Abstract
In “Gedankengefüge” Frege says that any two sentences of the form “A and B” and “B and A” have the same sense. In a 1906 letter to Husserl he says that sentences with the same sense should be represented in a perfect notation by one and the same formula. Frege’s own notation, just like any linear notation for sentential logic, is not perfect in this sense, because in it “A and B” and “B and A” are represented by distinct formulas, as is any pair of logically equivalent compound conditionals. A notation for the sentential calculus that meets Frege’s worries about conjunction, and indeed about any symmetric relation that there may be occasion to symbolize, is Peirce’s Alpha graphs.
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Notes
- 1.
It is debatable that this is how Frege would have explained sameness of sense. In a famous 1906 letter to Husserl he seems to be saying that logical equivalence is a sufficient criterion for sameness of sense. But this is in neat contrast with his principle of sense composition; see Dummett (1981, ch. 17). However, whether or not Frege did in fact think that logical equivalent sentences express the same sense does not affect in the least my argument. But see Bellucci (2020) for a discussion.
- 2.
Peirce sometimes considers the Alpha graphs to be based on the scroll, corresponding to the material implication, as the sole primitive, from which the meaning of the single cut (negation) is derived as the implication of the false. If so considered, the only symmetric relation that Alpha graphs represent is that between the antecedents of conditionals of the form of (3) and (4). The comparison between the “Begriffsschriftt” and such scroll-based Alpha graphs awaits investigation.
- 3.
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Bellucci, F. (2020). Fregean Logical Graphs. In: Pietarinen, AV., Chapman, P., Bosveld-de Smet, L., Giardino, V., Corter, J., Linker, S. (eds) Diagrammatic Representation and Inference. Diagrams 2020. Lecture Notes in Computer Science(), vol 12169. Springer, Cham. https://doi.org/10.1007/978-3-030-54249-8_34
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