formal_proofs_FOL_

formal_proofs_FOL_ - CS245 Winter 2010 Lecture of March 1st...

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Unformatted text preview: CS245 - Winter 2010, Lecture of March. 1st and 3rd Formal proofs in FOL Shai Ben-David This lecture is devoted to presenting formal proofs in first order logic (FOL). The structure of FOL formal proofs is similar to what we saw in propositional calculus. we have a set of axioms and use the Modus Ponens (MP) operation to generate the set of provable formulas from these axioms. More formally, a formula α is a formal theorem if it is a member of the inductive set I ( Axioms,MP ). We denote the statement that α is a formal theorems by ‘ α . Given a set of formulas, Γ, a formula α is a formal consequence of Γ if it is a member of the inductive set I ( Axioms ∪ Γ ,MP ). We denote the statement that α is a formal consequence of Γ by Γ ‘ α . So far, the definitions are identical to the definitions we have sen in propo- sitional calculus. To turn the above definitions into a concrete proof system, we need to define the set of axioms. the axioms of first order logic I Any substitution in a propositional tautology. That is, if φ ( p 1 ,...p n ) is a propositional tautology whose variables are among { p 1 ,...p n } and α 1 ,...α n are FOL formulas, then the formula φ ( α 1 ,...α n ), obtained by replacing each occurrence of any p i in φ by the formula α i is an axiom of type I . For example, the formula ∀ x ∃ y ( R ( x,y ) → ( P ( x )) ∨ ¬∀ x ∃ y ( R ( x,y ) → ( P ( x ))) is such an axiom. It is obtained by substituting the formula ∀ x ∃ y ( R ( x,y ) → ( P ( x )) for A in the tautology A ∨ ¬ A ). II α → ∀ xα whenever the variable x does not occur free in the formula α . For example, P ( x ) → ∀ yP ( x ). Note that the formula P ( x ) → ∀ xP ( x ) is not a case of this axiom, since x does occur free in the formula P ( x ). III ∀ x ( α → β ) → ( ∀ xα → ∀ xβ ). This is an axiom for any pair of formulas α,β ....
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formal_proofs_FOL_ - CS245 Winter 2010 Lecture of March 1st...

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