3 0 0 0 OA 矛盾は矛盾か

岡田 光弘
科学哲学 (ISSN:02893428)
vol.36, no.2, pp.79-102, 2003-12-30 (Released:2009-05-29)

In this paper we show some logical presumptions for the contradiction-form to really mean contradiction. We first give an introductory note that the same argument-form of Russell paradox could be interpreted to derive a contradiction (as Russell did) and to derive some positive non-contradictory results (such as Gödel's lemma on incompleteness and Cantor's lemma on cardinality), depending on the context. This surprisingly suggests that a logical argument of a contradiction itself is rather independent of interpreting it as contradiction or non-contradiction. In the main section (Section 2) we investigate further in the hidden logical assumptions underlying a usual derivation of contradiction (such as the last step from the Russell argument to conclude a contradiction). We show the logical form of contradiction does not always mean a contradiction in a deep structure level of logic. We use the linear logical analysis for this claim. Linear logic, in the author's opinion, provides fundamental logical structures of the traditional logics (such as classical logic and intuitionistic logic). The each traditional logical connectives split into two different kinds of connective, corresponding to the fundamental distinction, parallel or choice, of the fundamental level of logic; more precisely the connectives related to parallel-assertings and the connectives related to choice-assertions. We claim that (1) the law of contradiction is indisputable for the parallel-connectives, but (2) the law of contradiction is not justified for the choice-connectives. (In fact, the law of contradiction has the same meaning as other Aristotlean laws (the indentity, the excluded middle) from the view point of the duality principle in linear logic, and the disputability of the law of contradiction is exactly the same as the disputability of the law of excluded middle, in the linear logic level.) Here, although (1) admits the law of contradiction, the meaning of contradiction is quite diferrent and, in the author opinion, more basic than the traditional sense of contradiction. (2) tells us that the disputability of the law of contradiction for the choice-connectives is equivalent to the disputability of the law of excluded middle. However, this disputability is more basic than the traditional logicist-intuitionist issue on the excluded middle, since admitting the traditional law of excluded middle (from the classical or logicist viewpoint) is compatible with this disputability of the excluded middle (and equivalently the law of contradiction) with respect to the choice-connectives of the linear logic. Then, the traditional logics (classical and intuitionistic logics) are perfectly constituted from this fundamental level of logic by the use of reconstructibity or re-presentation operator, (which is the linear-logical modal operators). With the use of modal operator the originally splitted two groups of logical connectives merge into a single group, which makes the traditional logical connectives. (The use of slightly different modalities results in the difference between the traditional classical logic and the traditional intuitionistic logic.) With the use of modal operator, the contradiction-form becomes to get the traditional sense of contradiction. This situation shows that the traditional sense of contradiction presumes re-presentation or reconstruction of the inference-resources, which is now explicit by the use of linear logical modal operator(s), and which also makes possible the denotational or objectivity interpretation of logical language. The merge of the two different aspects (the parallel-connectives and the choice-connectives) into one, by the presence of the modal operators, also eliminates the original conflict (on the indisputability of the law of contradiction in the parallel-connectives side and the disputability of that in the choice-connectives side.)


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