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On a theory of weak implications

  • Mitsuhiro Okada (a1) (a2) (a3)

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The purpose of this paper is to study logical implications which are much weaker than the implication of intuitionistic logic.

In §1 we define the system SI (system of Simple Implication) which is obtained from intuitionistic logic by restricting the inference rules of intuitionistic implication. The implication of the system SI is called the “simple implication” and denoted by ⊃, where the simple implication ⊃ has the following properties:

(1) The simple implication ⊃ is much weaker than the usual intuitionistic implication.

(2) The simple implication ⊃ can be interpreted by the notion of provability, i.e., we have a very simple semantics for SI so that a sentence A ⊃ B is interpreted as “there exists a proof of B from A”.

(3) The full-strength intuitionistic implication ⇒ is definable in a weak second order extension of SI; in other words, it is definable by help of a variant of the weak comprehension schema and the simple implication ⊃. Therefore, though SI is much weaker than the intuitionistic logic, the second order extension of SI is equivalent to the second order extension of the intuitionistic logic.

(4) The simple implication is definable in a weak modal logic MI by the use of the modal operator and the intuitionistic implication ⇒ with full strength. More precisely, AB is defined as the strict implication of the form ◽(AB).

In §1, we show (3) and (4). (2) is shown in §2 in a more general setting.

Semantics by introduction rules of logical connectives has been studied from various points of view by many authors (e.g. Gentzen [4], Lorentzen [5], Dummett [1], [2], Prawitz [8]. Martin-Löf [7], Maehara [6]). Among them Gentzen (in §§10 and 11 of [4]) introduced such a semantics in order to justify logical inferences and the mathematical induction rule. He observed that all of the inference rules of intuitionistic arithmetic, except for those on implication and negation, are justified by means of his semantics, but justification of the inference rules on implication and negation contains a circular argument for the interpretation by introduction rules, where the natural interpretation of AB by ⊃-introduction rule is “there exists a proof of B from A ” (cf. §11 of Gentzen [4]).

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References

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[1]Dummett, M., Elements of intuitionism, Oxford University Press, Oxford, 1977.
[2]Dummett, M., The philosophical basis of intuitionistic logic, Logic Colloquum ’73, North-Holland, Amsterdam, 1975, pp. 540.
[3]Gentzen, G., Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39 (1935), pp. 176–210, 405431; English translation in his Collected papers, North-Holland, Amsterdam, 1969.
[4]Gentzen, G., Die Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen, vol. 112 (1936), pp. 493565; English translation in his Collected papers, North-Holland, Amsterdam, 1969.
[5]Lorentzen, P., Einführung in die operative Logik and Mathematik, Springer-Verlag, Berlin, 1969.
[6]Maehara, S., Semi-formal finitist proof of the transfinite induction in an initial segment of Cantor's second number class, Logic symposia, Hakone 1979, 1980, Lecture Notes in Mathematics, vol. 891, Springer-Verlag, Berlin, 1981, pp. 6779.
[7]Martin-Löf, P., On the meaning of the logical constants and the justifications of the logical laws, Lecture notes, Siena, 04 1983.
[8]Okada, M., A formalization of Gentzen's standpoint on consistency proofs of arithmetic (to appear).
[9]Prawitz, D., Natural deduction, Almqvist & Wiksell, Stockholm, 1965.
[10]Prawitz, D., Ideas and results in proof theory, Proceedings of the second Scandinavian logic symposium (Oslo, 1970), North-Holland, Amsterdam, 1971, pp. 235307.
[11]Prawitz, D., Proofs and the meaning and completeness of the logical constants, Essays on mathematical and philosophical logic (Hintikka, J.et al., editors), Reidel, Dordrecht, 1978, pp. 2540.
[12]Schütte, K., Proof theory, Springer-Verlag, Berlin, 1977.
[13]Schwichtenberg, H., Proof theory: some applicatioons of cut-elimination, Handbook of mathematical logic, North-Holland, Amsterdam, 1977, pp. 867895.
[14]Takeuti, G., Proof theory, North-Holland, Amsterdam, 1975.

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