Pirates & Revolutionaries: 2014

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[Tarski’s “Semantic Conception of Truth”, Entry Directory]

[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Alfred Tarski

“The Semantic Conception of Truth and the Foundations of Semantics”

Part I. Exposition

10. Conditions for a Positive Solution of the Main Problem

Brief Summary:

We are semantically defining truth by using a meta-language to talk about an object-language. We use the term ‘true’ in that meta-language in order to define parts of the object language. But we do not want to resort to an additional higher meta-language to define semantically this term ‘true’. We want instead to define it within that meta-language itself. We will be able to, so long as the meta-language is essentially richer than the object-language (meaning that the object language cannot say everything in the meta-language). In the next section we learn how that is so.

Summary

Previously Tarski provided

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.
(343, from this current section).

Solving the problem of how to define truth in detail would require “the whole machinery of contemporary logic”. So here Tarski will just give a rough outline of the solution. (343)

The solution may be positive or negative, depending on whether or not the meta-language is “essentially richer” than the object-language. (434d). Although it is not easy to define essential richness, Tarski says one way is to say that the meta-language is essentially richer if it is of a higher logical type. (343-344)

So, if the object language can say everything in the meta-language, then the meta-language is not essentially richer than the object-language. This can result in the liar paradox.

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.)
(344)

Without the essential richness of the meta-language, we encounter inconsistency, hence it is “necessary for the possibility of a satisfactory definition of truth in the meta-language” (344). [Semantically defining a term of a metalanguage may require a higher order meta-language to explain that to which this term is equated. Thus even the word ‘true’ cannot have a semantic meaning for 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.
(344)

But, so long as the meta-language is essentially richer than the object-language, it will be able to construct a satisfactory definition of truth. Tarski will show how in the next section.

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.
(344)

Text:

Tarski, Alfred. “The Semantic Conception of Truth and the Foundations of Semantics”. In The Nature of Truth: Classic and Contemporary Perspectives. Michael P. Lynch, ed. Cambridge, Massachusetts / London: MIT, 2001, pp.331-363.

A hyperlinked online version can be found here:

The Lynch edited book writes this in the acknowledgments:

Alfred Tarski. “The Semantic Conception of Truth and the Foundations of Semantics.” Philosophy and Phenomenological Research 4 (1944). Copyright 1992 by the Estate of Alfred Tarski. Reprinted by permission of Jan Tarski.

Further bibliographical information from
http://dingo.sbs.arizona.edu/~hharley/courses/522/522/MPPLecture4.html:

Alfred Tarski (1944) The semantic conception of truth and the foundations of semantics (Reprinted as Chapter 4 of Martinich’s anthology). This is an abridged and updated version of his 1935 long paper Der Wahrheitsbegriff in den formalisierten Sprache (The concept of truth in formalized languages), itself a translation from his book in Polish of 1933.

And yet further bibliographical information from the German wiki page for Tarski

http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf

Der Wahrheitsbegriff in den formalisierten Sprachen. In: Studia Philosophica. [Lemberg] 1 (1936), S. 261–405 (Vorabdruck datiert 1935).[4] Der Artikel ist eine deutsche Übersetzung der erstmals 1933 gedruckten polnischen Arbeit, die aber schon 1931 der Öffentlichkeit präsentiert wurde. Nachdruck in Karel Berka, Lothar Kreiser (Hrsg.): Logik-Texte. Kommentierte Auswahl zur Geschichte der modernen Logik. Akademie-Verlag, Berlin 1983, S. 445–546, in englischer Sprache in Tarski: Logic, Semantics and Metamathematics - papers from 1923 to 1938 by Alfred Tarski. Oxford 1956, 1983.

The German text can be found here:

Tarski (§9) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘9. Object-Language and Meta-Language’

[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Alfred Tarski

“The Semantic Conception of Truth and the Foundations of Semantics”

Part I. Exposition

9. Object-Language and Meta-Language

Brief Summary:

We are semantically defining truth using the (T) scheme:

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

p is the sentence we are defining, while X is the name for that sentence, often given in quotation marks. For example,

“snow is white” if and only if snow is white.

The text in the quotation marks is in the language that the statement is talking about, and that language is called the object-language. The parts of that expression which are not in quotes and which are talking about the object-language are parts of what is called the meta-language.

Summary

Previously Tarski concluded that in order for our semantic theory to avoid the liar’s paradox, we cannot “employ semantically closed languages.” Instead, “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” (341, both quotes from this section). The first language, called the object language, is the one we are talking about, while the second one, called the metalanguage, is doing that talking.

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.”
(341-342)

The terms ‘object-language’ and ‘meta-language’ have a relative sense, since if we in turn talk about some meta-language with another meta-language, then the first becomes the object-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.
(342)

Tarski then explains how we obtain the vocabulary of the metalanguage. [It seems he is saying that the metalanguage is mostly what is contained in the (T) formulation. So it also includes everything as well in the object language, which will be stated in the (T) formulations. It is not explained yet what to do with the fact that ‘is true if and only if’ could easily be found in an object language like English or formal logic.]

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.)
(342)

Because the (T) scheme needs names for sentences in the object-language, the meta-language should be rich enough to create names for all such sentences. As well, the meta-language needs basic logical terms like “if, and only if.”

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.”
(342)

The meta-language should not have undefined terms except ones in the object language, ones that refer to the object language expressions’ form, ones used to create names for those expressions, and logical ones.

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.
(343)

We do not need further requirements for the object and meta-languages. They should be like other known formalized languages.

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.
(343)

Text:

A hyperlinked online version can be found here:

The Lynch edited book writes this in the acknowledgments:

Further bibliographical information from
http://dingo.sbs.arizona.edu/~hharley/courses/522/522/MPPLecture4.html:

And yet further bibliographical information from the German wiki page for Tarski

http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf

The German text can be found here:

Tarski (§8) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘8. The Inconsistency of Semantically Closed Languages’

[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Alfred Tarski

“The Semantic Conception of Truth and the Foundations of Semantics”

Part I. Exposition

8. The Inconsistency of Semantically Closed Languages

Brief Summary:

The liar’s paradox arises because we considered the language in which it arose as being semantically closed, meaning that within it are expressible all its components and generative rules. Thus the languages we examine using Tarski’s method cannot be considered semantically closed, or else they will be inconsistent.

Summary

Previously Tarski explained how his (T) scheme for giving a semantic definition of truth – (T) X is true if, and only if, p, – brings about the liar’s paradox when the sentence in question refers to itself and is predicated as being not true:

‘s’ is true if and only if ‘s’ is not true.

He said that we need to address this problem, because it can help us uncover foundational principles in theoretical semantics, and to do so, we need to

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.
(339-340, from the prior section)

Now Tarski will examine “the assumptions which lead to the antinomy of the liar.” (340, current section) In do so, we notice three things. Firstly, we assumed that the paradox occurs in a certain language. We assumed that everything in the formulation

‘s’ is true if and only if ‘s’ is not true.

are also in the same language in which the paradox arises. We also assumed that all the sentences that determine how ‘true’ can be used are as well found in this language. A system with these two features is called “semantically closed” [perhaps because all the semantic properties are described within in and do not need a metalanguage for that description.] Secondly, we assumed that this language operates according to the normal laws of logic. And thirdly we assumed this language is capable of saying such sentences as:

‘s’ is true if and only if ‘s’ is not true.

[Note, my interpretation of the third assumption below needs revision, as I seem to misunderstand the terminology, particularly the term ‘empirical premise’ and perhaps I do not know the usage here of ‘argument’]

I. 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.”

II. We have assumed that in this language the ordinary laws of logic hold.

III. 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.
(340)

Tarski says that the third assumption is not essential, because we can create the liar paradox by other means within the language. He writes in footnote 11:

This can roughly be done in the following way. Let S be any sentence beginning with the words “Every sentence.” We correlate with S a new sentence S* by subjecting S to the following two modifications: we replace in S the first word, “Every,” by “The”; and we insert after the second word, “sentence,” the whole sentence S enclosed in quotation marks. Let us agree to call the sentence S “(self-)applicable” or “non-(self-)applicable” dependent on whether the correlated sentence S* is true or false. Now consider the following sentence:

Every sentence is non-applicable.

It can easily be shown that the sentence just stated must be both applicable and non-applicable; hence a contradiction. It may not be quite clear in what sense this formulation of the antinomy does not involve an empirical premiss; however, I shall not elaborate on this point.
(358, endnote 11)

But although the third assumption is not essential, the first two are. [The second assumption is that the language follows normal laws of logic. The first is that the language is semantically closed, meaning that all its semantic and logical properties can be described within it.] Because the first two assumptions together would make a language inconsistent, we must reject at least one of them.

Consider if we said that the language does not follow the normal laws of logic. This would have gravely detrimental consequences [for example, we would be unable to formulate it consistently, perhaps. A paraconsistent logic might not have such problems, but Tarski here is not specific about them, since they are apparently too obvious.]

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.
(340)

Thus we must reject assumption 1 and say that we are not using a semantically closed system.

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.
(340)

Some people might object saying that we cannot work with semantically closed languages. Such people might think that either there is one genuine language [and all natural languages are variations of it] or that all languages are mutually translatable [and thus there is no language that can be conceived of as standing above and beyond the others.] Tarski’s reply is that this objection is not a problem in science at least, which does not need semantically closed language. (341)

Another objection would be that our everyday language is closed [perhaps because it can say anything we want it too, and we do not need a separate language to describe the first one. English can be fully defined and described in English dictionaries and grammars. We do not need French for example to explain all the features of English.] But Tarski notes that everyday language does not have a specified structure. The problem of inconsistency only arises for languages with specified structures. We do our best to approximate everyday language in a formalized one, and we can only surmise that the inconsistency of the approximation is indicative of inconsistency in the original. (341)

Text:

A hyperlinked online version can be found here:

The Lynch edited book writes this in the acknowledgments:

Further bibliographical information from
http://dingo.sbs.arizona.edu/~hharley/courses/522/522/MPPLecture4.html:

And yet further bibliographical information from the German wiki page for Tarski

http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf

The German text can be found here:

Tarski (§7) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘7. The Antinomy of the Liar’

[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Alfred Tarski

“The Semantic Conception of Truth and the Foundations of Semantics”

Part I. Exposition

7. The Antinomy of the Liar

Brief Summary:

Tarksi has provided the (T) scheme for designating the truth of sentences: (T) X is true if, and only if, p. In this formulation, p is the sentence in question, and X is the name for it, often times being that same sentence with quotations around it. We encounter the liar paradox, however, when X refers to its own self and is predicated as not true. For example: This sentence is not true. The formulation would then read “This sentence is not true” is true, if and only if, this sentence is not true, or, after symbolic substitution, ‘s’ is true if and only if ‘s’ is not true. [The substitution is clearer when considering the more precise formulation that refers to where the sentence is on the page. See below.] Tarski thinks it is important to deal with this paradox, as its solution could play a central role in the foundation of theoretical semantics.

Summary

Previously Tarsky noted that in order to give a definition of truth, we cannot use any natural language, as they structurally speaking will cause too many difficulties [on account of ambiguities for example]. Instead we need to make a formalized language that approximates a natural language as close as possible. Part of this project of creating a suitable formalized language is determining the parts and generative operations of that language. We need primitive or undefined terms along with rules for defining new terms. We also need axioms (primitive sentences) as well as rules for inferring new sentences from them. Now in this section, Tarski says that

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.
(339)

Tarski then creates a liar paradox using the (T) formulation: (T) X is true if, and only if, p. In this case, he will make p be an actual sentence on the page. In this publication, the page this text is on is 339, and the stated sentence is on line 11. So Tarksi formulates it like this

To obtain this antinomy in a perspicuous form, consider the following sentence:

The sentence printed in this paper on p. 339, l. 11, is not true.
(339)

Tarski abbreviates this above sentence to the letter ‘s’. Then he places this sentence and its name into his (T) scheme.

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. 339, l. 11, is not true.
(339)

Yet, as we know, a name is identified with what it is a name of. So

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. 339, l. 11.

(339)

So since they are identical, we can substitute one for the other, which will produce a contradiction.

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. 339, l. 11” 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.
(339)

Tarski thinks that this is more than a joke. Since our structure produces it, we must take it seriously and deal with it. Specifically

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.
(339-340)

In fact, this antinomy of the liar can play a central role in semantics just as other antinomies have played central roles in other areas of philosophy.

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.
(340)

Text:

A hyperlinked online version can be found here:

The Lynch edited book writes this in the acknowledgments:

Further bibliographical information from
http://dingo.sbs.arizona.edu/~hharley/courses/522/522/MPPLecture4.html:

And yet further bibliographical information from the German wiki page for Tarski

http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf

The German text can be found here:

30 Dec 2014

Tarski (§6) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘6. Languages with a Specified Structure’

[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Alfred Tarski

“The Semantic Conception of Truth and the Foundations of Semantics”

Part I. Exposition

6. Languages with a Specified Structure

Brief Summary:

Although we are doing semantics, we need to look more formally at language’s structures. We need specifically to determine primitive or undefined terms, rules for defining new terms, and as well axioms (primitive sentences) and the inference rules that we use to derive other sentences from these axioms (both of which being called ‘theorems’).

Summary

Previously Tarski noted that the field of semantics for long has been unclear with its concepts, with one consequence being the production of such paradoxes as ‘the antinomy of the liar’. Given this danger, Tarski turns now to “the problem of specifying the formal structure and the vocabulary of a language in which definitions of semantic concepts are to be given.” (337)

Our task will be to specify the structure of a language. This involves us designating classes of words and expressions, with special attention to those classes that we consider meaningful. First among these meaningful expressions are undefined (or primitive) terms. We as well need to provide rules for defining new terms.

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. (337-338)

We also need to “set up criteria for distinguishing within the class of expressions those which we call ‘sentences.’” (338) Then, once knowing which expressions are sentences, we need to say how sentences may be generated systematically. For this we need certain sentences, axioms (primitive sentences) and inference rules describing how to produce new sentences on the basis of these given ones. Both these axioms and the sentences derived from them are called ‘theorems’ or ‘provable 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.”
(388)

[We might ignore actual contents to our sentences and be concerned mostly with their structures. In this case we would be dealing mostly with symbols, and this is a formalized language.]

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.
(338)

At this time, only deductive logic deals with formalized languages. However these could be developed in other branches of science like mathematics and theoretical physics. (338)

Tarksi then adds parenthetically that it is possible and potentially useful to construct languages that “have an exactly specified structure without being formalized.” (338)

Although we normally use natural languages, they are riddled with ambiguities, and so giving a precise meaning of truth for them is tricky. We can only accomplish this with formalized languages. The best we can do with natural languages is creating the closest approximation for them in formal languages.

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.”
(338)

Text:

A hyperlinked online version can be found here:

The Lynch edited book writes this in the acknowledgments:

Further bibliographical information from
http://dingo.sbs.arizona.edu/~hharley/courses/522/522/MPPLecture4.html:

And yet further bibliographical information from the German wiki page for Tarski

http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf

The German text can be found here:

Tarski (§5) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘5. Truth as a Semantic Concept’

[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Alfred Tarski

“The Semantic Conception of Truth and the Foundations of Semantics”

Part I. Exposition

5. Truth as a Semantic Concept

Brief Summary:

Although we normally think of truth as a logical concept rather than a semantic one, Tarksi has shown that it is in fact semantic, because we understand it more clearly by means of his semantic formulation.

Summary

Previously Tarski provided the ‘material conditions’ for a sentence to be true, which is that its name can take a truth predicate and be equated with its true articulation in some language. Specifically, a sentence p in some language needs to be equated with its name X, which is predicated as being true:

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

Now in this section, Tarski proposes a name for this conception of truth: “the semantic conception of truth”.

Semantics is a field that is concerned with expressions and their meanings.

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 “2 ● x = 1” defines (uniquely determines) the number 1/2.
(336)

‘Designates’, ‘satisfies’, and ‘defines’ express relations, but ‘true’ has a different logical nature. It “expresses a property (or denotes a class) of certain expressions, viz., of sentences.” (336) But even though truth is a logical property unlike these other semantic notions, Tarski thinks that truth is still a matter for semantics. He provides a couple of reasons: 1) our formulations for truth refer to sentences or to the objects ‘talked about’ by these sentences, and 2) the simplest and most natural way to provide an exact definition of truth involves using such semantic notions as 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.
(336)

Tarski acknowledges that semantics cannot do everything. (337)

Semantical notions have been a part of philosophy, logic, and philology from the beginning. However they “have been treated for a long time with a certain amount of suspicion.” (337) This suspicion has been warranted, since all efforts to present semantic notions clearly have been “miscarried”. To make matters worse, these semantic concepts have led to such paradoxes and antinomies as “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.” (337)

Despite these shortcomings in the field of semantics up to this point in history, Tarski believes “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.” (337)

Text:

A hyperlinked online version can be found here:

The Lynch edited book writes this in the acknowledgments:

Further bibliographical information from
http://dingo.sbs.arizona.edu/~hharley/courses/522/522/MPPLecture4.html:

And yet further bibliographical information from the German wiki page for Tarski

http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf

The German text can be found here:

Tarski (§4) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘4. A Criterion for the Material Adequacy of the Definition’

[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Alfred Tarski

“The Semantic Conception of Truth and the Foundations of Semantics”

Part I. Exposition

4. A Criterion for the Material Adequacy of the Definition

Brief Summary:

The conditions for our definition of truth are that it must be formally correct and materially adequate. It is formally correct if takes the form of the (T) scheme:

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

Here, X is the name for a sentence, conventionally placed within quotation marks, and p is the sentence expressed in some language (given without quotations). For example:

The sentence “snow is white” is true if, and only if, snow is white.

The definition then is materially adequate if this equivalence is fulfilled [that is to say, if the content or material of the left side of the equation and on the right side are fully adequate to one another.]

Summary

Previously Tarski examined some correspondence theory definitions of truth, and he found them to be too unclear and imprecise. So we must formulate the definition in a different way. Tarski will do so here first by beginning with an example. He has us consider the sentence:

“snow is white”
(334)

We wonder, under what conditions would this sentence be true or false? If we stick with the classic correspondence theory, we would 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.

(334)

[We above have an equivalence, since we have a biconditional, if and only if.] Tarski notes that “snow is white” on the left side of the equivalence has quotation marks around it, while on the right side it does not.

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.
(334)

Tarski provides two reasons for why we need to have the name for the sentence on the left and the sentence itself on the right. Recall again our formulation.

The sentence “snow is white” is true if, and only if, snow is white.

The first part,

The sentence “snow is white” is true

has the form “X is true”. In this grammatical structure, we need to replace X with a name, for otherwise it would not be meaningful, “since the subject of a sentence may be only a noun or an expression functioning like a noun.” (334) The second reason is that “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” (334).

Here we are using quotations around a sentence to indicate its name, but we can use other methods as well. For example, we could arbitrarily assign it some letter symbol. Or we can be more mechanical and

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.
(335)

Tarski will now generalize this procedure. Consider some sentence, and call it ‘p’. We then form the name of this sentence (in the above we did so with quotation marks), and we name it with another letter, for example, X. Now we have two sentences. We have p, which is the sentence in question. (Above the example was: snow is white). And we also form this sentence: X is true. (Above it was: “snow is white” is true). We now want to know, what is the logical relation between X is true and p? (Or as above, what is the logical relation between “Snow is white” is true and snow is white?) Tarski says that given the way we conceive of their truth, they are equivalent.

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

We shall call any such equivalence (with 'p' replaced by any sentence of the language to which the word “true” refers, and 'X' replaced by a name of this sentence) an “equivalence of the form (T).”
(335)

[Perhaps another way of saying the above sentence is that a sentence is true if its equivalent form in a certain language is true.] The above equivalence is the material adequacy for the truth definition. [I am not sure why we use the term ‘material’. Perhaps this is because it concerns the adequacy of the content of the right side of the formulation to the content of the left side.] [So our usage of term “true” is adequate from the material perspective when we use it apply in the above manner. In other words, it seems that the material conditions for the truth of a sentence are that a) it is true in some language and that b) we can declare that truth by predicating truth to its name while c) designating its expression in some language and d) affirming its truth in that language.]

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.
(335)

We note that expression (T) is not itself a sentence but rather it is a schema of a sentence. [Perhaps this is because it is not filled in yet with content, or perhaps for some reason even if it is filled in with content it should not be considered a sentence.] Tarski claims that neither (T) itself nor any instantiation of it suffices for 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.
(335)

Tarski closes with the following parenthetical remark:

(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.)
(336)

Text:

A hyperlinked online version can be found here:

The Lynch edited book writes this in the acknowledgments:

Further bibliographical information from
http://dingo.sbs.arizona.edu/~hharley/courses/522/522/MPPLecture4.html:

And yet further bibliographical information from the German wiki page for Tarski

http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf

The German text can be found here:

29 Dec 2014

Tarski (§3) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘3. The Meaning of the Term “True”’

[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Alfred Tarski

“The Semantic Conception of Truth and the Foundations of Semantics”

Part I. Exposition

3. The Meaning of the Term “True”

Brief Summary:
Truth often is defined using the ‘correspondence theory’ in the following way: “A sentence is true if it designates an existing state of affairs.” But this and similar correspondence theory definitions are not clear and precise enough to suffice as definitions for truth, and thus we must seek something better.

Summary

Previously Tarski discussed the extension of the term true. We will now encounter more difficulties with the term’s meaning (or intension).

We use the term “ true” with many meanings and usages. So as philosophers we must specify the meaning we want to give it. (333)

Tarski will follow in the tradition of Aristotle, who wrote in his 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.
(Aristotle, qt in Tarski, 333; Aristotle, gamma 7, 27)

Tarski reformulates this using modern terminology to say:

The truth of a sentence consists in its agreement with (or to) reality.
(333)

This is sometimes called the “correspondence theory of truth.” (333)

[Normally we might use the term ‘ designate’ to mean that some name or term stands for some object of reference. But also] we can think of whole sentences as designating states of affairs, and thus we could say:

A sentence is true if it designates an existing state of affairs.
(334)

Yet none of these definitions is precise and clear enough to suffice to define truth. Thus “It is up to us to look for a more precise expression of our intuitions. “ (334)

Text:

A hyperlinked online version can be found here:

The Lynch edited book writes this in the acknowledgments:

Further bibliographical information from
http://dingo.sbs.arizona.edu/~hharley/courses/522/522/MPPLecture4.html:

And yet further bibliographical information from the German wiki page for Tarski

http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf

The German text can be found here:

Tarski (§2) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘2. The Extension of the Term “True”’

[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Alfred Tarski

“The Semantic Conception of Truth and the Foundations of Semantics”

Part I. Exposition

2. The Extension of the Term “True”

Brief Summary:
We will apply the term ‘truth’ only to declarative sentences articulated in some language.

Summary

The concept of truth would have an extension [the set of things to which it refers.] Tarski begins with some remarks about truth’s extension.

The predicate “true” is often applied to many things, including

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.” (332)

“Sentence” will mean what in grammar we call a declarative sentence. “Proposition” is notoriously difficult to define, so Tarski will stick with applying the term ‘true’ only to sentences. (332-333)

[Since we are dealing with real grammatically formed sentences, we are dealing with real natural language.] Thus we always related the notion of truth 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.” (333)

However, the notion of truth could also be extended to apply to things other than sentences. (333)

Text:

A hyperlinked online version can be found here:

The Lynch edited book writes this in the acknowledgments:

Further bibliographical information from
http://dingo.sbs.arizona.edu/~hharley/courses/522/522/MPPLecture4.html:

And yet further bibliographical information from the German wiki page for Tarski