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?

scholarly journals Nonexistence, Vague Existence, Merely Possible Existence

Disputatio ◽  
2012 ◽  
Vol 4 (33) ◽  
pp. 427-443
Author(s):  
Iris Einheuser
Keyword(s):  
Possible Worlds ◽  
Target Object ◽  
Possible World ◽  
Plausible Assumption ◽  
De Re ◽  
Modal Semantics ◽  
Vague Existence

Abstract This paper explores a new non-deflationary approach to the puzzle of nonexistence and its cousins. On this approach, we can, under a plausible assumption, express true de re propositions about certain objects that don’t exist, exist indeterminately or exist merely possibly. The defense involves two steps: First, to argue that if we can actually designate what individuates a nonexistent target object with respect to possible worlds in which that object does exist, then we can express a de re proposition about “it”. Second, to adapt the concept of outer truth with respect to a possible world – a concept familiar from actualist modal semantics – for use in representing the actual world.

Constructing the Impossible

2021 ◽  
pp. 141-163
Author(s):  
Kit Fine
Keyword(s):  
Natural Language ◽  
Possible Worlds ◽  
Formal Languages ◽  
Impossible Worlds ◽  
Possible World ◽  
Limited Role ◽  
Wide Range ◽  
Degree Of Acceptance ◽  
Broad Role ◽  
Natural Way

Please keep the original abstract. A number of philosophers have flirted with the idea of impossible worlds and some have even become enamored of it. But it has not met with the same degree of acceptance as the more familiar idea of a possible world. Whereas possible worlds have played a broad role in specifying the semantics for natural language and for a wide range of formal languages, impossible worlds have had a much more limited role; and there has not even been general agreement as to how a reasonable theory of impossible worlds is to be developed or applied. This chapter provides a natural way of introducing impossible states into the framework of truthmaker semantics and shows how their introduction permits a number of useful applications.

Liberated Brouwerian Modal Logic

Dialogue ◽  
1974 ◽  
Vol 13 (3) ◽  
pp. 505-514 ◽  
Author(s):  
Charles G. Morgan
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Possible World ◽  
Modal Systems ◽  
Relationship Of ◽  
The Relationship

In an attempt to “purify” logic of existential presuppositions, attention has recently focused on modal logics, where one usually assumes that at least one possible world exists. Systems very analogous to some of the standard modal systems have been developed which drop this presupposition. We will here treat the removal of the existential assumption from Brouwerian modal logic and discuss the relationship of the system so derived to other modal systems.

scholarly journals Crossworld Predication in Natural Language

2021 ◽  
pp. 260-272
Author(s):  
Евгений Васильевич Борисов
Keyword(s):  
Natural Language ◽  
Possible Worlds ◽  
Possible World ◽  
Partial Functions ◽  
Possible World Semantics

Некоторые предложения естественного языка, такие как «Джон мог быть выше, чем Мэри, как она есть», не допускают адекватного анализа в терминах стандартной семантики возможных миров, поскольку содержат кросс-мировую предикацию, которая в стандартной семантике не отображается. Для логического анализа такого рода предложений автором была разработана (и представлена в других публикациях) логика для кросс-мировой предикации (СРL). В статье приведен ряд примеров, демонстрирующих широкую распространенность феномена кросс-мировой предикации в естественном языке и описаны главные особенности семантики СРL (кросс-мировая интерпретация предикатов и использование частичных функций от переменных к возможным мирам в истинностной оценке формул), а также охарактеризована специфика синтаксиса СРL и онтологии, лежащей в ее основе. Some sentences of natural language cannot be adequately analyzed in terms of standard possible world semantics because they involve cross-world predication that cannot be reflected by means of standard semantics. An instance is ‘John might be taller than Mary is’. In some other papers the author proposed a logic for cross-word predication (CPL) that can be used to logically analyze sentences of this sort. In this paper, some examples are adduced that show that cross-world predication is highly widespread in natural language. The main features of the semantics of CPL are described, namely cross-world interpretation of predicates, and using partial functions from variables to possible worlds in the evaluation of formulas. Finally, the specificity of the syntax of CPL, and the ontology behind the semantics of CPL is characterized.

Meaning, Modality, and Possible Worlds Semantics

2010 ◽  
Author(s):  
Scott Soames
Keyword(s):  
Natural Language ◽  
Classical Logic ◽  
Possible Worlds ◽  
David Lewis ◽  
Formal Languages ◽  
Possible Worlds Semantics ◽  
Natural Languages ◽  
Counterfactual Conditionals ◽  
Richard Montague ◽  
Close Parallel

This chapter begins with a discussion of Kripke-style possible worlds semantics. It considers one of the most important applications of possible worlds semantics, the account of counterfactual conditionals given in Robert Stalnaker and David Lewis. It then goes on to examine the work of Richard Montague. Montague specified syntactic rules that generate English, or English-like, structures directly, while pairing each such rule with a truth-theoretic rule interpreting it. This close parallel between syntax and semantics is what makes the languages of classical logic so transparently tractable, and what they were designed to embody. Montague's bold contention is that we do not have to replace natural language natural languages with formal substitutes to achieve such transparency. The same techniques employed to create formal languages can be used to describe natural languages in mathematically revealing ways.

Modal logic

2018 ◽  
Author(s):  
Steven T. Kuhn
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Predicate Logic ◽  
Modus Ponens ◽  
Possible World ◽  
Metaphysical Necessity ◽  
Truth Values ◽  
Classical Counterpart ◽  
Modern Development ◽  
Modal Predicate Logic

Modal logic, narrowly conceived, is the study of principles of reasoning involving necessity and possibility. More broadly, it encompasses a number of structurally similar inferential systems. In this sense, deontic logic (which concerns obligation, permission and related notions) and epistemic logic (which concerns knowledge and related notions) are branches of modal logic. Still more broadly, modal logic is the study of the class of all possible formal systems of this nature. It is customary to take the language of modal logic to be that obtained by adding one-place operators ‘□’ for necessity and ‘◇’ for possibility to the language of classical propositional or predicate logic. Necessity and possibility are interdefinable in the presence of negation: □A↔¬◊¬A and  ◊A↔¬□¬A hold. A modal logic is a set of formulas of this language that contains these biconditionals and meets three additional conditions: it contains all instances of theorems of classical logic; it is closed under modus ponens (that is, if it contains A and A→B it also contains B); and it is closed under substitution (that is, if it contains A then it contains any substitution instance of A; any result of uniformly substituting formulas for sentence letters in A). To obtain a logic that adequately characterizes metaphysical necessity and possibility requires certain additional axiom and rule schemas: K □(A→B)→(□A→□B) T □A→A 5 ◊A→□◊A Necessitation A/□A. By adding these and one of the □–◇ biconditionals to a standard axiomatization of classical propositional logic one obtains an axiomatization of the most important modal logic, S5, so named because it is the logic generated by the fifth of the systems in Lewis and Langford’s Symbolic Logic (1932). S5 can be characterized more directly by possible-worlds models. Each such model specifies a set of possible worlds and assigns truth-values to atomic sentences relative to these worlds. Truth-values of classical compounds at a world w depend in the usual way on truth-values of their components. □A is true at w if A is true at all worlds of the model; ◇A, if A is true at some world of the model. S5 comprises the formulas true at all worlds in all such models. Many modal logics weaker than S5 can be characterized by models which specify, besides a set of possible worlds, a relation of ‘accessibility’ or relative possibility on this set. □A is true at a world w if A is true at all worlds accessible from w, that is, at all worlds that would be possible if w were actual. Of the schemas listed above, only K is true in all these models, but each of the others is true when accessibility meets an appropriate constraint. The addition of modal operators to predicate logic poses additional conceptual and mathematical difficulties. On one conception a model for quantified modal logic specifies, besides a set of worlds, the set Dw of individuals that exist in w, for each world w. For example, ∃x□A is true at w if there is some element of Dw that satisfies A in every possible world. If A is satisfied only by existent individuals in any given world ∃x□A thus implies that there are necessary individuals; individuals that exist in every accessible possible world. If A is satisfied by non-existents there can be models and assignments that satisfy A, but not ∃xA. Consequently, on this conception modal predicate logic is not an extension of its classical counterpart. The modern development of modal logic has been criticized on several grounds, and some philosophers have expressed scepticism about the intelligibility of the notion of necessity that it is supposed to describe.

Possible worlds

2018 ◽  
Author(s):  
Joseph Melia
Keyword(s):  
Modal Logic ◽  
Actual World ◽  
Possible Worlds ◽  
Abstract Objects ◽  
Causal Powers ◽  
Possible World ◽  
Space And Time ◽  
Wide Range ◽  
Concrete Objects

The concept of Possible worlds arises most naturally in the study of possibility and necessity. It is relatively uncontroversial that grass might have been red, or (to put the point another way) that there is a possible world in which grass is red. Though we do not normally take such talk of possible worlds literally, doing so has a surprisingly large number of benefits. Possible worlds enable us to analyse and help us understand a wide range of problematic and difficult concepts. Modality and modal logic, counterfactuals, propositions and properties are just some of the concepts illuminated by possible worlds. Yet, for all this, possible worlds may raise more problems than they solve. What kinds of things are possible worlds? Are they merely our creations or do they exist independently of us? Are they concrete objects, like the actual world, containing flesh and blood people living in alternative realities, or are they abstract objects, like numbers, unlocated in space and time and with no causal powers? Indeed, since possible worlds are not the kind of thing we can ever visit, how could we even know that such things exist? These are but some of the difficult questions which must be faced by anyone who wishes to use possible worlds.

An interpretation of “finite” modal first-order languages in classical second-order languages

1976 ◽  
Vol 41 (2) ◽  
pp. 337-340
Author(s):  
Scott K. Lehmann
Keyword(s):  
Possible Worlds ◽  
Second Order ◽  
Universal Quantifier ◽  
Possible World ◽  
Simple Interpretation ◽  
First Order ◽  
Order Language ◽  
Finite Set ◽  
Modal Semantics ◽  
Individual Variables

This note describes a simple interpretation * of modal first-order languages K with but finitely many predicates in derived classical second-order languages L(K) such that if Γ is a set of K-formulae, Γ is satisfiable (according to Kripke's 55 semantics) iff Γ* is satisfiable (according to standard (or nonstandard) second-order semantics).The motivation for the interpretation is roughly as follows. Consider the “true” modal semantics, in which the relative possibility relation is universal. Here the necessity operator can be considered a universal quantifier over possible worlds. A possible world itself can be identified with an assignment of extensions to the predicates and of a range to the quantifiers; if the quantifiers are first relativized to an existence predicate, a possible world becomes simply an assignment of extensions to the predicates. Thus the necessity operator can be taken to be a universal quantifier over a class of assignments of extensions to the predicates. So if these predicates are regarded as naming functions from extensions to extensions, the necessity operator can be taken as a string of universal quantifiers over extensions.The alphabet of a “finite” modal first-order language K shall consist of a non-empty countable set Var of individual variables, a nonempty finite set Pred of predicates, the logical symbols ‘¬’ ‘∧’, and ‘∧’, and the operator ‘◊’. The formation rules of K generate the usual Polish notations as K-formulae. ‘ν’, ‘ν1’, … range over Var, ‘P’ over Pred, ‘A’ over K-formulae, and ‘Γ’ over sets of K-formulae.

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 Goedel's Property Abstraction and Possibilism

2014 ◽  
Vol 11 (2) ◽  
Author(s):  
Randoph Rubens Goldman
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Formal Semantics ◽  
Ontological Argument ◽  
Careful Examination ◽  
Third Order ◽  
Aesthetic Sense ◽  
Completeness Result ◽  
Positive Property ◽  
Modal Semantics

Gödel’s Ontological argument is distinctive because it is the most sophisticated and formal of ontological arguments and relies heavily on the notion of positive property. Gödel uses a third-order modal logic with a property abstraction operator and property quantification into modal contexts. Gödel describes positive property as "independent of the accidental structure of the world"; "pure attribution," as opposed to privation; "positive in the 'moral aesthetic sense.'" Pure attribution seems likely to be related to the Leibnizian concept of perfection.By a careful examination of the formal semantics of third-order modal logic with property abstraction together with a Completeness result for third-order modal logic with property abstraction for faithful models that I previously developed in 2000 in my work, Gödel’s Ontological Argument, I argue that it is not possible to develop a sufficient applied third-order modal semantics for Gödel’s ontological argument. As I explore possible approaches for an applied semantics including anti-Realist accounts of the semantics of modal logic compatible with Actualism, I argue that Gödel makes implicit philosophical assumptions which commit him to both possibilism (the belief in merely possible objects) and modal realism (the belief in possible worlds).

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