The Semantics of Value-Range Names and Frege's Proof of Referentiality

In this article, I try to shed some new light on Grundgeselze 10, 29-31 with special emphasis on Frege's criteria and proof of referentiality and his treatment of the semantics of canonical value-range names. I begin by arguing against the claim, recently defended by several Frege scholars, that the first-order domain in Grundgesetze is restricted to value-ranges (including the truth-values), but conclude that there is an irresolvable tension in Frege's view. The tension has a direct impact on the semantics of the concept-script, not least on the semantics of value-range names. I further argue that despite first appearances truth-value names (sentences) play a distinguished role as semantic "target names" for "test names" in the criteria of referentiality (29) and do not figure themselves as "test names" regarding referentiality. Accordingly. I show in detail that Frege's attempt to demonstrate that by virtue of his stipulations "regular" value-range names have indeed been endowed with a unique reference, can plausibly be regarded as a direct application of the context principle. In a subsequent section, I turn to some special issues involved in 10. 10 is closely intertwined with 31 and in my and Richard Heck's view would have been better positioned between 30 and 31. In a first step, I discuss the piecemeal strategy which Frege applies when he attempts to bestow a unique reference on value-range names in 3, 10-12. In a second step, I critically analyze his tentative, but predictably unsuccessful proposal (in a long footnote to 10) to identify all objects whatsoever, including those already clad in the garb of value-ranges, with their unit classes. In conclusion, I present two arguments for my claim that Frege's identification of the True and the False with their unit classes in 10 is illicit even if both the permutation argument and the identifiability thesis that he states in 10 are regarded as formally sound. The first argument is set out from the point of view of the syntax of his formal language. It suggests though that a reorganization of the exposition of the concept-script would have solved at least one of the problems to which the twin stipulations in 10 give rise. The second argument rests on semantic considerations. If it is sound, it may call into question, if not undermine the legitimacy of the twin stipulations

Second-order logic is logic

"Second-order logic" is the name given to a formal system. Some claim that the formal system is a logical system. Others claim that it is a mathematical system. In the thesis, I examine these claims in the light of some philosophical criteria which first motivated Frege in his logicist project. The criteria are that a logic should be universal, it should reflect our intuitive notion of logical validity, and it should be analytic. The analysis is interesting in two respects. One is conceptual: it gives us a purchase on where and how to draw a distinction between logic and other sciences. The other interest is historical: showing that second-order logic is a logical system according to the philosophical criteria mentioned above goes some way towards vindicating Frege's logicist project in a contemporary context

Classical Opacity

Philosophy and Phenomenological Research, EarlyView