This paper consists of two parts; the first has an expository character, and the second is rather polemical.

In the first part I want to summarize in an informal way the main results of my investigations concerning the definition of truth and the more general problem of the foundations of semantics. These results have been embodied in a work which appeared in print several years
ago.^{1} Although my investigations concern
concepts dealt with is classical philosophy, they happen to be comparatively little known in philosophical circles, perhaps because of their strictly technical character. For this reason
I hope I shall be excused for taking up the matter once again.^{2}

Since my work was published, various objections, of unequal value, have been raised to my
investigations; some of these appeared in print, and others were made in public and private
discussions in which I took part.^{3} In the
second part of the paper I should like to express my views regarding these objections. I
hope that the remarks which will be made in this context will not be considered as purely
polemical in character, but will be found to contain some constructive contributions to the
subject.

In the second part of the paper I have made extensive use of material graciously put at my disposal by Dr. Marja Kokoszynska (University of Lwow). I am especially indebted and grateful to Professors Ernest Nagel (Columbia University) and David Rynin (University of California, Berkeley) for their help in preparing the final text and for various critical remarks.

1. THE MAIN PROBLEM -- A SATISFACTORY DEFINITION OF TRUTH.

Our discussion will be
centered around the notion^{4} of *truth*.
The main problem is that of giving a *satisfactory definition* of this notion, i.e., a
definition which is *materially adequate* and *formally correct*. But such a
formulation of the problem, because of its generality, cannot be considered unequivocal, and
requires some further comments.

In order to avoid any ambiguity, we must first specify the conditions under which the definition of truth will be considered adequate from the material point of view. The desired definition does not aim to specify the meaning of a familiar word used to denote a novel notion; on the contrary, it aims to catch hold of the actual meaning of an old notion. We must then characterize this notion precisely enough to enable anyone to determine whether the definition actually fulfills its task.

Secondly, we must determine on what the formal correctness of the definition depends. Thus, we must specify the words or concepts which we wish to use in defining the notion of truth; and we must also give the formal rules to which the definition should conform. Speaking more generally, we must describe the formal structure of the language in which the definition will be given.

The discussion of these points will occupy a considerable portion of the first part of the paper.

2. THE EXTENSION OF THE TERM "TRUE."

We begin with some remarks regarding the extension of the concept of truth which we have in mind here.

The predicate "*true*" is sometimes used to refer to psychological phenomena
such as judgments or beliefs, sometimes to certain physical objects, namely, linguistic
expressions and specifically sentences, and sometimes to certain ideal entities called
"propositions." By "sentence" we understand here what is usually meant
in grammar by "declarative sentence"; as regards the term "proposition,"
its meaning is notoriously a subject of lengthy disputations by various philosophers and
logicians, and it seems never to have been made quite clear and unambiguous. For several
reasons it appears most convenient to *apply the term "true" to sentences*,
and we shall follow this course.^{5}

Consequently, we must always relate the notion of truth, like that of a sentence, to a specific language; for it is obvious that the same expression which is a true sentence in one language can be false or meaningless in another.

Of course, the fact that we are interested here primarily in the notion of truth for sentences does not exclude the possibility of a subsequent extension of this notion to other kinds of objects.

3. THE MEANING OF THE TERM "TRUE."

Much more serious difficulties are connected with the problem of the meaning (or the intension) of the concept of truth.

The word "*true*," like other words from our everyday language, is certainly
not unambiguous. And it does not seem to me that the philosophers who have discussed this
concept have helped to diminish its ambiguity. In works and discussions of philosophers we
meet many different conceptions of truth and falsity, and we must indicate which conception
will be the basis of our discussion.

We should like our definition to do justice to the intuitions which adhere to the *classical
Aristotelian conception of truth* -- intuitions which find their expression in the well-known
words of Aristotle's *Metaphysics*:

To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, or of what is not that it is not, is true.

If we wished to adapt ourselves to modern philosophical terminology, we could perhaps express this conception by means of the familiar formula

(For a theory of truth which is to be based upon the latter formulation the term "correspondence theory" has been suggested.)The truth of a sentence consists in its agreement with (or correspondence to) reality.

If, on the other band, we should decide to extend the popular usage of the term
"*designate*" by applying it not only to names, but also to sentences, and if
we agreed to speak of the designata of sentences as "states of affairs," we could
possibly use for the same purpose the following phrase:

A sentence is true if it designates an existing state of affairs.^{6}

However, all these formulations can lead to various misunderstandings, for none of them is sufficiently precise and clear (though this applies much less to the original Aristotelian formulation than to either of the others); at any rate, none of them can be considered a satisfactory definition of truth. It is up to us to look for a more precise expression of our intuitions.

4. A CRITERION FOR THE MATERIAL ADEQUACY OF THE DEFINITION.^{7}

Let us start with a concrete example. Consider
the sentence "*snow is white*." We ask the question under what conditions this
sentence is true or false. It seems clear that if we base ourselves on the classical conception
of truth, we shall say that the sentence is true if snow is white, and that it is false if snow
is not white. Thus, if the definition of truth is to conform to our conception, it must imply
the following equivalence:

The sentence "snow is white" is true if, and only if, snow is white.

Let me point out that the phrase "*snow is white*" occurs on tile left side of
this equivalence in quotation marks, and on the right without quotation marks. On the right
side we have the sentence itself, and on the left the name of the sentence. Employing the
medieval logical terminology we could also say that on the right side the words "*snow
is white*" occur in *suppositio formalis*, and on the left in *suppositio
materialis*. It is hardly necessary to explain why we must have the name of the sentence,
and not the sentence itself, on the left side of the equivalence. For, in the first place, from
the point of view of the grammar of our language, an expression of the form "*X is
true*" will not become a meaningful sentence if we replace in it '*X*' by a
sentence or by anything other than a name -- since the subject of a sentence may be only
a noun or an expression functioning like a noun. And, in the second place, the fundamental
conventions regarding the use of any language require that in any utterance we make about
an object it is the name of the object which must be employed, and not the object itself. In
consequence, if we wish to say something about a sentence, for example, that it is true, we
must use the name of this sentence, and not the sentence itself.^{8}

It may be added that enclosing a sentence in quotation marks is by no means the only way
of forming its name. For instance, by assuming the usual order of letters in our alphabet, we
can use the following expression as the name (the description) of the sentence "*snow
is white*":

the sentence constituted by three words, the first of which consists of the 19th, 14th, 15th, and 23rd letters, the second of the 9th and 19th letters, and the third of the 23rd, 8th, 9th, 20th, and 5th letters of the English alphabet.

We shall now generalize the procedure which we have applied above. Let us consider an
arbitrary sentence; we shall replace it by the letter '*p*.' We form the name of this
sentence and we replace it by another letter, say '*X*.' We ask now what is the logical
relation between the two sentences "*X is true*" and '*p*.' It is clear that
from the point of view of our basic conception of truth these sentences are equivalent. In
other words, the following equivalence holds:

(T)We shall call any such equivalence (with 'X is true if, and only if, p.

Now at last we are able to put into a precise form the conditions under which we will
consider the usage and the definition of the term "*true*" as adequate from
the material point of view: we wish to use the term "*true*" in such a way that
all equivalences of the form (T) can be asserted, and *we shall call a definition of truth
"adequate" if all these equivalences follow from it.*

It should be emphasized that neither the expression (T) itself (which is not a sentence, but
only a schema of a sentence) nor any particular instance of the form (T) can be regarded as
a definition of truth. We can only say that every equivalence of the form (T) obtained by
replacing '*p*' by a particular sentence, and '*X*' by a name of this sentence, may
be considered a partial definition of truth, which explains wherein the truth of this one
individual sentence consists. The general definition has to be, in a certain sense, a logical
conjunction of all these partial definitions.

(The last remark calls for some comments. A language may admit the construction of infinitely many sentences; and thus the number of partial definitions of truth referring to sentences of such a language will also be infinite. Hence to give our remark a precise sense we should have to explain what is meant by a "logical conjunction of infinitely many sentences"; but this would lead us too far into technical problems of modern logic.)

5. TRUTH AS A SEMANTIC CONCEPT.

I should like to propose the name *"the semantic
conception of truth"* for the conception of truth which has just been discussed.

*Semantics* is a discipline which, speaking loosely, *deals with certain relations between
expressions of a language and the objects* (or "states of affairs")
*"referred to" by those expressions*. As typical examples of semantic concepts
we may mention the concepts of *designation, satisfaction,* and *definition* as these
occur in the following examples:

the expression "the father of his country" designates (denotes) George Washington;snow satisfies the sentential function (the condition) "2 is white";

the equation "21/2.^{.}x = 1" defines (uniquely determines) the number

While the words "*designates*," "*satisfies*," and
"*defines*" express relations (between certain expressions and the objects
"referred to" by these expressions), the word "*true*" is of a
different logical nature: it expresses a property (or denotes a class) of certain expressions,
viz, of sentences. However, it is easily seen that all the formulations which were given earlier
and which aimed to explain the meaning of this word (cf. Sections 3 and 4) referred not only
to sentences themselves, but also to objects "talked about" by these sentences, or
possibly to "states of affairs" described by them. And, moreover, it turns out that
the simplest and the most natural way of obtaining an exact definition of truth is one which
involves the use of other semantic notions, e.g., the notion of satisfaction. It is for these
reasons that we count the concept of truth which is discussed here among the concepts of
semantics, and the problem of defining truth proves to be closely related to the more general
problem of setting up the foundations of theoretical semantics.

It is perhaps worth while saying that semantics as it is conceived in this paper (and in former papers of the author) is a sober and modest discipline which has no pretensions of being a universal patent-medicine for all the ills and diseases of mankind, whether imaginary or real. You will not find in semantics any remedy for decayed teeth or illusions of grandeur or class conflicts. Nor is semantics a device for establishing that everyone except the speaker and his friends is speaking nonsense.

From antiquity to the present day the concepts of semantics have played an important role
in the discussions of philosophers, logicians, and philologists. Nevertheless, these concepts
have been treated for a long time with a certain amount of suspicion. From a historical
standpoint, this suspicion is to be regarded as completely justified. For although the meaning
of semantic concepts as they are used in everyday language seems to be rather clear and
understandable, still all attempts to characterize this meaning in a general and exact way
miscarried. And what is worse, various arguments in which these concepts were involved, and
which seemed otherwise quite correct and based upon apparently obvious premises, led
frequently to paradoxes and antinomies. It is sufficient to mention here the *antinomy of
the liar*, Richard's *antinomy of definability* (by means of a finite number of words),
and Grelling-Nelson's *antinomy of heterological terms*.^{9}

I believe that the method which is outlined in this paper helps to overcome these difficulties and assures the possibility of a consistent use of semantic concepts.

6. LANGUAGES WITH A SPECIFIED STRUCTURE.

Because of the possible occurrence of antinomies, the problem of specifying the formal structure and the vocabulary of a language in which definitions of semantic concepts are to be given becomes especially acute; and we turn now to this problem.

There are certain general conditions under which the structure of a language is regarded as
*exactly specified*. Thus, to specify the structure of a language, we must characterize
unambiguously the class of those words and expressions which are to be considered
*meaningful*. In particular, we must indicate all words which we decide to use without
defining them, and which are called "*undefined* (or *primitive*)
*terms*"; and we must give the so-called *rules of definition* for introducing
new or *defined terms*. Furthermore, we must set up criteria for distinguishing within
the class of expressions those which we call "*sentences*." Finally, we must
formulate the conditions under which a sentence of the language can be *asserted*. In
particular, we must indicate all *axioms* (or *primitive sentences*), i.e., those
sentences which we decide to assert without proof; and we must give the so-called *rules
of inference* (or *rules of proof*) by means of which we can deduce new asserted
sentences from other sentences which have been previously asserted. Axioms, as well as
sentences deduced from them by means of rules of inference, are referred to as
"*theorems*" or "*provable sentences*."

If in specifying the structure of a language we refer exclusively to the form of the
expressions involved, the language is said to be *formalized*. In such a language
theorems are the only sentences which can be asserted.

At the present time the only languages with a specified structure are the formalized languages of various systems of deductive logic, possibly enriched by the introduction of certain non-logical terms. However, the field of application of these languages is rather comprehensive; we are able, theoretically, to develop in them various branches of science, for instance, mathematics and theoretical physics.

(On the other hand, we can imagine the construction of languages which have an exactly specified structure without being formalized. In such a language the assertability of sentences, for instance, may depend not always on their form, but sometimes on other, non-linguistic factors. It would be interesting and important actually to construct a language of this type, and specifically one which would prove to be sufficient for the development of a comprehensive branch of empirical science; for this would justify the hope that languages with specified structure could finally replace everyday language in scientific discourse.)

*The problem of the definition of truth obtains a precise meaning and can be solved in a
rigorous way only for those languages whose structure has been exactly specified.* For
other languages -- thus, for all natural, "spoken" languages -- the meaning of the
problem is more or less vague, and its solution can have only an approximate character.
Roughly speaking, the approximation consists in replacing a natural language (or a portion
of it in which we are interested) by one whose structure is exactly specified, and which
diverges from the given language "as little as possible."

In order to discover some of the more specific conditions which must be satisfied by languages in which (or for which) the definition of truth is to be given, it will be advisable to begin with a discussion of that antinomy which directly involves the notion of truth, namely, the antinomy of the liar.

To obtain this antinomy in a perspicuous form,^{10} consider the following sentence:

For brevity we shall replace the sentence just stated by the letter 'The sentence printed in this paper on p. 347, l. 31, is not true.

According to our convention concerning the adequate usage of the term
"*true*," we assert the following equivalence of the form (T):

(1)'s' is true if, and only if, the sentence printed in this paper on p. 347, l. 31, is not true.

On the other hand, keeping in mind the meaning of the symbol '*s*,' we establish
empirically the following fact:

(2)'s' is identical with the sentence printed in this paper on p. 347, l. 31.

Now, by a familiar law from the theory of identity (Leibniz's law), it follows from (2) that we
may replace in (1) the expression "*the sentence printed in this paper on p. 347, l.
31*" by the symbol " '*s*.' " We thus obtain what follows:

(3)'s' is true if, and only if, 's' is not true.

In this way we have arrived at an obvious contradiction.

In my judgment, it would be quite wrong and dangerous from the standpoint of scientific progress to depreciate the importance of this and other antinomies, and to treat them as jokes or sophistries. It is a fact that we are here in the presence of an absurdity, that we have been compelled to assert a false sentence (since (3), as an equivalence between two contradictory sentences, is necessarily false). If we take our work seriously, we cannot be reconciled with this fact. We must discover its cause, that is to say, we must analyze premises upon which the antinomy is based; we must then reject at least one of these premises, and we must investigate the consequences which this has for the whole domain of our research.

It should be emphasized that antinomies have played a preeminent role in establishing the foundations of modern deductive sciences. And just as class-theoretical antinomies, and in particular Russell's antinomy (of the class of all classes that are not members of themselves were the starting point for the successful attempts at a consistent formalization of logic and mathematics, so the antinomy of the liar and other semantic antinomies give rise to the construction of theoretical semantics.

8. THE INCONSISTENCY OF SEMANTICALLY CLOSED LANGUAGES.^{7}

If we now analyze the assumptions which lead to the antinomy of the liar, we notice the following:

- We have implicitly assumed that the language in which the antinomy is constructed
contains, in addition to its expressions, also the names of these expressions, as well as
semantic terms such as the term "
*true*" referring to sentences of this language; we have also assumed that all sentences which determine the adequate usage of this term can be asserted in the language. A language with these properties will be called "*semantically closed*." - We have assumed that in this language the ordinary laws of logic hold.
- We have assumed that we can formulate and assert in our language an empirical premise such as the statement (2) which has occurred in our argument.

It turns out that the assumption (III) is not essential, for it is possible to reconstruct the
antinomy of the liar without its help.^{11} But
the assumptions (I) and (II) prove essential. Since every language which satisfies both of
these assumptions is inconsistent, we must reject at least one of them;

It would be superfluous to stress here the consequences of rejecting the assumption (II), that
is, of changing our logic (supposing this were possible) even in its more elementary and
fundamental parts. We thus consider only the possibility of rejecting the assumption (I).
Accordingly, we decide *not to use any language which is semantically closed* in the
sense given.

This restriction would of course be unacceptable for those who, for reasons which are not clear to me, believe that there is only one "genuine" language (or, at least, that all "genuine" languages are mutually translatable). However, this restriction does not affect the needs or interests of science in any essential way. The languages (either the formalized languages or -- what is more frequently the case -- the portions of everyday language) which are used in scientific discourse do not have to be semantically closed. This is obvious in case linguistic phenomena and, in particular, semantic notions do not enter in any way into the subject-matter of a science; for in such a case the language of this science does not have to be provided with any semantic terms at all. However, we shall see in the next section how semantically closed languages can be dispensed with even in those scientific discussions in which semantic notions are essentially involved.

The problem arises as to the position of everyday language with regard to this point. At first blush it would seem that this language satisfies both assumptions (I) and (II), and that therefore it must be inconsistent. But actually the case is not so simple. Our everyday language is certainly not one with an exactly specified structure. We do not know precisely, which expressions are sentences, and we know even to a smaller degree which sentences are to be taken as assertible. Thus the problem of consistency has no exact meaning with respect to this language. We may at best only risk the guess that a language whose structure has been exactly specified and which resembles our everyday language as closely as possible would be inconsistent.

9. OBJECT-LANGUAGE AND META-LANGUAGE.

Since we have agreed not to employ semantically
closed languages, we have to use two different languages in discussing the problem of the
definition of truth and, more generally, any problems in the field of semantics. The first of
these languages is the language which is "talked about" and which is the subject
matter of the whole discussion; the definition of truth which we are seeking applies to the
sentences of this language. The second is the language in which we "talk about"
the first language, and in terms of which we wish, in particular, to construct the definition
of truth for the first language. We shall refer to the first language as "*the object
language*," and to the second as "*the meta-language*."

It should be noticed that these terms "object-language" and "meta- language" have only a relative sense. If, for instance, we become interested in the notion of truth applying to sentences, not of our original object-language, but of its meta-language, the latter becomes automatically the object-language of our discussion; and in order to define truth for this language, we have to go to a new meta-language so to speak, to a meta-language of a higher level. In this way we arrive at a whole hierarchy of languages.

The vocabulary of the meta-language is to a large extent determined by previously stated conditions under which a definition of truth will be considered materially adequate This definition, as we recall, has to imply all equivalences of the form (T):

(T)X is true if, and only if, p.

The definition itself and all the equivalences implied by it are to be formulated in the meta-language. On the other hand, the symbol '*p*' in (T) stands for an arbitrary sentence
of our object-language. Hence it follows that every sentence which occurs in the object-
language must also occur in the meta-language; in other words, the meta-language must
contain the object-language as a part. This is at any rate necessary for the proof of the
adequacy of the definition -- even though the definition itself can sometimes be formulated
in a less comprehensive meta-language which does not satisfy this requirement.

(The requirement in question can be somewhat modified, for it suffices to assume that the
object-language can be translated into the meta-language; this necessitates a certain change
in the interpretation of the symbol '*p*' in (T). In all that follows we shall ignore the
possibility of this modification.)

Furthermore, the symbol '*X*' in (T) represents the name of the sentence which '*p*'
stands for. We see therefore that the meta-language must be rich enough to provide
possibilities of constructing a name for every sentence of the object-language.

In addition, the meta-language must obviously contain terms of a general logical character,
such as the expression "if, and only if."^{12}

It is desirable for the meta-language not to contain any undefined terms except such as are
involved explicitly or implicitly in the remarks above, i.e.: terms of the object-language; terms
referring to the form of the expressions of the object-language, and used in building names
for these expressions; and terms of logic. In particular, we desire *semantic terms*
(referring to the object-language) *to be introduced into the meta-language only by
definition*. For, if this postulate is satisfied, the definition of truth, or of any other
semantic concept, will fulfill what we intuitively expect from every definition; that is, it will
explain the meaning of the term being defined in terms whose meaning appears to be
completely clear and unequivocal. And, moreover, we have then a kind of guarantee that the
use of semantic concepts will not involve us in any contradictions.

We have no further requirements as to the formal structure of the object-language and the meta-language; we assume that it is similar to that of other formalized languages known at the present time. In particular, we assume that the usual formal rules of definition are observed in the meta-language.

10. CONDITIONS FOR A POSITIVE SOLUTION OF THE MAIN PROBLEM.

Now, we have already a clear idea both of the conditions of material adequacy to which the definition of truth is subjected, and of the formal structure of the language in which this definition is to be constructed. Under these circumstances the problem of the definition of truth acquires the character of a definite problem of a purely deductive nature.

The solution of the problem, however, is by no means obvious, and I would not attempt to give it in detail without using the whole machinery of contemporary logic. Here I shall confine myself to a rough outline of the solution and to the discussion of certain points of a more general interest which are involved in it.

The solution turns out to be sometimes positive, sometimes negative. This depends upon some formal relations between the object-language and its meta-language; or, more specifically,
upon the fact whether the meta-language in its logical part is "*essentially
richer*" than the object-language or not. It is not easy to give a general and precise definition of this notion of "essential richness." If we restrict ourselves to
languages based on the logical theory of types, the condition for the meta-language to be "essentially richer" than the object-language is that it contain variables of a higher
logical type than those of the object-language.

If the condition of "essential richness" is not satisfied, it can usually be shown that an interpretation of the meta-language in the object-language is possible; that is to say, with any given term of the meta-language a well determined term of the object-language can be correlated in such a way that the assertible sentences of the one language turn out to be correlated with assertible sentences of the other. As a result of this interpretation, the hypothesis that a satisfactory definition of truth has been formulated in the meta-language turns out to imply the possibility of reconstructing in that language the antinomy of the liar; and this in turn forces us to reject the hypothesis in question.

(The fact that the meta-language, in its non-logical part, is ordinarily more comprehensive than the object-language does not affect the possibility of interpreting the former in the latter. For example, the names of expressions of the object-language occur in the meta-language, though for the most part they do not occur in the object-language itself; but, nevertheless, it may be possible to interpret these names in term of the object-language.)

Thus we see that the condition of "essential richness" is necessary for the
possibility of a satisfactory definition of truth in the meta-language. If we want to develop
the theory of truth in a meta-language which does not satisfy this condition, we must give
up the idea of defining truth with the exclusive help of those terms which were indicated
above (in Section 8). We have then to include the term
"*true*," or some other semantic term, in the list of undefined terms of the
meta-language, and to express fundamental properties of the notion of truth in a series of
axioms. There is nothing essentially wrong in such an axiomatic procedure, and it may prove
useful for various purposes.^{13}

It turns out, however, that this procedure can be avoided. For *the condition of the
"essential richness" of the meta-language proves to be, not only necessary, but also
sufficient for the construction of a satisfactory definition of truth;* i.e., if the meta-language satisfies this condition, the notion of truth can be defined in it. We shall now
indicate in general terms how this construction can be carried through.

11. THE CONSTRUCTION (IN OUTLINE) OF THE DEFINITION.^{14}

A definition of truth can be obtained in a very simple way from that of
another semantic notion, namely, of the notion of *satisfaction*.

Satisfaction is a relation between arbitrary objects and certain expressions called "*sentential
functions*." These are expressions like "*x is white*," "*x is
greater than y*," etc. Their formal structure is analogous to that of sentences;
however, they may contain the so-called free variables (like '*x*' and '*y*' in
"*x is greater than y*"), which cannot occur in sentences.

In defining the notion of a sentential function in formalized languages, we usually apply what
is called a "recursive procedure"; i.e., we first describe sentential functions of the
simplest structure (which ordinarily presents no difficulty), and then we indicate the
operations by means of which compound functions can be constructed from simpler ones. Such
an operation may consist, for instance, in forming the logical disjunction or conjunction of
two given functions, i.e., by combining them by the word "*or*" or
"*and*." A sentence can now be defined simply as a sentential function which
contains no free variables

As regards the notion of satisfaction, we might try to define it by saying that given objects
satisfy a given function if the latter becomes a true sentence when we replace in it free
variables by names of given objects. In this sense, for example, snow satisfies the sentential
function "*x is white*" since the sentence "*snow is white*" is
true. However, apart from other difficulties, this method is not available to us, for we want
to use the notion of satisfaction in defining truth.

To obtain a definition of satisfaction we have rather to apply again a recursive procedure.
We indicate which objects satisfy the simplest sentential functions; and then we state the
conditions under which given objects satisfy a compound function -- assuming that we know
which objects satisfy the simpler functions from which the compound one has been
constructed. Thus, for instance, we say that given numbers satisfy the logical disjunction
"*x is greater than y or x is equal to y*" if they satisfy at least one of the
functions "*x is greater than y*" or "*x is equal to y*."

Once the general definition of satisfaction is obtained, we notice that it applies automatically
also to those special sentential functions which contain no free variables, i.e., to sentences.
It turns out that for a sentence only two cases are possible: a sentence is either satisfied
by all objects, or by no objects. Hence we arrive at a definition of truth and falsehood simply
by saying that *a sentence is true if it is satisfied by all objects, and false otherwise*.^{15}

(It may seem strange that we have chosen a roundabout way of defining the truth of a sentence, instead of trying to apply, for instance, a direct recursive procedure. The reason is that compound sentences are constructed from simpler sentential functions, but not always from simpler sentences; hence no general recursive method is known which applies specifically to sentences.)

From this rough outline it is not clear where and how the assumption of the "essential
richness" of the meta-language is involved in the discussion; this becomes clear only
when the construction is carried through in a detailed and formal way.^{16}

12. CONSEQUENCES OF THE DEFINITION.

The definition of truth which was outlined above has many interesting consequences.

In the first place, the definition proves to be not only formally correct, but also materially
adequate (in the sense established in Section 4); in other words, it
implies all equivalences of the form (T). In this connection it is important to notice that the
conditions for the material adequacy of the definition determine uniquely the extension of the
term "*true*."^{17} Therefore,
every definition of truth which is materially adequate would necessarily be equivalent to that
actually constructed. The semantic conception of truth gives us, so to speak, no possibility
of choice between various non-equivalent definitions of this notion.

Moreover, we can deduce from our definition various laws of a general nature. In particular,
we can prove with its help the *laws of contradiction and of excluded middle*, which are
so characteristic of the Aristotelian conception of truth; i.e., we can show that one and only
one of any two contradictory sentences is true. These semantic laws should not be identified
with the related logical laws of contradiction and excluded middle; the latter belong to the
sentential calculus, i.e., to the most elementary part of logic and do not involve the term
"*true*" at all.

Further important results can be obtained by applying the theory of truth to formalized
languages of a certain very comprehensive class of mathematical disciplines; only disciplines
of an elementary character and a very elementary logical structure are excluded from this
class. It turns out that for a discipline of this class *the notion of truth never coincides
with that of provability*; for all provable sentences are true, but there are true sentences
which are not provable.^{17} Hence it follows
further that every such discipline is consistent but incomplete; that is to say, of two
contradictory sentences at most one is provable, and -- what is more -- there exists a pair
of contradictory sentences neither of which is provable.^{18}

13. EXTENSION OF THE RESULTS TO OTHER SEMANTIC NOTIONS.

Most of the results at which
we arrived in the preceding sections in discussing the notion of truth can be extended with
appropriate changes to other semantic notions, for instance, to the notion of satisfaction
(involved in our previous discussion), and to those of *designation* and *definition*.

Each of these notions can be analyzed along the lines followed in the analysis of truth. Thus,
criteria for an adequate usage of these notions can be established; it can be shown that each
of these notions, when used in a semantically closed language according to those criteria,
leads necessarily to a contradiction;^{19} a
distinction between the object-language and the meta-language becomes again indispensable;
and the "essential richness" of the meta-language proves in each case to be a
necessary and sufficient condition for a satisfactory definition of the notion involved. Hence
the results obtained in discussing one particular semantic notion apply to the general problem
of the foundations of theoretical semantics.

Within theoretical semantics we can define and study some further notions, whose intuitive
content is more involved and whose semantic origin is less obvious; we have in mind, for
instance, the important notions of *consequence, synonymity*, and *meaning*.^{20}

We have concerned ourselves here with the theory of semantic notions related to an individual
object-language (although no specific properties of this language have been involved in our
arguments). However, we could also consider the problem of developing *general
semantics* which applies to a comprehensive class of object-languages. A considerable part
of our previous remarks can be extended to this general problem; however, certain new
difficulties arise in this connection, which will not be discussed here. I shall merely observe
that the axiomatic method (mentioned in Section 10) may prove the most
appropriate for the treatment of the problem.^{21}