Reduction and Elimination in Philosophy and the Sciences

The Comprehension Principle and Arithmetic in Fuzzy Logic
Shunsuke Yatabe, Toyonaka, Osaka, Japan
1 Introduction
Shapiro and Weir showed that Hume’s principle has a
finite model (so it can not deduce Peano arithmetic PA) in
Aristotelian logic which does not admit the existence of the
empty property [Shapiro, Weir 2000]. This shows that
Hume’s principle itself is not enough to develop arithmetic,
and we should have a certain theory of properties to realize
Frege’s program [Hale 2008]. In this paper, we investigate
a theory of property which satisfies what Myhill called
Frege’s principle [Myhill 1984], that “every formula with one
free variable determines a property (not a set) which holds
of all those and only those things which satisfy the formula”,
and we examine how much arithmetic we can develop
by it.
It is well-known that it is impossible in classical logic:
the comprehension principle, ∀ x[ x ∈ { y : P(
y)}
≡ P(
x)]
for
any P, implies a contradiction by the Russell paradox.
However, in non-classical logic, the situation is slightly
different. Many first order logics, including fuzzy logic, have
been known to imply no contradiction when the
comprehension principle is assumed. So we can regard
any singular term as an object in such logic (and this plays
an important role in Fregean Platonism). Therefore, as
Myhill, we can say:
402
So if we want to take Frege’s principle seriously, we
must begin to look at some kind of non-classical logic.
Let us consider the case of the set theory H with the comprehension
principle in Lukasiewicz infinite-valued predicate
logic ∀ L . It is a version of fuzzy logic, and is a nonclassical
logic weaker than classical logic which only has a
weak fragment of the contraction rule. White showed that
H is consistent [White 1979], and it is known as the
strongest theory among set theories with the comprehension
principle. We highlight two special features of sets (or
properties) in H which might provide a clue of analysis:
non-extensionality and full circularity.
First, the basic law V does not hold in H. Let us
remember the case of classical logic: Frege’s basic law V,
( ∀ P)( ∀Q)[
ext(
P)
= ext(
Q)
≡ ( ∀x)[
P(
x)
≡ Q(
x)]]
and the definition of membership relation imply the comprehension
principle and it implies a contradiction. Neo-
Fregeans modified the basic law V and adopted Restricted-V
(RV) when they developed a set theory in classical
logic [Shapiro 2003]. In H, the basic law V is equivalent
to the axiom of extensionality (which insists the extensional
equality is equivalent to the Leibniz equality), and it implies
a contradiction (it is called Grisin’s paradox [Grisin 1982]).
Furthermore, any version of RV has not been known to be
consistent to H yet.
Second, H forgives circular definitions. It is because
H proves a general form of recursive definition, which is a
circular definition of a very strong shape, is permitted:
( ∀ x)( ∃z)[
x ∈ z ≡ φ(
x,
z)]
for any formula φ [Cantini 2003]. Here, we define a set z by
using z itself. Since the recursive definition is an essence
of computation, a certain amount of arithmetic can be
developed: we can define a graph of any recursive
function in H. However, the arithmetic developed in H is
not a conservative extension of PA: the mathematical
induction scheme implies a contradiction in H [Yatabe
2007]. In fact, we can show, in any model of H, the
sentence which can be interpreted as “ω contains a nonstandard
natural number” is truth-value 1.
On a final note we remark about an adaptation of
non-classical logic. As for an antecedent, Dummett lapsed
classical logic for his own philosophical purposes, antirealism.
In this sense, adherence to classical logic is not
what is should be. Furthermore, this suggests that, we may
argue that some rule (the law of excluded middle in his
case) of classical logic corresponds a certain philosophical
viewpoint (as realism). We might ask what kind of a
philosophical assumption corresponds to the contraction
rule as a future task 1 .
2 The set theory H, extensionality and the
basic law V
The comprehension principle will derive a contradiction in
classical logic. However, the contraction rule is essential to
derive it. Grisin proved that the comprehension principle
derives no contradiction in the system Grisin logic which is
classical logic minus the contraction rule [Grisin 1982].
So, the next question is, where the limit is: what is
the strongest logic, between Girisin logic and classical
logic, which does not derive a contradiction? Currently, the
strongest logic is known to be Lukasiewicz infinite-valued
predicate logic ∀ L [White 1979]. This system is known to
be impossible to recursively axiomatize, so we introduce
the definition of its model (because of the luck of space,
here we introduce the quite informal one: we note that, this
is a simplification of ( [ 0,
1],
*, ⇒, 0,
1)
-structure where
( [ 0,
1],
*, ⇒ , 0,
1)
forms a standard MV algebra [Hajek
2001]).
(1) The truth value set is [0, 1] of real numbers (it is
a kind of fuzzy logic).
(2) ⊥ = 0 , φ → ϕ = min( 11,
− φ + ϕ )
(3) ( ∀x) ϕ ( x)
= inf{ ϕ(
a)
: a ∈ M }
The rest connectives are defined by using →, ⊥ (for example
¬ A is A →⊥ and A ∧ B is ¬ ( A → ¬ ( A → B))
). We
note that, we have the multiplicative conjunction ⊗ in ∀ L :
A ⊗ B = max{ 0,
A + B −1}
. It is easy to see that
A → A ⊗ A is truth value 1 if and only if A is truth value 0
or 1. We write A ≡ B instead of ( A → B)
⊗ ( B → A)
.
Let H be a set theory whose only axiom scheme is the
comprehension principle (i.e. ∀ x[ x ∈ { y : P(
y)}
≡ P(
x)]
is
truth value 1 for any P in any its model). We introduce two
kinds of equality in H.
Leibniz equality: x = y if and only if ( ∀ z)[ x ∈ z ≡ y ∈ z]
,
Extensional equality: x = ext y if and only if
( ∀ z)[ z ∈ x ≡ z ∈ y]
1 To answer this, we note that the contraction rule plays an essential role to
imply a contradiction not only in the Russell paradox but also in the liar paradox
and in the sorites paradox. Therefore we seem to need a unified framework
to analyze them.