chapter10 - CHAPTER 10 Introduction to Intuitionistic Logic...

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CHAPTER 10 Introduction to Intuitionistic Logic Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism . Intuitionism was orig- inated by L. E. J. Brouwer in 1908. The first Hilbert style formalization of the intuitionistic logic, formulated as a proof system, is due to A. Heyting (1930). In this chapter we present a Hilbert style proof system that is equivalent to the Heyting’s original formalization and discuss the relationship between intuition- istic and classical logic. There have been, of course, several successful attempts at creating models for the intuitionistic logic, and hence to define formally a notion of the intuitionistic tautology. The most known are Kripke models and topological and algebraic models. Kripke models were defined by Kripke in 1964. The topological and algebraic models were initiated by Stone and Tarski in 1937, 1938, respectively. An uniform theory and presentation of topological and algebraic models was given by Rasiowa and Sikorski in 1964. We present both approaches, Kripke and Rasiowa and Sikorski in the respective chapters on Kripke and Algebraic Models. We also give there the respective proofs of the Completeness Theorem for the proof systems presented in this section. The goal of this chapter is to give a presentation of the intuitionistic logic formulated as a proof system and discuss the basic theorems that establish the relationship between classical and intuitionistic logics. 1 Philosophical Motivation Intuitionists’ view-point on the meaning of the basic logical and set theoretical concepts used in mathematics is different from that of most mathematicians in their research. The basic difference lies in the interpretation of the word exists . For exam- ple, let A ( x ) be a statement in the arithmetic of natural numbers. For the mathematicians the sentence xA ( x ) (1) is true if it is a theorem of arithmetic, i.e. if it can be deduced from the axioms of arithmetic by means of classical logic. If a mathematician proves sentence 1
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( 1), this does not mean that he is able to indicate a method of construction of a natural number n such that A ( n ) holds. For the intuitionist the sentence ( 1) is true only he is able to provide a con- structive method of finding a number n such that A ( n ) is true. Moreover, the mathematician often obtains the proof of the existential sentence ( 1), i.e. of the sentence xA ( x ) by proving first a sentence ¬∀ x ¬ A ( x ) . (2) Next he makes use of a classical tautology ( ¬∀ x ¬ A ( x )) ⇒ ∃ xA ( x )) . (3) By applying Modus Ponens to ( 2) and ( 3) he obtains ( 1). For the intuitionist such method is not acceptable, for it does not give any method of constructing a number n such that A ( n ) holds. For this reason the in- tuitionist do not accept the classical tautology ( 3) i.e. ( ¬∀ x ¬ A ( x )) ⇒ ∃ xA ( x )) as intuitionistic tautology, or as as an intuitionistically provable sentence. Let us denote by
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chapter10 - CHAPTER 10 Introduction to Intuitionistic Logic...

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