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Published online by Cambridge University Press: 12 March 2014
Classical syntax and semantics make fundamental use of variables ranging over classes, relations, etc. In syntax, one speaks exclusively of such objects as symbols, formulae, terms, in short of what may be called (following Carnap) sign-designs. A sign-design is a class of similar concrete marks or sign-events or inscriptions or sequences of typographical characters. In semantics, in addition to sign-designs, one also speaks of the objects denoted or designated by the sign-designs called constants or of the objects which are values for the sign-designs called variables.
Interest in the possibility of a syntax and semantics based upon sign-events or inscriptions rather than upon sign-designs appears to have originated with Lesniewski and Tarski. Not until recently, however, in a paper by Goodman and Quine, has a purely inscriptional syntax been explicitly formulated. The discussion in that paper is confined to the syntax of one object language, but the methods developed are applicable to other systems as well and should suffice for many of the purposes of syntax.
In the present paper we make an attempt to construct a purely inscriptional semantics, in which, as with Goodman and Quine, sign-designs or classes of similar inscriptions in no way figure as values for variables. In rejecting such entities it might appear that the resulting semantics would be drastically curtailed. That the methods used here are of sufficient power to be of interest, however, is shown by constructing a definition of a semantical concept of truth for a given sample elementary object language L. The definition of a predicate for truth in L is given, and the predicate is shown to be adequate in essentially the sense due to Leśniewski, Kotarbiński, and Tarski.
1 See Carnap, R., Introduction to semantics, Cambridge, Mass., 1942, pp. 5–8Google Scholar.
2 In this connection, see Tarski, A., Der Wahrheitsbegriff in den formalisierten Sprachen Studio philosophica, vol. I (1936), pp. 261–405, esp. pp. 290–291 and footnote 23Google Scholar.
3 See Goodman, N. and Quine, W. V., Steps toward a constructive nominalism, this Journal, vol. 12 (1947), pp. 105–122Google Scholar.
4 The fundamental paper on the semantical concept of truth is Tarski's Der Wahrheitsbegriff. See also his popular account, The semantic conception of truth and the foundations of semantics, Philosophy and phenomenological research, vol. 4 (1944), pp. 341–376CrossRefGoogle Scholar.
5 See Tarski, loc. cit. Also Kotarbiński, T., Elementy teorji poznania, logiki formalnej i metodologji nauk, Lwów, 1929Google Scholar.
6 The words ‘intension’ and ‘extension’ are used here roughly in the sense of Carnap, R., Meaning and necessity, Chicago, 1947Google Scholar.
7 See Carnap, , Meaning and necessity, p. 98Google Scholar.
8 See Quine, W. V., Designation and existence, Journal of philosophy, vol. 36 (1939), pp. 701–709CrossRefGoogle Scholar, and Notes on existence and necessity, Journal of philosophy, vol. 40 (1943), pp. 113–127CrossRefGoogle Scholar. See also Quine's, On universals, this Journal, vol. 12 (1947), pp. 74–84Google Scholar, and Martin, R. M., A homogeneous system for formal logic, this Journal, vol. 8 (1943), pp. 1–23Google Scholar (referred to here as H.L.), and A note on nominalism and recursive functions, this Journal, vol. 14 (1949), pp. 27–31Google Scholar.
9 Goodman and Quine, loc. cit.
10 See, e.g., Carnap, R., Introduction to semantics, esp. pp. 49–55Google Scholar.
11 The use of ‘applies’ in this context is suggested by Carnap, , Meaning and necessity, footnote 1 on p. 97Google Scholar. But because ‘application of function to argument’ is already a technical phrase, it is perhaps better not to use ‘applies’ in this new semantical meaning.
12 The middle domain of a three-placed relation R is, familiarly, 
.
13 See Goodman and Quine, loc. cit., esp. p. 112.
14 Goodman and Quine, p. 113.
15 See Carnap, R., The two concepts of probability, Philosophy and phenomenological research, vol. 5 (1945), pp. 513–532CrossRefGoogle Scholar, and On inductive logic, Philosophy of science, vol. 12 (1945), pp. 72–79CrossRefGoogle Scholar; and Hempel, C. G. and Oppenheim, P., Studies in the logic of explanation, Philosophy of science, vol. 15 (1948), pp. 135–175CrossRefGoogle Scholar.
16 See Tarski, A., Der Wahrheitsbegriff, pp. 285–286Google Scholar.
17 See Goodman and Quine, p. 112.
18 Quasi-quotes or corners are used here in Quine's sense. See, e.g., his Mathematical logic, New York, 1940, pp. 33–37Google Scholar. If we think of the metalanguage of ISM as in turn being concerned with inscriptions, the result of applying quasi-quotes to a given sequence of symbols must be reconstrued appropriately as designating an inscription or inscriptions. In talking about ISM, however, we frequently allow ourselves the convenience of the classical terminology.
19 The word ‘nominatum’ is used by Carnap, . See Meaning and necessity, p. 97Google Scholar. See also the definition of ‘synonymous’ in Introduction to semantics, p. 55.
20 See Carnap, R., Introduction to semantics, esp. pp. 24–29, pp. 32–33, and pp. 44–48Google Scholar.
21 See Tarski, A., Der Wahrheitsbegriff, esp. Konvention 
, p. 305Google Scholar. Carnap, Alan R., Introduction to semantics, pp. 26–27Google Scholar.
