Weak liberated versions of T and S4

1975 ◽  
Vol 40 (1) ◽  
pp. 25-30 ◽  
Author(s):  
Charles G. Morgan
Keyword(s):  
Possible Worlds ◽  
Accessibility Relation ◽  
Independence Results ◽  
Modal Systems

AbstractThe usual semantics for the modal systems T, S4, and S5 assumes that the set of possible worlds contains at least one member. Recently versions of these modal systems have been developed in which this assumption is dropped. The systems discussed here are obtained by slightly weakening the liberated versions of T and S4. The semantics does not assume the existence of possible worlds, and the accessibility relation between worlds is only required to be quasi-reflexive instead of reflexive. Completeness and independence results are established.

Modal logic, philosophical issues in

2018 ◽  
Author(s):  
Thomas J. McKay
Keyword(s):  
Modal Logic ◽  
Natural Language ◽  
Possible Worlds ◽  
Possible World ◽  
Modal Claim ◽  
Knowing That ◽  
Modal Semantics ◽  
Bull's Eye ◽  
Modal Systems ◽  
Richard Montague

In reasoning we often use words such as ‘necessarily’, ‘possibly’, ‘can’, ‘could’, ‘must’ and so on. For example, if we know that an argument is valid, then we know that it is necessarily true that if the premises are true, then the conclusion is true. Modal logic starts with such modal words and the inferences involving them. The exploration of these inferences has led to a variety of formal systems, and their interpretation is now most often built on the concept of a possible world. Standard non-modal logic shows us how to understand logical words such as ‘not’, ‘and’ and ‘or’, which are truth-functional. The modal concepts are not truth-functional: knowing that p is true (and what ‘necessarily’ means) does not automatically enable one to determine whether ‘Necessarily p’ is true. (‘It is necessary that all people have been people’ is true, but ‘It is necessary that no English monarch was born in Montana’ is false, even though the simpler constituents – ‘All people have been people’ and ‘No English monarch was born in Montana’– are both true.) The study of modal logic has helped in the understanding of many other contexts for sentences that are not truth-functional, such as ‘ought’ (‘It ought to be the case that p’) and ‘believes’ (‘Alice believes that p’); and also in the consideration of the interaction between quantifiers and non-truth-functional contexts. In fact, much work in modern semantics has benefited from the extension of modal semantics introduced by Richard Montague in beginning the development of a systematic semantics for natural language. The framework of possible worlds developed for modal logic has been fruitful in the analysis of many concepts. For example, by introducing the concept of relative possibility, Kripke showed how to model a variety of modal systems: a proposition is necessarily true at a possible world w if and only if it is true at every world that is possible relative to w. To achieve a better analysis of statements of ability, Mark Brown adapted the framework by modelling actions with sets of possible outcomes. John has the ability to hit the bull’s-eye reliably if there is some action of John’s such that every possible outcome of that action includes John’s hitting the bull’s-eye. Modal logic and its semantics also raise many puzzles. What makes a modal claim true? How do we tell what is possible and what is necessary? Are there any possible things that do not exist (and what could that mean anyway)? Does the use of modal logic involve a commitment to essentialism? How can an individual exist in many different possible worlds?

Impossible Worlds

Philosophy ◽  
2012 ◽  
Author(s):  
Ira Kiourti
Keyword(s):  
Possible Worlds ◽  
Impossible Worlds ◽  
Impossible World ◽  
Possible World ◽  
Accessibility Relation ◽  
Possible World Semantics ◽  
Potential Applicability ◽  
Controversial Topic ◽  
Truth Value ◽  
Modern Work

Impossible worlds constitute an increasingly popular yet controversial topic in logic and metaphysics. The term “impossible worlds” parallels the term “possible worlds” and commonly refers to setups, situations, or totalities (“worlds”) that are inconsistent, incomplete, non-classical, or non-normal in possible-world semantics and metaphysics. These may verify a proposition and its negation, be silent as to the truth value of a proposition, or somehow fail to conform to the (classical) laws of logic. Some authors object to the term “impossible world,” preferring to talk of nonstandard worlds or partial situations instead. While the term “impossible world” is sometimes used to refer to a world that is inaccessible from another relative to some specified accessibility relation, impossible worlds are often conceived of as absolutely impossible in a broadly logical, conceptual, or metaphysical sense. As in the case of possible worlds, modern talk of impossible worlds originates with semantic interpretations of modal and non-classical logics, yet the potential applicability of these worlds to logical, metaphysical, and semantic philosophical puzzles has allowed them to permeate the wider philosophical arena. Arguments for impossible worlds often parallel those for possible worlds (see From Possible Worlds to Impossible Worlds) and focus largely on the proposed applications for such worlds (see Applications). As with possible worlds, there are various metaphysical conceptions of impossible worlds (see the Metaphysics of Impossible Worlds), and objections to such worlds are often theory specific (see Objections to Applications and Objections to Impossible Worlds). This article focuses on modern work on impossible worlds and its critics.

EVIDENCE MEASURES INDUCED BY KRIPKE'S ACCESSIBILITY RELATIONS

1999 ◽  
Vol 07 (06) ◽  
pp. 589-613 ◽  
Author(s):  
ELENA TSIPORKOVA ◽  
VESELKA BOEVA ◽  
BERNARD DE BAETS
Keyword(s):  
Modal Logic ◽  
Multivalued Mapping ◽  
Possible Worlds ◽  
Accessibility Relation

Modal logic interpretations of plausibility and belief measures are developed based on the observation that the accessibility relation in a model of modal logic, regarded as a multivalued mapping, induces a plausibility measure and a belief measure on the set of possible worlds.

An incomplete decidable modal logic

1984 ◽  
Vol 49 (2) ◽  
pp. 520-527 ◽  
Author(s):  
M. J. Cresswell
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Order Theory ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
General Frame ◽  
Propositional Modal Logic ◽  
Modal Systems ◽  
Modal Propositional Logic

The most common way of proving decidability in propositional modal logic is to shew that the system in question has the finite model property. This is not however the only way. Gabbay in [4] proves the decidability of many modal systems using Rabin's result in [8] on the decidability of the second-order theory of successor functions. In particular [4, pp. 258-265] he is able to prove the decidability of a system which lacks the finite model property. Gabbay's system is however complete, in the sense of being characterized by a class of frames, and the question arises whether there is a decidable modal logic which is not complete. Since no incomplete modal logic has the finite model property [9, p. 33], any proof of decidability must employ some such method as Gabbay's. In this paper I use the Gabbay/Rabin technique to prove the decidability of a finitely axiomatized normal modal propositional logic which is not characterized by any class of frames. I am grateful to the referee for suggesting improvements in substance and presentation.The terminology I am using is standard in modal logic. By a frame is understood a pair 〈W, R〉 in which W is a class (of “possible worlds”) and R ⊆ W2. To avoid confusion in what follows, a frame will henceforth be referred to as a Kripke frame. By contrast, a general frame is a pair 〈, Π〉 in which is a Kripke frame and Π is a collection of subsets of W closed under the Boolean operations and satisfying the condition that if A is in Π then so is R−1 “A. A model on a frame (of either kind) is obtained by adding a function V which assigns sets of worlds to propositional variables. In the case of a general frame we require that V(p) ∈ Π.

scholarly journals Four-valued modal logic: Kripke semantics and duality

10.29007/12bb ◽  
2018 ◽  
Author(s):  
Achim Jung ◽  
Umberto Rivieccio
Keyword(s):  
Modal Logic ◽  
Independent Interest ◽  
Kripke Semantics ◽  
Kripke Models ◽  
Consequence Relations ◽  
Accessibility Relation ◽  
Kripke Frames ◽  
Modal Systems ◽  
Global Consequence

Along the lines of recent investigations combining many-valued and modal systems, we address the problem of defining and axiomatizing the least modal logic over the four-element Belnap lattice. By this we mean the logic determined by the class of all Kripke frames where the accessibility relation as well as semantic valuations are four-valued. Our main result is the introduction of two Hilbert-style calculi that provide complete axiomatizations for, respectively, the local and the global consequence relations associated to the class of all four-valued Kripke models. Our completeness proofs make an extensive and profitable use of algebraic and topological techniques; in fact, our algebraic and topological analyses of the logic have, in our opinion, an independent interest and contribute to the appeal of our approach.

Combining the Best of Two Possible Worlds

1991 ◽  
Vol 36 (12) ◽  
pp. 1057-1058
Author(s):  
Marvin R. Goldfried ◽  
Douglas A. Vakoch
Keyword(s):  
Possible Worlds

Multiple Strands and Possible Worlds

2002 ◽  
Vol 11 (2) ◽  
pp. 44-61
Author(s):  
Ruth Perlmutter
Keyword(s):  
Possible Worlds

The Failure of Conceptual Analysis

2018 ◽  
Author(s):  
Jennifer McKitrick
Keyword(s):  
Conceptual Analysis ◽  
Possible Worlds ◽  
Causal Power ◽  
Possible Worlds Semantics ◽  
Conditional Statements ◽  
Reductive Analysis

Dispositions are often regarded with suspicion. Consequently, some philosophers try to semantically reduce disposition ascriptions to sentences containing only non-dispositional vocabulary. Typically, reductionists attempt to analyze disposition ascriptions in terms of conditional statements. These conditional statements, like other modal claims, are often interpreted in terms of possible worlds semantics. However, conditional analyses are subject to a number of problems and counterexamples, including random coincidences, void satisfaction, masks, antidotes, mimics, altering, and finks. Some analyses fail to reduce disposition ascriptions to non-modal vocabulary. If reductive analysis of disposition ascriptions fails, then perhaps there can be metaphysical reduction of dispositions without semantic reduction. However, the reductionist still owes us an account of what makes disposition ascriptions true. But to posit a causal power for every unreduced dispositional predicate is an overreaction to the failure of conceptual analysis.

Is There Life in Possible Worlds?

2017 ◽  
Author(s):  
Mark Wilson
Keyword(s):  
Possible Worlds ◽  
Machine Design ◽  
Lagrangian Mechanics ◽  
Philosophical Investigations ◽  
Search Spaces ◽  
Dimensional Modeling ◽  
Lower Dimensional

Scientists have developed various collections of specialized possibilities to serve as search spaces in which excessive reliance upon speculative forms of lower dimensional modeling or other unwanted details can be skirted. Two primary examples are discussed: the search spaces of machine design and the virtual variations utilized within Lagrangian mechanics. Contemporary appeals to “possible worlds” attempt to imbed these localized possibilities within fully enunciated universes. But not all possibilities are made alike and these reductive schemes should be resisted, on the grounds that they render the utilities of everyday counterfactuals and “possibility” talk incomprehensible. The essay also discusses whether Wittgenstein’s altered views in his Philosophical Investigations reflect similar concerns.

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