Modal Logic Research Articles

Counterparts, Essences and Quantified Modal Logic

It is commonplace to formalize propositions involving essential properties of objects in a language containing modal operators and quantifiers. Assuming David Lewis’s counterpart theory as a semantic framework for quantified modal logic, I will show that certain statements discussed in the metaphysics of modality de re, such as the sufficiency condition for essential properties, cannot be faithfully formalized. A natural modification of Lewis’s translation scheme seems to be an obvious solution but is not acceptable for various reasons. Consequently, the only safe way to express some intuitions regarding essential properties is to use directly the language of counterpart theory without modal operators.

Pooling Modalities and Pointwise Intersection: Semantics, Expressivity, and Dynamics

We study classical modal logics with pooling modalities, i.e. unary modal operators that allow one to express properties of sets obtained by the pointwise intersection of neighbourhoods. We discuss salient properties of these modalities, situate the logics in the broader area of modal logics (with a particular focus on relational semantics), establish key properties concerning their expressive power, discuss dynamic extensions of these logics and provide reduction axioms for the latter.

Remarks about the unification types of some locally tabular normal modal logics

AbstractIt is already known that unifiable formulas in normal modal logic $\textbf {K}+\square ^{2}\bot $ are either finitary or unitary and unifiable formulas in normal modal logic $\textbf {Alt}_{1}+\square ^{2}\bot $ are unitary. In this paper, we prove that for all $d{\geq }3$, unifiable formulas in normal modal logic $\textbf {K}+\square ^{d}\bot $ are either finitary or unitary and unifiable formulas in normal modal logic $\textbf {Alt}_{1}+\square ^{d}\bot $ are unitary.

Phenomenological investigation of spatial structures in mathematics and modal logic

Phenomenological investigation of spatial structures in mathematics and modal logic

Logic of the ontological argument

In his ontological argument Gödel says nothing about its underlying logic. The argument is modal and at least of second-order and since S5 axiom is used so it is widely accepted that the logic of the argument is the S5 second-order modal logic. However, there is a step in the proof in which Gödel applies the necessitation rule on the assumptions of the argument (see [3]). This is repeated by all of his followers (see [1] and [5]). This application of the necessitation rule can seriously harm the consequence relation of the logic of the ontological argument. It seems that the only way to preserve the modal logic S5 for the ontological argument is to assume some of its axioms in the necessitated form.

The history variations and modal logic

The history variations and modal logic

Modalitäten und der Raum der Möglichkeiten

With the comeback of metaphysics on the agenda of analytical philosophy a renewed interest for modalities and their specific properties was not lagging far behind. Yet, their indispensability in debates, say for example on identity and essence, stands in marked contrast to the explanatory meagerness when modalities are introduced by reference to ‘possible worlds’. However, what is required for an appropriate understanding of modalities is not, or so I try to argue, a philosophical account in any ambitious sense of the word but rather a specification of their ‘structural’ properties, which allow us to differentiate between various types of modalities. The proper framework for such a specification is conveniently offered by proof theory, and here specifically by Gentzen methods. Therein we find not only all that is needed to understand modalities, modalities are thereby also handed back to where once they originated: logic. Keywords: modalities, modal logic, possibilities, possible worlds, proof-theoretic meaning

An Augustinian–Edwardsian Metaphysics of Possibility for the Barcan Formula

The Barcan formula is a theorem of quantified modal logic. Its most straightforward interpretation appears to commit one to “possibilism,” the view that merely possible things exist. Alternative systems of logic revise the formal semantics to preclude the theorem and its consequences. The crux, however, is the modal metaphysics presupposed by the formal semantics. This paper presents an alternative metaphysics of possibility that follows Augustine’s suggestion that God’s plan is only one of a range of alternative histories for a creation. The metaphysics is a version of “trace actualism”—neither pure possibilism nor pure actualism.

The Mind of Charles Hartshorne: A Critical Examination

The Mind of Charles Hartshorne: A Critical Examination

The predicate version of the joint logic of problems and propositions

We consider the joint logic of problems and propositions $\mathrm{QHC}$ introduced by Melikhov. We construct Kripke models with audit worlds for this logic and prove the soundness and completeness of $\mathrm{QHC}$ with respect to this type of model. The conservativity of the logic $\mathrm{QHC}$ over the intuitionistic modal logic $\mathrm{QH4}$, which coincides with the ‘lax logic’ $\mathrm{QLL}^+$, is established. We construct Kripke models with audit worlds for the logic $\mathrm{QH4}$ and prove the corresponding soundness and completeness theorems. We also prove that the logics $\mathrm{QHC}$ and $\mathrm{QH4}$ have the disjunction and existence properties. Bibliography: 33 titles.

The Actuality of a World: What Ceases Not to Be Written

“There is no longer any world,” wrote the late philosopher Jean-Luc Nancy in 1993, and in this paper, the sense of this loss of world is analysed in terms of the modal notions of necessity, impossibility, and possibility. Modal differentiation can illuminate what constitutes the sense of actuality in a world, and hence, what it is that has been lost regarding this actuality of being in a world. Modal thinking does not rely on knowledge of the true state of affairs, nor on having a constant grasp on necessity: modal thinking enables us to discern the actual by way of the relations between necessity, possibility, and impossibility. It is on the basis of these relations that a mode of rethinking actuality is suggested. In order to pursue this line of thought, I rely on Jacques Lacan’s understanding of modalities as writing operations, concentrating mostly on certain sections from his Seminar XXI (1973–1974), wherein he refers to Aristotelian modal logic and mentions the analytic philosopher Jaakko Hintikka (as one of Aristotle’s commentators). This ternary relation between Aristotle, Hintikka, and Lacan suggests a different outlook on the actual being-such of a world. The being-such of a world, its actuality, is understood as a contingency (what ceases not to be written), a possibility emerging from the relation between what is declared necessary and what is impossible. The paper shows the implications of such a modal understanding of actuality, as demonstrated in Lacan’s view of human sexuality and sexual difference.

On Formalizing Logical Modalities

This paper is in the scope of the philosophy of modal logic; more precisely, it concerns the semantics of modal logic, when the modal elements are interpreted as logical modalities. Most authors have thought that the logic for logical modality—that is, the one to be used to formalize the notion of logical truth (and other related notions)—is to be found among logical systems in which modalities are allowed to be iterated. This has raised the problem of the adequacy, to that formalization purpose, of some modal schemes, such as S4 and S5 . It has been argued that the acceptance of S5 leads to non-normal modal systems, in which the uniform substitution rule fails. The thesis supported in this paper is that such a failure is rather to be attributed to what will be called “Condition of internalization.” If this is correct, there seems to be no normal modal logic system capable of formalizing logical modality, even when S5 is rejected in favor of a weaker system such as S4, as recently proposed by McKeon.

Correction to: Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter

Correction to: Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter

Modular non-deterministic semantics for T, TB, S4, S5 and more

Abstract In this paper, a modular approach for non-deterministic semantics for (non-normal) modal logics is developed. In particular, our aim is to improve and reinterpret some results from Omori and Skurt (2016, IfCoLog J. Logics Appl., 3, 815–845) and Coniglio et al. (2015, J. Appl. Non-Class. Log., 25, 20–45) regarding modal systems T, TB, S4 and S5. More economical axiomatizations make the rule of necessitation modular, thus providing non-deterministic semantics for (NEC)-free fragments for all the investigated systems. Moreover, by fixing the interpretation of all connectives but the modal ones, a combinatorial outlook at their matrices is provided to the effect that a new modal system and simplification of those for T and S4 are achieved.

Monadic NM-algebras: an algebraic approach to monadic predicate nilpotent minimum logic

AbstractIn this paper, we further study the variety of monadic nilpotent minimum (NM)-algebras and their corresponding logic. In order to solve the drawback of monadic NM-algebras, we review some well-known classes of monadic t-norm-based fuzzy logical algebras and then revise the axiomatic system of monadic NM-algebras. Then we show that the variety of monadic NM-algebras is the equivalent algebraic semantics of monadic predicate fuzzy logic $\textbf {mNM}_{\forall }$, which is equivalent to the modal fuzzy logic $\textbf {S5(NM)}$. Moreover, we show that the propositional case of the modal fuzzy logic $\textbf {S5(NM)}$, which is $\textbf{S5}^{\prime}\textbf{(NM),}$ is also complete with respect to the variety of monadic NM-algebras in the sense of Blok and Pigozzi and obtain a necessary and sufficient condition for this logic to be semilinear. Finally, we give some representations of monadic NM-algebras. In particular, we give some characterizations of representable and directly indecomposable monadic NM-algebras.

Expressive Logics for Coinductive Predicates

The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas in a certain modal logic. In this paper we study this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. We formulate when a logic fully characterises a coinductive predicate on coalgebras, by providing suitable notions of adequacy and expressivity, and give sufficient conditions on the semantics. The approach is illustrated with logics characterising similarity, divergence and a behavioural metric on automata.

Truth tables for modal logics T and S4, by using three-valued non-deterministic level semantics

AbstractNovel three-valued non-deterministic level semantics for modal logics $\textbf {T}$ and $\textbf {S4}$ are presented. A criterion for partial level valuations is given, making it possible to create truth tables. Additionally, semantics and truth tables for $\textbf {0}$ (defined as $\textbf {PC}$ plus rule of necessitation) and $\textbf {0T}$ with only two values are based on Ivlev’s work. We need Kearns’ notion of level valuations: a generalization of Dugundji’s theorem shows that there is no non-deterministic semantics for modal logics up to $\textbf {S5}$, containing the rule of necessitation.

Finitary unification in locally tabular modal logics characterized

We provide necessary and sufficient conditions for finitary unification in locally tabular modal logics, solely in terms of Kripke models. We apply the conditions to establish the unification types of logics determined by simple finite frames. In particular, we show that unification is finitary (or unitary) in the logic determined by the fork (frame F4, see Fig. 6), the rhombus (frame F5), in GL.3m, GrzBd2, S4Bd2 and other logics; whereas it is nullary in the logic of F6, and of the pentagon FN5. In Appendix analogous results are given for superintuitionistic logics.

Memory in Terms of Logic and Semiotics

The phenomenon of memory is considered as a component implicitly present in the process of information communication. A short typology of memory kinds that form a referential field around so-called semantic memory is given. Through the approaches of Yu. Lotman and R. Jacobson, the classical notion of time is considered through the relationship "Past-Present-Future", which is closely related to the problems of memory. The focus is on how could the memory be considered within logic and by means of logic? As one of the ways of solving the point, to apply the tools of tense logic to the problem through the analysis of the works of Anatoly Ishmuratov (1946–2017), a prominent Ukrainian logician, is proposed. The classifications of the tenses by O. Jespersen, H. Reichenbach, W. Bull are considered. The subjective and objective axes of orientation, which form the basis of calendars, charts and scales, as fragments of time, are analyzed. According to Ishmuratov, these instruments – schemes, diagrams, etc. can be considered languages. The possibility of language objectification of meaning determines the relationship of logical inference, and thus the structuring of semantic areas, which are memories, by means the language of logic and in accordance with its structures. Through the study of logical and cognitive conditions of action, A. Ishmuratov continued to develop ideas of tense logic and their application to explain the psychological perception of objective time. He construed a scheme of semantic connections of memory as a mental act that reproduces the life path of the individual; he distinguished between memories and "pseudo-memories", which together influence the reassessment of past events, shape the individual`s experience and his ability to construe alternatives to the future and reasoning about it. A special place in A. Ishmuratov's research is occupied by his explication of temporal three-valued logic and application of temporal modalities to the analysis of so-called transient states. Further study of such approaches could help to make sense of rational explication of memory, testimonies, reminiscences of past events and give interesting results.

The Evil That Free Will Does: Plantinga’s Dubious Defense

Abstract Alvin Plantinga’s controversial free will defense (FWD) for the problem of evil is an important attempt to show with certainty that moral evils are compatible and justifiable with God’s omnipotence and omniscience. I agree with critics who argue that it is untenable and the FWD fails. This paper proposes new criticisms by analyzing Plantinga’s presuppositions and objectionable assumptions in God, Freedom and Evil. Notably, his limited concept of omnipotence, and possible worlds theory lack rigorous argument and are subjectively biased with irrelevant weak examples. My ontological possible worlds theory (Possible Conditional Timelines) shows that it is very likely that the omnipotent God exists of necessity in some worlds but perhaps not this one. Omnipotence is total and absolute, and should imply the freedom of will to actualize all worlds God chooses. Plantinga’s position regarding God’s omniscience of future counterfactuals is implausible based on modal logic conjecture.