 22 / 49

Hilbertian System: Prove A ⇒ A A. Demaille Natural Deduction 23 / 49

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 A. Demaille Natural Deduction 23 / 49

Natural Deduction 1 Logical Formalisms 2 Natural Deduction Syntax Normalization 3 Additional Material A. Demaille Natural Deduction 24 / 49

Syntax 1 Logical Formalisms 2 Natural Deduction Syntax Normalization 3 Additional Material A. Demaille Natural Deduction 25 / 49

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

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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