数学教育における記号表現の問題(II) : 「論理記号」考(その1)

In this paper, I investigate the sources and the developments on the "Logical Symbols", through adaptations of logical thoughts, within our school mathematics.
Giuseppe Peano, a mathematician in Italy, carried out an important part in symbolism of logic, because logical symbols we use generally are led by him. His ideas was expressed by uses of ε, C , = , ∩, ∪, ～, Λ. as the primitive ideas. (Still, he introduced the existence-quantifier Ea in 1897.)
His symbolism-formal language does not symbolize a contraction, an abbreviation, and a substance in short hand. That is, it does not mean "words", but "expressions of ideas".
B. Russell has developed the ideas and the symbols of Peano. The other side, D. Hilbert adopted his own symbols which he formulated and systematized logical ideas from a mathematical standpoint. For instance, he deviced p^^￣ in place of the negation -p, p & q in place of the conjunction p・q, p→q in place of the implication p⊃q, p～q (→←,⇔) in place of the equivalence (logical connective) p≡q, and (Ex) in splace of (Ex) etc. His symbol (Ex) represent "Es gibt ein x von . . . . . . . ".
Further, A. Heyting (and G. Gentzen) adopted ¬p as the negation, G. Gentzen adopted (Ax) as the all-quantifier, and Skolem, Hermes, and Freudenthal use ∧x, ∨x, the recent symbols, as the quantifiers.