This article investigates the prospect of giving de dicto- and de re-necessity a uniform treatment. The historical starting point is a puzzle raised by Aristotle's claim, advanced in one of the modal chapters of his Prior Analytics, that universally privative apodeictic premises simply convert. As regards the Prior and the Posterior Analytics, the data suggest a representation of propositions of the type in question by doubly modally qualified formulae of modal predicate logic that display a necessity operator (...) in two distinct positions. Can the N-operator occurring in these positions be given a unified semantical treatment (which would justify dispensing with a notational differentiation)? A positive answer, based on a suitably shaped truth condition for N-formulae, is given, and is supported in the final section with an alternative proof theoretically based conception of a property's essential belonging to an individual. (shrink)

This paper is concerned with a propositional modal logic with operators for necessity, actuality and apriority. The logic is characterized by a class of relational structures defined according to ideas of epistemic two-dimensional semantics, and can therefore be seen as formalizing the relations between necessity, actuality and apriority according to epistemic two-dimensional semantics. We can ask whether this logic is correct, in the sense that its theorems are all and only the informally valid formulas. This (...) paper gives outlines of two arguments that jointly show that this is the case. The first is intended to show that the logic is informally sound, in the sense that all of its theorems are informally valid. The second is intended to show that it is informally complete, in the sense that all informal validities are among its theorems. In order to give these arguments, a number of independently interesting results concerning the logic are proven. In particular, the soundness and completeness of two proof systems with respect to the semantics is proven (Theorems 2.11 and 2.15), as well as a normal form theorem (Theorem 3.2), an elimination theorem for the actuality operator (Corollary 3.6), and the decidability of the logic (Corollary 3.7). It turns out that the logic invalidates a plausible principle concerning the interaction of apriority and necessity; consequently, a variant semantics is briefly explored on which this principle is valid. The paper concludes by assessing the implications of these results for epistemic two-dimensional semantics. (shrink)

Prior to Kripke's seminal work on the semantics of modal logic, McKinsey offered an alternative interpretation of the necessity operator, inspired by the Bolzano-Tarski notion of logical truth. According to this interpretation, `it is necessary that A' is true just in case every sentence with the same logical form as A is true. In our paper, we investigate this interpretation of the modal operator, resolving some technical questions, and relating it to the logical interpretation of modality and some (...) views in modal metaphysics. In particular, we present an hitherto unpublished solution to problems 41 and 42 from Friedman's 102 problems, which uses a different method of proof from the solution presented in the paper of Tadeusz Prucnal. (shrink)

We consider a version of so called T x W logic for historical necessity in the sense of R.H. Thomason (1984), which is somewhat special in three respects: (i) it is explicitly based on two-dimensional modal logic in the sense of Segerberg (1973); (ii) for reasons of applicability to interesting fields of philosophical logic, it conceives of time as being discrete and finite in the sense of having a beginning and an end; and (iii) it utilizes (...) the technique of systematic frame constants in order to handle the problem of irreflexivity in tense logics, well known since Gabbay (1981). Axiomatizations are given for two infinite hierarchies of two-dimensional modal tense logics, one without and one with the characteristic operators for historical necessity and possibility. Strong and weak completeness results are obtained for both hierarchies as well as a result to the effect that two approaches to their semantics are equivalent, much in the spirit of Di Maio and Zanardo (1996) and von Kutschera (1997). (shrink)

By the lights of a central logical positivist thesis in modal epistemology, for every necessary truth that we know, we know it a priori and for every contingent truth that we know, we know it a posteriori. Kripke attacks on both flanks, arguing that we know necessary a posteriori truths and that we probably know contingent a priori truths. In a reflection of Kripke's confidence in his own arguments, the first of these Kripkean claims is far more widely accepted than (...) the second. Contrary to received opinion, the paper argues, the considerations Kripke adduces concerning truths purported to be necessary a posteriori do not disprove the logical positivist thesis that necessary truth and a priori truth are co-extensive. (shrink)

In spite of a powerful tradition, more than two thousand years old, that in a valid argument the premises must be relevant to the conclusion, twentieth-century logicians neglected the concept of relevance until the publication of Volume I of this monumental work. Since that time relevance logic has achieved an important place in the field of philosophy: Volume II of Entailment brings to a conclusion a powerful and authoritative presentation of the subject by most of the top people working (...) in the area. Originally the aim of Volume II was simply to cover certain topics not treated in the first volume--quantification, for example--or to extend the coverage of certain topics, such as semantics. However, because of the technical progress that has occurred since the publication of the first volume, Volume II now includes other material. The book contains the work of Alasdair Urquhart, who has shown that the principal sentential systems of relevance logic are undecidable, and of Kit Fine, who has demonstrated that, although the first-order systems are incomplete with respect to the conjectured constant domain semantics, they are still complete with respect to a semantics based on "arbitrary objects." Also presented is important work by the other contributing authors, who are Daniel Cohen, Steven Giambrone, Dorothy L. Grover, Anil Gupta, Glen Helman, Errol P. Martin, Michael A. McRobbie, and Stuart Shapiro. Robert G. Wolf's bibliography of 3000 items is a valuable addition to the volume. Originally published in 1992. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

The idea that the epistemology of modality is in some sense a priori is a popular one, but it has turned out to be difficult to precisify in a way that does not expose it to decisive counterexamples. The most common precisifications follow Kripke’s suggestion that cases of necessary a posteriori truth that can be known a priori to be necessary if true ‘may give a clue to a general characterization of a posteriori knowledge of necessary truths’. The idea is (...) that whether it is contingent whether p can be known a priori for at least some broad range of sentences ‘p’. Recently, Al Casullo and Jens Kipper have discussed restrictions of such principles to atomic sentences. We show that decisive counterexamples even to such dramatically restricted Kripke-style principles can be constructed using minimal logical resources. We then consider further restrictions, and show that the counterexamples to the original principles can be turned into counterexamples to the further restricted principles. We conclude that, if there are any true restrictions of Kripke-style principles, then they are so weak as to be of little epistemological interest. (shrink)

We study a logic for deontic necessity and sufficiency, as originally proposed in van Benthem :36–41, 1979). Building on earlier work in modal logic, we provide a sound and complete axiomatization for it, consider some standard extensions, and study other important properties. After that, we compare this logic to the logic of “obligation as weakest permission” from Anglberger et al. :807–827, 2015).

We study a logic for deontic necessity and sufficiency, as originally proposed in van Benthem :36–41, 1979). Building on earlier work in modal logic, we provide a sound and complete axiomatization for it, consider some standard extensions, and study other important properties. After that, we compare this logic to the logic of “obligation as weakest permission” from Anglberger et al. :807–827, 2015).

1. Introductory. The notions of existence and necessity have held the interest of philosophers longer than many other problems in the philosophy of logic. The nature of necessity has been debated since the ancient Greeks; and many philosophers have pronounced their opinions on whether ‘existence is a predicate’. In this essay, I shall discuss the notion of existence. In its planned sequel, the same methods will be applied to the concepts of necessity and possibility.

It is often remarked that Indian logic (IL) has no conception of necessity. But what kind of necessity is absent in this system? Logical necessity is presumably absent: the structure of the logical argument in IL is often given as a reason for this claim. However even a cursory understanding of IL illustrates an abiding attempt to formulate the idea of necessity. In Dharmakīrti's classification of inferences, one can detect the formal process of entailment in (...) the inferences arising from class inclusion. In Western philosophy, Leibniz's invocation of 'contingent necessity' as distinguished from the 'necessary necessity' is part of a tradition that finds value in the idea of contingent necessity. In contemporary philosophy, this has been championed by Armstrong, specifically in the context of understanding the necessity inherent in scientific laws. In IL, the analysis of 'invariable concomitance' (vyāpti) is of crucial importance and its definitions are very complex. This paper argues how vyāpti can be understood in terms of contingent necessity and also how the complex definitions can be interpreted as an attempt to define contingent necessity in terms of 'logical' necessity. (shrink)

This book, one of the first full-length studies of the modalities to emerge from the debate to which Saul Kripke, David Lewis, Ruth Marcus, and others are contributing, is an exploration and defense of the notion of modality de re, the idea that objects have both essential and accidental properties. Plantinga develops his argument by means of the notion of possible worlds and ranges over such key problems as the nature of essence, transworld identity, negative existential propositions, and the existence (...) of unactual objects in other possible worlds. He also applies his logical theories to the elucidation of two problems in the philosophy of religion: the problem of evil and the ontological argument. (shrink)

Interest in the metaphysics and logic of possible worlds goes back at least as far as Aristotle, but few books address the history of these important concepts. This volume offers new essays on the theories about the logical modalities held by leading philosophers from Aristotle in ancient Greece to Rudolf Carnap in the twentieth century. The story begins with an illuminating discussion of Aristotle's views on the connection between logic and metaphysics, continues through the Stoic and mediaeval traditions, (...) and then moves to the early modern period with particular attention to Locke and Leibniz. The views of Kant, Peirce, C. I. Lewis and Carnap complete the volume. Many of the essays illuminate the connection between the historical figures studied, and recent or current work in the philosophy of modality. The result is a rich and wide-ranging picture of the history of the logical modalities. (shrink)

In `Essence and Modality', Kit Fine proposes that for a proposition to be metaphysically necessary is for it to be true in virtue of the nature of all objects whatsoever. Call this view Fine's Thesis. This paper is a study of Fine's Thesis in the context of Fine's logic of essence (LE). Fine himself has offered his most elaborate defense of the thesis in the context of LE. His defense rests on the widely shared assumption that metaphysical necessity (...) obeys the laws of the modal logic S5. In order to get S5 for metaphysical necessity, he assumes a controversial principle about the nature of all objects. I will show that the addition of this principle to his original system E5 leads to inconsistency with an independently plausible principle about essence. In response, I develop a theory that avoids this inconsistency while allowing us to maintain S5 for meta- physical necessity. However, I conclude that our investigation of Fine's Thesis in the context of LE motivates the revisionary conclusion that metaphysical necessity obeys the principles of the modal logic S4, but not those of S5. I argue that this constitutes a distinctively essentialist challenge to the received view that the logic of metaphysical necessity is S5. (shrink)

Operators can be compared in at least two respects: expressive strength and deductive strength. Inspired by Hintikka’s treatment of question embedding verbs, the variations of noncontingency operator, and also the various combinations of modal operators and Boolean connectives, we propose a logic with strong noncontingency operator as the only primitive modality. The novel operator is deductively but not expressively stronger than both noncontingency operator and essence operator, and expressively but not deductively weaker than the necessity operator. The frame-definability (...) power of this new logic is in between standard modal logic and noncontingency logic. A notion of bisimulation is proposed to characterize this logic within standard modal logic and first-order logic. Axiomatizations over various frame classes are presented, among which the minimal logic is related to the treatment of an alternative semantics of the agreement operator proposed by Lloyd Humberstone. (shrink)

The title of this and proposed second volume presents the basic idea which unifies the wide variety of topics developed and investigated by the principal authors, major contributing authors, J. M. Dunn and Robert K. Meyer, and eleven other contributors. The other contributors are: J. R. Chidgey, J. A. Coffa, Dorthy L. Grover, Bas van Fraassen, H. Leblanc, Storrs McCall, A. Parks, G. Pottinger, R. Routley, A. Urquhart, and R. G. Wolf. From both the useful analytic table of contents and (...) from the section titles it is clear which contributing author has written each section. Chapter I, "The Pure Calculus of Entailment," suggests how logics of entailment are logics of relevance and necessity. When we explain why a large number of systems are investigated, it will be clear why we write of logics of entailment instead of the logic of entailment. The principal authors build a case in Chapter I, and elsewhere, that a genuinely valid formal inference, viz., a formal entailment, from a formula A to a formula B requires that A be relevant to B in addition to it being impossible that there be an interpretation making A true and B false. So, a claim that A entails B tells us, explicitly or implicitly, that A is relevant to B and that "if A then B" is necessarily true, i.e., A strictly implies B. Consequently since a logic of entailment presents allegedly unfalsifiable claims, i.e., theses, using entailment claims, a logic of entailment presents theses about relevance between antecedents and consequents and necessity. It should be noted that the systems of logic developed are for formal entailments; they may not be logics for material entailments such as: being a cube entails having six surfaces. (shrink)

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. (...) What George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency. (shrink)

We construct a Hilbert style system RPL for the notion of plausibility measure introduced by Halpern J, and we prove the soundness and completeness with respect to a neighborhood style semantics. Using the language of RPL, we demonstrate that it can define well-studied notions of necessity, conditionals and propositional identity.

We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of *being a necessity*, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, (...) and questions about their logical behaviour. We also wield the framework to probe the conditions under which a logicist account of necessities is possible, in which the theory is completely reducible to logic. (shrink)

In the Posterior Analytics (I 6, 75a18–27) Aristotle discusses a puzzle which endangers the possibility of inferring a non-necessary conclusion. His solution relies on the distinction between the necessity of the conclusion's being the case and the necessity of admitting the conclusion once one has admitted the premisses. The former is a factual necessity, whereas the latter is meant to be a normative or deontic necessity that is independent of the facts stated by the premisses and (...) the conclusion. This paper maintains that Aristotle resorts to this distinction because he thinks that, as long as it is conceived as a factual relation, logical consequence cannot exist independently of the facts expressed by the premisses and the conclusion. As a corollary, the necessity of such a consequence relation always requires the necessity of these facts. Aristotle holds this factual conception of logical consequence responsible for the puzzle, since it cannot account for valid syllogisms with contingent or false premisses. The alternative conception of necessity is then introduced by him in order to make good this deficiency. The distinction between the necessity of being and the necessity of saying was revived by the Oxford logician E. W. B. Joseph, and taken over by Frank Ramsey in his seminal Truth and Probability, but has not received attention from recent interpreters of Aristotle's logic. This paper, however, argues that, in spite of its intrinsic interest, the distinction bore no significant fruit in Aristotle's logical doctrine. (shrink)

I investigate two asymmetrical approaches to knowledge of absolute possibility and of necessity--one which treats knowledge of possibility as more fundamental, the other according epistemological priority to necessity. Two necessary conditions for the success of an asymmetrical approach are proposed. I argue that a possibility-based approach seems unable to meet my second condition, but that on certain assumptions--including, pivotally, the assumption that logical and conceptual necessities, while absolute, do not exhaust the class of absolute necessities--a necessity-based approach (...) may be able to do so. (shrink)

We consider the modal logic of non-contingency in a general setting, without making special assumptions about the accessibility relation. The basic logic in this setting is axiomatized, and some of its extensions are discussed, with special attention to the expressive weakness of the language whose sole modal primitive is non-contingency , by comparison with the usual language based on necessity.

In his influential paper ‘‘Essence and Modality’’, Kit Fine argues that no account of essence framed in terms of metaphysical necessity is possible, and that it is rather metaphysical necessity which is to be understood in terms of essence. On his account, the concept of essence is primitive, and for a proposition to be metaphysically necessary is for it to be true in virtue of the nature of all things. Fine also proposes a reduction of conceptual and logical (...) necessity in the same vein: a conceptual necessity is a proposition true in virtue of the nature of all concepts, and a logical necessity a proposition true in virtue of the nature of all logical concepts. I argue that the plausibility of Fine's view crucially requires that certain apparent explanatory links between essentialist facts be admitted and accounted for, and I make a suggestion about how this can be done. I then argue against the reductions of conceptual and logical necessity proposed by Fine and suggest alternative reductions, which remain nevertheless Finean in spirit. (shrink)

I introduce a proof system for the logic of relative fundamentality, as well as a natural semantics with respect to which the system is both sound and complete. I then “modalise” the logic, and finally I discuss the properties of grounding given a suggested account of this notion in terms of necessity and relative fundamentality.

This book sets out a Kant-inspired theory of modality, driven by a methodology which takes seriously questions about the function of modal judgment as a guide to a metaphysics of modality. It argues that we need logical modal concepts as a condition on our ability to think, and metaphysical modal concepts as a condition on our ability to think objectively. Concordant with this, it argues that logical necessity has its source in the laws of thought and that metaphysical (...) class='Hi'>necessity is relative to conditions on objective thought. This this account of metaphysical necessity, which is termed “Modal Transcendentalism”, is then further developed, covering questions concerning necessary and contingent existence, de re necessity, essentialism, and modal epistemology. (shrink)