On the meaning of truth degrees

Background
Formalizing truth degrees in PALTr
counterexample of the formalized truth degree theory
Conclusion

On the meaning of truth degrees

Shunsuke Yatabe

Research Center for Veriﬁcation and Semantics,
National Institute of Advanced Industrial Science and Technology,
Japan

February 20, 2010

1 / 20

Background
Conclusion

Abstract

Primary objective: to introduce an a conﬂict of semantic
account of truth and axiomatic account of truth
Secondary objective: to analyze the truth conception in
fuzzy logics by formalizing “truth degrees”
Motivation: try to explain how truth degrees relate to truth
conception
Methodology: to formalize truth degree theory in axiomatic
truth theory PAŁTr2 .
Discussion: Since PAŁTr2 is ω-inconsistent, the formalized
truth degree theory fails.

2 / 20

Background
On the meaning of fuzzy truth values
Axiomatic approach
Transparency of the truth conception
Conclusion

The framework logic

Łukasiewicz inﬁnite-valued predicate logic ∀Ł is deﬁned as follows:

(1) Truth values are real numbers in [0, 1],
(2) ϕ0 → ϕ1 = min{1, 1 − ϕ0 + ϕ1 }, ⊥ = 0,
¬A ≡ A → ⊥, etc.
(3) (∀x)ϕ(x) = inf{ ϕ(a) M : a ∈ |M|}.

∀Ł is a sublogic of classical logic (i.e. ∀Ł ϕ implies CL ϕ).

3 / 20

Background
Axiomatic approach
Conclusion

Motivation: semantic account of truth in fuzzy logics

Often said: [0, 1] are truth values,
due to historical reason (e.g. Łukasiewicz, Zadeh)
We can deﬁne truth degrees
we can construct degrees of all sentences in any algebra from
a viewpoint of metatheory:
for any sentence A, B,

A ≤ B if and only if A → B = 1

we call this ordering “truth degrees”, and often think that they
represent “degrees of truthhood” of sentences.

4 / 20

Background
Axiomatic approach
Conclusion

Motivation: fuzzy truth values and algebra (1)

Not all semantic objects are called “truth values”:
example: sets of possible worlds, situations, etc.
Many fuzzy logics are characterized by their algebras, but it is
not trivial to say “such algebraic values are truth values of
such fuzzy logics”,

Analogy: intutionistic logic case
it is not complete for Tarskian semantics with truth values
{0, 1}: two truth values are not appropriate for interpretting
intutionistic logic.
To say “the Heyting algebra is a truth value of intuitionistic
logic” is controversial: no constructivist agrees this.

5 / 20

Background
Axiomatic approach
Conclusion

Motivation: fuzzy truth values and algebra (2)

The problem: [0, 1] are not enough to interpret many fuzzy
logics.
some fuzzy predicate logics (as BL∀) are not complete for
[0, 1],
What are truth values of such logics?
Sticking around “fuzzy truth values” comes with a heavy price:
we can’t give a unified account to explain the meaning of
all fuzzy logics.

The first proposal: to have the unified account has the first
priority.

6 / 20

Background
Axiomatic approach
Conclusion

Motivation the relationship between “truth degrees” and truth

Instead of asking the meaning of truth values [0, 1] ....
Question: what is a relationship between the conception of
truth and the so called “truth degrees”?
if [0,1] are truth values, it is trivial,
if not: the ordering of truth degrees in fuzzy logics are
regarded as an abstraction of the order relation in the
algebraic semantics (e.g. Paoli).
but we never call the chain in Heyting algebra “truth degrees”....
truth degrees should be about truth, but relationship between
algebraic value and truth is not trivial.
Asking the meaning of truth degrees is asking the meaning of
“truth values” of fuzzy logic in a roundabout way.

The next goal:
to ﬁnd a framework to formalize truth degree theory without
mentioning truth values,
to formalize truth degree theory within it.
7 / 20

Background
Axiomatic approach
Conclusion

T-scheme and disquotation

Alternative framework: axiomatic truth theory
First we introduce its motivation.
Tarskian deﬁnition of truth:

Tr( ϕ ) ≡ ϕ

for any formula ϕ.
sentence “Snow is white.” is true iff snow is white.
“disquotation view of truth” (Quine, etc.):
The role of truth predicate seems to “disquote” quoted
sentences ϕ (then we get ϕ).
According to them, truth does not have a signiﬁcant role in
semantics.
8 / 20

Background
Axiomatic approach
Conclusion

The liar paradox

However, we can define the liar sentence:
“This sentence is false” (if the language has an indexical)
L ≡ ¬Tr( L )
in case truth predicate is contained in the language, or it is
definable in that theory,
we can define λ in arithmetic by diagonalization argument.
L ↔ ¬L L ↔ ¬L
[v : L] L → ¬L [w : ¬L] ¬L → L
[v : L] ¬L [w : ¬L] L
L ∨ ¬L ⊥ ⊥ −
⊥ ∨
LC
P

9 / 20

Background
Axiomatic approach
Conclusion

Two ways out

The solution 1: to sustain classical logic:
basic strategy:
to restrict the domain of truth predicate (to exclude the liar
sentence),
axiomatic truth theory case:
to restrict T-scheme not to prove L, e.g. (McGee [M85])

Tr( ϕ ) → ϕ

Such restrictions prevents that the theory implies the liar
sentence as a theorem.
The solution 2: to sustain totality (and full T-scheme) of truth
predicate
abandon classical logic

10 / 20

Background
Axiomatic approach
Conclusion

Our framework PAŁTr2

The framework to analyze truth conception of ∀Ł: PAŁTr2
[HPS00] over ∀Ł
whose axioms are all axioms of classical PA,
the induction scheme for formulae possibly containing the truth
predicate Tr and
T-schemata for a total truth predicate Tr(x)

ϕ ≡ Tr( ϕ )

where ϕ is the Godel code of ϕ.
¨
The total truth predicate is not contradictory in PAŁTr2 .
The liar sentence, L ≡ ¬Tr( L ), dose not imply a
contradiction in ∀Ł: L = 0.5,
We can have a semantically closed language of arithmetic in
∀Ł.
11 / 20

Background
Axiomatic approach
Conclusion

Transparency of the truth conception in PAŁTr2

Since the total truth predicate exists, their truth conceptions
seem to be transparent,
i.e. no theoretical restriction on the domain of Tr (as ”Tr can’t
be applied to the liar sentence) are made
philosophically, they are successors of disquotational view of
truth.

12 / 20

Background
Conclusion

Formalization: “truth degrees” in terms of axiomatic truth
theory

We define ≤ as follows (call this “degree theoretic ordering”):
ϕ ≤ ψ ≡ϕ→ψ
We define ≺ as follows (call this “ordering of truthhood”):
define an ordering ≺⊆ ω × ω as follows:
ϕ ≺ ψ ≡ Tr( ϕ ) → Tr( ψ )
≺ is defined by the conditionals of the form “a truthhood of
some formula implies a truthhood of another formula”.
Therefore these conditionals definitely represent “degrees of
truthhood” in the sense of truth theory [Fl08].
Truth degree theory says two orderings are identical: for any
formula ϕ, ψ,
ϕ ≤ ψ ≡ ϕ ≺ ψ
13 / 20

Background
Conclusion

Formzlization: the formalized truth degree theory in PAŁTr2

Before we answer the question, we formalize the truth degree
theory in PAŁTr2 .
We deﬁne ≤, ≺ between Godel codes of formulae of PAŁTr2 .
¨
(∀x, y)(Form(x)&Form(y) → [x ≤ y ≡ (Tr(x→y))],
˙
(∀x, y)(Form(x)&Form(y) → [x ≺ y ≡ (Tr(x) → Tr(y))],

The formalized truth degree theory identity degree
theoretic ordering (≤) with degrees of truthhood (≺) :

(∀x, y)(Form(x)&Form(y) → [x ≤ y ≡ x ≺ y])

The ordering need not to be linearly ordered: e.g. Paoli’s
“really fuzzy” truth degrees.

14 / 20

Background
Conclusion

A (pathological) counter example

We ﬁx PAŁTr2 as a object theory and metatheory.

PAŁTr2 shows that ≤ and ≺ are not identical:

(∀x, y)(Form(x)&Form(y) → [x ≤ y ≡ x ≺ y]) → ⊥

if Tr(x→y) ≡ (Tr(x) → Tr(y)), then mathematical induction
˙
implies the contradictory sentences [HPS00].
In this sense, the assumption of truth degree theory need not
hold.

15 / 20

Background
Conclusion

Remarks

For any formula ϕ, ψ, PAŁTr2 proves the following:

ϕ ≤ ψ ≡ ϕ ≺ ψ

However, PAŁTr2 proves, the following formalized
commutativity implies a contradiction:

(∀x, y)(Form(x)&Form(y) → [Tr(x→y) ≡ (Tr(x) → Tr(y))])
˙

This fails when x or y is a non-standard natural number
(ω-inconsistency! [R93]).

16 / 20

Background
Conclusion

An objection and replies (1)

Objection: HPS paradox merely shows that PAŁTr2 is not
suitable framework to analyze the conception of “Truth
degrees”.
ω-inconsistency is a crucial crux [Fl08],
This is not a failure of truth degree theory, but a failure of
axiomatic theory (hajek).

Reply:
Defensive: Even though PAŁTr2 are pathological, it provides
a precise distinction of two concepts (as constructive
mathematics gives a distinction between propositions which
are equivalent in classical logic).

Offensive: Since T-scheme is the key concept of truth, and
induction is essential to arithmetic, we must think the
consequence of PAŁTr2 seriously.
17 / 20

Background
Conclusion

An objection and replies (2)

Objection: What is a meaning of non-standard elements of
Form?
Reply:
In PAŁTr2 , we can represent an infinite operation on a formula
by some formula: taking a sup of
A, ¬A → A, ¬A → (¬A → A), · · · .
PAŁTr2 is a theory based on an extension of PA to represent
infinite processes in PAŁTr2 itself.
we can define an arithmetical function which corresponds an
infinite operation on codes,
it is interpreted to the real operation on formulae by using Tr,
this enables to treat an infinite process as an object in PAŁTr2 ,
non-standard numbers and non-standard elements of Form
represent such infinite processes.

18 / 20

Background
Conclusion

Conclusion

We try to formalize truth degree theory without mentioning
truth values.
We can deﬁne degree theoretic ordering ≤, but it is not trivial
that how such degree relate truth.
If we want to formalize “truth degree”, we needed a truth
predicate and truth theoretic machinery.
Truth degree theory can be formalized as supposing that
degree theoretic ordering ≤ is isomorphic to degrees of
truthhood ≺.
However some truth theory provides its counterexample
because of ω-inconsistency.
This means that, sometimes semantic anaysis and axiomatic
analysis have differing opinions.
19 / 20

Background
Conclusion

Reference

Hartry Field. “Saving Truth From Paradox” Oxford (2008)

´
Petr Hajek, Jeff B. Paris, John C. Shepherdson. “ The Liar Paradox
and Fuzzy Logic” Journal of Symbolic Logic, 65(1) (2000) 339-346.

Hannes Leitgeb. “Theories of truth which have no standard models”
Studia Logica, 68 (2001) 69-87.

Vann McGee. “How truthlike can a predicate be? A negative result”
Journal of Philosophical Logic, 17 (1985): 399-410.

Robin Milner, Mads Tofte. “Co-induction in relational semantics”
Theoretical computer science 87 (1991) 209-220.

Greg Restall “Arithmetic and Truth in Łukasiewicz’s Inﬁnitely Valued
Logic” Logique et Analyse 36 (1993) 25-38.

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On the meaning of truth degrees

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