Modal operators and functional completeness, II

1977 ◽  
Vol 42 (3) ◽  
pp. 391-399 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Kripke Semantics ◽  
Functional Completeness ◽  
Possible World ◽  
States Of Affairs ◽  
Propositional Modal Logic ◽  
Modal Operators ◽  
Fixed Frame ◽  
State Of Affairs

In the Kripke semantics for propositional modal logic, a frame W = (W, ≺) represents a set of “possible worlds” and a relation of “accessibility” between possible worlds. With respect to a fixed frame W, a proposition is represented by a subset of W (regarded as the set of worlds in which the proposition is true), and an n-ary connective (i.e. a way of forming a new proposition from an ordered n-tuple of given propositions) is represented by a function fw: (P(W))n → P(W). Finally a state of affairs (i.e. a consistent specification whether or not each proposition obtains) is represented by an ultrafilter over W. {To avoid possible confusion, the reader should forget that some people prefer the term “states of affairs” for our “possible worlds”.}In a broader sense, an n-ary connective is represented by an n-ary operatorf = {fw∣ W ∈ Fr}, where Fr is the class of all frames and each fw: (P(W))n → P(W). A connective is modal if it corresponds to a formula of propositional modal logic. A connective C is coherent if whether C(P1,…, Pn) is true in a possible world depends only upon which modal combinations of P1,…,Pn are true in that world. (A modal combination of P1,…,Pn is the result of applying a modal connective to P1,…, Pn.) A connective C is strongly coherent if whether C(P1, …, Pn) obtains in a state of affairs depends only upon which modal combinations of P1,…, Pn obtain in that state of affairs.

Computational Principles

1990 ◽  
Author(s):  
John L. Pollock
Keyword(s):  
Mathematical Theory ◽  
Possible Worlds ◽  
Mathematical Structure ◽  
Possible World ◽  
States Of Affairs ◽  
Classical Probability ◽  
Probability Calculus ◽  
Logical Operators ◽  
Necessary Truth ◽  
State Of Affairs

Much of the usefulness of probability derives from its rich logical and mathematical structure. That structure comprises the probability calculus. The classical probability calculus is familiar and well understood, but it will turn out that the calculus of nomic probabilities differs from the classical probability calculus in some interesting and important respects. The purpose of this chapter is to develop the calculus of nomic probabilities, and at the same time to investigate the logical and mathematical structure of nomic generalizations. The mathematical theory of nomic probability is formulated in terms of possible worlds. Possible worlds can be regarded as maximally specific possible ways things could have been. This notion can be filled out in various ways, but the details are not important for present purposes. I assume that a proposition is necessarily true iff it is true at all possible worlds, and I assume that the modal logic of necessary truth and necessary exemplification is a quantified version of S5. States of affairs are things like Mary’s baking pies, 2 being the square root of 4, Martha’s being smarter than John, and the like. For present purposes, a state of affairs can be identified with the set of all possible worlds at which it obtains. Thus if P is a state of affairs and w is a possible world, P obtains at w iff w∊P. Similarly, we can regard monadic properties as sets of ordered pairs ⧼w,x⧽ of possible worlds and possible objects. For example, the property of being red is the set of all pairs ⧼w,x⧽ such that w is a possible world and x is red at w. More generally, an n-place property will be taken to be a set of (n+l)-tuples ⧼w,x1...,xn⧽. Given any n-place concept α, the corresponding property of exemplifying a is the set of (n + l)-tuples ⧼w,x1,...,xn⧽ such that x1,...,xn exemplify α at the possible world w. States of affairs and properties can be constructed out of one another using logical operators like conjunction, negation, quantification, and so on.

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 EVERYONE KNOWS THAT SOMEONE KNOWS: QUANTIFIERS OVER EPISTEMIC AGENTS

2019 ◽  
Vol 12 (2) ◽  
pp. 255-270 ◽  
Author(s):  
PAVEL NAUMOV ◽  
JIA TAO
Keyword(s):  
Modal Logic ◽  
Epistemic Logic ◽  
Possible Worlds ◽  
Completeness Theorem ◽  
Kripke Semantics ◽  
Possible Worlds Semantics ◽  
Distributed Knowledge ◽  
First Order ◽  
Soundness And Completeness ◽  
First Order Modal Logic

AbstractModal logic S5 is commonly viewed as an epistemic logic that captures the most basic properties of knowledge. Kripke proved a completeness theorem for the first-order modal logic S5 with respect to a possible worlds semantics. A multiagent version of the propositional S5 as well as a version of the propositional S5 that describes properties of distributed knowledge in multiagent systems has also been previously studied. This article proposes a version of S5-like epistemic logic of distributed knowledge with quantifiers ranging over the set of agents, and proves its soundness and completeness with respect to a Kripke semantics.

Representability in second-order propositional poly-modal logic

2002 ◽  
Vol 67 (3) ◽  
pp. 1039-1054 ◽  
Author(s):  
G. Aldo Antonelli ◽  
Richmond H. Thomason
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Second Order ◽  
Order Logic ◽  
Multiple Modalities ◽  
Modal Operators ◽  
Second Order Systems ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Propositional System

AbstractA propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.In this paper we generalize this framework by allowing multiple modalities. While this does not affect the undecidability of K, B, T, K4 and S4, poly-modal second-order S5 is dramatically more expressive than its mono-modal counterpart. As an example, we establish the definability of the transitive closure of finitely many modal operators. We also take up the decidability issue, and, using a novel encoding of sets of unordered pairs by partitions of the leaves of certain graphs, we show that the second-order propositional logic of two S5 modalitities is also equivalent to full second-order logic.

On Necessary Gratuitous Evils

2020 ◽  
Vol 12 (3) ◽  
pp. 117
Author(s):  
Michael James Almeida
Keyword(s):  
Necessary Condition ◽  
Possible World ◽  
States Of Affairs ◽  
Gratuitous Evil ◽  
Moral Perfection ◽  
Standard Position ◽  
State Of Affairs

The standard position on moral perfection and gratuitous evil makes the prevention of gratuitous evil a necessary condition on moral perfection. I argue that, on any analysis of gratuitous evil we choose, the standard position on moral perfection and gratuitous evil is false. It is metaphysically impossible to prevent every gratuitously evil state of affairs in every possible world. No matter what God does—no matter how many gratuitously evil states of affairs God prevents—it is necessarily true that God coexists with gratuitous evil in some world or other. Since gratuitous evil cannot be eliminated from metaphysical space, the existence of gratuitous evil presents no objection to essentially omnipotent, essentially omniscient, essentially morally perfect, and necessarily existing beings.

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 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.

Prior’s Three-Valued Modal Logic Q and its Possible Applications

2007 ◽  
Vol 11 (1) ◽  
pp. 105-110
Author(s):  
Seiki Akama ◽  
◽  
Yasunori Nagata ◽  
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Completeness Theorem ◽  
Kripke Semantics ◽  
Formal Properties

Prior proposed a three-valued modal logic Q as a “correct” modal logic from his philosophical motivations. Unfortunately, Prior’s Q and many-valued modal logic have been neglected in the tradition of many-valued and modal logic. In this paper, we introduce a version of three-valued Kripke semantics for Q, which aims to establish Prior’s ideas based on possible worlds. We investigate formal properties of Q and prove the completeness theorem of Q. We also compare our approach with others and suggest possible applications.

scholarly journals An algebraic generalization of Kripke structures

2008 ◽  
Vol 145 (3) ◽  
pp. 549-577 ◽  
Author(s):  
SÉRGIO MARCELINO ◽  
PEDRO RESENDE
Keyword(s):  
Modal Logic ◽  
Operator Algebras ◽  
Inverse Semigroups ◽  
Kripke Semantics ◽  
Algebraic Semantics ◽  
Propositional Modal Logic ◽  
Kripke Structures ◽  
Algebraic Generalization ◽  
Foliated Manifolds ◽  
Topological Groupoids

AbstractThe Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4 and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL and the ramified temporal logic CTL.

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