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1

The lattice of Belnapian modal logics: Special extensions and counterparts

100%
Odintsov S. P., Speranski S. O.
Logic and Logical Philosophy
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2016
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vol. 25
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issue 1
3-33
EN
Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on: • introducing the classes (or rather sublattices) of so-called explosive, complete and classical Belnapian modal logics; • assigning to every normal modal logic three special conservative extensions in these classes; • associating with every Belnapian modal logic its explosive, complete and classical counterparts. We investigate the relationships between special extensions and counterparts, provide certain handy characterisations and suggest a useful decomposition of the lattice of logics containing BK.
2

Paranormal modal logic – Part II: K?, K and Classical Logic and other paranormal modal systems

100%
Silvestre R. S.
Logic and Logical Philosophy
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2013
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vol. 22
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issue 1
89-130
EN
In this two-part paper we present paranormal modal logic: a modal logic which is both paraconsistent and paracomplete. Besides using a general framework in which a wide range of logics - including normal modal logics, paranormal modal logics and classical logic - can be defined and proving some key theorems about paranormal modal logic (including that it is inferentially equivalent to classical normal modal logic), we also provide a philosophical justification for the view that paranormal modal logic is a formalization of the notions of skeptical and credulous plausibility.
3

Some new results on PCL1 and its related systems

100%
Waragai T., Omori H.
Logic and Logical Philosophy
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2010
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vol. 19
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issue 1-2
129–158
EN
In [Waragai & Shidori, 2007], a system of paraconsistent logic called PCL1, which takes a similar approach to that of da Costa, is proposed. The present paper gives further results on this system and its related systems. Those results include the concrete condition to enrich the system PCL1 with the classical negation, a comparison of the concrete notion of “behaving classically” given by da Costa and by Waragai and Shidori, and a characterisation of the notion of “behaving classically” given by Waragai and Shidori.
4
Content available remote

A Brief Note on Béziau’s “Rather Trivial Theorem” About LP

100%
Estrada-González L.
Logic and Logical Philosophy
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2019
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vol. 28
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issue 2
355-361
EN
Béziau has recently argued that the logic LP commits dialetheists to trivialism and Martin has pointed out very clearly the main problems with that alleged result. My sole purpose here is to make the spirit of Martin’s reply more concise, exhibiting as clearly as possible the logical defects in Béziau’s reasoning. Additionally, I want to make some remarks on LP qua logic and not only as an interpreted language.
5

Paranormal modal logic – Part I. The system K? and the foundations of the logic of skeptical and credulous plausibility

100%
Silvestre R. S.
Logic and Logical Philosophy
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2012
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vol. 21
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issue 1
65-95
EN
In this two-parts paper we present paranormal modal logic: a modal logic which is both paraconsistent and paracomplete. Besides using a general framework in which a wide range of logics - including normal modal logics, paranormal modal logics and classical logic - can be defined and proving some key theorems about paranormal modal logic (including that it is inferentially equivalent to classical normal modal logic), we also provide a philosophical justification for the view that paranormal modal logic is a formalization of the notions of skeptical and credulous plausibility.
6

Algebraization of Jaśkowski’s Paraconsistent Logic D2

88%
Ciuciura J.
Studies in Logic, Grammar and Rhetoric
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2015
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vol. 42
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issue 1
173-193
EN
The aim of this paper is to present an algebraic approach to Jaśkowski’s paraconsistent logic D2. We present: a D2-discursive algebra, Lindenbaum- Tarski algebra for D2 and D2-matrices. The analysis is mainly based on the results obtained by Jerzy Kotas in the 70s.
7

Nicephorus Blemmydes on the Holy Trinity and the Paraconsistent Notion of Numbers: A Logical Analysis of a Byzantine Approach to the Filioque

75%
Lourié B.
Studia Humana
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2016
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vol. 5
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issue 1
40-54
EN
The paper deals with the most controversial - in the modern scholarly discussion - episode within the Byzantine polemics on the Filioque, Nicephorus Blemmydes‘ acknowledgement of proceeding of the Spirit through the Son providing that the Son be considered as generated through the Spirit. The logical intuition behind this theological idea is explicated in the terms of paraconsistent logic and especially of a kind of paraconsistent numbers called by the author “pseudo-natural numbers”. Such numbers could not be interpreted via the notion of ordered pair. Instead, they imply a known (first described by Emil Post in 1941) but still little studied logical connective ternary exclusive OR.
8

The Logic of God

63%
Heller M.
Edukacja Filozoficzna
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2019
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issue 68
227-244
EN
When speaking on the “logic of God”, we can understand the logic of our reasoning about God or the logic as it is supposedly employed by God. It is rather obvious, at least for believers in God, that we can infer something about God’s logic in the latter meaning, from how the logic operates in the world created by Him. In the present essay, my strategy is to use this narrow window through which we can grasp some glimpses of “God’s ways of thinking”. There are strong reasons to believe that it is category theory that best displays the role of logic in the system of our mathematical and physical knowledge. It gives us a refreshingly new perspective on logic and its various applications, and could be a good starting point for our speculations concerning the “logic of God”. A quick look at category theory and its applications to physics shows that logic can change from theory to theory, or from level to meta-level. This poses the question of the existence of “superlogic” to which all other logics would somehow be subordinated. The fact that this question remains unanswered forces us to face the problem of plurality of logics. Usually, it is tacitly assumed that the role of “superlogic” is played by classical logic with its non-contradiction law as the most obvious tautology. We briefly discuss paraconsistent logic as an example of a logical system in which contradictions are allowed, albeit under the condition that they do not make the system to explode, i.e. that they do not spill over the whole system. Such logic is an internal logic in categories called cotopoi (or complement topoi). I refer to some theological discussions, both present and from the past, that associate “God’s logic” with classical logic, in particular with the non-contradiction principle. However, we argue that this principle should not be absolutized. The only thing we can, with some certainty, assert on “God’s logic” is that it is not an exploding logic, i.e. that it is not an “anything goes logic”. God is a Source-of-All-Rationality but His rationality need not to conform to our standards of what is rational. This “principle of logical apophaticism” is formulated and briefly discussed. In the history of theology at least one attempt is known to reconstruct the “process of God’s thinking”, namely Leibniz’s idea of God’s selecting the best world to be created from among all possible worlds. Some modifications are suggested which we believe Leibniz would have introduced in his reconstruction, if he knew present developments in categorical logic.
PL
Mówiąc o „logice Boga”, możemy mieć na myśli logikę naszego myślenia o Bogu albo logikę, jaką w naszym wyobrażeniu posługuje się Bóg. Jest dość oczywiste, przynajmniej dla wierzących, że to i owo na temat logiki Boga w tym drugim znaczeniu możemy wywnioskować z logiki obowiązującej w stworzonym przez Niego świecie. W tym eseju stosuję właśnie tę strategię, by zidentyfikować niektóre przebłyski „Boskiego sposobu myślenia”. Istnieją dobre powody, by przyjąć, że rolę logiki w systemie naszej matematycznej i fizycznej wiedzy najlepiej obrazuje teoria kategorii. Daje nam ona nowe, świeże spojrzenie na logikę i jej różne zastosowania, i może być dobrym punktem wyjścia dla badań nad „logiką Boga”. Pobieżne nawet przyjrzenie się zastosowaniom teorii kategorii w fizyce pozwala się przekonać, że logika może się zmieniać przy przejściu od teorii do teorii i z jednego poziomu ogólności na drugi. Nasuwa się pytanie o istnienie „superlogiki”, której w jakimś sensie podlegałyby wszystkie logiki niższego rzędu. Fakt, że nie potrafimy na to pytanie odpowiedzieć, stawia nas w obliczu problemu logicznego pluralizmu. Zwykle przyjmuje się milcząco, że rolę „superlogiki” odgrywa logika klasyczna z zasadą niesprzeczności jako najbardziej oczywistą tautologią. Artykuł omawia krótko logikę parakonsystentną jako przykład logiki, w której sprzeczności są dozwolone, choć jedynie pod warunkiem, że nie eksplodują, tzn., nie rozlewają się na całość systemu. Taka logika jest wewnętrzną logiką w kategoriach zwanych ko-toposami (complement topoi). Przywołuję niektóre teologiczne dyskusje, zarówno dawne, jak i współczesne, w których „logikę Boga” utożsamiano z logiką klasyczną, a zwłaszcza z zasadą niesprzeczności, starając się pokazać, że zasady tej jednak nie powinno się absolutyzować. Jedyne, co z jakąś pewnością możemy powiedzieć o „logice Boga”, to że nie jest to logika bez żadnych reguł, w której wszystko jest dozwolone. Bóg jest Źródłem-Wszelkiej-Racjonalności, ale Jego racjonalność nie musi spełniać naszych standardów. Formułuję zatem i krótko omawiam tę „zasadę logicznej apofatyczności”. Historia teologii zna przynajmniej jedną próbę zrekonstruowania „procesu Boskiego namysłu” – wizję Leibniza, u którego Bóg ze wszystkich światów, które mógłby stworzyć, wybiera najlepszy. Sugeruję na koniec kilka zmian, które moim zdaniem Leibniz wprowadziłby do swojej rekonstrukcji, gdyby znał współczesną logikę kategorialną.
9
Content available remote

Partial and paraconsistent three-valued logics

63%
Degauquier V.
Logic and Logical Philosophy
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2016
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vol. 25
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issue 2
143–171
EN
On the sidelines of classical logic, many partial and paraconsistent three-valued logics have been developed. Most of them differ in the notion of logical consequence or in the definition of logical connectives. This article aims, firstly, to provide both a model-theoretic and a proof-theoretic unified framework for these logics and, secondly, to apply these general frameworks to several well-known three-valued logics. The proof-theoretic approach to which we give preference is sequent calculus. In this perspective, several results concerning the properties of functional completeness, cut redundancy, and proof-search procedure are shown. We also provide a general proof for the soundness and the completeness of the three sequent calculi discussed.
10

The critics of paraconsistency and of many-valuedness and the geometry of oppositions

51%
Moretti A.
Logic and Logical Philosophy
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2010
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vol. 19
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issue 1-2
63–94
EN
In 1995 Slater argued both against Priest’s paraconsistent system LP (1979) and against paraconsistency in general, invoking the fundamental opposition relations ruling the classical logical square. Around 2002 Béziau constructed a double defence of paraconsistency (logical and philosophical), relying, in its philosophical part, on Sesmat’s (1951) and Blanche’s (1953) “logical hexagon”, a geometrical, conservative extension of the logical square, and proposing a new (tridimensional) “solid of opposition”, meant to shed new light on the point raised by Slater. By using n-opposition theory (NOT) we analyse Beziau’s anti-Slater move and show both its right intuitions and its technical limits. Moreover, we suggest that Slater’s criticism is much akin to a well-known one by Suszko (1975) against the conceivability of many-valued logics. This last criticism has been addressed by Malinowski (1990) and Shramko and Wansing (2005), who developed a family of tenable logical counter-examples to it: trans-Suszkian systems are radically many-valued. This family of new logics has some strange logical features, essentially: each system has more than one consequence operator. We show that a new, deeper part of the aforementioned geometry of logical oppositions (NOT), the “logical poly-simplexes of dimension m”, generates new logical-geometrical structures, essentially many-valued, which could be a very natural (and intuitive) geometrical counterpart to the “strange”, new, non-Suszkian logics of Malinowski, Shramko and Wansing. By a similar move, the geometry of opposition therefore sheds light both on the foundations of paraconsistent logics and on those of many-valued logics.
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