chapter13(rs)

# chapter13(rs) - CHAPTER 13 Gentzen Style Proof System for...

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Unformatted text preview: CHAPTER 13 Gentzen Style Proof System for Classical Predicate Logic - The System QRS Part One 1 System QRS Definition Let F denote a set of formulas of a Predicate (first Order) Logic Language L ( P , F , C ) = L {∩ , ∪ , ⇒ , ¬} ( P , F , C ) for P, F, C countably infinite sets of predicate, functional, and constant symbols respectively. The rules of inference of our system QRS will operate, as in the propositional case, on finite sequences of formulas , i.e. elements of F * , instead of just plain formulas F , as in Hilbert style formalizations. We will denote the sequences of formulas by Γ , Δ , Σ, with indices if necessary. The intuitive meaning of a sequence Γ ∈ F * is that the truth assignment v makes it true if and only if it makes the formula of the form of the disjunction of all formulas of Γ true. If Γ is a sequence A 1 ,A 2 ,...,A n then by δ Γ we will understand the disjunction of all formulas of Γ. As we know, the disjunction in classical logic is commutative, i.e., for any for- mulas A,B,C , A ∪ ( B ∪ C ) ≡ ( A ∪ B ) ∪ C , we w will denote any of those formulas by A ∪ B ∪ C = δ { A,B,C } . Similarly, we will write δ Γ = A 1 ∪ A 2 ∪ ..., ∪ A n . The sequence Γ is said to be satisfiable ( falsifiable ) if the formula δ Γ = A 1 ∪ A 2 ∪ ..., ∪ A n is satisfiable (falsifiable). The sequence Γ is said to be a tautology if the formula δ Γ = A 1 ∪ A 2 ∪ ..., ∪ A n is a tautology. The system QRS consists of one axiom and eleven rules of inference. They form two groups. First is similar to the propositional case and called propo- sitional connectives group. Each rule of this group introduces a new logical 1 connective or its negation, so we will name them, as in the propositional case: ( ∪ ) , ( ¬∪ ) , ( ∩ ) , ( ¬∩ ) , ( ⇒ ) , ( ¬ ⇒ ) , and ( ¬¬ ). The second group deals with the quantifiers. It consists of four rules. Two of them introduce the universal and existential quantifiers, and are named ( ∀ ) and ( ∃ ), respectively. The two others correspond to the De Morgan Laws and deal with the negation of the universal and existential quantifiers, and ere named ( ¬∀ ) and ( ¬∃ ), respectively. As the axiom we adopt, as in propositional case, any sequence which contains any formula and its negation, i.e any sequence of the form Γ 1 ,A, Γ 2 , ¬ A, Γ 3 or of the form Γ 1 , ¬ A, Γ 2 ,A, Γ 3 , for any formula A ∈ F and any sequences of formulas Γ 1 , Γ 2 , Γ 3 ∈ F * . We will denote the axioms by AX * . The proof system QRS = ( F * , AX * , ( ∪ ) , ( ¬∪ ) , ( ∩ ) , ( ¬∩ ) , ( ⇒ ) , ( ¬ ⇒ ) , ( ¬¬ ) , ( ¬∀ ) , ( ¬∃ ) , ( ∀ ) , ( ∃ )) will be called a Gentzen- style formalization of classical predicate calculus....
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## This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.

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chapter13(rs) - CHAPTER 13 Gentzen Style Proof System for...

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