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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
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:
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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