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We also suppose that Carnap's sign ‘≡’ of identity of individuals is a predicator constant, and that when A and B are individual expressions, A ≡ B is to be understood merely as an abbreviation or alternate way of writing ≡ AB. This modification of S1 serves to simplify the discussion but is not otherwise essential to the conclusions we reach.
I follow Carnap's terminology, in spite of my own preference for a somewhat different terminology—e.g., ‘well-formed formula’ instead of ‘designator matrix.’
The definition of ‘L-true’ need not be repeated here. But notice should be taken of two necessary corrections to the definition as it is developed in §§ 1–2 of Carnap's book. In 2-2 the correction of Kemeny must be adopted (Journal of Symbolic Logic, 16:206 (1951)); i.e., in place of “every state-description” the restriction must be made to non-contradictory state-descriptions. Otherwise consequences will follow that are certainly not intended by Carnap, for instance that no two different atomic sentential matrices (and no two different predicator constants) can be L-equivalent. In the rules of designation 1-1 and 1–2, the way in which the English language and certain phrases of the English language are mentioned, rather than used, is inadmissible—as may be seen by the fact that it forces the tacit use, in 1–3 and 1–4, of certain rules of designation of theEnglish language, which, if stated, would have a quite different form from 1-1 and 1–2. For example, Carnap's rule of designation, “‘s’ is a symbolic translation [i.e., from English] of ‘Walter Scott’,” should be changed to a rule which mentions the man Walter Scott rather than the words ‘Walter Scott’; perhaps it should be simply “‘s’ refers to Walter Scott,” in order to justify the inference from 1–3 to 1–4. These corrections are not directly relevant to the present paper, but our discussion presupposes that suitable corrections have been made.
Carnap uses ‘≡’ not only between sentential matrices as a sign of material equivalence, but also between other designator matrices as a sign of identity (in place of the usual ‘≡’).
Because of the restriction to the single language S1 and to designator matrices containing the same free variables, we have been able to give a simplified form to Carnap's definitions of ‘L-equivalent’ and ‘intensionally isomorphic.’
In this form, as applied to the construction of a new language and the determination of what its expressions shall mean, the Principle of Tolerance is hardly open to doubt. The attempt to apply the Principle of Tolerance to the transformation rules of a language after the meaning of the expressions of the language has already been determined (whether by explicit semantical rules or in some looser way) is another matter, and certainly doubtful, but is not at issue here. In fact Carnap (if he ever did) does not now maintain the Principle of Tolerance in this latter and more doubtful form (see § 39 of hisIntroduction to Semantics).
There is no condition to the effect that a predicator constant must express a simple property, rather than such a comparatively complex property as that which is here expressed by Q. In fact some of Carnap's examples of predicator constants express properties which are evidently not especially simple. And it is moreover not clear how the distinction between a simple and a complex property could be made precise in any satisfactory way (except by making it relative to the choice of a particular language).
This is the account of definition which is given, for example, by Hilbert and Bernays inGrundlagen der Mathematik. In constructing formalized languages, others (including myself) have often preferred to avoid definitions in this sense, which change the object language by adding new notations to it. But such avoidance is on the same ground of economy that underlies the avoidance of synonymous primitive constants, and need not be demanded when economy is not the objective.
Compare Carnap,The Logical Syntax of Language, § 22, 1(b).
In particular to any of the languages considered in my paper, “A Formulation of the Logic of Sense and Denotation” (Structure Method and Meaning, pp. 3–24), and to languages obtained from these by adding constants of any types, with specified meanings. It is necessary to explain that the statement on page 5 of that paper, that Alternative (0) “may be described roughly by saying that it makes the notion of sense correspond to Carnap's notion of intensional structure” is an error (unless “roughly” is understood in a very liberal sense). The intention of Alternative (0) is rather that two well-formed formulas shall have the same sense if and only if they are synonymously isomorphic.
See my “The Need for Abstract Entities in Semantic Analysis,”Proceedings of the American Academy of Arts and Sciences, 80 (No. 1): 100–12 (1951).
Compare the “rules of definition,” originally Aristotelian, which are often included in books on traditional logic.
“Synonymity, and the Analysis of Belief Sentences,”Analysis, 14(No. 5):114-22 (1954).
In a forthcoming paper, “On Belief Sentences: Reply to Alonzo Church.”
“Synonymity,”University of California Publications in Philosophy, vol. 25 (1950), see the lower half of page 215.
To treat the English language as a language for which syntactical and semantical rules have been fully given is of course to make a supposition contrary to fact, but it is one which is very convenient for illustrative purposes and has in fact been adopted in informal discussion by Carnap, Mates, Putnam, and many others. Use of this device has the effect that it may be necessary in the course of the illustration just to invent a rule of English, either to fill a gap in the rules as found in existing grammars and dictionaries or to remove an equivocacy. In the present context, for instance, we have been obliged to decide arbitrarily (or on the basis of mere plausibility) that ‘fortnight’ is synonymous with ‘a period of fourteen days’ rather than with ‘a period of two weeks’; existing English dictionaries either fail to decide this point or disagree among themselves, probably because universal familiarity with the multiplication table tends to obscure the fact that the two latter (quoted) phrases are not synonymous with each other.
A context of doubting is of course a belief context, since to doubt is to withhold belief. And a criterion of identity of belief must also be a criterion of identity of doubt.
Doubt being one of the fundamentals of philosophical method, it would be hard indeed to find a proposition that some philosopher might not be found to doubt.
The two occurrences of the phrase ‘in English’ would usually be omitted, but strictly they are necessary; for the semantical relation of satisfaction (or fulfillment) is a ternary relation among an individual, a sentential matrix, and a language.
The point is that names of two different sentences are not synonymous in any sense, and in particular not synonymously isomorphic, even though the sentences themselves be synonymously isomorphic.
(Added August 4, 1954.) The existence of more than one language is not usually to be thought of as a fundamental ground of the conclusions reached by this method. Its role is rather as a useful device to separate those features of a statement which are essential to its meaning from those which are merely accidental to its expression in a particular language, the former but not the latter being invariant under translation. And distinctions (e.g., of use and mention) which are established by this method it should be possible also to see more directly. The point is well illustrated by a paper of Wilfrid Sellars, “Putnam on Synonymity and Belief,” forthcoming inAnalysis, in which conclusions the same as or similar to those of Part II of this paper are reached by a more direct analysis. Professor Sellars's paper and mine were written independently, but I saw a copy of it by return mail when my own was submitted toPhilosophical Studies.
Analysis, 10(No. 5): 97–99 (1950).
The object of the doubt must still be a proposition, but a proposition about certain sentential matrices.
The shorter translation ‘vierzehn Tage’ would be more usual, but is not quite literal, as may be seen by considering the question of translating the phrase ‘three fortnights’ into German.
Of course ‘(14)’ and ‘(15)’ are here used, not as names of the sentences which we have so numbered, but just as convenient abbreviations. The reader must imagine the full sentences written out in place of the ‘(14)’ and ‘(15).’ Indeed throughout the paper such parenthetic numerals are to be understood asabbreviations when preceded by the word ‘that’—but elsewhere asnames of their sentences.
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Church, A. Intensional isomorphism and identity of belief. Philos Stud 5, 65–73 (1954). https://doi.org/10.1007/BF02221771
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DOI: https://doi.org/10.1007/BF02221771