Priest, (8) ‘Dialectic and Dialetheic’, section 8, “Identity in Difference”, summary

 

by Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary are my own, unless otherwise indicated.]

Graham Priest

“Dialectic and Dialetheic”

8 Identity in Difference

Brief Summary:

Hegel’s dialectic takes the form of identity in difference, formulable as (a=b)&(a≠b). This is a variation on the dialetheic formulation A&~A.

Summary 

At the end of section 4, Priest noted that although dialetheic logic says that there are true cases of A&~A, this formulation might not apply to Hegel’s dialectics, which calls for a more unified, intimate, internal, intensional relation/contradiction between the terms. He returns now to this issue so to better describe “the exact nature of dialectical contradictions” (410). [First recall what Priest says regarding the dialetheias of nature and spirit in Hegel:

For spirit, s, is | then both spirit and not spirit. In the notation of section 3: (s=s)&(s≠s). Alternatively, nature, n, which is not spirit, is spirit: (n≠s)&(n=s). The existence (truth) of this contradiction allows spirit to think (understand) what it is: spirit and nature, spirit and not spirit; and thus to achieve its telos, in which form it is the Absolute.
(401-402)

] We previously have encountered many contradictions taking the form (a=b)&(a≠b), “something’s being identical with, and different from something (else?)” (410). This is Hegel’s notion of identity in difference. Priest “will now argue that this is the form of a dialectical contradiction, to which all others reduce” (410).

We have also encountered contradictions of the form: (a=a)&(a≠a). “They are obviously of this form” (411) [(a=b)&(a≠b), the form of ‘identity in difference’.] We also encountered the contradiction of something

being identical with its opposite: ^A=^~A. Thus for example, that something, a is free (Fa) is identical to its being bound (not free): ^Fa=^~Fa. This, too is a special form of identity in difference. For, as we noted in section 3, it is always true that ^A≠^~A. Thus, identity of opposites is just the identity in difference (^A=^~A)&(^A≠^~A).
(411)

[Perhaps Priest understands “identity in difference” to mean that both something equals its negation and it does not equal its negation.]

[In this next paragraph, Priest refers to the Tarski T-scheme. Recall that Tarski uses quotations around a sentence to mean the name for that sentence. For example:

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

In our article here, (see sect.3) Priest is doing something similar with the ^ symbol, meaning we would put ‘that’ in front of the sentence. The basic operations Priest seems he is doing is that if we have a sentence, let us say: snow is white; we can then make it: that snow is white is true iff snow is white. Since snow is white, then we can just say: that snow is white is true. However, we can substitute the  ‘quoted’ or ‘that-ed’ term with its negation, since we equated the two. Thus this means that the negation is true as well.]

In fact, the identity of opposites ^A=^~A is doubly contradictory, since it also gives rise to the contradiction A&~A. For either A or ~A; without loss of generality, suppose the former. Then ^A is true, by the T-scheme (^A is true iff A). But if ^A=^~A, ^A is true implies ^~A is true  (by the substitutivity of identicals). Hence ^~A is true too. It follows that ~A, again by the T-scheme. Thus, both A and ~A.
(411)

[In the above, we began by assuming either only A or not-A. No matter which we assume, the other holds by consequence (when one is also equated to the other). We did not assume a third possibility. It is not completely clear to me how this factors in. It might have something to do with truth gaps, in which a proposition is neither true nor false, and so if A were valueless, we would not say A or not-A. For, it need not be that one is true, since A is valueless. See discussion of valueless statements, and Priest’s defense of the law of excluded middle in these situations, in section 1.3 and section 4.7 of In Contradiction.] One objection to the above reasoning is that it uses the law of excluded middle, but Hegel spoke against it. Priest replies that it is still valid given our semantics (outlined in section 3 of this article), and also that Hegel’s objection to the law is not that it is untrue but rather that it is trivial; “it may be false (as well)!” (411).

Priest will now address two particular cases of identity in difference. The first is “something’s being identical with itself is its being different from itself. This is just the identity of opposites (^a=a)=(^a≠a)” (411). The second is the dialectics of motion that he discussed in section 4. This as well can be understood as the identity of opposites. [[The reasoning here seems to be as follows. Consider an instantaneous state of contradiction. For an object to be in one position is for it to be in another. If it were only in the same position, it would not be in motion. Thus we can equate its being in a position with its not being in that position. Because this affirms both terms, we can conjoin them even though one can be seen as the negation of the other. Priest furthermore says that any such equation of a term with its negation can be seen as one term ‘changing’ into its opposite, perhaps because the equation brings one term upon the other. This is important, because it could explain the collapsing of one term upon the other. In logic this seems to be a mode of substitution. ~A passes into A in the sense that it supplants A, or overlays it. It is also important to note that Priest here distinguishes the equality of opposites with the conjunction. The conjunction does not imply the passing of one into the other, even though the passing of one into the other implies the conjunction of the terms.]]

we may take the instantaneous contradiction produced in a state of motion to be that the body’s being in a certain place is its not being in that place, ^A=^~A. This will imply that it both is and is not in that place, A&~A, as I have just observed. Moreover, because this type of contradiction is identified as a state of change, it is natural to describe any state of the form ^A=^~A as a state where ^A is changing into its opposite ^~A, or vice versa. Thus, the identity of opposites is frequently described in this way, as, for example, the opposites going over into each other.
(411)

So recall again the objection that A&~A does not do justice to the intimacy of the terms in dialectical contradiction. Priest shows how dialectical contradictions take the form (a=b)&(a≠b) [which is a variation on A&~A]. The intimacy here is that the terms are identical.

We have now seen that all the dialectical contradictions we have met are instances of identity in difference: (a=b)&(a≠b). We may therefore take this to be the general form of a dialectical contradiction. This is an excellent way of doing justice to the point we noted in section 4, that the poles of a dialectical contradiction must have a tighter relation than mere extensional conjunction. For the poles of the identity in difference (a=b)&(a≠b), a and b, are actually identical with (though different from) each other; (dialectical) identity is therefore the relationship between the poles of a dialectical contradiction.
(412)

Citations from:

Priest, Graham. ‘Dialectic and Dialetheiç’. Science & Society, 1989/1990, 53 (4) 388–415.

 

 

 

 

 

 

 

Priest (1.2) In Contradiction, ‘The Semantic Paradoxes’, summary

 

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

Graham Priest

In Contradiction:
A Study of the Transconsistent

Part I. The Logical Paradoxes

Ch.1. Semantic Paradoxes

1.2 The Semantic Paradoxes

Brief Summary:

Tarski tried to give a semantic definition of truth for formal languages, and these formal languages would approximate natural languages. There are certain conditions in his system for doing this that lead to a dialetheia (a true contradiction). This is a sort of liar paradox. Priest here shows how Tarski’s conditions necessarily lead to the inconsistency. In the next section, he will argue that they apply to English, thus dialetheias are inherent to both natural and formal languages.

Summary

Previously Priest looked at the logical paradoxes of self-reference, and he distinguished semantic ones from set theoretical ones. Now he will focus on the semantic ones. We noted how these paradoxes generate dialetheias, or true contradictions. Many people are against saying there exists true contradictions. But to maintain that stance, they must show what is invalid in the arguments producing the paradoxes. Priest’s “aim here is to defend the view that the semantic paradoxes are bona fide sound arguments.” (10) Priest will first “state a set of conditions sufficient for contradiction and then defend against all comers the view that natural language satisfies these conditions (or if not these, then others which have the same effect).” (10) One benefit from the following is that we will better see that solutions to the paradox fail, and we will also learn a bit as to why they do.

Priest explains that “Tarski (1936) located the root of the semantic paradoxes” in closure conditions. [Recall that a language is semantically closed when within it are contained all its components and generative rules. Such semantically closed languages prove inconsistent, as seen in the liar paradox.] Priest will now give the three Tarski conditions for a formal theory (and for simplicity we concern ourselves just with formulas with a single free variable).

(1) For every formula, α, there is a term of the language, α, its name.
(Priest, 11)

[Recall again the (T) scheme:

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

with the example:

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

As we can see, the ‘meta-language’ on the right side of the equation contains the formula: snow is white, or instead, p. Its name, given in quotes “snow is white” or symbolized as X, is the name for that formula, articulated in the object language. Priest uses underline for quotation and α instead of p.]

(2) There is a formula of two free variables, Sat(x y), such that every instance of the scheme

Sat(t α) ↔ α(v/t)               (*)

is a theorem, where t is any term, α is any formula of one free variable, v, and α(v/t) is α with all free occurrences of ‘v’ replaced by ‘t’ (with the usual precautions concerning the binding of variables free in t).
(Priest, 11)

[Recall the formula and our discussion from  the prior section:

x satisfies α ↔ α(y/x)

We said that means, x satisfies the formula named α (that is, makes its value true when substituted into that sentential function) if and only if we can rightly substitute x in for the free variable y in the formula having that name. Let us look again at Priest’s formulation: “There is a formula of two free variables, Sat(x y)”. In the above, we said there was one free variable in α, which is y. But given the construction, which seemingly means, x satisfies y, this might be equivalent to: x satisfies α . And thus the ‘y’ here might stand for the formula name α . So again:

Sat(t α) ↔ α(v/t)  

Here, α is a formula with one free variable, v. Then t is the term that substitutes for v. So again our formulation says, t satisfies the formula named α (that is to say, the formulation with that name is true), if and only if t can be rightly substituted into the actual formulation.]

(3) The rule of inference {α ↔ ¬α} ⊢ α ∧ ¬α is valid in the logic underlying the theory.
(Priest 11)

[It seems simply that the third condition is that there is a particular rule of inference involved. Consider if we have a formula if and only if we have its negation. From this it is provable (we may infer) that we have both that formula and its negation. I am not sure how better to understand this. Using logical equivalences (as I find them on this wiki page) we might go about it like this:

α ↔ ¬α ≡ (α → ¬α) ∧ (¬α → α)

{from the first logical equivalence of biconditionals:
p↔q≡(p→q)∧(q→p)}

α ↔ ¬α ≡ (¬α ∨ ¬α) ∧ (¬¬α ∨ α)

{from the first logical equivalence of conditionals:
p→q≡¬p∨q}

α ↔ ¬α ≡ (¬α ∨ ¬α) ∧ (α ∨ α)

{double negation law: ¬(¬p)≡p}

α ↔ ¬α ≡ ¬α ∧ α

{from the idempotent laws: p∨p≡p}

]

Now Priest will show that “any theory which satisfies the Tarski closure conditions is inconsistent.” (11)

[So recall again our formulation in step 2:

Sat(t α) ↔ α(v/t)               (*)

It seems we will make a complicated sort of substitution. So we first consider a formulation like

v does not satisfy itself

or perhaps, 

v does not satisfy formula v

We would write it:

¬Sat(v v)

This would be the α formulation. Then t would be the name for this

“v does not satisfy formula v”

And we would write this:

¬Sat(v v)

Then we make our substitions into

Sat(t α) ↔ α(v/t)

and we obtain

Sat(¬Sat(v v) ¬Sat(v v)) ↔ ¬Sat(¬Sat(v v) ¬Sat(v v))

I am not sure how to make this more conceivable with an example. I would think that it should be something like:

“This sentence is false” is true if and only if this sentence is false.

Let us begin with:

“v is white” is true if and only if v is white.

Now, we use satisfaction.

t satisfies “v is white” if and only if t can be substituted for v in the formula with this name (v is white).

Instead of ‘v is white’, we have ‘v does not satisfy v’. So

t satisfies “v does not satisfy v” if and only if t can be substituted for v in the formula: v does not satisfy v.

Then we substitute “v does not satisfy v” for t.

“v does not satisfy v” satisfies “v does not satisfy v” if and only if “v does not satisfy v” can be substituted for v in the formula:  v does not satisfy v (hence making: “v does not satisfy v” does not satisfy “v does not satisfy v”)

Thus we have

“v does not satisfy v” satisfies “v does not satisfy v” if and only if  “v does not satisfy v” does not satisfy “v does not satisfy v”.

Now consider again:

This sentence is false.

Since it is self referential, it is equivalent to saying in our formulations,

This sentence does not satisfy itself.

When we put the liar sentence into the (T) scheme using our different term for satisfaction, we then get:

“This sentence is false” is true if and only if “this sentence is false” is false.

]

Priest then discusses some other issues that complicate the matter [see page 11], but concludes: “the point is shown: these closure conditions give rise to contradiction.” (11)

Now consider the English language, and ask if the three conditions hold for it. It would seem they do. However, since English is not a formal language, the jargon “formula”, “term”, “theorem” and so on will not apply to it, since they only apply to formal languages. Yet,

Still, it is easy enough to rephrase the conditions while retaining their spirit. A natural language satisfies the Tarski conditions iff:
(1) For every phrase  α, there is a noun phrase α, its name.

(2) There is a phrase Sat, requiring two noun phrases to be inserted to make a sentence, such that every sentence of the form

Sat(t α) iff α(t)

is true, where a is any phrase requiring a noun phrase, t, to be inserted to make a sentence, and parentheses mark insertion.

(3) The following rule of inference is truth preserving:
α iff it is not the case that aα:
Hence, α and it is not the case that α.
(Priest, 12)

Thus “We can now proceed, exactly as before, to establish that any natural language that satisfies the Tarski conditions contains true sentences of the form ‘α and it is not the case that α.’ ” (11) And so “A natural language which satisfies the Tarski conditions therefore contains true contradictions.” (11) Next Priest will turn to the issue of whether or not natural languages can satisfy these Tarski conditions.

 

 

Citations from:
Priest, Graham. In Contradiction: A Study of the Transconsistent. Oxford/New York: Clarendon/Oxford University, 2006 [first published 1987].

 

 

 

Tarski (§11) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘11. The Construction (in Outline) of the Definition’

by Corry Shores

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Alfred Tarski

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

Part I. Exposition

11. The Construction (in Outline) of the Definition

 

 

Brief Summary:

We want an axiomatic, and not a semantic, way to define truth in a metalanguage. We must do this recursively and with the notion of satisfaction. Our basic units are sentences, like “snow is white”. The recursive method begins first by establishing the simplest sentential functions, which have the structure of sentences, but they have free variables, for example, “x is white”. Next, the recursive method indicates the operations which allow for more complex sentence functions to be built upon the simpler ones. For example, we might make such combinations as “x is white and y is on the mat”. In this functional form, the formulations are neither true nor false, because they have not yet been giving affirmable or deniable content. They become ‘satisfied’ if when replacing the variables with determinate contents they become true. But in our system, we need to define ‘satisfaction’. Yet, so far our semantic definition is problematic [for, it is circular: truth is what results from satisfaction, but satisfaction is what produces truth.] Instead, we need again to give a recursive definition for satisfaction. (So thirdly,) we do this by first stating the conditions under which the simplest sentential functions would be satisfied by certain objects [for example, ‘snow is white’ is true if and only if snow is something that really is white, and ‘the cat is on the mat’ is true if and only if the cat is indeed on the mat.] Then (fourthly) we state those conditions under which objects would satisfy a compound function [for example, the above compound is true if and only if it really is the case that both snow is white and the cat is on the mat.] [Now, since a sentence does not have variables, there is only one ‘object’ that can satisfy it, that being, the meaning on the right side of the equation given without quotes. Thus] “a sentence is true if it is satisfied by all objects, and false otherwise.”

Summary

Previously Tarski claimed that, so long as the meta-language is essentially richer than the object-language, we can adequately define ‘true’ in the meta-language axiomatically without resorting to a semantic definition provided in yet a higher meta-language. Now he will explain that we can do so by means of the notion of satisfaction.

Satisfaction is a relation between sentential functions, like ‘x is white’ and ‘x is greater than y.’ As we see, sentential functions have something like the structure of a sentence. However [being functions] they, unlike normal sentences, can contain free variables like x and y.

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.
(345)

We will define sentential functions in formalized languages using a recursive procedure, which means we first give sentential functions with the simplest structure, and then we explain how to make more complicated sentential functions on their basis. For example, we might use such logical operations as conjunction and disjunction.

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.
(345)

Tarksi then considers one way to understand satisfaction, but he also explains why it will not be of use to us here. [It seems he is saying that for example, the sentential function ‘x is white’ can be satisfied if we substitute in for x white things, like snow. But what about ‘X is true’? We would need already to establish the substitutions as true before correctly making that substitution. Thus given this circularity, it does not help us to define truth using the notion of satisfaction. Put another way, we would encounter this problematic formulation: satisfaction happens when a substitution is true, and true is what happens when a substitution is satisfactory.]

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.
(345)

Tarski again says that satisfaction will have to be defined more mechanically using a recursive method, which would say what are the simplest sentential functions and then explain how more complex ones are built correctly upon them.

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.”
(345)

[A sentential function we said is a formulation with the structure of a sentence but with free variables instead of being made entirely of just words. So ‘x is greater than y’ is a sentential function, as we noted. As soon as we designate actual values or contents to those variables, it is no longer a function, but is now instead just a sentence. So ‘2 is greater than 1’ is a sentence and not a function. Were it still a function, then some objects would satisfy it, like the one just given, and other objects would produce a false sentence, for example, ‘3 is greater than 4’. So we would not say that a function is either true or false. However, sentences like ‘2 is greater than 1’ are true, because it is satisfied in the (T) scheme:

“2 is greater than 1” is true if and only if 2 is greater than 1.

There are no variables, so there is a finite number of objects which can satisfy the sentence. thus all objects satisfy it. (The value 2 and the value 1 are the only objects that could satisfy the terms in the quoted formulation, and thus all objects satisfy the sentence.) However, consider again:

“3 is greater than 4” is true if and only if 3 is greater than 4.

Here, the only objects that could substitute for the terms in the formulated quotation are 3 and 4. But this does not satisfy its meaning, since 3 is not greater than 4. Therefore, since no objects (none that can be substituted in for the terms in the name) can satisfy the quoted sentence, then not all objects satisfy it, and thus it is false. (Another interpretation would be that instead of ‘3’ and ‘4’ being the objects, that either the quoted name for the sentence or its unquoted meaning are singularly the object. Please read the following text to generate your own better interpretation, as mine is based on a partial misunderstanding.)]

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
(345-346)

[Endnote 15:] In carrying through this idea a certain technical difficulty arises. A sentential function may contain an arbitrary number of free variables; and the logical nature of the notion of satisfaction varies with this number. Thus, the notion in question when applied to functions with one variable is a binary relation between these functions and single objects; when applied to functions with two variables it becomes a ternary relation between functions and couples of objects; and soon. Hence, strictly speaking, we are confronted, not with one notion of satisfaction, but with infinitely many notions; and it turns out that these notions cannot be defined independently of each other, but must all be introduced simultaneously.

To overcome this difficulty, we employ the mathematical notion of an infinite sequence (or, possibly, of a finite sequence with an arbitrary number of terms). We agree to regard satisfaction, not as a many-termed relation between sentential functions and an indefinite number of objects, but as a binary relation between functions and sequences of objects. Under this assumption the formulation of a general and precise definition of satisfaction no longer presents any difficulty; and a true sentence can now be defined as one which is satisfied by every sequence.
(359)

Tarski then adds parenthetically:

(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.)
(346)

So far, Tarski has been leaving out many technical details. He would need to include them in order to explain the role of “essential richness”, so for now the concept will seem a bit vague.

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
(346)

[Endnote 16:] To define recursively the notion of satisfaction, we have to apply a certain form of recursive definition which is not admitted in the object-language. Hence the "essential richness" of the meta-language may simply consist in admitting this type of definition. On the other hand, a general method is known which makes it possible to eliminate all recursive definitions and to replace them by normal, explicit ones. If we try to apply this method to the definition of satisfaction, we see that we have either to introduce into the meta-language variables of a higher logical type than those which occur in the object-language; or else to assume axiomatically in the meta-language the existence of classes that are more comprehensive than all those whose existence can be established in the object- language. See here Tarski [2], pp. 393 ff., and Tarski [5], p. 110.
(359)

[From the Bibliography:]

Tarski, A. [2]. “Der Wahrheitsbegriff in den formalisierten Sprachen.” (German translation of a book in Polish, 1933.) Studia philosophica, vol. I, 1935, pp. 261-405.

Tarski, A. [5]. “On Undecidable Statements in Enlarged Systems of Logic and the Concept of Truth.” The Journal of Symbolic Logic, vol. IV, 1939, pp. 105-112.
(

363)

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:

http://www.ditext.com/tarski/tarski.html

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://de.wikipedia.org/wiki/Alfred_Tarski:

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:

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

 

Tarski (§10) of “The Semantic Conception of Truth and the Foundations of Semantics”, entitled ‘10. Conditions for a Positive Solution of the Main Problem’

by Corry Shores

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[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:

http://www.ditext.com/tarski/tarski.html

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://de.wikipedia.org/wiki/Alfred_Tarski:

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:

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

 

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

by Corry Shores

[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]
[Logic & Semantics, Entry Directory]
[Alfred Tarski, Entry Directory]
[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

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:

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:

http://www.ditext.com/tarski/tarski.html

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://de.wikipedia.org/wiki/Alfred_Tarski:

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:

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

 

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

by Corry Shores

[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]
[Logic & Semantics, Entry Directory]
[Alfred Tarski, Entry Directory]
[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

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:

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:

http://www.ditext.com/tarski/tarski.html

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://de.wikipedia.org/wiki/Alfred_Tarski:

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:

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

 

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

by Corry Shores

[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]
[Logic & Semantics, Entry Directory]
[Alfred Tarski, Entry Directory]
[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

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:

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:

http://www.ditext.com/tarski/tarski.html

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://de.wikipedia.org/wiki/Alfred_Tarski:

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:

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