Metamath Proof Explorer Frege Notation |
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Frege used Greek letters to stand for concrete propositions, propositional formulae with one or more indeterminate slots (which he called functions), and concepts similar to our classes and relations; Latin letters to serve as placeholders within the scope of a judgment similar to the metavariables in the axiom schema of Metamath; and introduced Fraktur (German blackletter) for the variables introduced by quantifiers. In 1910, Frege wrote to Philip Jourdain that these conventions were not a barrier to substitution but more of an aid to exposition.
In the following, parenthesized expressions like 𝐹 (𝑥) and 𝑓 (𝑥, 𝑦) have no fixed meaning in the notation, but are placeholders for indeterminate propositions where terms 𝑥 (and 𝑦) may appear. If we interpret 𝑥 and 𝑦 as sets, then 𝐹 (𝑥) may be interpreted as membership in a class while 𝑓 (𝑥, 𝑦) may be interpreted as membership in a relation. Alternately, they may be interpreted as placeholders for an arbitrary proposition following proper substitution for the symbols 𝑥 and 𝑦.
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Colors of variables: wff set class |
Additional Notation Having laid the foundations of notation for propositions and classes that are sometimes required to be sets, Frege invents additional notation to introduce definitions and proceeds to define new arrangements of symbols. Unlike left-to-right notation like , Frege does not privilege the ordering of the slots in a two-slot meta-notation like 𝑓 (δ, α) as having fixed meaning regarding the natural ordering such a relation indicates. Instead Frege indicated the intended ordering of the slots with lowercase Greek letters. Instead of having separate notation for the converse of a relation, Frege would reverse the ordering of the lowercase Greek letters relative to the vertically oriented marker which introduces them.
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Colors of variables: wff set class |
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