Abstract
There are 16 bivalent binary truth functions, but only two fundamental logical relations between propositions: implying and precluding. In addition, some higher-level concepts express the scope and nature of inferential relations. In this chapter, I will explain Frege’s treatment of these two notions, which are the semantic and pragmatic support for conditionality and negation. I will also offer an explanation of Frege’s semantic account of universal and existential quantifiers, and the relations between them. The richness and originality of Frege’s semantics manifest in his treatment of the role of logical notions and related expressions. Frege’s logic is nothing more (and nothing less) than his semantics for logical terms together with an appropriate method for representing inferential transitions. The deductive system that converts a language into a calculus is the representation of the commitments and entitlements that individuate concepts and propositional contents. The very notation that Frege introduces shows that logical notions do not affect judgeable contents, which systematically occur as arguments of the former. In this chapter, I will argue that Frege defended an expressivist approach to the meaning of logical notions. Frege’s insights at this point can help in the as-yet unfinished task of giving a satisfactory semantics for logical notions. I consider Frege’s views and suggestions entirely correct.
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Notes
- 1.
As I will explain in the next section, Begriffsschrift explains the meaning of the conditional stroke (§5) and negation (§7) on expressivist lines. Some of Frege’s Logical Investigations, where he deals with generality (Frege, 1923) and negation (Frege, 1918–9b), also suggest this expressivist approach.
- 2.
In the Begriffsschrift, §5. One might retort at this point that the conditional and implication are different notions. In one case, the relation between the two propositions involved is contingent, and in the other it is necessary. This point will be discussed in the Chap. 6. But here it suffices to say that conditional terms (‘if’, ‘when’) are natural language tools for expressing implication (in a broad sense).
- 3.
For an up-to-date survey, see e.g. (Stanley & Szabó, 2003).
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Frápolli, M.J. (2023). Implying, Precluding, and Quantifying Over: Frege’s Logical Expressivism. In: The Priority of Propositions. A Pragmatist Philosophy of Logic. Synthese Library, vol 475. Springer, Cham. https://doi.org/10.1007/978-3-031-25229-7_4
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