数学教育における記号表現の問題(III) : 「論理記号」考(その2)

We know that the logic we use now in our schools is based on the idea of D. Hilbert and W. Ackermann.
Their theory say : "In dem Aussagenkalkul wird auf die feinere logische Struktur der Aussagen, die etwa in der Beziehung zwischen Pradikat und Subjekt zum Ausdruck kommt, nicht eingegangen, sondern die Aussagen werden als Ganzes in ihrer logischen Verknupfung mit anderen Aussagen betrachtet. "(Hilbert.D. & W. Ackermann : Grundzlige der theoretischen Logik, Springer, 1928. s. 3) and"... Richtigkeit order Falschheit einer Aussagenverknupfung nur von der Richtigkeit und Falschheit der verkntipften Aussagen, nicht aber von ihrem Inhalt abhangig ist. "(ibid.s.4)
Then their theory can be signified by the logic that is syntactically constructional or theoretical one.
But syntactical questions are not come out of semantic affairs. Then as the result of introducing the truth-function to the components of connectives, they try to give the connective a interpretation on semantrcs. Now, its production in this manner should be found out senses and significances as a simple logic (or prototype).
For beginners, this logic, nevertheless, makes often to find difficulty in realizing and corresponding, that is, it has a distance of projection into daily logic. Hence we must know that students will make a bid for support in the case to learn such a logic.
Some steps to understanding logic we shall read out of the thoughts of G. Peano (and/or out of a way of some logisticians too). For an intention like this, one will say that there are heterogeneous studies between logrstic and formalistic one. Certainly there are. They are, however, no qinte incongruous in a situation of instruction, rather to be laid claim to students.
We shall believe that this intention has a inevitability in the situation of logical instruction.
To put it in the concrete, there are the following :
(1) to describe singular terms to the form (ιx) Fx.
i. e. the object open sentence Fx (of which the predicate represented by F) - to quantify Fx.
(2) to make possible a translation through negation between universal quantification and existential quantifrcation.
(3) to come to the front and to do abstracting the logical physique, through stylizing and /or symbolising of statements. - this task makes us to become aware of the intensive aspects of mathematical thinking.
(4) to manage the schematized or symbolized statements. - this work makes us to open our eyes to the extensive aspects of mathematical thinking.

In addition to the above, we must point out how to operate the logical diagram of J.Venn. His diagrams were occured in examination (in 1880) of Eulerean representation (Lettres a une Princesse d'Allemagne, Lettres 102-105, 1761).
Venn diagrams actually fulfill its function in regard to categorical propositions ("A", "E", "I", "O") and their syllogism, not to truth-function.

記号ιは，ギリシャ語ισοζの頭文字イオータを天地逆にしたものである。