Hegel: Formal Logic

By the beginning of the 19th century Hegel's dialectic turned contradiction (conceived as unity of a concept with its determined negation) into distinguished inference. In the course of 20th century a family of systems known as "paraconsistent" formalized dialectical logic according to the contemporary paradigm of inference, oriented to truth-preserving, and not powered anymore solely by contradiction. In this way, nevertheless, Hegel's idea of logic as unfolding of concepts ordered by degree of "determination" reached at every step of the process, (...) was put aside. We attempt to recover the importance of that idea for the hegelian concept of "logical necessity", pointing out in addition that it may well play the role of an order relation for dialectical inference. (shrink)

The concept of the subject, of what Hegel calls absolute negativity, already appears early in the logic of being.1 Absolute negativity, negation of the negation, occurs throughout the logic as identity in difference understood as self-identification under different descriptions. First, the subject refers to itself merely under an incomplete description. Secondly, it refers to something other than itself under a second description which is logically required by the first. (For example, the description of being in general requires some determinate description (...) of being in particular). But this second description is dialectically excluded by the assumption that the first description is complete. Thirdly, the subject negates its negation of the other. It discovers itself in the other, under the other description, and thus comes to refer to itself less incompletely. This is Hegel in the analytic mode. (shrink)

I take advantage of two recent results: 1) the recognition of an alternative theoretical organization to the deductive-axiomatic one; it is characterized by a sequence of four logical steps belonging to intuitionist logic; 2) the recognition of the logical content of Cusanus’ philosophical works; also this content pertains to intuitionist logic, which Cusanus anticipated by even identifying some its logical laws. Many Cusanus’ books present the alternative theoretical organization; whose yet he did not apply in a clear way its last (...) two steps, the ad absurdum arguing and the application of the principle of sufficient reason. For this reason in the last books his thinking transcended to a metaphysical level. Hegel’s philosophy of logic shares Cusanus’ philosophical effort for exiting out classical logic. He seems to have assumed Cusanus’ anticipation of the new theoretical organization; all goes as if Hegel wanted to improve Cusanus’ poor method, by introducing a process of additions of two negations which however does not agree with the intuitionist logic. Moreover, Hegel presupposes, like Cusanus in his last years, the free application of the principle of sufficient reason to the entire subject of his philosophy of logic without awareness of the logical constraints to which is subject a correct application of this principle; hence, he introduced a metaphysical arguing which according to modern mathematical logic has no consistent content. Also Hegel’s supposed prolongation of Cusanus’ logical discoveries was an essentially metaphysical view. (shrink)