| Frege'**s** Logic, Theorem, and Foundations for Arithmetic |

| | The claim that **Hume**'**s** **Principle** is an analytic **principle** of logic is subject to the same problem just posed for Basic Law V. The equinumerosity of F and G does not, as a matter of meaning, imply (identity claims that entail) the existence of numbers. |

| | This consistent **principle**, known in the literature as "**Hume**'**s** **Principle**", asserts that for any concepts F and G, the number of F-things is equal to the number G-things if and only if there is a one-to-one correspondence between the F-things and the G-things. |

| | So by setting aside the derivation of **Hume**'**s** **Principle** from the inconsistent Basic Law V and focusing on Frege'**s** proofs of the basic propositions of arithmetic, his theoretical accomplishment emerges much more clearly, for his work shows us how to prove the Dedekind/Peano axioms for number theory from **Hume**'**s** **Principle** in second-order logic. |

| plato.stanford.edu /entries/frege-logic (15095 words) |