The purpose of this study is to propose new similarity measures namely rough variational coefficient similarity measure under the rough neutrosophic environment. The weighted rough variational coefficient similarity measure has been also defined. The weighted rough variational coefficient similarity measures between the rough ideal alternative and each alternative are xxxxx calculated to find the best alternative. The ranking order of all the alternatives can be determined by using the numerical values of similarity measures. Finally, an illustrative example has been provided (...) to show the effectiveness and validity of the proposed approach. Comparisons of decision results of existing rough similarity measures have been provided. (shrink)

This paper is devoted to present Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method for multi-attribute group decision making under rough neutrosophic environment. The concept of rough neutrosophic set is a powerful mathematical tool to deal with uncertainty, indeterminacy and inconsistency. In this paper, a new approach for multi-attribute group decision making problems is proposed by extending the TOPSIS method under rough neutrosophic environment. Rough neutrosophic set is characterized by the upper and lower approximation operators and the (...) pair of neutrosophic sets that are characterized by truth-membership degree, indeterminacy membership degree, and falsity membership degree. In the decision situation, ratings of alternatives with respect to each attribute are characterized by rough neutrosophic sets that reflect the decision makers’ opinion. Rough neutrosophic weighted averaging operator has been used to aggregate the individual decision maker’s opinion into group opinion for rating the importance of attributes and alternatives. Finally, a numerical example has been provided to demonstrate the applicability and effectiveness of the proposed approach. (shrink)

In this paper, I will argue that there is a version of possibilism—inspired by the modal analogue of Kit Fine’s fragmentalism—that can be combined with a weakening of actualism. The reasons for analysing this view, which I call Modal Fragmentalism, are twofold. Firstly, it can enrich our understanding of the actualism/possibilism divide, by showing that, at least in principle, the adoption of possibilia does not correspond to an outright rejection of the actualist intuitions. Secondly, and more specifically, it can enrich (...) our understanding of concretism, by proving that, at least in principle, the idea that objects have properties in an absolute manner is compatible with transworld identity. (shrink)

For a novice this book is a mathematically-oriented introduction to modal logic, the discipline within mathematical logic studying mathematical models of reasoning which involve various kinds of modal operators. It starts with very fundamental concepts and gradually proceeds to the front line of current research, introducing in full details the modern semantic and algebraic apparatus and covering practically all classical results in the field. It contains both numerous exercises and open problems, and presupposes only minimal knowledge in mathematics. A specialist (...) can use the book as a source of references. Results and methods of many directions in propositional modal logic, from completeness and duality to algorithmic problems, are collected and systematically presented in one volume. (shrink)

Timothy Williamson has argued that in the debate on modal ontology, the familiar distinction between actualism and possibilism should be replaced by a distinction between positions he calls contingentism and necessitism. He has also argued in favor of necessitism, using results on quantified modal logic with plurally interpreted second-order quantifiers showing that necessitists can draw distinctions contingentists cannot draw. Some of these results are similar to well-known results on the relative expressivity of quantified modal logics with so-called inner and outer (...) quantifiers. The present paper deals with these issues in the context of quantified modal logics with generalized quantifiers. Its main aim is to establish two results for such a logic: Firstly, contingentists can draw the distinctions necessitists can draw if and only if the logic with inner quantifiers is at least as expressive as the logic with outer quantifiers, and necessitists can draw the distinctions contingentists can draw if and only if the logic with outer quantifiers is at least as expressive as the logic with inner quantifiers. Secondly, the former two items are the case if and only if all of the generalized quantifiers are first-order definable, and the latter two items are the case if and only if first-order logic with these generalized quantifiers relativizes. (shrink)

A textbook on modal logic, intended for readers already acquainted with the elements of formal logic, containing nearly 500 exercises. Brian F. Chellas provides a systematic introduction to the principal ideas and results in contemporary treatments of modality, including theorems on completeness and decidability. Illustrative chapters focus on deontic logic and conditionality. Modality is a rapidly expanding branch of logic, and familiarity with the subject is now regarded as a necessary part of every philosopher's technical equipment. Chellas here (...) offers an up-to-date and reliable guide essential for the student. (shrink)

This paper examines "moderate modal skepticism", a form of skepticism about metaphysical modality defended by Peter van Inwagen in order to blunt the force of certain modal arguments in the philosophy of religion. Van Inwagen’s argument for moderate modal skepticism assumes Yablo's (1993) influential world-based epistemology of possibility. We raise two problems for this epistemology of possibility, which undermine van Inwagen's argument. We then consider how one might motivate moderate modal skepticism by relying on a different epistemology of possibility, (...) which does not face these problems: Williamson’s (2007: ch. 5) counterfactual-based epistemology. Two ways of motivating moderate modal skepticism within that framework are found unpromising. Nevertheless, we also find a way of vindicating an epistemological thesis that, while weaker than moderate modal skepticism, is strong enough to support the methodological moral van Inwagen wishes to draw. (shrink)

Timothy Williamson gives an original and provocative treatment of deep metaphysical questions about existence, contingency, and change, using the latest resources of quantified modal logic. Contrary to the widespread assumption that logic and metaphysics are disjoint, he argues that modal logic provides a structural core for metaphysics.

It is widely agreed that the intelligibility of modal metaphysics has been vindicated. Quine's arguments to the contrary supposedly confused analyticity with metaphysical necessity, and rigid with non-rigid designators.2 But even if modal metaphysics is intelligible, it could be misconceived. It could be that metaphysical necessity is not absolute necessity – the strictest real notion of necessity – and that no proposition of traditional metaphysical interest is necessary in every real sense. If there were nothing otherwise “uniquely metaphysically significant” about (...) metaphysical necessity, then paradigmatic metaphysical necessities would be necessary in one sense of “necessary”, not necessary in another, and that would be it. The question of whether they were necessary simpliciter would be like the question of whether the Parallel Postulate is true simpliciter – understood as a pure mathematical conjecture, rather than as a hypothesis about physical spacetime. In a sense, the latter question has no objective answer. In this article, I argue that paradigmatic questions of modal metaphysics are like the Parallel Postulate question. I then discuss the deflationary ramifications of this argument. I conclude with an alternative conception of the space of possibility. According to this conception, there is no objective boundary between possibility and impossibility. Along the way, I sketch an analogy between modal metaphysics and set theory. (shrink)

The paper begins with a clarification of the notions of intuition (and, in particular, modal intuition), modal error, conceivability, metaphysical possibility, and epistemic possibility. It is argued that two-dimensionalism is the wrong framework for modal epistemology and that a certain nonreductionist approach to the theory of concepts and propositions is required instead. Finally, there is an examination of moderate rationalism’s impact on modal arguments in the philosophy of mind -- for example, Yablo’s disembodiment argument and Chalmers’s zombie argument. A less (...) vulnerable style of modal argument is defended, which nevertheless wins the same anti-materialist conclusions sought by these other arguments. (shrink)

This paper motivates and develops a novel semantic framework for deontic modals. The framework is designed to shed light on two things: the relationship between deontic modals and substantive theories of practical rationality and the interaction of deontic modals with conditionals, epistemic modals and probability operators. I argue that, in order to model inferential connections between deontic modals and probability operators, we need more structure than is provided by classical intensional theories. In particular, we need probabilistic structure that interacts directly (...) with the compositional semantics of deontic modals. However, I reject theories that provide this probabilistic structure by claiming that the semantics of deontic modals is linked to the Bayesian notion of expectation. I offer a probabilistic premise semantics that explains all the data that create trouble for the rival theories. (shrink)

The paper argues against Peacocke's moderate rationalism in modality. In the first part, I show, by identifying an argumentative gap in its epistemology, that Peacocke's account has not met the Integration Challenge. I then argue that we should modify the account's metaphysics of modal concepts in order to avoid implausible consequences with regards to their possession conditions. This modification generates no extra explanatory gap. Yet, once the minimal modification that avoids those implausible consequences is made, the resulting account cannot (...) support Peacocke's moderate rationalism. (shrink)

This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.

The paper compares the suitability of two different epistemologies of counterfactuals—(EC) and (W)—to elucidate modal knowledge. I argue that, while both of them explain the data on our knowledge of counterfactuals, only (W)—Williamson’s epistemology—is compatible with all counterpossibles being true. This is something on which Williamson’s counterfactual-based account of modal knowledge relies. A first problem is, therefore, that, in the absence of further, disambiguating data, Williamson’s choice of (W) is objectionably biased. A second, deeper problem is that (W) cannot satisfactorily (...) elucidate modal knowledge. Third, from a naturalistic perspective, the nature of this second problem favours (EC) against (W). (shrink)

Lange argues that some natural phenomena can be explained by appeal to mathematical, rather than natural, facts. In these “distinctively mathematical” explanations, the core explanatory facts are either modally stronger than facts about ordinary causal law or understood to be constitutive of the physical task or arrangement at issue. Craver and Povich argue that Lange’s account of DME fails to exclude certain “reversals”. Lange has replied that his account can avoid these directionality charges. Specifically, Lange argues that in legitimate DMEs, (...) but not in their “reversals,” the empirical fact appealed to in the explanation is “understood to be constitutive of the physical task or arrangement at issue” in the explanandum. I argue that Lange’s reply is unsatisfactory because it leaves the crucial notion of being “understood to be constitutive of the physical task or arrangement” obscure in ways that fail to block “reversals” except by an apparent ad hoc stipulation or by abandoning the reliance on understanding and instead accepting a strong realism about essence. (shrink)

This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.

Normal propositional modal systems This first chapter has two main aims. One is to give a general account of the propositional modal systems that we shall ...

Epistemic modal operators give rise to something very like, but also very unlike, Moore's paradox. I set out the puzzling phenomena, explain why a standard relational semantics for these operators cannot handle them, and recommend an alternative semantics. A pragmatics appropriate to the semantics is developed and interactions between the semantics, the pragmatics, and the definition of consequence are investigated. The semantics is then extended to probability operators. Some problems and prospects for probabilistic representations of content and context are explored.

Provisional draft, pre-production copy of my book “The Modal Future” (forthcoming with Cambridge University Press).

Abstract Timothy Williamson has recently proposed to undermine modal skepticism by appealing to the reducibility of modal to counterfactual logic ( Reducibility ). Central to Williamson’s strategy is the claim that use of the same non-deductive mode of inference ( counterfactual development , or CD ) whereby we typically arrive at knowledge of counterfactuals suffices for arriving at knowledge of metaphysical necessity via Reducibility. Granting Reducibility, I ask whether the use of CD plays any essential role in a Reducibility-based reply (...) to two kinds of modal skepticism. I argue that its use is entirely dispensable, and that Reducibility makes available replies to modal skeptics which show certain propositions to be metaphysically necessary by deductive arguments from premises the modal skeptic accepts can be known. Content Type Journal Article Pages 1-19 DOI 10.1007/s11098-011-9784-4 Authors Juhani Yli-Vakkuri, Wolfson College, Oxford University, Oxford, OX2 6UD UK Journal Philosophical Studies Online ISSN 1573-0883 Print ISSN 0031-8116. (shrink)

In a recent article forthcoming in *Mind*, Leech (2020) presents a challenge for essentialist accounts of metaphysical modality: why should it be that essences imply corresponding necessities? Leech’s main focus is to argue that one cannot overcome the challenge by utilizing an account of essence in terms of generalized identity due to Correia and Skiles (2019), on pain of circularity. In this reply, we will show how to use identity-based essentialism to bridge ‘epistemic’ and ‘explanatory’ understandings of this alleged (...) essence-to-necessity gap without circularity, Leech’s arguments notwithstanding. We do so by first presenting a novel proof that generalized identities imply corresponding necessities. We then propose several substantive identity-based explanations of how it is, exactly, that essences imply necessities. -/- . (shrink)

The practice of scientific modelling often resorts to hypothetical, false, idealised, targetless, partial, generalised, and other types of modelling that appear to have at least partially non-actual targets. In this paper, I will argue that we can avoid a commitment to non-actual targets by sketching a framework where models are understood as having networks of possibilities as their targets. This raises a further question: what are the truthmakers for the modal claims that we can derive from models? I propose that (...) we can find truthmakers for the modal claims derived from models in actuality, even in the case of supposedly non-actual targets. I then put this framework to use by examining a case study concerning the modelling of superheavy elements. (shrink)

This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.

According to the simple modal account of essence, an object has a property essentially just in case it has it in every world in which it exists. As many have observed, the simple modal account is implausible for a number of reasons. This has led to various proposals for strengthening the account, for example, by adding a restriction to the intrinsic or sparse properties. I argue, however, that these amendments to the simple modal account themselves fail. Drawing on lessons from (...) these failures, I propose a new version of a modal account, inspired by Ruth Barcan Marcus's defense of the coherence of quantified modal logic, according to which an object has a property essentially just in case (i) it has it in every world in which it exists, (ii) the property is discriminating (or non-trivial), and (iii) the property is qualitative. The resulting account of essence does not face any of the standard objections other accounts face, and I defend it from other potential objections. (shrink)

We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written φ) if φ is true in some extension of that structure, and φ is necessary (written φ) if it is true in all extensions of the structure. A principal case for us will be the class Mod(T) (...) of all models of a given theory T—all graphs, all groups, all fields, or what have you—considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, k-colorability, finiteness, countability, size continuum, size ℵ1, ℵ2, ℵω, ℶω, first ℶ-fixed point, first ℶ-hyper-fixed point, and much more. A graph obeys the maximality principle φ(a)→φ(a) with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs. (shrink)

We discuss the semantic significance of a puzzle concerning ‘ought’ and conditionals recently discussed by Kolodny and MacFarlane. We argue that the puzzle is problematic for the standard Kratzer-style analysis of modality. In Kratzer’s semantics, modals are evaluated relative to a pair of conversational backgrounds. We show that there is no sensible way of assigning values to these conversational backgrounds so as to derive all of the intuitions in Kolodny and MacFarlane’s case. We show that the appropriate verdicts can (...) be derived by extending Kratzer’s framework to feature a third conversational background and claiming that the relevant reading of ‘ought’ is sensitive to this parameter. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of (...) large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μ-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This volume contains eighteen papers, three with new postscripts, that were written over the past 35 years. Five of the papers have not been previously published. Together they provide a comprehensive account of modal reality—the realm of possible worlds—from a Humean perspective, with excursions into neighboring topics in metaphysics. Part 1 sketches an account of reality as a whole, both the mathematical and the modal, defending a form of plenitudinous realism: every consistent proposition is true of some portion of reality. (...) Part 2 presents and defends a realist theory of concrete possible worlds with an absolute ontological distinction between the actual and the merely possible. Part 3 presents and defends a Humean account of modal plenitude, formulating and endorsing principles of recombination, of plenitude of possible structures, and of plenitude of alien contents. Part 4 applies the Humean account to truthmaking, mereology, spacetime, and quantities. I argue that holding fast to Humean strictures leads to views that differ in radical ways from those put forth by contemporary metaphysicians. (shrink)

Recently, Stefan Wintein published an article in which he presents four objections to my modal-epistemic argument for the existence of God. His first objection is an alleged counterexample to the argument’s first premise, and the second objection is an alleged counterexample to the argument’s second premise. Wintein’s third objection attempts to show that the modal-epistemic argument is circular. Finally, the fourth objection is a parody objection. In this paper, I show that Wintein’s four objections all fail.

This book treats modal logic as a theory, with several subtheories, such as completeness theory, correspondence theory, duality theory and transfer theory and is intended as a course in modal logic for students who have had prior contact with modal logic and who wish to study it more deeply. It presupposes training in mathematical or logic. Very little specific knowledge is presupposed, most results which are needed are proved in this book.

This book presents the first study of the development of the theory of modal syllogistic in the Middle Ages.

According to the modal view, essence admits of reductive analysis in exclusively modal terms. Fine (1994) argues that modal view delivers an inadequate analysis of essence. This paper defends the modal view from Fine's challenge. This defense proceeds by examining the disagreement between Finean primitivists and Quinean eliminativists about essence. In order to model this disagreement, a distinction between essence and a separable concept, nature, is required. This distinction is then used to show that Fine's challenge is misdirected and therefore (...) unsuccessful. (shrink)

Kit Fine famously objected against the idea that essence can be successfully analyzed in terms of de re necessity. In response, I want to explore a novel, interesting, but controversial modal account of essence in terms of intrinsicality and grounding. In the first section, I will single out two theoretical requirements that any essentialist theory should meet—the essentialist desideratum and the essentialist challenge—in order to clarify Fine’s objections. In the second section, I will assess Denby’s improved modal account, which appeals (...) to the notion of intrinsicality, and argue that it is untenable. In the third section, I will explain how, when combined with a modal-existential criterion, a hyperintensional account of intrinsicality—in the same vein as Bader (J Philos 110(10): 525–563, 2013) and Rosen (in: Hale and Hoffman (eds) Modality: Metaphysics, logic, and epistemology, Oxford, OUP, 2010)—can help successfully address Fine’s counterexamples. In the fourth section, I will evaluate how this novel analysis of essence stands with respect to sortal, origin, and natural kinds essentialism and discuss potential objections and difficulties. (shrink)

This paper targets a series of potential issues for the discussion of, and modal resolution to, the alethic paradoxes advanced by Scharp (2013). I proffer four novel extensions of the theory, and detail five issues that the theory faces.

This book explores noteworthy approaches to modal syllogistic adopted by medieval logicians including Abélard, Albert the Great, Avicenna, Averröes, Jean Buridan, Richard Campsall, Robert Kilwardby, and William of Ockham. The book situates these approaches in relation to Aristotle's discussion in the Prior and Posterior Analytics, and other parts of the Organon, but also in relation to the thought of Alexander of Aphrodisias and Boethius on the one hand, and to modern interpretations of the modal syllogistic on the other. Problems explored (...) include: Aristotle's doctrine of modal conversion, the pure and mixed necessity-moods, modal ecthesis, the pure and mixed contingency-moods, and Aristotle's use of counter-examples. Medieval logicians brought various concepts to bear on these problems, including the distinction between per se and per accidens terms, the notion of essential predication, the distinction between ut nunc and simpliciter propositions, the distinction between de dicto and de re modals, and the notion of ampliation. All these are examined in this book. (shrink)

There are three primary aims of the book. The first, set out in the book's introduction, is to explain how two fairly recent developments in philosophy, externalism and modalism, provide the basis for a promising account of knowledge - an account that achieves anti-skeptical results and avoids Gettier-style counterexamples that are based on an agent having warranted beliefs that are merely luckily true. Epistemological externalism is the thesis that not all the factors that make a true belief a case of (...) knowledge need to be cognitively accessible or "internal" to the agent. Modalism, as the author construes it, is the use of possible worlds talk in clarifying what would or might be the case and what is necessarily so." "The second objective is to work through the different modalized epistemologies extant in the literature, assessing their strengths and weaknesses, with a view toward constructing a working theory. Finally, the author proposes the theory that knowledge is reliably formed, sensitive true belief, and defends the theory against objections."--Jacket. (shrink)

After introductory reminder of and comments on Gödel’s ontological proof, we discuss the collapse of modalities, which is provable in Gödel’s ontological system GO. We argue that Gödel’s texts confirm modal collapse as intended consequence of his ontological system. Further, we aim to show that modal collapse properly fits into Gödel’s philosophical views, especially into his ontology of separation and union of force and fact, as well as into his cosmological theory of the nonobjectivity of the lapse of time. As (...) a result, modal collapse should not be conceived in Gödel so much as a deficit, but rather as a kind of the rise of modalities to the “perfect” being. We further show that, in accordance with Gödel’s ontology, the concepts of modality and time should be derived in terms of the “fundamental philosophical concept” of cause. To give an example of how a formalization of such causative Gödelian ontology and ontological proof might look, we propose the transformation of GO into a kind of causally re-interpreted justification logic. (shrink)