![]() ![]() Fanden frtiher die wenigen Forscher, die sich emsthaft mit dieser 'Frage beschaftigten, wenig Be achtung, heute ist die Teilnahme sowohl von mathematischer wie von philosophischer Seite fast allgemein. ![]() In den letzten Jahrzehntel! hat sich das Interesse an der Grund legung der Mathematik immer gesteigert. Mathematics comes into being, when the two-ity created by a move of time is divested of all quality by the subject, and when the remaining empty form of the common substratum of all two-ities, as basic intuition of mathematics, is left to an unlimited unfolding, creating new mathematical entities in the shape of predeterminately, or more or less freely proceeding infinite sequences of mathematical entities acquired, and in the shape of mathematical species. By a move of time, a present sensation gives way to another present sensation in such a way that consciousness retains the former one as a past sensation, and moreover, through this distinction between present and past, recedes from both and from stillness, and becomes mind. This initial phenomenon is a move of time. It seems that only the status of sensation allows the initial phenomenon of the said transition. Consciousness in its deepest home seems to oscillate slowly, will-lessly, and reversibly between stillness and sensation. This chapter describes the importance of consciousness, philosophy, and mathematics. Adequate semantics and a decision method are presented for both BLE and LETj, as well as some technical results that fit the intended interpretation. LETj is anti-dialetheist in the sense that, according to the intuitive interpretation proposed here, its consequence relation is trivial in the presence of any true contradiction. The latter is a logic of formal inconsistency and undeterminedness that is able to express not only preservation of evidence but also preservation of truth. The paper defines a paraconsistent and paracomplete natural deduction system, called the Basic Logic of Evidence (BLE), and extends it to the Logic of Evidence and Truth (LETj). Contradictions are, instead, epistemically understood as conflicting evidence, where evidence for a proposition A is understood as reasons for believing that A is true. The purpose of this paper is to present a paraconsistent formal system and a corresponding intended interpretation according to which true contradictions are not tolerated. The result is the loss of at least one among reflexivity, monotonicity, and the deduction theorem in a Brouwerian intuitionistic logic, which seems to be an undesirable result. Although the same argument against explosion can be also applied against weak explosion, rejecting the latter requires the rejection of ex quodlibet verum. Given ex quodlibet verum, the inference we call weak explosion, according to which any negated proposition follows from a contradiction, is proved in a few steps. ![]() The principle known as ex quodlibet verum, according to which a valid formula follows from anything, should also be rejected by a relevantist. We agree that explosion should not hold in intuitionistic logic, but a relevance logic requires more than the invalidity of explosion. Indeed, van Atten (2009) argues that a formal system in line with Brouwer's ideas should be a relevance logic. However, it is not clear that explosion is in accordance with Brouwer's views on the nature of mathematics and its relationship with logic. The inference called ex falso quodlibet, or the principle of explosion, according to which anything follows from a contradiction, holds in intuitionistic logic. ![]() The formal system proposed by Heyting (1930, 1956) became the standard formulation of intuitionistic logic. ![]()
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