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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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 Spring '08
 Bachmair,L
 Computer Science, Logic, decomposition tree

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