Semantic holism

A bivalent valuation is snt iff sound (standard PC inference rules take truths only into truths) and non-trivial (not all wffs are assigned the same truth value). Such a valuation is normal iff classically correct for each connective. Carnap knew that there were nion-normal snt valuations of PC, and that the gap they revealed between syntax and semantics could be "jumped" as follows. Let VALS.t be the set of snt valuations, and VALnrm be the set of normal ones. The bottom row in the table for the wedge 'v' is not semantically determined by VALS,n but if one deletes from VALsnt all those valuations that are not classically correct at the aforementioned row, one jumps straights to VALnVm and thus to classical semantics. The conjecture we call semantic holism claims that the same thing happens for any semantic indeterminacy in any row in the table of any connective of PC, i.e., to remove it is to jump straight to classical semantics. We show (i) why semantic holism is plausible and (ii) why it is nevertheless false. And (iii) we pose a series of questions concerning the number of possible steps or jumps between the indeterminate semantics given by VALSnt and classical semantics given by VALnrm. 1. Historical Context Rudolf Carnap's influential 1942 book Introduction to Semantics has so fascinated logicians that, for the most part, they have overlooked or forgotten its provenance. According to his own account, Carnap wrote it to set the stage for the ideas and results to be found in his 1943 book Formalization of Logic. Though published later, the 1943 book was drafted earlier than the 1942 one, in the autumn of 1938. What were the ideas and results that Carnap deemed so important that he composed another substantial book to pave the way for them? In a word, misalignment. Carnap had discovered that classical syntax and classical semantics don't line up quite so neatly as nearly everyone seemed to think. For example, the consistency and completeness metatheorems for classical proposi tional logic (PC) were taken uncritically, even unconsciously, to entail that sound (in the sense that the inference rules of PC permit only truths to be deduced from truths) bivalent valuations (functions that map the wffs into the two truth values) of the wffs of PC must conform every-where to the classical truth tables for the connectives. (Throughout this paper we will be concerned only with bivalent valuations, so we will hence-forth make bivalence part of the concept of a valuation.) By a normal valuation Carnap meant a sound valuation in which all sentential connectives conform everywhere to the prescriptions of their classical truth tables. By a non-normal valuation Carnap meant a sound valuation in 68 N. D. Belnap, Jr., G. J. Massey which at least one sentence connective deviates in at least one instance from the prescriptions of its classical truth table. (Hereafter, we will use the terms normal valuation and non-normal valuations in the foregoing Carnapian sense. Note in particular that both normal and non-normal valuations are sound.) What had profoundly unsettled Carnap was his discovery of non-normal valuations, i.e., sound valuations that here and there violate the prescriptions of classical truth