Philosophy as Adventures of Ideas

Week 9

Truth“This sentence is not true.”

Kazuyoshi KAMIYAMA

NIT, Ibaraki College

2016/2/18

CONTENTS

The Liar Paradox

Tarski’s Definition of Truth

Theories of Truth

Overview of Truth Theories

Ⅰ Positive views

Ⅰ-1. Inflationism

Ⅰ-2. Deflationism

Ⅱ Negative views

References

To say of what is, that it is not– or of what is not,

that it is– is false. While to say of what is, that it is–

and of what is not, that it is not– is true.

（Aristotle, Metaphysics）

THE LIAR PARADOX(OR EPIMENIDES PARADOX)

A Cretan states that “all Cretans are liars.“

* Epimenides (Cretan philosopher, alive circa 600 BC)

A PARADOX OF SELF-REFERENCE

“This sentence is not true.” (1)

if (1) is true, then (1) says, truly, that (1) is not true so

that (1) is not true; on the other hand, if (1) is not true,

then what (1) says is the case, i.e., (1) is true.

Note: “this sentence" refers to (1) (the sentence itself).

RUSSELL'S PARADOX(1901)

Naive set theory(G.Cantor)：For any property there

exists the set which consists of all things that satisfies

the property

A property：being not members of themselves

ex. The set of all Americans

R＝ the set of all sets that are not members of

themselves

R= [x: ¬(x∈x)]

If R∈R, then ¬(R∈R). Contradiction.

If ¬(R∈R), then R∈R. Contradiction.

Contradiction.

Kurt Gödel (1906 - 1978)

"On Formally Undecidable Propositions of Principia

Mathematica and Related Systems," (1931)

Gödel's Incompleteness Theorem

Any consistent axiomatic system of mathematics will

contain theorems which cannot be proven.

If all the theorems of an axiomatic system can be

proven then the system is inconsistent, and thus has

theorems which can be proven both true and false.

DOUBTS ON THE PREDICATE “TRUE”

Given such a paradox, one might be skeptical of the

notion of truth, or at least of the prospects of giving a

scientifically respectable account of truth.

THE DEFINITION OF TRUTH BY TARSKI

Alfred Tarski, 1935, “Der Wahrheitsbegriff in den

formalisierten Sprachen”, Studia Philosophica, 1: 261–405.

（“The concept of truth in formalized languages”）

Convention T:

An adequate theory of truth for L must imply, for each

sentence φ of L ⌈ φ ⌉ is true if and only if φ .

Base clauses

reference and satisfaction:

‘Snow’ refers to snow.

‘Grass’ refers to grass.

a satisfies ‘is white’ if and only if a is white.

a satisfies ‘is green’ if and only if a is green.

For any atomic sentence ⌈ t is P ⌉ : ⌈ t is P ⌉ is true if and

only if the referent of t satisfies the referent of P.

Recursion clauses

For any sentences φ and ψ of L:

⌈ φ∨ ψ ⌉ is true if and only if ⌈ φ ⌉ is true or ⌈ ψ ⌉ is

true.

⌈ ¬φ ⌉ is true if and only if it is not the case that ⌈ φ ⌉ is

true.

This theory satisfies Convention T.

PROBLEMS OF TARSKI’S THEORY OF TRUTH (FOR PHILOSOPHERS)

(1) It is the definition of truth for formalized languages

(not for ordinary languages)

(2) It doesn't actually tell us why/when/how a sentence

is true.

WHAT ARE TRUTHS？

THEORIES OF TRUTH

OVERVIEW OF TRUTH THEORIES

Ⅰ Positive views: Truth exists.

Ⅰ-1. Inflationism： To define “true” with thick ontology

Ⅰ-2. Deflationism： To define “true” with thin ontology, or

without ontology

Ⅱ Negative views： Truth doe not exist. Knowing truth is not

possible. : Agnosticism, Nihilism

Ⅰ POSITIVE VIEWS

AN EARLY PROBLEM

Metaphysics (ontology) - theory about what exists in

the world : When you define "true," you seam to need

an ontology, what kind of ontology do you need? Is it

possible to dispense with ontology?

Correspondence theory: the correspondence

between a sentence and the fact. This theory assumes

a "fact" (the world consists of facts.)

Early Moore, Russell: true proposition = fact

(same theory or identity theory)

In this definition, what is false proposition?

(Is it a "shadow fact"? )

Ontology: The world consists of the facts and shadow facts.

→ They gave up this undesirable theory.

Ⅰ-1 INFLATIONISM膨張主義

A. CORRESPONDENCE THEORY

A belief is true if there exists an appropriate

entity—a fact—to which it corresponds.

If there is no such entity, the belief is false.

B. COHERENCE THEORY

A proposition is true if it coheres (or agrees) with

other propositions we already hold to be true.

C. PRAGMATIST THEORIES

C.S. Pierce (1839–1914)

William James(1842–1910)

A proposition is true if it is useful to believe it. those

propositions that best justify what we do and help us

to achieve what we are aiming at are true.

NOTE: PRAGMATISM

Pragmatism, school of philosophy, dominant in the United

States in the first quarter of the 20th century, based on the

principle that the usefulness, workability, and practicality of

ideas, policies, and proposals are the criteria of their merit. It

stresses the priority of action over doctrine, of experience over

fixed principles, and it holds that ideas borrow their meanings

from their consequences and their truths from their verification.

Thus, ideas are essentially instruments and plans of action.

(Britannica.com)

Ⅰ-2 DEFLATIONISM縮小主義

G. Frege (1848 –1925)

the founder of modern logic

and analytic philosophy

Ⅰ-2A. REDUNDANCY THEORY

F. Ramsey (1927)：the equivalence thesis:

「「 φ 」 is true」has the same meaning as φ.

P.F.Strawson (1949; 1950)

To assert that 「 φ 」 is true is just to assert that φ.

Ⅰ-2B. MINIMALIST THEORIES

P.Horwich (1990)

For a given language L and every φ in L, the biconditionals

「「 φ 」 is true if and only if φ」 hold by definition (or

analytically, or trivially, or by stipulation …).

This is all there is to say about the concept of truth.

Ⅰ-2C. DISQUOTATIONALISM

W.Quine(1970)

An attribution of truth to a sentence undoes the effects

of the quotation marks that have been used to form

sentences: Sentence "S" is true if and only if S.

Ⅰ-2D. PROSENTENTIAL THEORY代文説

Grover, et al.(1975), Grover (1992), Brandom (1994)

noun – pronoun 名詞―代名詞

sentence – prosentence 文―代文

Prosentential theorists claim that sentences such as

"That is true" are prosentences that function

analogously to their better known cousins—pronouns.

For example, just as we might use the pronoun 'he' in

place of 'James' to transform "James went to the

supermarket" into "He went to the supermarket," so

we might use the prosentence-forming operator 'is

true' to transform "Snow is white" into "'Snow is white'

is true." According to the prosentential theory of truth,

ｔo assert that a sentence is true is simply to assert or

reassert that sentence.

PLURALISM ABOUT TRUTH

Lynch (2001b; 2009), Wright (1992; 1999)

In certain domains of discourse what we say is true in virtue of a

correspondence-like relation, while in others it is its true in

virtue of a kind of assertibility relation that is closer in spirit to

the anti-realist views.

：there are multiple concepts of truth, or that the term ‘true’ is

itself ambiguous.

TENTATIVE PROPOSAL

If "Φ" is true and known as true, then we may use the "φ" as a

presupposition for subsequent proofs, or "φ" may be used as a

presuppositions for the subsequent discussions.

(If "snow is white" is true and known among us, we may freely say

that anow is white in the conversation. Conversely, if you can not

use the "snow white" freely in the conversation, it is not true that

"snow is white, " or it is not known as true. "Pluto is a planet

"cannot be used freely as a premise of the current conversation.

Thus,"Pluto is a planet "is not true.)

As a thesis:

Only if "φ" can be used as a presupposition for subsequent

proofs or discussions, "φ" is true.

「 φ 」 is true only if we can use 「 φ 」 as a presupposition of

proofs (or arguments).

Ⅱ NEGATIVE VIEWS

AGNOSTICISM

I. Kant（1724-1804）

We cannot know about the thing-in-itself

(das “Ding an sich”).

NIHILISM

No theory of truth is possible. Truth is too fundamental to our

thought to be understood in any other terms.

[It should be mentioned that no serious philosopher ever denies

the existence of Truth, fully aware that such a denial would be

self-defeating.]

Friedrich W. Nietzsche(1844-1900)

REFERENCES

M. Glanzberg, “Truth”, Stanford Encyclopedia of Philosophy, 2013

(http://plato.stanford.edu/entries/truth/)

“Theories of Truth,” Truth-Defined com.

(http://www.truthdefined.com/TheoriesOfTruth.htm#top)

Gödel's Incompleteness Theorem (1931)

(http://www.sscc.edu/home/jdavidso/math/goedel.html)