Abstract
This paper considers various objections to Carnap’s logical syntax definition of ’logical expression’, including those by Saunders Mac Lane and W. V. O. Quine. While the specific objections of these two authors can be answered, if necessary by a slight modification of Carnap’s definition, there are other objections that I do not see how to meet. I also consider the proposal by Denis Bonnay for avoiding the objections to Carnap’s definition. In light of the unresolved problems with Carnap’s definition, I go on to consider what Carnap’s assumptions must have been in framing his definition and to assess how much damage is caused by this failure of Carnap’s definition. This damage is not as much as might be assumed.
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Notes
The English word ‘concurrent’ contains ten letters, but we are not forced to say that it contains ten component expressions any more than the fact that the letter ‘a’ can divided into right and left halves forces us to say that it has at least two component expressions. Now ‘concurrent’ can be subdivided into three English words. Again this should not be determinative any more than printers’ practice forces us to say that ‘concurrent’ is a single elementary expression. My own preference would be to say that ‘con’ is separately meaningful as is ‘current’ for an expression total of two. I am even prepared to have it turn out that some of what we typically call expressions, such as quotation marks, could turn out to be font-forming devices rather than fully fledged expressions. All these examples taken together do not go very far toward a real definition of ‘basic expression’. For this reason I take ‘basic expression of language L’ as a new primitive term and revise Carnap’s definition of ‘logical’ accordingly. There may be ways to avoid it, but I am far from confident.
For example, Carnap seems to be led by the idea that the logical vocabulary “lacks content” (where the “content” of a vocabulary is the set of non-determinate but meaningful sentences that can be said with it) to think that the logical terms could be added to any vocabulary without adding to the content of the original vocabulary. This is the only account I can think of for why Carnap takes the intersection of the various classes, \(\hbox {K}_\mathrm{i}\), to be the logical vocabulary. The various counter-examples suggest that this is not likely to be correct.
Strictly, Carnap defines ‘analytic’ in terms of ‘L-valid’ which is defined in terms of ‘L-consequence’ which is where the devices of the elimination of defined terms and essential occurrence appear directly, though the effect is the same. Also the definition calls for the elimination of every defined term. Where a set of terms is mutually interdefinable, one can hardly eliminate all of them, and so a maximum of elimination is probably what Carnap had in mind.
More precisely the analytic claim would be the conditional whose antecedent is the observational content of the reduction sentences and whose consequent is the conjunction of the reduction sentences. This idea anticipates what Carnap would later develop into an account of analyticity in the theoretical language, an account that appeals to the Ramsey sentence of a theory as the antecedent of such a conditional.
Much the same view animates Formalization of Logic (Carnap 1943). This is not surprising because, though published after (Carnap 1942), it was actually written first. Moreover, one of the central results of (Carnap 1943) is that there are non-standard interpretations of logical symbols such as negation. This would complicate any attempt to delimit the logical symbols as those for which only one interpretation is available given the rules of the language.
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