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Philosophers are interested in a constellation of issues involving the concept of truth. A preliminary issue, although somewhat subsidiary, is to decide what sorts of things can be true. Is truth a property of sentences (which are linguistic entities in some language or other), or is truth a property of propositions (nonlinguistic, abstract and timeless entities)? The principal issue is: What is truth? It is the problem of being clear about what you are saying when you say some claim or other is true. The most important theories of truth are the Correspondence Theory, the Semantic Theory, the Deflationary Theory, the Coherence Theory, and the Pragmatic Theory. They are explained and compared here. Whichever theory of truth is advanced to settle the principal issue, there are a number of additional issues to be addressed:
Table of Contents (Clicking on the links below will take you to that part of this article)
The principal problem is to offer a viable theory as to what truth
itself consists in, or, to put it another way, "What is the nature of
truth?" To illustrate with an example the problem is not: Is
it true that there is extraterrestrial life? The problem
is: What does it mean to say that it is true that there is
extraterrestrial life? Astrobiologists study the former problem;
philosophers, the latter.
This philosophical problem of truth has been with us for a
long time. In the first century AD, Pontius Pilate (John
18:38) asked "What is truth?" but no answer was forthcoming. The
problem has been studied more since the turn of the twentieth century
than at any other previous time. In the last one hundred or so years,
considerable progress has been made in solving the problem.
The three most widely accepted contemporary theories of
truth are [i] the Correspondence
Theory ; [ii] the Semantic
Theory of Tarski and Davidson; and [iii] the Deflationary Theory of Frege and Ramsey.
The competing theories are [iv] the Coherence
Theory , and [v] the Pragmatic
Theory . These five theories will be examined after addressing the
following question.
What Sorts of Things are True (or False)?
Although we do
speak of true friends and false identities, philosophers believe these are
derivative uses of 'true' and 'false'. The central use of 'true', the more
important one for philosophers, occurs when we say, for example, it's true
that Montreal is north of Pittsburgh. Here,"true" is contrasted with "false",
not with "fake" or "insincere". When we say that Montreal is north of
Pittsburgh, what sort of thing is it that is true? Is it a statement or a
sentence or something else, a 'fact', perhaps? More generally, philosophers
want to know what sorts of things are true and what sorts of things are false.
This same question is expressed by asking: What sorts of things have (or bear)
truth-values?
Ontological Issues
What sorts of things are these candidates? In particular, should
the bearers of truth-values be regarded as being linguistic items (and,
as a consequence, items within specific languages), or are they
non-linguistic items, or are they both? In addition, should they be
regarded as being concrete entities, i.e. things which have a determinate
position in space and time, or should they be regarded as abstract
entities, i.e. as being neither temporal nor spatial
entities?
Constraints on Truth and Falsehood
There are two commonly accepted constraints on truth and falsehood:
Which Sentences Express Propositions?
Not all sentences express propositions. The interrogative sentence "Who
won the World Series in 1951?" does not; neither does the imperative sentence
"Please close the window." Declarative (i.e. indicative) sentences
rather than interrogative or imperative sentences typically are used to
express propositions.
Problem Cases
But do all declarative sentences express propositions? The
following four kinds of declarative sentences have been suggested as not being
typically used to express propositions, but all these suggestions are
controversial.
Correspondence Theory
We return to the principal question, "What is
truth?" Truth is presumably what valid reasoning preserves. It is the goal of
scientific inquiry, historical research, and business audits. We understand much
of what a sentence means by understanding the conditions under which what it
expresses is true. Yet the exact nature of truth itself is not wholly revealed
by these remarks.
Tarski's Semantic Theory
Extending the Semantic Theory Beyond "Simple" Propositions
Tarski's complete theory is intended to work for (just about) all
propositions, expressed by non-problematic declarative
sentences, not just "Snow is white." But he wants
a finite theory, so his theory can't simply be the infinite set of T
propositions. Also, Tarski wants his truth theory to reveal the logical
structure within propositions that permits valid reasoning to preserve truth. To
do all this, the theory must work for more complex propositions by showing how
the truth-values of these complex propositions depend on their parts, such as
the truth-values of their constituent propositions. Truth tables show how this
is done for the simple language of Propositional Logic (e.g. the complex
proposition expressed by "A or B" is true, according to the truth
table, if and only if proposition A is true, or proposition B is true, or both
are true).
Can the Semantic Theory Account for Necessary Truth?
Many philosophers divide the class of propositions into two mutually
exclusive and exhaustive subclasses: viz. propositions that are
contingent (i.e. those that are neither necessarily-true nor
necessarily-false) and those that are noncontingent (i.e. those
that are necessarily-true or necessarily-false).
The Linguistic Theory of Necessary Truth
Does the Semantic Theory need to be supplemented in this manner? If one
were to adopt the Semantic Theory of Truth, would one also need to adopt a
complementary theory of truth, viz. a theory of linguistic truth (for
noncontingentpropositions)? Or, can the Semantic Theory of Truth be used to
explain the truth-values of all propositions, the contingent and
noncontingent alike? If so, how?
Coherence Theories
The Correspondence Theory and
the Semantic Theory account for the truth of a proposition as arising out of a
relationship between that proposition and features or events in the world.
Coherence Theories (of which there are a number), in contrast, account for the
truth of a proposition as arising out of a relationship between that proposition
and other propositions.
Postmodernism: The Most Recent Coherence Theory
In recent years, one particular Coherence Theory has attracted a lot of
attention and some considerable heat and fury. Postmodernist philosophers ask us
to carefully consider how the statements of the most persuasive or politically
influential people become accepted as the 'common truths'. Although everyone
would agree that influential people the shakers and the movers have profound
effects upon the beliefs of other persons, the controversy revolves around
whether the acceptance by others of their beliefs is wholly a matter of their
personal or institutional prominence. The most radical postmodernists do not
distinguish acceptance as true from being true; they claim that
the social negotiations among influential people "construct" the truth. The
truth, they argue, is not something lying outside of human collective decisions;
it is not, in particular, a 'reflection' of an objective reality. Or, to put it
another way, to the extent that there is an objective reality it is nothing more
nor less than what we say it is. We human beings are, then, the ultimate
arbiters of what is true. Consensus is truth. The 'subjective' and the
'objective' are rolled into one inseparable compound.
Pragmatic Theories
A Pragmatic Theory of Truth holds (roughly) that a
proposition is true if it is useful to believe. Peirce and James were its
principal advocates. Utility is the essential mark of truth. Beliefs that lead
to the best 'payoff', that are the best justification of our actions, that
promote success, are truths, according to the
pragmatists.
Deflationary Theories
What all the theories of truth discussed so far
have in common is the assumption that a proposition is true just in case the
proposition has some property or other correspondence with the facts,
satisfaction, coherence, utility, etc. Deflationary theories deny this
assumption.
Redundancy Theory
The principal deflationary theory is the Redundancy Theory advocated by
Frege, Ramsey, and Horwich. Frege expressed the idea this way:
Performative Theory
The Performative Theory is a deflationary theory that is not a redundancy
theory. It was advocated by Strawson who believed Tarski's Semantic Theory of
Truth was basically mistaken.
Prosentential Theory of Truth
The Prosentential Theory of Truth suggests that the grammatical predicate "is
true" does not function semantically or logically as a predicate. All uses
of "is true" are prosentential uses. When someone asserts "It's true that
it is snowing", the person is asking the hearer to consider the sentence
"It is snowing" and is saying "That is true" where the remark "That is true"
is a taken holistically as a prosentence, in analogy to a pronoun. A
pronoun such as "she" is a substitute for the name of the person being
referred to. Similarly, "That is true" is a substitute for the
proposition being considered. Likewise, for the expression "it is true."
According to the Prosentential
Theory, all uses of "true" can be reduced to uses either of "That is true"
or "It is true" or variants of these with other tenses. Because these
latter prosentential uses of the word "true" cannot be eliminated from our
language during analysis, the Prosentential Theory is not a redundancy
theory.
Related Issues
Beyond Truth to Knowledge
For generations, discussions of truth have been bedeviled by the question,
"How could a proposition be true unless we know it to be true?"
Aristotle's famous worry was that contingent propositions
about the future, such as "There will be a sea battle tomorrow",
couldn't be true now, for fear that this would deny free will to the sailors
involved. Advocates of the Correspondence Theory and the Semantic Theory have
argued that a proposition need not be known in order to be true. Truth, they
say, arises out of a relationship between a proposition and the way the world
is. No one need know that that relationship holds, nor for that matter need
there even be any conscious or language-using creatures for that relationship to
obtain. In short, truth is an objective feature of a proposition, not a
subjective one.
Algorithms for Truth
An account of what "true" means does not have to tell us what is true, nor
tell us how we could find out what is true. Similarly, an account of what
"bachelor" means should not have to tell us who is a bachelor, nor should it
have to tell us how we could find out who is. However, it would be fascinating
if we could discover a way to tell, for any proposition, whether it is
true.
Can "is true" Be Eliminated?
Can "is true" be defined so that it can be replaced by its definition?
Unfortunately for the clarity of this question, there is no one concept of
"definition". A very great many linguistic devices count as definitions. These
devices include providing a synonym, offering examples, pointing at objects that
satisfy the term being defined, using the term in sentences, contrasting it with
opposites, and contrasting it with terms with which it is often confused. (For
further reading, see Definitions,
Dictionaries, and Meanings .)
Can a Theory of Truth Avoid Paradox?
The brief answer is, "Not if it contains its own concept of truth." If the
language is made precise by being formalized, and if it contains its own
so-called global truth predicate, then Tarski has shown that the language will
enable us to reason our way to a contradiction. That result shows that we do not
have a coherent concept of truth (for a language within that language). Some of
our beliefs about truth, and about related concepts that are used in the
argument to the contradiction, must be rejected, even though they might seem to
be intuitively acceptable.
Is The Goal of Scientific Research to Achieve Truth?
Except in special cases, most scientific researchers would agree that
their results are only approximately true. Nevertheless, to make sense of this,
philosophers need adopt no special concept such as "approximate truth." Instead,
it suffices to say that the researchers' goal is to achieve truth, but they
achieve this goal only approximately, or only to some
approximation.
Bradley, Raymond and Norman Swartz . Possible Worlds: an Introduction to Logic and Its Philosophy, Hackett Publishing Company, 1979.
Davidson, Donald. Inquiries into Truth and Interpretation, Oxford University Press, 1984.
Davidson, Donald. "The Structure and Content of Truth", The Journal of Philosophy, 87 (1990), 279-328.
Horwich, Paul. Truth, Basil Blackwell Ltd., 1990.
Mates, Benson. "Two Antinomies", in Skeptical Essays, The University of Chicago Press, 1981, 15-57.
McGee, Vann. Truth, Vagueness, and Paradox: An Essay on the Logic of Truth, Hackett Publishing, 1991.
Kirkham, Richard. Theories of Truth: A Critical Introduction, MIT Press, 1992.
Kripke, Saul. "Outline of a Theory of Truth", Journal of Philosophy, 72 (1975), 690-716.
Quine, W. V. "Truth", in Quiddities: An Intermittently Philosophical Dictionary, The Belknap Press of Harvard University Press, 1987.
Ramsey, F. P. "Facts and Propositions", in Proceedings of the Arisotelian Society, Supplement, 7, 1927.
Russell, B. The Problems of Philosophy, Oxford University Press, 1912.
Strawson, P. F. "Truth", in Analysis, vol. 9, no. 6, 1949.
Tarski, Alfred, "The Semantic Conception of Truth and the Foundations of Semantics", in Philosophy and Phenomenological Research, 4 (1944).
Tarski, Alfred. "The Concept of Truth in Formalized Languages", in Logic, Semantics, Metamathematics, Clarendon Press, 1956.
Bradley Dowden
Norman Swartz
The term "truth-value" has been coined by logicians as a generic term for
"truth or falsehood". To ask for the truth-value of P, is to ask whether P is
true or whether P is false. "Value" in "truth-value" does not mean "valuable".
It is being used in a similar fashion to "numerical value" as when we say that
the value of "x" in
There are many candidates for the sorts of things
that can bear truth-values:
statements
assertions
sentence-tokens
utterances
sentence-types
beliefs
propositions
opinions
theories
doctrines
facts
etc.
Sentences are linguistic items: they exist in some
language or other, either in a natural language such as English or in an
artificial language such as a computer language. However, the term "sentence"
has two senses: sentence-token and sentence-type. These three
English sentence-tokens are all of the same sentence-type:
Sentence-tokens are concrete objects. They are composed of ink marks on
paper, or sequences of sounds, or patches of light on a computer monitor, etc.
Sentence-tokens exist in space and time; they can be located in space and can be
dated. Sentence-types cannot be. They are abstract objects. (Analogous
distinctions can be made for letters, for words, for numerals, for musical notes
on a stave, indeed for any symbols whatsoever.)
Might sentence-tokens be the bearers of truth-values?
One reason
to favor tokens over types is to solve the problems involving so-called
"indexical" (or "token reflexive") terms such as "I" and "here" and "now". Is
the claim expressed by the sentence-type "I like chocolate" true or false? Well,
it depends on who "I" is referring to. If Jack, who likes chocolate, says "I
like chocolate", then what he has said is true; but if Jill, who dislikes
chocolates, says "I like chocolate", then what she has said is false. If it were
sentence-types which were the bearers of truth-values, then the sentence-type "I
like chocolate" would be both true and false an unacceptable contradiction.
The contradiction is avoided, however, if one argues that sentence-tokens are
the bearers of truth-values, for in this case although there is only one
sentence-type involved, there are two distinct sentence-tokens.
A
second reason for arguing that sentence-tokens, rather than sentence-types, are
the bearers of truth-values has been advanced by nominalist philosophers.
Nominalists are intent to allow as few abstract objects as possible. Insofar as
sentence-types are abstract objects and sentence-tokens are concrete objects,
nominalists will argue that actually uttered or written sentence-tokens are the
proper bearers of truth-values.
But the theory that
sentence-tokens are the bearers of truth-values has its own problems. One
objection to the nominalist theory is that had there never been any
language-users, then there would be no truths. (And the same objection can be
leveled against arguing that it is beliefs that are the bearers of
truth-values: had there never been any conscious creatures then there would be
no beliefs and, thus, no truths or falsehoods, not even the truth that there
were no conscious creatures an unacceptably paradoxical
implication.)
And a second objection to the theory that
sentence-tokens are the bearers of truth-values is that even though there
are language-users, there are sentences that have never been uttered and
never will be. (Consider, for example, the distinct number of different ways
that a deck of playing cards can be arranged. The number,
8×1067 [the digit "8" followed by sixty-seven zeros], is so
vast that there never will be enough sentence-tokens in the world's past or
future to describe each unique
arrangement. And there are countless other examples as well.) Sentence-tokens,
then, cannot be identified as the bearers of truth-values there simply are too
few sentence-tokens.
Thus both theories (i) that
sentence-tokens are the bearers of truth-values, and (ii) that sentence-types
are the bearers of truth-values encounter difficulties.
Might propositions be the bearers of truth-values?
To
escape the dilemma of choosing between tokens and types, propositions have been
suggested as the primary bearer of truth-values.
The following
five sentences are in different languages, but they all are typically used to
express the same proposition or statement.
The truth of the proposition that Jupiter is the sixth planet from the Sun
depends only on the physics of the solar system, and not in any obvious way on
human convention. By contrast, what these five sentences say does depend
partly on human convention. Had English speakers chosen to adopt the word
"Jupiter" as the name of a different particular planet, the first sentence would
have expressed something false. By choosing propositions rather than sentences
as the bearers of truth-values, this relativity to human conventions does not
apply to truth, a point that many philosophers would consider to be a virtue
in a theory of truth.
-- [Hebrew]
Propositions are abstract entities; they do not exist in
space and time. They are sometimes said to be 'timeless', 'eternal', or
'omnitemporal' entities. Terminology aside, the essential point is that
propositions are not concrete (or material) objects. Nor, for that matter, are
they mental entities; they are not 'thoughts' as Frege had suggested in the
nineteenth century.
The theory that propositions are the bearers of truth-values also has been
criticized. Nominalists object to the abstract character of
propositions. Another complaint is that it's not sufficiently clear when
we have a case of the same propositions as opposed to similar
propositions. This is much like the complaint that we can't determine when
two sentences have exactly the same meaning. The relationship between
sentences and propositions is a serious philosophical
problem.
Because it is the more favored theory, and for the sake
of expediency and consistency, the theory that propositions and not sentences
are the bearers of truth-values will be adopted in this article. When we speak
below of "truths", we are referring to true propositions. But it should be
pointed out that virtually all the claims made below have counterparts in
nominalistic theories which reject propositions.
These constraints require that every proposition has exactly one
truth-value.
Although the point is controversial, most philosophers add the further
constraint that a proposition never changes its truth-value in space or time.
Consequently, to say "The proposition that it's raining was true yesterday but
false today" is to equivocate and not actually refer to just one
proposition. Similarly, when someone at noon on Jan. 15, 2000 in
Vancouver says that the proposition that it is raining is true in Vancouver
while false in Sacramento, that person is really talking of two different
propositions: (i) that it rains in Vancouver at noon on Jan. 15, 2000 and (ii)
that it rains in Sacramento at noon on Jan. 15, 2000. The person is saying
proposition (i) is true and (ii) is false.
1. Sentences containing non-referring
expressions
In light of the fact that France has no king,
Strawson argued that the sentence, "The present king of France is bald", fails
to express a proposition. In a famous dispute, Russell disagreed with Strawson,
arguing that the sentence does express a proposition, and more exactly, a false
one.
2. Predictions of future events
What about declarative sentences that refer to events
in the future? For example, does the sentence "There will be a sea battle
tomorrow" express a proposition? Presumably, today we do not know whether
there will be such a battle. Because of this, some philosophers (including
Aristotle who toyed with the idea) have argued that the sentence, at the present
moment, does not expresses anything that is now either true or false. Another,
perhaps more powerful, motivation for adopting this view is the belief that if
sentences involving future human actions were to express propositions, i.e. were
to express something that is now true or false, then humans would be determined
to perform those actions and so humans would have no free will. To defend free
will, these philosophers have argued, we must deny truth-values to
predictions.
This complicating restriction that sentences about
the future do not now express anything true or false has been attacked
by Quine and others. These critics argue that the restriction upsets the logic
we use to reason with such predictions. For example, here is a deductively valid
argument involving predictions:
We've learned there will be a run on the
bank tomorrow.
Without assertions in this argument having truth-values, regardless of
whether we know those values, we could not assess the argument using the
canons of deductive validity and invalidity. We would have to say contrary to
deeply-rooted philosophical intuitions that it is not really an argument at
all. (For another sort of rebuttal to the claim that propositions about the
future cannot be true prior to the occurrence of the events described, see
Logical
Determinism .)
If there will be a run on the bank tomorrow, then the CEO
should be awakened.
So, the CEO should be awakened.
3. Liar Sentences
"This very sentence expresses a false proposition"
and "I'm lying" are examples of so-called liar sentences. A liar sentence can be
used to generate a paradox when we consider what truth-value to assign it. As a
way out of paradox, Kripke suggests that a liar sentence is one of those rare
declarative sentences that does not express a proposition. The sentence falls
into the truth-value gap. See the article Liar
Paradox .
4. Sentences that state moral, ethical, or
aesthetic values
Finally, we mention the so-called 'fact/value
distinction'. Throughout history, moral philosophers have wrestled with the
issue of moral realism. Do sentences such as "Torturing children is wrong"
which assert moral principles assert something true (or false), or do they
merely express (in a disguised fashion) the speaker's opinions, or feelings or
values? Making the latter choice, some philosophers argue that these declarative
sentences do not express propositions.
Historically, the most popular theory of truth
was the Correspondence Theory. First proposed in a vague form by Plato and by
Aristotle in his Metaphysics, this realist theory says truth is what
propositions have by corresponding to a way the world is. The theory says that a
proposition is true provided there exists a fact corresponding to it. In other
words, for any proposition p,
p is true if and only if p corresponds to a
fact.
The theory's answer to the question, "What is truth?" is that truth is a
certain relationshipthe relationship that holds between a proposition and its
corresponding fact. Perhaps an analysis of the relationship will reveal what all
the truths have in common.
Consider the proposition that snow is white. Remarking that the
proposition's truth is its corresponding to the fact that snow is white leads
critics to request an acceptable analysis of this notion of correspondence.
Surely the correspondence is not a word by word connecting of a sentence to its
reference. It is some sort of exotic relationship between, say, whole
propositions and facts. In presenting his theory of logical atomism early in the
twentieth century, Russell tried to show how a true proposition and its
corresponding fact share the same structure. Inspired by the notion that
Egyptian hieroglyphs are stylized pictures, his student Wittgenstein said the
relationship is that of a 'picturing' of facts by propositions, but his
development of this suggestive remark in his Tractatus
Logico-Philosophicus did not satisfy many other philosophers, nor after
awhile, even Wittgenstein himself.
And what are facts? The notion of a
fact as some sort of ontological entity was first stated explicitly in the
second half of the nineteenth century. The Correspondence Theory does permit
facts to be mind-dependent entities. McTaggart, and perhaps Kant, held such
Correspondence Theories. The Correspondence theories of Russell, Wittgenstein
and Austin all consider facts to be mind-independent. But regardless of their
mind-dependence or mind-independence, the theory must provide answers to
questions of the following sort. "Canada is north of the U.S." can't be a fact.
A true proposition can't be a fact if it also states a fact, so what is
the ontological standing of a fact? Is the fact that corresponds to "Brutus
stabbed Caesar" the same fact that corresponds to "Caesar was stabbed by
Brutus", or is it a different fact? It might be argued that they must be
different facts because one expresses the relationship of stabbing but the other
expresses the relationship of being stabbed, which is different. In addition to
the specific fact that ball 1 is on the pool table and the specific fact
that ball 2 is on the pool table, and so forth, is there the specific fact
that there are fewer than 1,006,455 balls on the table? Is there the
general fact that many balls are on the table? Does the existence of
general facts require there to be the Forms of Plato or Aristotle? What about
the negative proposition that there are no pink elephants on the table?
Does it correspond to the same situation in the world that makes there be no
green elephants on the table? The same pool table must involve a great
many different facts. These questions illustrate the difficulty in counting
facts and
distinguishing them. The difficulty is well recognized by advocates of the
Correspondence Theory, but critics complain that characterizations of facts too
often circle back ultimately to saying facts are whatever true propositions must
correspond to in order to be true. Davidson has criticized the notion of fact,
arguing that "if true statements correspond to anything, they all correspond to
the same thing" (in "True to the Facts", Davidson
[1984]). Davidson also has argued that facts really
are the true statements themselves; facts are not named by them, as the
Correspondence Theory mistakenly supposes.
Defenders of the Correspondence Theory have
responded to these criticisms in a variety of ways. Sense can be made of the
term 'correspondence', some say, because speaking of propositions corresponding
to facts is merely making the general claim that summarizes the remark that
(i) The sentence, "Snow is white", means that
snow is white, and (ii) snow actually is white,
and so on for all the other propositions. Therefore, the Correspondence
theory must contain a theory of 'means that' but otherwise is not at fault.
Other defenders of the Correspondence Theory attack Davidson's identification of
facts with true propositions. Snow is a constituent of the fact that snow is
white, but snow is not a constituent of a linguistic entity, so facts and true
statements are different kinds of entities.
Recent work in possible world
semantics has identified facts with sets of possible worlds. The fact that the
cat is on the mat contains the possible world in which the cat is on the mat and
Adolf Hitler converted to Judaism while Chancellor of Germany. The motive for
this identification is that, if sets of possible worlds are metaphysically
legitimate and precisely describable, then so are facts.
To capture what he considered to be the essence of the Correspondence Theory,
Tarski created his Semantic Theory of Truth. In Tarski's theory, however, talk
of correspondence and of facts is eliminated. (Although in early versions of his
theory, Tarski did use the term "correspondence" in trying to explain his
theory, he later regretted having done so, and dropped the term altogether since
it plays no role within his theory.) The Semantic Theory is the successor
to the Correspondence Theory. It seeks to preserve the core concept of that
earlier theory but without the problematic conceptual baggage.
For an illustration of the theory, consider the German sentence
'Schnee is weiss' which means that snow is white. Tarski asks for the
truth-conditions of the proposition expressed by that
sentence: "Under what conditions is that proposition true?" Put another way:
"How shall we complete the following in English: 'The proposition
expressed by the German sentence 'Schnee is
weiss' is true ...'?" His answer:
We can rewrite Tarski's T-condition on three lines:
T:
The proposition expressed by
the German sentence 'Schnee ist weiss' is
true if and only if snow is white.
Line 1 is about truth. Line 3 is not about truth it asserts a
claim about the nature of the world. Thus T makes a substantive claim.
Moreover, it avoids the main problems of the earlier Correspondence Theories
in that the terms "fact" and "correspondence" play no role whatever.
1.
The proposition
expressed by the German sentence
'Schnee ist weiss' is true
2.
if and only if
3.
snow is white
A theory is a Tarskian truth theory for language L if and only if,
for each sentence S of L, if S expresses the proposition that p,
then the theory entails a true
In the example we have been using, viz. 'Schnee is weiss', it is quite clear
that the
(T)
The proposition expressed
by
There are, we see, sentences in two distinct languages involved in
this
T:
The proposition expressed
by the German sentence 'Schnee is weiss' is true if and
only if snow is white.
In this latter case, it looks as if only one language (English), not
two, is involved in expressing the
T:
The proposition expressed
by the English sentence 'Snow is white' is true if and
only if snow is white.
Tarski discovered that in order to avoid contradiction in his semantic
theory of truth, he had to restrict the object language to a limited
portion of the metalanguage. Among other restrictions, it is the metalanguage
alone that contains the truth-predicates, "true" and "false"; the object
language does not contain truth-predicates.
It is essential to see that Tarski's
This latter claim is certainly true (it is a tautology), but it is no
significant part of the analysis of the concept of truth indeed it does
not even use the words "true" or "truth", nor does it involve an object language
and a metalanguage. Tarski's T-condition does both.
X:
Snow is white is true if and
only if snow is white.
Tarski's goal is to define truth for even more complex
languages. Tarski's theory does not explain (analyze) when a name
denotes an object or when an object falls under a predicate; his theory begins
with these as given. He wants what we today call a model theory for quantified predicate
logic. His actual theory is very technical.
It uses the notion of Gödel numbering, focuses on satisfaction rather than
truth, and approaches these via the process of
recursion.
The idea of using satisfaction treats the truth of a simple proposition such
as expressed by "Socrates is mortal" by saying:
If "Socrates" is a name and "is mortal" is a predicate, then "Socrates is
mortal" expresses a true proposition if and only if there exists an object
x such that "Socrates" refers to x and "is mortal" is satisfied by x.
For Tarski's formal language of predicate logic, he'd put this more generally
as follows:
If "a" is a name and "Q" is a predicate, then "a is Q" expresses a true
proposition if and only if there exists an object x such that "a" refers
to x and "Q" is satisfied by x.
The idea is to define the predicate "is true" when it
is applied to the simplest (i.e. the non-complex or
atomic) sentences in the object language (a language, see above, which does not, itself,
contain the truth-predicate "is true"). The predicate "is true" is a predicate
that occurs only in the metalanguage, i.e. in the language we use to describe
the object language. At the second stage, his theory shows how
the truth predicate, when it has been defined for propositions
expressed by sentences of a certain degree
of grammatical complexity, can be defined for
propositions of the next greater
degree of complexity.
According to Tarski, his theory applies only to artificial languages
in particular, the classical formal languages of symbolic logic because
our natural languages are vague and unsystematic. Other philosophers for
example, Donald Davidson have not been as pessimistic as Tarski about
analyzing truth for natural languages. Davidson has made progress in extending
Tarski's work to any natural language. Doing so, he says, provides at the same
time the central ingredient of a theory of meaning for the language. Davidson
develops the original idea Frege stated in his Basic Laws of Arithmetic
that the meaning of a declarative sentence is given by certain conditions under
which it is truethat meaning is given by truth conditions.
As part of the larger program of research begun by Tarski and Davidson,
many logicians, linguists, philosophers, and cognitive scientists, often
collaboratively, pursue research programs trying to elucidate the
truth-conditions (i.e. the 'logics' or semantics for) the propositions
expressed by such complex sentences as:
Each of these research areas contains its own intriguing problems. All
must overcome the difficulties involved with ambiguity, tenses, and indexical
phrases.
"It is possible that snow is white."
[modal propositions]
"Snow is white because sunlight is
white."
[causal propositions]
"If snow were yellow, ice would melt at -
4°C."
[contrary-to-fact conditionals]
"Napoleon believed that snow is
white."
[intentional propositions]
"It is obligatory that one provide care for
one's children."
[deontological propositions]
etc.
On the Semantic Theory of Truth, contingent
propositions are those that are true (or false) because of
some specific way the world happens to be. For example all of the following
propositions are contingent:
The contrasting class of propositions comprises those whose truth (or
falsehood, as the case may be) is dependent, according to the Semantic Theory, not
on some specific way the world happens to be, but on any way the world
happens to be. Imagine the world changed however you like (provided, of course,
that its description remains logically consistent [i.e. logically possible]
Snow is white.
Snow is purple.
Canada belongs to the U.N.
It is false that Canada belongs to the
U.N.
).
Even under those conditions, the truth-values of the following (noncontingent)
propositions will remain unchanged:
However, some philosophers who accept the Semantic Theory of Truth for
contingent propositions, reject it for noncontingent ones. They have argued that
the truth of noncontingent propositions has a different basis from the truth of
contingent ones. The truth of noncontingent propositions comes about, they
say not through their correctly describing the way the world is but as a
matter of the definitions of terms occurring in the sentences expressing those
propositions. Noncontingent truths, on this account, are said to be true by
definition, or as it is sometimes said, in a variation of this theme as
a matter of conceptual relationships between the concepts at play within
the propositions, or yet another (kindred) way as a matter of the
meanings of the sentences expressing the propositions.
Truths
Falsehoods
Snow is white or it is false that snow is
white.
Snow is white and it is false that snow is
white.
All squares are rectangles.
Not all squares are
rectangles.
2 + 2 = 4
2 + 2 = 7
It
is apparent, in this competing account, that one is invoking a kind of theory of
linguistic truth. In this alternative theory, truth for a certain class
of propositions, namely the class of noncontingent propositions, is to be accounted
for not in their describing way the world is, but rather because of certain
features of our human linguistic constructs.
To see how one can argue that the Semantic Theory of Truth can be used to
explicate the truth of noncontingent propositions, consider the following
series of propositions, the first four of which are contingent, the
fifth of which is noncontingent:
Each of these propositions, as we move from the second to the fifth, is
slightly less specific than its predecessor. Each can be regarded as
being true under a greater range of variation (or circumstances) than its
predecessor. When we reach the fifth member of the series we have a proposition
that is true under any and all sets of circumstances. (Some philosophers a few
in the seventeenth century, a very great many more after the mid-twentieth
century use the idiom of "possible worlds", saying that noncontingent
truths are true in all possible worlds [i.e. under any logically possible
circumstances].) On this view, what distinguishes noncontingent truths from
contingent ones is not
that their truth arises as a consequence of facts about our language or of
meanings, etc.; but that their truth has to do with the scope (or number) of
possible circumstances under which the proposition is true. Contingent
propositions are true in some, but not all, possible circumstances (or possible
worlds). Noncontingent propositions, in contrast, are true in all possible
circumstances or in none. There is no difference as to the nature of
truth for the two classes of propositions, only in the ranges of
possibilities in which the propositions are true.
An adherent of the Semantic Theory will allow that there is, to be sure,
a powerful insight
in the theories of linguistic truth. But, they will counter, these linguistic
theories are really shedding no light on the nature of truth itself. Rather,
they are calling attention to how we often go about ascertaining the
truth of noncontingent propositions. While it is certainly possible to
ascertain the truth experientially (and inductively) of the noncontingent
proposition that all aunts are females for example, one could
knock on a great many doors asking if any of the residents were aunts and if
so, whether they were female it would be a needless exercise. We need
not examine the world carefully to figure out the truth-value of the
proposition that all aunts are females. We might, for example, simply consult
an English dictionary. How we ascertain, find out,
determine the truth-values of noncontingent propositions may (but
need not invariably) be by nonexperiential means; but from that it does not
follow that the nature of truth of noncontingent propositions is
fundamentally different from that of contingent ones.
On this latter view, the Semantic Theory of Truth is adequate for both
contingent propositions and noncontingent ones. In neither case is the
Semantic Theory of Truth intended to be a theory of
how we might go about finding out what the truth-value is of any specified
proposition. Indeed, one very important consequence of the Semantic Theory of
Truth is that it allows for the existence of propositions whose truth-values
are in principle unknowable to human beings.
And there is a second motivation for promoting the Semantic Theory of Truth
for noncontingent propositions. How is it that mathematics is able to be used
(in concert with physical theories) to explain the nature of the world? On the
Semantic Theory, the answer is that the noncontingent truths of mathematics
correctly describe the world (as they would any and every possible world). The
Linguistic Theory, which makes the truth of the noncontingent truths of
mathematics arise out of features of language, is usually thought to have
great, if not insurmountable, difficulties in grappling with this
question.
Coherence Theories are valuable because
they help to reveal how we arrive at our truth claims, our knowledge. We
continually work at fitting our beliefs together into a coherent system. For
example, when a drunk driver says, "There are pink elephants dancing on the
highway in front of us", we assess whether his assertion is true by
considering what other beliefs we have already accepted as true, namely,
But perhaps the most important reason for rejecting the drunk's claim is
this:
In short, the drunk's claim fails to cohere with a great many other claims
that we believe and have good reason not to abandon. We, then, reject the
drunk's claim as being false (and take away the car
keys).
Specifically, a Coherence Theory of Truth will claim that a
proposition is true if and only if it coheres
Coherence Theories have their critics, too. The proposition
that bismuth has a higher melting point than tin may cohere with my beliefs but
not with your beliefs. This, then, leads to the proposition being both 'true for
me' but 'false for you'. But if "true for me" means "true" and "false for you"
means "false" as the Coherence Theory implies, then we have a violation of the
law of non-contradiction, which plays havoc with logic. Most philosophers prefer
to preserve the law of non-contradiction over any theory of truth that requires
rejecting it. Consequently, if someone is making a sensible remark by saying,
"That is true for me but not for you," then the person must mean simply, "I
believe it, but you do not." Truth is not relative in the sense that something
can be true for you but not for me.
A second difficulty with
Coherence Theories is that the beliefs of any one person (or of any group) are
invariably self-contradictory. A person might, for example, believe both
"Absence makes the heart grow fonder" and "Out of sight, out of mind." But under
the main interpretation of "cohere", nothing can cohere with an inconsistent
set. Thus most propositions, by failing to cohere, will not have truth-values.
This result violates the law of the excluded middle.
And there is
a third objection. What does "coheres with" mean? For x to 'cohere with' y, at
the very least x must be consistent with y. All right, then, what does
"consistent with" mean? "x is consistent with y" means "it is possible for x and
y both to be true together". This analysis of truth is circular because it is
presupposing the very concept of truth that it is supposed to be
analyzing.
Some defenders of the Coherence Theory will respond that
"coheres with" means instead "is harmonious with". Opponents, however, are
pessimistic about the prospects for explicating the concept "is harmonious
with" without at some point or other having to invoke the concept of joint
truth.
A fourth objection is that Coherence theories focus on the
nature of verifiability and not truth. They focus on the holistic character of
verifying that a proposition is true but don't answer the principal problem,
"What is truth itself?"
These
postmodernist views have received a more sympathetic reception among social
scientists than among physical scientists. Social scientists will more easily
agree, for example, that the proposition that human beings have a superego is a
'construction' of (certain) politically influential psychologists, and that as a
result, it is (to be regarded as) true. In contrast, physical scientists are
for the most part rather unwilling to regard propositions in their own field
as somehow merely the product of consensus among eminent physical scientists.
They are inclined to believe that the proposition that protons are composed of
three quarks is true (or false) depending on whether
(or not) it accurately describes an
objective reality. They are disinclined to believe that the truth of such a
proposition arises out of the pronouncements of eminent physical scientists. In
short, physical scientists do not believe that prestige and social influence
trump reality.
The problems with Pragmatic accounts of truth are
counterparts to the problems seen above with Coherence Theories of
truth.
First, it may be useful for someone to believe a
proposition but also useful for someone else to disbelieve it. For example,
Freud said that many people, in order to avoid despair, need to believe there is
a god who keeps a watchful eye on everyone. According to one version of the
Pragmatic Theory, that proposition is true. However, it may not be useful
for other persons to believe that same proposition. They would be crushed if
they believed that there is god who keeps a watchful eye on everyone. Thus, by
symmetry of argument, that proposition is false. In this way, the
Pragmatic theory leads to a violation of the law of non-contradiction, say its
critics.
Second, certain beliefs are undeniably useful, even
though on other criteria they are judged to be objectively false. For
example, it can be useful for a person to believe that they live in a world
surrounded by people who love or care for them. According to this criticism, the
Pragmatic Theory of Truth overestimates the strength of the connection between
truth and usefulness.
Truth is what an ideally rational inquirer would in
the long run come to believe, say some pragmatists. Truth is the ideal outcome
of rational inquiry. The criticism that we don't now know what happens in the
long run merely shows we have a problem with knowledge, but it doesn't show that
the meaning of 'true' doesn't now involve hindsight from the perspective of the
future. Yet, as a theory of truth, does this reveal what 'true' means?
It is worthy of notice that the sentence "I smell the scent of
violets" has the same content as the sentence "It is true that I smell the
scent of violets." So it seems, then, that nothing is added to the thought by
my ascribing to it the property of truth. (Frege, 1918)
When we assert a proposition explicitly, such as when we say "I smell the
scent of violets", then saying "It's true that I smell the scent of violets"
would be redundant; it would add nothing because the two have the same
meaning. Today's more minimalist advocates of the Redundancy Theory retreat from
this remark about meaning and say merely that the two are necessarily
equivalent.
Where the concept of truth
really pays off is when we do not, or can not, assert a proposition explicitly,
but have to deal with an indirect reference to it. For instance, if we wish to
say, "What he will say tomorrow is true", we need the truth predicate "is true".
Admittedly the proposition is an indirect way of saying, "If he says tomorrow
that it will snow, then it will snow; if he says tomorrow that it will rain,
then it will rain; if he says tomorrow that 7 + 5 = 12, then
7 + 5 = 12; and so forth." But the phrase "is true" cannot
be eliminated from "What he will say tomorrow is true" without producing an
unacceptable infinite conjunction. The truth predicate "is true" allows us to
generalize and say things more succinctly (indeed to make those claims with only
a finite number of utterances!). In short, the Redundancy Theory may work
for certain cases, say its critics, but it is not generalizable to all; there
remain recalcitrant cases where "is true" is not redundant.
Advocates of the Redundancy Theory respond that
their theory recognizes the essential point about needing the concept of truth
for indirect reference. The theory says that this is all that the concept
of truth is needed for, and that otherwise its use is redundant.
The Performative Theory of Truth
argues that ascribing truth to a proposition is not really characterizing the
proposition itself, nor is it saying something redundant. Rather, it is telling
us something about the speaker's intentions. The speaker through his or
her agreeing with it, endorsing it, praising it, accepting it, or perhaps
conceding it is licensing our adoption of (the belief in) the
proposition. Instead of saying, "It is true that snow is white", one could
substitute "I embrace the claim that snow is white." The key idea is that saying
of some proposition, P, that it is true is to say in a disguised fashion
"I commend P to you", or "I endorse P", or something of the sort.
The
case may be likened somewhat to that of promising. When you promise to
pay your sister five dollars, you are not making a claim about the proposition
expressed by "I will pay you five dollars"; rather you are performing the
action of promising her something. Similarly, according to the Performative
Theory of Truth, when you say "It is true that Vancouver is north of
Sacramento", you are performing the act of giving your listener license to
believe [and to act upon the belief] that Vancouver is north of
Sacramento.
Critics of the Performative Theory charge that it
requires too radical a revision in our logic. Arguments have premises that are
true or false, but we don't consider premises to be actions, says Geach. Other
critics complain that, if all the ascription of "is true" is doing is gesturing
consent, as Strawson believes, then, when we say
"Please shut the door" is true,
we would be consenting to the door's being shut. Because that is absurd,
says Huw Price, something is wrong with Strawson's Performative Theory.
Critics of the theory remark that it can give no account of what is common
to all our uses of the word "true", such as those in the unanalyzed
operators "it-will-be-true-that" and "it-is-true-that" and
"it-was-true-that".
For a true proposition to be known, it must (at the very
least) be a justified belief. Justification, unlike truth itself, requires a
special relationship among propositions. For a proposition to be justified it
must, at the very least, cohere with other propositions that one has
adopted. On this account, coherence among propositions plays a critical role in
the theory of knowledge. Nevertheless it plays no role in a theory of truth,
according to advocates of the Correspondence and Semantic Theories of
Truth.
Finally, should coherence which plays such a central role
in theories of knowledge be regarded as an objective relationship or as a
subjective one? Not surprisingly, theorists have answered this latter question
in divergent ways. But the pursuit of that issue takes one beyond the theories
of truth.
Perhaps some machine could do this, philosophers have speculated.
For any formal language, we know in principle how to generate all the
sentences of that language. If we were to build a machine that produces one
by one all the many sentences, then eventually all those that express truths
would be produced. Unfortunately, along with them, we would also generate all
those that express false propositions. We also know how to build a machine that
will
generate only sentences that express truths. For example, we might
program a computer to generate "1 + 1 is not 3", then
"1 + 1 is not 4", then "1 + 1 is not 5", and so
forth. However, to generate all and only those sentences that express
truths is quite another matter.
Leibniz (1646-1716) dreamed of
achieving this goal. By mechanizing deductive reasoning he hoped to build a
machine that would generate all and only truths. As he put it, "How much better
will it be to bring under mathematical laws human reasoning which is the most
excellent and useful thing we have." This would enable one's mind to "be freed
from having to think directly of things themselves, and yet everything will turn
out correct." His actual achievements were disappointing in this regard, but his
dream inspired many later investigators.
Some progress on the general problem of capturing all and only those
sentences which express true propositions can be made by limiting the focus
to a specific domain. For instance, perhaps we can find some procedure that
will produce all and only the truths of arithmetic, or of chemistry, or of
Egyptian political history. Here, the key to progress is to appreciate that
universal and probabilistic truths 'capture' or 'contain' many more specific
truths. If we know the universal and probabilistic laws of quantum mechanics,
then (some philosophers have argued) we thereby indirectly (are in a position
to) know the more specific scientific laws about chemical bonding. Similarly,
if we can axiomatize an area of mathematics, then we indirectly have captured
the infinitely many specific theorems that could be derived from those axioms,
and we can hope to find a decision procedure for the truths, a procedure that
will guarantee a correct answer to the question, "Is that true?"
Significant progress was
made in the early twentieth century on the problem of axiomatizing arithmetic
and other areas of mathematics. Let's consider arithmetic. In the 1920s, David
Hilbert hoped to represent the sentences of arithmetic very precisely in a
formal language, then to generate all and only the theorems of arithmetic from
uncontroversial axioms, and thereby to show that all true propositions of
arithmetic can in principle be proved as theorems. This would put the concept of
truth in arithmetic on a very solid basis. The axioms would 'capture' all and
only the truths. However, Hilbert's hopes would soon be dashed. In 1931, Kurt
Gödel (1906-1978), in his First Incompleteness Theorem, proved that any
classical self-consistent formal language capable of expressing arithmetic must
also contain sentences of arithmetic that cannot be derived within that system,
and hence that the propositions expressed by those sentences could not be proven
true (or false) within that system. Thus the concept of truth transcends the
concept of proof in classical formal languages. This is a remarkable, precise
insight into the nature of truth.
However, modern theories
about definition have not been especially recognized, let alone adopted, outside
of certain academic and specialist circles. Many persons persist with the
earlier, naive, view that the role of a definition is only to offer a
synonym for the term to be defined. These persons have in mind such
examples as: "'hypostatize' means (or, is a synonym for) 'reify'
".
If one
were to adopt this older view of definition, one might be inclined to demand of
a theory of truth that it provide a definition of "is true" which permitted its
elimination in all contexts in the language. Tarski was the first person to show
clearly that there could never be such a strict definition for "is true" in its
own language. The definition would allow for a line of reasoning that produced
the Liar Paradox (recall above)
and thus would lead us into self contradiction. (See the discussion, in the
article The Liar
Paradox , of Tarski's Undefinability Theorem of 1936.)
Kripke has attempted to avoid this theorem by using only a 'partial'
truth-predicate so that not every sentence has a truth-value. In effect,
Kripke's 'repair' permits a definition of the truth-predicate within its
own language but at the expense of allowing certain violations of the law
of excluded middle.
There is no reason to believe that paradox is
to be avoided by rejecting formal languages in favor of natural languages. The
Liar Paradox first appeared in natural languages. And there are other paradoxes
of truth, such as Löb's Paradox, which follow from principles that are
acceptable in either formal or natural languages, namely the principles of modus
ponens and conditional proof.
The best solutions to the paradoxes use a
similar methodology, the "systematic approach". That is, they try to remove
vagueness and be precise about the ramifications of their solutions, usually by
showing how they work in a formal language that has the essential features of
our natural language. The Liar Paradox and Löb's Paradox represent a serious
challenge to understanding the logic of our natural language. The principal
solutions agree that to resolve a paradox we must go back and
systematically reform or clarify some of our original beliefs. For
example, the solution may require us to revise the meaning of "is true".
However, to be acceptable, the solution must be presented systematically and be
backed up by an argument about the general character of our language. In short,
there must be both systematic evasion and systematic explanation. Also, when it
comes to developing this systematic approach, the goal of establishing a
coherent basis for a consistent semantics of natural language is much more
important than the goal of explaining the naive way most speakers use the terms
"true" and "not true". The later Wittgenstein did not agree. He rejected the
systematic approach and elevated the need to preserve ordinary language, and our
intuitions about it, over the need to create a coherent and consistent
semantical theory.
Other philosophers believe it's a mistake to say the
researchers' goal is to achieve truth. These 'scientific anti-realists'
recommend saying that research in, for example, physics, economics, and
meteorology, aims only for usefulness.
When they aren't overtly identifying truth with usefulness, the instrumentalists
Peirce, James and Schlick take this anti-realist route, as does Kuhn. They would
say atomic theory isn't true or false but rather is useful for predicting
outcomes of experiments and for explaining current data.
Giere recommends saying science aims for the best available
'representation', in the same sense that maps are representations of the
landscape. Maps aren't true; rather, they fit to a better or worse degree.
Similarly, scientific theories are designed to fit the world. Scientists should
not aim to create true theories; they should aim to construct theories whose
models are representations of the world.
Sources
Author Information:
California State University Sacramento
Email: dowden@csus.edu
Simon Fraser University
Email: swartz@sfu.ca
HomePage: http://www.sfu.ca/philosophy/swartz.htm