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chapter11 - CHAPTER 11 Two Gentzen Style Proof Systems for...

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CHAPTER 11 Two Gentzen Style Proof Systems for Classical Logic Hilbert style systems are easy to define and admit a simple proof of the Com- pleteness Theorem but they are difficult to use. By humans, not mentioning computers. Their emphasis is on logical axioms, keeping the rules of inference at a minimum. Gentzen systems reverse this situation by emphasizing the importance of infer- ence rules, reducing the role of logical axioms to an absolute minimum. They may be less intuitive then the Hilbert-style systems, but they will allow us to give an effective automatic procedure for proof search, what was impossible in a case of the Hilbert-style systems. The first idea of this type was presented by G. Gentzen in 1934. He dealt with a complicated structure, called sequents . We present here a version (without structural rules) of his original formalization. His exact formalizations for clas- sical logic and for intuitionistic logic are presented in the next chapter. The other automated proof system presented here is due to H. Rasiowa and R. Sikorski and appeared for the first time in 1961. It is inspired by Gentzen original system and is equivalent to it (like may others), so all of them are called Gentzen style proof systems . The Rasiowa and Sikorski system (RS System) is more elegant and easier to understand, so we present it first. 1 The Gentzen Style System RS Language Let F denote a set of formulas of L = L , , , ∩} . The rules of inference of our system RS will operate on finite sequences of formulas , i.e. elements of F * , instead of just plain formulas F , as in Hilbert style formalizations. It means that we adopt as the set of expressions E of RS the set F * , i.e E = F * . 1
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We will denote the finite sequences of formulas by Γ , Δ , Σ, with indices if nec- essary. Meaning of Sequences 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. As we know, the disjunction in classical logic is associative and commutative, i.e., for any formulas A, B, C ∈ F , the formulas A ( B C ), ( A B ) C , A ( C B ), ( B A ) C , C ( B A ), C ( A B ), ( C A ) B , etc... are logically equivalent. We adopt hence a notation δ { A,B,C } = A B C to denote any disjunction of formulas A, B, C . In a general case, for any sequence Γ ∈ F * , if Γ is of a form A 1 , A 2 , ..., A n (1) then by δ Γ we will understand any disjunction of all formulas of Γ, i.e. δ Γ = A 1 A 2 ... A n . Formal Semantics for RS Let v : V AR -→ { T, F } be a truth assignment, v * its extension to the set of formulas F , we formally extend v to the set F * of all finite sequences of F as follows. For any sequence Γ ∈ F * , if Γ is of the form 1, then we define: v * (Γ) = v * ( δ Γ ) = v * ( A 1 ) v * ( A 2 ) ... v * ( A n ) .
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