chapter7 - CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction...

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Unformatted text preview: CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an infer- ence machine with special statements, called provable statements , or sometimes theorems being its final products. The starting points are called axioms of the system . We distinguish two kinds of axioms: logical LA and specific SX . When building a proof system for a given language and its semantics i.e. for a logic defined semantically we choose as a set of logical axioms LA some subset of tautologies, i.e. statements always true. This is why we call them logical axioms. A proof system with only logical axioms LA is also called a logic proof system . If we build a proof system for which there is no known semantics, like it has happened in the case of classical, intuitionistic, and modal logics, we think about the logical axioms as statements universally true. We choose as axioms (finite set) the statements we for sure want to be universally true, and whatever semantics follows they must be tautologies with respect to it. Logical axioms are hence not only tautologies under an established semantics, but they also guide us how to establish a semantics, when it is yet unknown. For the set of specific axioms SA we choose these formulas of the language that describe our knowledge of a universe we want to prove facts about. They are not universally true, they are true only in the universe we are interested to describe and investigate. This is why we call them specific axioms. A proof system with logical axioms LA and specific axioms SA is called a formal theory . The inference machine is defined by a finite set of rules, called inference rules . The inference rules describe the way we are allowed to transform the informa- tion within the system with axioms as a staring point. The process of this transformation is called a formal proof and can be depicted as follows: 1 AXIOMS RULES applied to AXIOMS Provable formulas RULES applied to any expressions above NEW Provable formulas ..................... . etc. .................... . The provable formulas are those for which we have a formal proof are called consequences of the axioms, or theorem, or just simple provable formulas . When building a proof system we choose not only axioms of the system, but also specific rules of inference. The choice of rules is often linked, as was the choice of axioms, with a given semantics. We want the rules to preserve the truthfulness of what we are proving, i.e. generating from axioms via the rules. Rules with this property are called sound rules and the system a sound proof system . The theorem establishing the soundness of a given proof system is called Soundness Theorem . It states in a case of a logic proof system S that for any formula A of the language of the system S , A is provable in a (logic) proof system S , then A is a tautology....
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chapter7 - CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction...

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