chapter8 - CHAPTER 8 Hilbert Proof Systems, Formal Proofs,...

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Unformatted text preview: CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction Theorem The Hilbert proof systems are systems based on a language with implication and contain a Modus Ponens rule as a rule of inference. They are usually called Hilbert style formalizations. We will call them here Hilbert style proof systems, or Hilbert systems, for short. Modus Ponens is probably the oldest of all known rules of inference as it was already known to the Stoics (3rd century B.C.). It is also considered as the most natural to our intuitive thinking and the proof systems containing it as the inference rule play a special role in logic. The Hilbert proof systems put major emphasis on logical axioms, keeping the rules of inference to minimum, often in propositional case, admitting only Modus Ponens, as the sole inference rule. 1 Hilbert System H 1 Hilbert proof system H 1 is a simple proof system based on a language with implication as the only connective, with two axioms (axiom schemas) which characterize the implication, and with Modus Ponens as a sole rule of inference. We define H 1 as follows. H 1 = ( L {} , F { A 1 ,A 2 } MP ) (1) where A 1 ,A 2 are axioms of the system, MP is its rule of inference, called Modus Ponens, defined as follows: A1 ( A ( B A )) , A2 (( A ( B C )) (( A B ) ( A C ))) , MP ( MP ) A ; ( A B ) B , 1 and A,B,C are any formulas of the propositional language L {} . Finding formal proofs in this system requires some ingenuity. Lets construct, as an example, the formal proof of such a simple formula as A A . Example 1 The formal proof of ( A A ) in H 1 is a sequence B 1 , B 2 , B 3 , B 4 ,B 5 (2) as defined below. B 1 = (( A (( A A ) A )) (( A ( A A )) ( A A ))) , axiom A2 for A = A , B = ( A A ), and C = A B 2 = ( A (( A A ) A )) , axiom A1 for A = A , B = ( A A ) B 3 = (( A ( A A )) ( A A ))), MP application to B 1 and B 2 B 4 = ( A ( A A )) , axiom A1 for A = A , B = A B 5 = ( A A ) MP application to B 3 and B 4 We have hence proved the following. Lemma 1.1 For any A F , H 1 ( A A ) and the sequence 2 constitutes its formal proof. It is easy to see that the above proof wasnt constructed automatically. The main step in its construction was the choice of a proper form (substitution) of logical axioms to start with, and to continue the proof with. This choice is far from obvious for un-experienced prover and impossible for a machine, as the number of possible substitutions is infinite. Observe that the systems S 1- S 4 from the previous chapter were syntactically decidable for one simple reason. Their inference rules were such that it was 2 possible to reverse their use; to use them in the reverse manner in order to search for proofs, and we were able to do so in a blind, fully automatic way. We were able to conduct an argument of the type: if this formula has a proof the only way to construct it is from such and such formulas by the means of one of...
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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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chapter8 - CHAPTER 8 Hilbert Proof Systems, Formal Proofs,...

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