Truth

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:

1. Can claims about the future be true now?
2. Can there be some algorithm for finding truth – some recipe or
procedure for deciding, for any claim in the system of, say,
arithmetic, whether the claim is true?
3. Can the predicate "is true" be completely defined in other terms
so that it can be eliminated, without loss of meaning, from any
context in which it occurs?
4. To what extent do theories of truth avoid paradox?
5. Is the goal of scientific research to achieve truth?

1. The Principal Problem

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.
2. 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?

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 "x + 3 = 7″
is 4. To ask "What is the truth-value of the statement that Montreal
is north of Pittsburgh?" is to ask whether the statement that Montreal
is north of Pittsburgh is true or whether it is false. (The
truth-value of that specific statement is true.)

There are many candidates for the sorts of things that can bear truth-values:

* statements
* sentence-tokens
* sentence-types
* propositions
* theories
* facts

* assertions
* utterances
* beliefs
* opinions
* doctrines
* etc.

a. 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?

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:

* Saturn is the sixth planet from the Sun.
* Saturn is the sixth planet from the Sun.
* Saturn is the sixth planet from the Sun.

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 chocolate,
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 bearers of
truth-values.

The following five sentences are in different languages, but they all
are typically used to express the same proposition or statement.

Saturn is the sixth planet from the Sun. [English]
Saturn je šestá planeta od slunce. [Czech]
Saturne est la sixième planète la plus éloignée du soleil. [French]
[Hebrew]
Saturn er den sjette planeten fra solen. [Norwegian]

The truth of the proposition that Saturn 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 "Saturn" 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.

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.
b. Constraints on Truth and Falsehood

There are two commonly accepted constraints on truth and falsehood:

Every proposition is true or false. [Law of the Excluded Middle.]
No proposition is both true and false. [Law of Non-contradiction.]

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 January 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 January 15,
2000 and (ii) that it rains in Sacramento at noon on January 15, 2000.
The person is saying proposition (i) is true and (ii) is false.
c. 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 (that is,
indicative) sentences – rather than interrogative or imperative
sentences – typically are used to express propositions.
d. 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.

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 express 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.
If there will be a run on the bank tomorrow, then the CEO should
be awakened.
So, the CEO should be awakened.

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.)

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.
3. 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.

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 relationship—the 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.
4. Tarski's Semantic Theory

tarskiTo capture what he considered to be the essence of the
Correspondence Theory, Alfred 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 ist 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 ist weiss" is true …'?" His answer:
T: The proposition expressed by the German sentence "Schnee ist
weiss" is true if and only if snow is white.

We can rewrite Tarski's T-condition on three lines:

1. The proposition expressed by the German sentence "Schnee ist
weiss" is true
2. if and only if
3. 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.

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 "T-proposition" of the bi-conditional form:
(T) The proposition expressed by S-in-L is true, if and only if p.

In the example we have been using, namely, "Schnee ist weiss", it is
quite clear that the T-proposition consists of a containing (or
"outer") sentence in English, and a contained (or "inner" or quoted)
sentence in German:
T: The proposition expressed by the German sentence "Schnee ist
weiss" is true if and only if snow is white.

There are, we see, sentences in two distinct languages involved in
this T-proposition. If, however, we switch the inner, or quoted
sentence, to an English sentence, e.g. to "Snow is white", we would
then have:
T: The proposition expressed by the English sentence "Snow is
white" 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-proposition. But, according to
Tarski's theory, there are still two languages involved: (i) the
language one of whose sentences is being quoted and (ii) the language
which attributes truth to the proposition expressed by that quoted
sentence. The quoted sentence is said to be an element of the object
language, and the outer (or containing) sentence which uses the
predicate "true" is in the metalanguage.

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 T-proposition is not saying:
X: Snow is white if and only if snow is white.

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.
a. 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).

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 (that is, 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 true—that 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 (that is, the "logics" or semantics
for) the propositions expressed by such complex sentences as:
"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.

Each of these research areas contains its own intriguing problems. All
must overcome the difficulties involved with ambiguity, tenses, and
indexical phrases.
b. Can the Semantic Theory Account for Necessary Truth?

Many philosophers divide the class of propositions into two mutually
exclusive and exhaustive subclasses: namely, propositions that are
contingent (that is, those that are neither necessarily-true nor
necessarily-false) and those that are noncontingent (that is, those
that are necessarily-true or necessarily-false).

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:
Snow is white. Snow is purple.
Canada belongs to the U.N. It is false that Canada belongs to the U.N.

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]). Even under those conditions,
the truth-values of the following (noncontingent) propositions will
remain unchanged:
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

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.

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 the way
the world is, but rather – because of certain features of our human
linguistic constructs.
c. 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, namely, a theory of linguistic
truth (for noncontingent propositions)? 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?

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:

1. There are fewer than seven bumblebees or more than ten.
2. There are fewer than eight bumblebees or more than ten.
3. There are fewer than nine bumblebees or more than ten.
4. There are fewer than ten bumblebees or more than ten.
5. There are fewer than eleven bumblebees or more than ten.

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.
5. 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.

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,

* Elephants are gray.
* This locale is not the habitat of elephants.
* There is neither a zoo nor a circus anywhere nearby.
* Severely intoxicated persons have been known to experience hallucinations.

But perhaps the most important reason for rejecting the drunk's claim is this:

* Everyone else in the area claims not to see any pink elephants.

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 with ___. For example,
one Coherence Theory fills this blank with "the beliefs of the
majority of persons in one's society". Another fills the blank with
"one's own beliefs", and yet another fills it with "the beliefs of the
intellectuals in one's society". The major coherence theories view
coherence as requiring at least logical consistency. Rationalist
metaphysicians would claim that a proposition is true if and only if
it "is consistent with all other true propositions". Some rationalist
metaphysicians go a step beyond logical consistency and claim that a
proposition is true if and only if it "entails (or logically implies)
all other true propositions". Leibniz, Spinoza, Hegel, Bradley,
Blanshard, Neurath, Hempel (late in his life), Dummett, and Putnam
have advocated Coherence Theories of truth.

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? It would be circular to
say that "X is consistent with Y" means "it is possible for X and Y
both to be true together" because this response 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?"
a. 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 movers and shakers – 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.

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.
6. 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.

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 a 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 some persons 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?
7. 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.
a. Redundancy Theory

The principal deflationary theory is the Redundancy Theory advocated
by Frege, Ramsey, and Horwich. Frege expressed the idea this way:

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.
b. 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.

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.
c. Prosentential Theory

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.

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".
8. Related Issues
a. 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.

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.
b. 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.

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.
c. 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.)

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 Udefinability 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.
d. 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.

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.
e. 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.

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.
9. References and Further Reading

* 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.