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1

Reflections on temporal and modal logic

100%
Epstein R. L.
Logic and Logical Philosophy
|
2015
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vol. 24
|
issue 1
111–139
EN
The most popular method of incorporating time into a formal logic is based on the work of Arthur Prior. It treats tenses as operators on sentences. In this essay I show a serious problem with that approach, a confusion of scheme versus proposition, which makes any system built in that way incoherent. I will compare how other formal logics deal with the scheme versus proposition distinction and find that only for formal modal logics does the same problem arise. I then compare Prior’s approach to other ways of taking time into account in formal logics.
2
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Najkrótsze aksjomaty modalnej logiki Łukasiewicza

80%
Dywan Z.
Roczniki Filozoficzne
|
2016
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vol. 64
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issue 2
5-9
PL
Pokazujemy, że formuła CLpEqLq aksjomatyzuje logikę modalną Łukasiewicza i nie istnieje krótszy aksjomat dla tej logiki.
EN
We show that the formula CLpEqLq axiomatizes Łukasiewicz’s modal logic and that there exists no shorter axiom for it.
3

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

80%
Silvestre R. S.
Logic and Logical Philosophy
|
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.
4

Hypersequent Calculi for S5: The Methods of Cut Elimination

80%
Bednarska K., Indrzejczak A.
Logic and Logical Philosophy
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2015
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vol. 24
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issue 3
277–311
EN
S5 is one of the most important modal logic with nice syntactic, semantic and algebraic properties. In spite of that, a successful (i.e. cut-free) formalization of S5 on the ground of standard sequent calculus (SC) was problematic and led to the invention of numerous nonstandard, generalized forms of SC. One of the most interesting framework which was very often used for this aim is that of hypersequent calculi (HC). The paper is a survey of HC for S5 proposed by Pottinger, Avron, Restall, Poggiolesi, Lahav and Kurokawa. We are particularly interested in examining different methods which were used for proving the eliminability/admissibility of cut in these systems and present our own variant of a system which admits relatively simple proof of cut elimination.
5

On Axiomatization of Łukasiewicz’s Four-Valued Modal Logic

80%
Tkaczyk M.
Logic and Logical Philosophy
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2011
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vol. 20
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issue 3
215-232
EN
Formal aspects of various ways of description of Jan Łukasiewicz’s four-valued modal logic are discussed. The original Łukasiewicz’s description by means of the accepted and rejected theorems, together with the four-valued matrix, is presented. Then the improved E. J. Lemmon’s description based upon three specific axioms, together with the relational semantics, is presented as well. It is proved that Lemmon’s axiomatics is not independent: one axiom is derivable on the base of the remanent two. Several axiomatizations, based on three, two or one single axiom are provided and discussed, including S. Kripke’s axiomatics. It is claimed that (a) all substitutions of classical theorems, (b) the rule of modus ponens, (c) the definition of “⋄” and (d) the single specific axiom schema: ⌐□A ∧ B→A ∧□ B⌐, , called the jumping necessity axiom, constitute an elegant axiomatics of the system.
6

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

80%
Silvestre R. S.
Logic and Logical Philosophy
|
2012
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vol. 21
|
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.
7

On the Polish “Via Modalization” Approach to Paraconsistency

70%
Nicolás-Francisco R. A.
Edukacja Filozoficzna
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2022
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issue 74
161-182
EN
In this paper, I survey the history of the Polish tradition of paraconsistency and its chronological development. I outline the features of this tradition to provide some insights into the more general notion of philosophical schools. The main features of the Polish tradition of paraconsistency are the continuation of research on a previous philosophical tradition and international collaboration.
8

Simulation Logic

70%
Allwein G., Harrison W. L., Andrews D.
Logic and Logical Philosophy
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2014
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vol. 23
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issue 3
277–299
EN
Simulation relations have been discovered in many areas: Computer Science, philosophical and modal logic, and set theory. However, the simulation condition is strictly a first-order logic statement. We extend modal logic with modalities and axioms, the latter’s modeling conditions are the simulation conditions. The modalities are normal, i.e., commute with either conjunctions or disjunctions and preserve either Truth or Falsity (respectively). The simulations are considered arrows in a category where the objects are descriptive, general frames. One can augment the simulation modalities by axioms for requiring the underlying modeling simulations to be bisimulations or to be p-morphisms. The modal systems presented are multi-sorted and both sound and complete with respect to their algebraic and Kripke semantics.
9

Neighborhood Semantics for Basic and Intuitionistic Logic

61%
Moniri M., Maleki F. S.
Logic and Logical Philosophy
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2015
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vol. 24
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issue 3
339–355
EN
In this paper we present a neighborhood semantics for Intuitionistic Propositional Logic (IPL). We show that for each Kripke model of the logic there is a pointwise equivalent neighborhood model and vice versa. In this way, we establish soundness and completeness of IPL with respect to the neighborhood semantics. The relation between neighborhood and topological semantics are also investigated. Moreover, the notions of bisimulation and n-bisimulation between neighborhood models of IPL are defined naturally and some of their basic properties are proved. We also consider Basic Propositional Logic (BPL), a logic weaker than IPL introduced by Albert Visser, and introduce and study its neighborhood models in the same manner.
10

I.M. Bocheński and Theophrastus’ Modal Logic

61%
Gili L.
Edukacja Filozoficzna
|
2020
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issue 70
23-38
EN
Innocenty Maria Bocheński expounded his interpretation of Theophrastus’ logic chiefly in his book La logique de Théophraste (1947). In Bocheński’s reconstruction, Theophrastus worked on the last insights of Aristotle’s syllogistic and systematized it, thereby opening the door to later (Stoic) developments in the history of logic. A closer look at Bocheński’s interpretation of Theophrastus’ logic can lead us to reassess the originality of the contribution of the philosopher of Eresus. As more recent studies have convincingly shown, Aristotle’s modal system is grounded on the theory of predication expounded in the Topics. The validity of Barbara LX-L rests on the essential predications that the major premise and the conclusion are descriptive. According to Bocheński, Theophrastus had an extensional understanding of logic, as is clear from his proof for the rules of conversion of categorical universal propositions. Bocheński also stresses that Theophrastus consistently avoids Aristotle’s two-sided possibility and this might also be read as an attempt to develop a self-contained logical system that is not merely seen as the deductive system of a theoretical discipline. Bocheński’s overall assessment of Theophrastus’ logic might be in need of revision, inasmuch as our understanding of Aristotle’s logical enterprise has radically changed in the last decades, but the minutiae of Bocheński’s reading of Theophrastus are compelling and can stimulate new studies on the successor of the Stagirite.
11
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All Properties are Divine or God Exists - The Sacred Thesis and its Ontological Argument

61%
Bjørdal F. A.
Logic and Logical Philosophy
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2018
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vol. 27
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issue 3
329-350
EN
A metaphysical system engendered by a third order quantified modal logic S5 plus impredicative comprehension principles is used to isolate a third order predicate D, and by being able to impredicatively take a second order predicate G to hold of an individual just if the individual necessarily has all second order properties which are D we in Section 2 derive the thesis (40) that all properties are D or some individual is G. In Section 3 theorems 1 to 3 suggest a sufficient kinship to Gödelian ontological arguments so as to think of thesis (40) in terms of divine property and Godly being; divine replaces positive with Gödel and others. Thesis (40), the sacred thesis, supports the ontological argument that God exists because some property is not divine. In Section 4 a fixed point analysis is used as diagnosis so that atheists may settle for the minimal fixed point. Theorem 3 shows it consistent to postulate theistic fixed points, and a monotheistic result follows if one assumes theism and that it is divine to be identical with a deity. Theorem 4 (the Monotheorem) states that if Gg and it is divine to be identical with g, then necessarily all objects which are G are identical with g. The impredicative origin of D suggests weakened Gaunilo-like objections that offer related theses for other second order properties and their associated diverse presumptive individual bearers. Nevertheless, in the last section we finesse these Gaunilo-like objections by adopting what we call an apathiatheistic opinion which suggest that the best concepts `God’ allow thorough indifference as to whether God exists or not.
12
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Krata podwójna: próba opisu wydarzeń przyszłych za pomocą narzędzi logicznych

57%
Bażyk A. P.
Semina Scientiarum
|
2012
|
vol. 11
EN
An unflagging interest in describing future events has continuously motivated investigations, particularly in the field of logic.Aristotle, universally acknowledged as the father of logic, proposed a set of certain bases from which we could depart with our investigations. However, these are tools in which, despite their great value, one can perceive certain shortcomings.Over the centuries many attempts have been made to discover a means of describing any sentence, expressed in any grammatical form. One of these attempts is bilattice theory, through which it has been attempted to describe future events. This theory makes use of tools such as vagueness and different forms of semantics including subvaluationism and supervaluationism.In lattice theory itself, the double lattice known as the FOUR lattice (four-valued bilattice logic) is directly employed.After analyzing the structure of a given theory one may easily make use of it in practice, providing examples of its usage.
13
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Temporal Alethic Dyadic Deontic Logic and the Contrary-to-Duty Obligation Paradox

51%
Rönnedal D.
Logic and Logical Philosophy
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2018
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vol. 27
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issue 1
3–52
EN
A contrary-to-duty obligation (sometimes called a reparational duty) is a conditional obligation where the condition is forbidden, e.g. “if you have hurt your friend, you should apologise”, “if he is guilty, he should confess”, and “if she will not keep her promise to you, she ought to call you”. It has proven very difficult to find plausible formalisations of such obligations in most deontic systems. In this paper, we will introduce and explore a set of temporal alethic dyadic deontic systems, i.e., systems that include temporal, alethic and dyadic deontic operators. We will then show how it is possible to use our formal apparatus to symbolise contrary-to-duty obligations and to solve the so-called contrary-to-duty (obligation) paradox, a problem well known in deontic logic. We will argue that this response to the puzzle has many attractive features. Semantic tableaux are used to characterise our systems proof theoretically and a kind of possible world semantics, inspired by the so-called T× W semantics, to characterise them semantically. Our models contain several different accessibility relations and a preference relation between possible worlds, which are used in the definitions of the truth conditions for the various operators. Soundness results are obtained for every tableau system and completeness results for a subclass of them.
14
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Teodycea a logika modalna. Bogusława Wolniewicza polemika z krytyką teodycei u Pierre’a Bayle’a

51%
Mirosławski M.
Racjonalia. Z punktu widzenia humanistyki
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2017
|
vol. 7
EN
The essay aims at presenting original ideas of Bogusław Wolniewicz in the field of modal logic, which improve the traditional theodicy, criticized by Pierre Bayle. The classical theodicy attempts to reconcile divine omnipotence and God’s goodness. According to classical theodicy both of these elements can be reconciled, because evil is not the work of God, but a work of man – freedom is in fact necessary for a man so that he could do good, but to do good, a person automatically has to be able to do the evil (understood morally, not physically). Bayle’s argument says that freedom implies the possibility of evil and it does not imply its existence, and this means that evil can be avoided even with the assumption of freedom. Wolniewicz tries to refute that argument. He quotes a little known definitions of possibility derived from the logic of Megarian. It turns out to be an inspiration for him to introduce an original definition of possibility – “what is possible = what happened or what will happen in the future.” With this interpretation of possibility classic theodicy seems to be efficient.
PL
Referat ma na celu zaprezentowanie oryginalnych pomysłów Bogusława Wolniewicza z dziedziny logiki modalnej, które służą usprawnieniu tradycyjnej teodycei, krytykowanej przez Pierre'a Bayle'a. Klasyczna teodycea stara się pogodzić boską wszechmoc z bożą dobrocią. Według klasycznej teodycei oba te elementy dają się pogodzić, ponieważ zło nie jest dziełem Boga, lecz człowieka – wolność jest bowiem konieczna dla człowieka, aby ten mógł czynić dobro, ale aby czynić dobro, człowiek automatycznie musi móc czynić także zło (rozumiane moralnie, nie fizycznie). Argumentacja Bayle'a mówi natomiast, że choć wolność implikuje możliwość zła, to nie implikuje jednak jego istnienia, a to oznacza, że zła można uniknąć nawet przy założeniu wolności. Wolniewicz stara się obalić tę argumentację. W tym celu przywołuje on mało znane definicje możliwości wywodzące się z logiki megarejskiej, które dostarczają inspiracji do wprowadzenia oryginalnej definicji możliwości – „to co możliwe = to, co się zdarzyło lub to co zdarzy się w przyszłości". Przy takiej interpretacji modalnej teodycea klasyczna wydaje się być sprawna
15
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Pragmatyczne znaczenia predykatu modalnego móc w konstrukcjach konwersacyjnych dzieci przedszkolnych

41%
Brzdęk E.
Socjolingwistyka
|
2016
|
issue 30
245-259
PL
W pracy koncentrujemy się głownie na określeniu zakresu użycia pragmatycznych wariantów znaczenia predykatu modalnego moc w konstrukcjach konwersacyjnych dzieci przedszkolnych. Opis konkretnych realizacji orzeczeń modalnych w wypowiedzeniach dotyczy niektórych aspektów modalności zdaniowej i fakultatywnej oraz wybranych zagadnień systemowych, sytuacyjnych i interakcyjnych języka. Biorąc pod uwagę kontekst pragmatycznojęzykowy i interakcyjny konstrukcji modalnych i innych, obecnych w tekstach, zwracamy uwagę na jakość kompetencji dyskursywnej dzieci przedszkolnych.
EN
The paper concerns the analysis of the modal logic and intentional constructions with the modal verb móc in preschool discourse. Some aspects of pragmalinguistic and systemic-functional theory provide tools to characterize the ways in which children use constructions in the colloquial-conversational language and text. Our analysis is focused on the sense of predicate móc in particular applications and on the process of creation of indirect acts of speech. It also provides information on the development of discourse competence of children.
16
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Knowability as De Re Modality: A Certain Solution to Fitch Paradox

31%
Jarmużek T., Krawczyk K., Palczewski R.
Roczniki Filozoficzne
|
2020
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vol. 68
|
issue 4
291-313
EN
In the paper, we try to find a new, intuitive solution to the Fitch paradox. We claim that traditional expression of Knowability Principle (p → ◊Kp) is based on erroneous understanding of knowability as de dicto modality. Instead, we propose to understand knowability as de re modality. In the paper we present the minimal logic of knowability in which Knowability Principle is valid, but Fitch Paradox does not hold anymore. We characterize the logic semantically as well as by an axiomatic and tableaux procedure approach.
PL
Poznawalność jako modalność de re: pewne rozwiązanie paradoksu Fitcha W artykule staramy się znaleźć nowe, intuicyjne rozwiązanie paradoksu Fitcha. Twierdzimy, że tradycyjne wyrażenie zasady poznawalności (p → ◊Kp) opiera się na błędnym rozumieniu poznawalności jako modalności de dicto. Zamiast tego proponujemy rozumieć poznawalność jako modalność de re. W artykule przedstawiamy minimalną logikę poznawalności, w której zasada poznawalności jest ważna, ale paradoks Fitcha już nie obowiązuje. Logikę charakteryzujemy semantycznie, a także poprzez podejście aksjomatyczne i tabelaryczne.
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