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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’ viewpoint 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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View Full Document( 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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 Spring '08
 Bachmair,L
 Computer Science

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