Paraconsistency and its Philosophical Interpretations

Many authors have considered that the notions of paraconsistency and dialetheism are intrinsically connected, in many cases, to the extent of confusing both phenomena. However, paraconsistency is a formal feature of some logics that consists in invalidating the rule of explosion, whereas dialetheism is a semantical/ontological position consisting in accepting true contradictions. In this paper, we argue against this connection and show that it is perfectly possible to adopt a paraconsistent logic and reject dialetheism, and, moreover, that there are examples of non-paraconsistent logics that can be interpreted in a dialetheic way

Andreas Kapsner, “Logics and Falsifications: A New Perspective on Constructivist Semantics”, Series Trends in Logic, vol. 40, Springer

Andreas Kapsner, Logics and Falsifications: A New Perspective on Constructivist Semantics, Series Trends in Logic, vol. 40, Springer, Cham, 215 pages, ISSN 1572-6126, ISBN 978-3-319-05205-2

The Case of Dialetheism

The concept of dialetheia and the claim of dialetheism has been examined and compared to such related concept as contradiction, antinomy, consistency and paraconsistency. Dialetheia is a true contradiction and dialetheism is the claim that there exists at least one dialetheia. It has been observed that dialetheism is equivalent to the negation of the traditional principle of contradiction. Hence, dialetheism itself is no new idea in whatsoever. The novelty of dialetheism consists in the arguments delivered for its case. Key justification the partisans deliver for dialetheism has been examined and evaluated: antinomies, an alleged Gödel’s paradox, and existence of limits of thought. The structure of those arguments has been analyzed. It has been claimed that they share one and the same simple structure which may be called reverse paradox. The vital content dialetheists add to the traditional paradoxes is only the thesis of reliability of the vernacular prima facie knowledge. Three objections have been raised against the justification of dialetheism: firstly, it has been claimed that exactly the same argument supports principle of contradiction, secondly, it has been questioned whether the arguments preserve their value when logic is subject to revision, and thirdly, it has been claimed that the underlying logic of dialetheism is classical

Loogikavigade lubatavusest

Loogikavigu on kahte liiki: (1) vastuolud; (2) kehtetud järeldused. Sooritada kehtetu järeldus on üldiselt väiksem viga kui väita vastuolu. On kaks erinevat printsiipi: (A) teadaolevalt väära väidet ei tohi esitada; (B) ei tohi esitada väidet, mille tõeväärtus pole teada. (A) keelab sooritada esimest liiki loogikavigu; (B) ka teist liiki loogikavigu. Printsiibi (B) lükkame me tagasi. Oletusi saab genereerida näiteks: (1) dedutseerides neid teadaolevalt vastuolulisest teooriast; (2) saades neid induktiivsete järelduste tulemeina. Ent ka printsiipi (A) tohib eirata. Võib esitada teadaolevalt vastuolulise teooria kui ligilähedaselt tõese ning seda kasutada - kui paremaid pole käepärast. Popper esitas tõesarnasuse teooria. Samuti nõudis ta vastuoluliste teadusteooriate elimineerimist. - Ent paremate teadusteooriate leiutamine võib ebaõnnestuda. Wittgensteini vaated tema Märkmetes matemaatika alustest on usutavamad. Implitsiitselt rakendas Witttgenstein mitteformaliseeritud tõesarnasuse mõistet vastuolulistele teooriatele, lubades selliseid teooriaid kasutada

Truth Values in t-norm based Systems Many-valued FUZZY Logic

In t-norm based systems many-valued logic, valuations of propositions form a non-countable set: interval [0,1]. In addition, we are given a set E of truth values p, subject to certain conditions, the valuation v is v=V(p), V reciprocal application of E on [0,1]. The general propositional algebra of t-norm based many-valued logic is then constructed from seven axioms. It contains classical logic (not many-valued) as a special case. It is first applied to the case where E=[0,1] and V is the identity. The result is a t-norm based many-valued logic in which contradiction can have a nonzero degree of truth but cannot be true; for this reason, this logic is called quasi-paraconsistent

Crònica

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Swyneshed, Paradox and the Rule of Contradictory Pairs

Roger Swyneshed, in his treatise on insolubles (logical paradoxes), dating from the early 1330s, drew three notorious corollaries of his solution. The third states that there is a contradictory pair of propositions both of which are false. This appears to contradict the Rule of Contradictory Pairs, which requires that in every such pair, one must be true and the other false. Looking back at Aristotle's treatise De Interpretatione, we find that Aristotle himself, immediately after defining the notion of a contradictory pair, gave counterexamples to the rule. Thus Swyneshed's solution to the logical paradoxes is not contrary to Aristotle's teaching, as many of Swyneshed's contemporaries claimed. Dialetheism, the contemporary claim that some propositions are both true and false, is wedded to the Rule, and in consequence divorces denial from the assertion of the contradictory negation