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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)