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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
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)
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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
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