Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method by Carlo Cellucci

Author:Carlo Cellucci
Language: eng
Format: epub
Publisher: Springer Netherlands, Dordrecht

10.13 Frege’s Primitive Laws of Logic

To carry out his logicist programme, Frege must indicate the primitive laws of logic from which all arithmetical laws are to be derived. His crucial primitive law of logic is the following:

This law – which is known as the Basic Law (V) because it occurs fifth in Frege’s list of primitive laws of logic – states that, for any functions F, G, the extension of F is identical to the extension of G if and only if, for every x, F(x) if and only if G(x).

For the reason explained above, Frege does not say that intellectual intuition permits us to acknowledge that the Basic Law (V) is true, he only says that the Basic Law (V) is “what people have in mind, for example, where they speak of the extensions of concepts.”89 For ∀x(F(x)↔G(x)) “expresses the same sense” as {x : F(x)} = {x : G(x)}, although “in a different way.”90 But Frege is so confident in intellectual intuition that he states that “it is prima facie improbable” that his logic “could be erected on a base that was uncertain or defective.”91 And he adds: “As a refutation in this I can only recognize someone’s actually demonstrating either that a better, more durable edifice can be erected” on another base, “or else that my principles lead to manifestly false conclusions. But no one will be able to do that.”92

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