Dialectical Contradictions and Classical Formal Logic

Abstract

A dialectical contradiction can be appropriately described within the framework of classical formal logic. It is in harmony with the law of noncontradiction. According to our definition, two theories make up a dialectical contradiction if each of them is consistent and their union is inconsistent. It can happen that each of these two theories has an intended model. A number of examples of this are to be found in the history of science.

Acknowledgements

I am deeply grateful to James W. McAllister for his encouraging suggestions. I am also in debt to two anonymous referees of this journal for their valuable comments.

Notes

[1] In spite of what was mentioned in section 1, Russell was never ignorant of Kant or Hegel. As to Kant he says: ‘I was much impressed by Kant's Metaphysische Anfangsgründe der Naturwissenschaft and made elaborate notes on it’, and as to Hegel: ‘I was at this time [about 1896] a full-fledged Hegelian, and I aimed at constructing a complete dialectic of the sciences’ (Russell 1959, 32).

[2] Any emphasis in a quotation is original, throughout.

[3] I do not imply that the Ethics is logically well organized. Actually, it bears only some superficial similarities to Euclid's Elements.

[4] Grim (2004, 49) names this ‘LNC2’.

[5] Grim (2004, 49) names this ‘LNC1’.

[6] We may as well say ‘point of view’, ‘standpoint’, ‘perspective’, etc.

[7] For the capability of first-order language to express various areas of mathematics, see Ebbinghaus, Flum, and Thomas (1984).

[8] If we apply this definition not only to mathematics but also to science in general, it might be criticized for disregarding the so-called semantic or model-theoretic view of scientific theories. It should be remembered, however, that Suppes, who is regarded as the founder of the semantic view, clearly says, ‘The important distinction that we shall need is that a theory is a linguistic entity consisting of a set of sentences and models are non-linguistic entities in which the theory is satisfied’ (Suppes 1960, 5). See also Halvorson (2012).

[9] I think again this idea is in accordance with Suppes, who says ‘[T]he meaning of the concept of model is the same in mathematics and the empirical sciences’ (Suppes 1960, 4).

[10] Actually, from our perspective, there was no need to mention the rotation of the moon for the purpose of illustrating the frame dependence of motion. Any body will be found to be at rest in some frames of reference, but not in others. Probably, the idea of absolute space prevented Kant from conceiving of such relativity.

[11] Actually, it might happen that χ ∈ S₁, ψ ∈ S₁, or the like, but this would not invalidate our argument.

[12] Often the standard model of a theory is identified with the intended model, but some authors distinguish them. See, for example, Gaifman (2004).

[13] Kneale and Kneale (1962, 355–356) criticize Kant's apparent trichotomy in his Critique of Pure Reason.

[14] It may appear that what Kuhn (1996) called a scientific revolution or a paradigm shift has something to do with what we call a dialectical contradiction. Certainly, most examples he cited there relate to some dialectical contradictions, and it seems likely that two theories are ‘incommensurable’ if they are dialectically contradictory. But here I refrain from discussing this problem too hastily.