Reduction and Elimination in Philosophy and the Sciences

Clearly x = y → x = ext y holds, but the converse does not
hold in H.
Theorem 1 The axiom of extensionality,
( ∀ , y )[ x = y ≡ x = y]
, does not hold in H.
x ext
For the proof, see [Hajek 2005]. Here, we introduce the
outline: this is by the following lemma.
Lemma 1 H proves that Leibniz equality is a crisp relation.
We note that formula P(x) is crisp (Crisp(P)) if, for any
object a, the truth value of P(a) is either 0 or 1. Since =ext
can have a fuzzy truth value for fuzzy sets in H, = and =ext
are different.
As for the basic law V, it implies that = is not the Leibniz
equality. For V defines = as the extensional equality, and if
= is Leibniz equality then the axiom of extensionality holds,
and this implies a contradiction in H. However, Frege’s
intension seems to define = as Leibniz equality. In this
sense the basic law V does not hold in H.
In this proof, we imply a contradiction when x =ext y
has a fuzzy truth value: this is possible when not less than
one of x and y is a fuzzy set. As for another paradox, it is
known that we can imply a contradiction if we assume the
empty set φ = { x : ⊥}
satisfies the extensionality axiom
[Cantini 2003]. So it has been unknown that the following
scheme is consistent:
(({ x : P(
x)}
≠ext φ ) ∧Crisp(
P))
∧ (({ x : Q(
x)}
≠ext
φ)
∧Crisp(
Q))
→ [{ x : P(
x)}
= { y : Q(
y)}
≡ ( ∀x)[
P(
x)
≡ Q(
x)]]
for any formula P(x), Q(x). This means that, the following
law might hold (this is a version of the RV [Shapiro 2003]):
∀ P∀Q( Good(
P)
∧Good(
Q))
→ [ ext(
P)
= ext(
Q)
≡ ( ∀x)[
P(
x)
≡ Q(
x)]]
Therefore if this is consistent, we can think that RV gives
an implicit definition of crisp sets, badness means fuzziness,
and any fuzzy set can be regarded to represent indefinitely
extensibility.
3 Circularity and arithmetic without the induction
scheme
When we mention the formalization of arithmetic, we often
come to axiom systems with the induction scheme as PA,
but it is not a unique way. We also have type systems
which are widely used in computer science. For example,
Gödel’s system T 2 is a simple type theory [Girard et al.
1989]. T has two types, Int (integers) and Bool (booleans).
As for Int, it has two type constructors, 0 (constant symbol)
and s : Int → Int (successor function). And T does not
have the induction scheme. Instead, it has a recursion
operator R for recursive definition whose type is
R : U → ( U → Int → U)
→ U → U for any type U. It satisfies
R uv0
= u and R uv(
sn)
= v(
Ruvn)
n (if we substitute U
for Int). This operator enables us to have the primitive
recursion on integer numbers: for example, the addition
Int Int
x + y is defined by Rx(
λ z . λz'
. sz)
y . For, we can calculate
as follows:
Int Int
x + 0 a Rx(
λ z . λz
. sz)
0 a x
Int Int
x + ( sy
) a Rx(
λz
. λz
. sz)(
x + y ) y a s(
x + y )
Here t a s represents that s is a result of the computation
whose input is t. So, the above computations show that
x + 0 = x and x + ( sy
) = s(
x + y ) hold. In this way, we can
2 We note that Gödel’s original system T has the induction scheme.
The Comprehension Principle and Arithmetic in Fuzzy Logic — Shunsuke Yatabe
represent primitive recursive functionals in T without using
the induction scheme.
The arithmetic developed in H is very similar to the
system T in that they do not have the induction scheme. H
allows the circular definition 3 (as in section 1), and it
enables us to use the general form of the recursive
definition as R. Here we introduce three points (for more
details, see [Cantini 2003][Hajek 2005]). First we can
define ordinal numbers in Zermelo style: 0 = φ and
s n = { x : x = n}
for any finite ordinal n. It is easy to see, we
can define Fregean term “the number of the conception P”
(NxPx) by the comprehension principle if P is crisp and
finite. Second, the set ω of all finite ordinals can be defined
as
ω = ext { x : x = 0 ∨ ( ∃y
∈ω
)[ x = sy
]}
(Because of the luck of extensionality, we can not
require the uniqueness of ω). Third, any recursive
function’s graph can be defined. In other words, any partial
recursive function is numerically representable in H
Let us give an example of the generalized recursive
definition in H. For example, we can define the graph P of
the addition as follows 4 :
x,
0,
z ∈ P ⇔ x = z,
x,
sy,
sz
∈ P ⇔ x,
y,
z ∈P
Both x + 0 = x and x + ( sy
) = s(
x + y)
are guaranteed by
very simple way. However, we do not know P satisfies the
following conditions:
1. P is a crisp relation,
2. P defines a function p(x, y) = z
(i.e. ( ∀ x , y )[ P(
x,
y,
z)
∧ P(
x,
y,
z'
) → z = z']
),
3. p(x, y) is a total function.
If x and y are standard natural numbers, then we can calculate
the unique value z satisfying P(x, y, z). However we
have a trouble when one of x, y is a non-standard natural
number: P(x, y, z) might be a fuzzy truth value. We neither
know, whether ω or any graph defined by recursion becomes
crisp or not. If ω is fuzzy, then we might think this is
another expression of Dummett’s “ω is indefinitely extensible”.
H develops a fair degree of arithmetic; however it
can not deduce Peano arithmetic PA. In fact, the following
theorem holds.
Theorem 2 H proves that the induction scheme implies
inconsistency.
This means that H is ω-inconsistent: in any model of H, the
sentence which can be interpreted as “ω contains a nonstandard
natural number” is truth-value 1. For more details,
see [Yatabe 2007].
Contrary, Girard showed that the weak version of
mathematical induction scheme is provable in LAST, a set
theory with the comprehension principle in light linear logic
[Girard 1998] [Terui 2004]. In LAST, the definition of
natural numbers is quite different: such definition seems to
enable to prove the weak induction.
3 Meanwhile, the comprehension the principle can be thought as a special
case of the recursive definition. Since it is the foundation placed in the calculation,
so we had better to say that it is the generalized recursive definition that
is essential principle in this theory.
4 For the definition of the ordered pair in H , see [Cantini 2003].
403