In ‘Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability’, Acta Philosophica Fennica16 (1963), 153–167, Richard Montague shows that the use of a single syntactic predicate (with a context-independent semantic value) to represent modalities of alethic necessity and idealized knowledge leads to inconsistency. In ‘A Note on Syntactical Treatments of Modality’, Synthese44 (1980), 391–395, Richmond Thomason obtains a similar impossibility result for idealized belief: under a syntactical treatment of belief, the assumption (...) that idealized belief is deductively closed, together with certain other plausible conditions on idealized belief, imply that an ideal believer with consistent beliefs cannot believe the truth of Robinson's Arithmetic. In this essay I show that an impossibility result similar to Thomason's can be obtained which does not assume that belief is deductively closed or ideal in any other way. (shrink)

[IMPORTANT CORRECTION - See end of abstract.] In Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability, Acta Philosophica Fennica 16 (1963), 153–167, Richard Montague shows that the use of a single syntactic predicate (with a context-independent semantic value) to represent modalities of alethic necessity and idealized knowledge leads to inconsistency. In A Note on Syntactical Treatments of Modality, Synthese 44 (1980), 391–395, Richmond Thomason obtains a similar impossibility result for idealized belief: (...) under a syntactical treatment of belief, the assumption that idealized belief is deductively closed, together with certain other plausible conditions on idealized belief, imply that an ideal believer with consistent beliefs cannot believe the truth of Robinson's Arithmetic. In this essay I assert an impossibility result similar to Thomason's but which does not assume that belief is deductively closed or ideal in any other way. There are technical mistakes in the formulations of Lemmas 1-4 and Theorem 2, however. Corrections can be found in the Erratum section of the author's website. (shrink)

The systems T N and T M show that necessity can be consistently construed as a predicate of syntactical objects, if the expressive/deductive power of the system is deliberately engineered to reflect the power of the original object language operator. The system T N relies on salient limitations on the expressive power of the language L N through the construction of a quotational hierarchy, while the system T Mrelies on limiting the scope of the modal axioms schemas to the (...) sublanguage L infM +, which corresponds exactly with the restrictive hierarchy of L N. The fact that L infM + is identical to the image of the metalinguistic mapping C + from the normal operator system into L M reveals that iterated operator modality is implicitly hierarchical, and that inconsistency is produced by applying the principles of the modal logic to formulas which have no natural analogues in the operator development. Thus the contradiction discovered by Montague can be diagnosed as the result of instantiating the axiom schemas with modally ungrounded formulas, and thereby adding radically new modal axioms to the predicate system.The predicate treatment of necessity differs significantly from that of the operator in that the cumulative models for the predicate system are strictly first-order. Possible worlds are not used as model-theoretic primitives, but rather alternate models are appealed to in order to specify the extension of N, which is semantically construed as a first-order predicate. In this manner, the intensional aspects of modality are built into the mode of specifying the particular set of objects which the denotation function assigns to N, rather than in the specification of the basic truth conditions for modal formulas. Intensional phenomena are thereby localised to the special requirements for determining the extension of a particular predicate, and this does not constitute a structural modification of the first-order models, but rather limits the relevant class of models to those which possess an appropriate denotation function. (shrink)

This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework (...) of first-order arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368–387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338–355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287–304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy. (shrink)

This paper introduces the notion of syntactic feature to provide a unified treatment of earlier model theoretic proofs of both the compactness and interpolation theorems for a variety of two valued logics including sentential logic, first order logic, and a family of modal sentential logic includingM,B,S 4 andS 5. The compactness papers focused on providing a proof of the consequence formulation which exhibited the appropriate finite subset. A unified presentation of these proofs is given by isolating their essential feature and (...) presenting it as an abstract principle about syntactic features. The interpolation papers focused on exhibiting the interpolant. A unified presentation of these proofs is given by isolating their essential feature and presenting it as a second abstract principle about syntactic features. This second principle reduces the problem of exhibiting the interpolant to that of establishing the existence of a family of syntactic features satisfying certain conditions. The existence of such features is established for a variety of logics (including those mentioned above) by purely combinatorial arguments. (shrink)

Syntactical treatments of propositional attitudes are attractive to artificial intelligence researchers. But results of Montague (1974) and Thomason (1980) seem to show that syntactical treatments are not viable. They show that if representation languages are sufficiently expressive, then axiom schemes characterizing knowledge and belief give rise to paradox. Des Rivières and Levesque (1988) characterize a class of sentences within which these schemes can safely be instantiated. These sentences do not quantify over the propositional objects of knowledge (...) and belief. We argue that their solution is incomplete, and extend it by characterizing a more inclusive class of sentences over which the axiom schemes can safely range. Our sentences do quantify over propositional objects. (shrink)

This paper discusses the account of alethic modality as presented by Wilfrid Sellars in his earlier work from 1947 to 1958. Its aim is twofold. First, I discuss Sellars' analysis by exploring its historical relationship to Carnap's account of modality. I argue that Carnap's early syntactic treatment of modality profoundly influenced Sellars' own so-called ‘regulist' account of modality in terms of rules of inference. Furthermore, it is suggested that Sellars' lesser-known possible worlds analysis was influenced by Carnap's later semantic account (...) of modality. The second aim of the paper is a critical one. I raise a number of objections to Sellars’ account of modality and argue that his account confronts some serious limitations. I also argue that Sellars does not provide a satisfying explanation of the relationship between his regulist account and his possible worlds analysis. I offer such an explanation by arguing that his regulist account should be understood as a pragmatic account of what is conveyed by the use of modal sentences, whereas his possible worlds analysis is a semantic account of the asserted content of modal statements. That is, the regulist account and the possible worlds analysis have different explanatory targets. (shrink)

In 1945 J.C.C. McKinsey produced a ‘semantics’ for modal logic based on necessity defined in terms of validity. The present papers looks at how to update F.R. Drake’s completeness proof for McKinsey’s semantics by comparing McKinsey ‘models’ with the now standard Kripke models. It also looks at the motivation behind the system McKinsey called S4.1, but which we now call S4M; and use this motivation to produce a McKinsey semantics for that system. One lesson which emerges from this work is (...) an appreciation of the superiority of the current possible worlds semantics based on frames and models, both in terms of an intuitive understanding of modality, and also in terms of the ease of working with particular systems. (shrink)

This thesis is a study on the nature of modality in Hegel’s Logic and contemporary modal metaphysics. The thesis has two aims: Firstly, it examines Lewisian modal realism, as well as the post-Lewisian modal metaphysical accounts of modal actualism and modal essentialism in order to reveal that each position appeals to a non-modal foundation to make modal concepts explicit. Each position thus falls under what Hegel regards as pre-critical metaphysics by suggesting a modally unaccountable ground for modal concepts. The very (...) idea of a foundation to modality is problematic because as a foundation, it is necessarily assumed without further justification, and it directs our attention away from the nature of modality itself to its supposed foundations. These non-modal foundational take the form of possible worlds for modal realism, the form of properties for modal actualism and the form of essence for modal essentialism. To eschew foundationalism in modal metaphysics and provide a less questionable basis for modality, the thesis suggests Hegel’s treatment of modality as a viable alternative in virtue of its being free from the concerns stemming from the pre-critical way of doing metaphysics. The thesis situates Hegel in contemporary modal metaphysics by undertaking two types of exposition. By examining Hegel's treatment of essence and property, the thesis will show negatively that essence and property cannot be a foundation for modality, while by examining Hegel’s treatment of modality, the thesis positively proves that modal concepts are analysable and explainable within the domain of actuality, a domain that can only be made explicit by modal determinations. These two expositions provide an idea of a self-sustaining and self-explanatory modal metaphysics without appealing to non-modal foundationalist metaphysical commitments. More precisely, I argue that Hegel’s logical derivation of modal concepts constitutes a Hegelian modal metaphysics, which contains a comprehensive account of modal concepts that does not rely on any non-modal primitive. (shrink)

What is meant under the genuine title of Aristotle’s ta Analytika is rarely properly understood. Presumably, his analytics was inspired by the method of geometric analysis. For Aristotle, this was a regressive or heuristic procedure, departing from a proposed conclusion and asking which premises could be found in order to syllogize, demonstrate or explain it. The terms that form categorical and modal propositions play a fundamental role in analytics. Aristotle introduces letters in lieu of the triples of terms constitut­ing the (...) propositions and the three syllogistic figures that schematize them. His formulation of the three syllogistic figures refers to a syntacti­cal and predicative order and position of the triples of terms, arranged in some diagrammed schemata, which, regrettably, are missing from the extant text of the Prior Analytics. Considering planar and graphic arrangements, both vertical and horizontal orders as well as the posi­tion of the three terms involved, we propose a reconstruction, at least to some extent, of these probable lettered diagrams. In such reconstructed diagrams, we can appropriately capture the definition of syllogism as a predicative connexion of terms, and easier survey a synoptic account of all valid predicative relations and transpositions, and also reduce the imperfect syllogisms into the moods of the first figure. Aristotle’s syllogistic is an analytical calculation of terms, understood as predicates and subjects within the categorical propositions, and more precisely of three terms schematized in three figures in predicative links such that, by means of a middle, follows from necessity a conclusion of the extreme terms. The necessity of the consequence is not based on the implication or inference of the propositions, but on a predictive transi­tivity through the middle term within the syllogistic figures. Syllogism must draw its conclusion through the way its terms are predicated of one another. Aristotle in his Prior Analytics developed also a complex account of modal syllogisms within necessity and possibility of belonging. This account involves also such an analyti­cal reduction to the syllogistic figures. In this analytical perspective, we try to throw some light on his modal syllogisms, although this difficult and nowadays thoroughly discussed topic would require a much wider treatment. (shrink)

Much work has been done on specific instances of residuated lattices with modal operators . In this paper, we develop a general framework that subsumes three important classes of modal residuated lattices: interior algebras, Abelian ℓ-groups with conuclei, and negative cones of ℓ-groups with nuclei. We then use this framework to obtain results about these three cases simultaneously. In particular, we show that a categorical equivalence exists in each of these cases. The approach used here emphasizes the role played by (...) reducts in the proofs of these categorical equivalences. Lastly, we develop a connection between translations of logics and images of modal operators. (shrink)

In this work Ockham proposes a theory of simple predication, which he uses in explicating the truth conditions of progressively more complicated kinds of propositions. His discussion includes what he takes to be the correct semantic treatment of quantified propositions, past tense and future tense propositions, and modal propositions, all of which are receiving much attention from contemporary philosophers. He also illustrates the use of exponential analysis to deal with propositions that prove troublesome in both semantic theory and other disciplines, (...) such as metaphysics, physics, and theology. This type of analysis plays an essential role in his substantive philosophical and theological works, and in many cases then can hardly be understood without a prior acquaintance with this section of the Summa. An introduction by Alfred J. Freddoso clarifies and summarizes Ockham's theory, discusses certain philosophical problems which it engenders, and proposes new interpretations for parts of it. The essay will be helpful both to those interested in Ockham as a historical figure and to those interested in his substantive systematic contributions to logical theory. (shrink)

Since Montague’s work it is well known that treating a single modality as a predicate may lead to paradox. In their paper “No Future”, Horsten and Leitgeb show that if the two temporal modalities are treated as predicates paradox might arise as well. In our paper we investigate whether paradoxes of multiple modalities, such as the No Future paradox, are genuinely new paradoxes or whether they “reduce” to the paradoxes of single modalities. In order to address this (...) question we develop a notion of reducibility based on a version of Smoryński Diagonalized Operator Logic. We show that there are reducible multimodal paradoxes as well as irreducible paradoxes of interaction. In particular, we show the No Future paradox to be an irreducible paradox according to our notion of reducibility. (shrink)

The question of which is the logic that underlies quantum physics does not have an absolute answer, but only in relation to a conventional choice of interpretation . Most of the interpretations that have been offered work within the framework of classical logic. In contrast to these, we examine the corpuscular interpretation which is assumed in the application of non-distributive logic . The experiment in which single photons pass through a Mach-Zehnder interferometer is examined, indicating the difficulty of employing a (...) realist corpuscular interpretation in this context. One way to save this interpretation would be to use non-distributive logic to analyze the experiment, but this is not satisfactory . However, the use of an alethic modal logic solves the problem, blocking the argument that put the aforementioned corpuscular interpretation into difficulty. In the discussion of the conceptual problems involved, we suggest that a stochastic corpuscular interpretation is well adapted to this logical description. The project of extending this modal logical approach to other experiments in quantum physics, and of providing a rigorous logical treatment, is left open. (shrink)

Quantification over individuals, times, and worlds can in principle be made explicit in the syntax of the object language, or left to the semantics and spelled out in the meta-language. The traditional view is that quantification over individuals is syntactically explicit, whereas quantification over times and worlds is not. But a growing body of literature proposes a uniform treatment. This paper examines the scopal interaction of aspectual raising verbs (begin), modals (can), and intensional raising verbs (threaten) with quantificational subjects in (...) Shupamem, Dutch, and English. It appears that aspectual raising verbs and at least modals may undergo the same kind of overt or covert scope-changing operations as nominal quantifiers; the case of intensional raising verbs is less clear. Scope interaction is thus shown to be a new potential diagnostic of object-linguistic quantification, and the similarity in the scope behavior of nominal and verbal quantifiers supports the grammatical plausibility of ontological symmetry, explored in Schlenker (2006). (shrink)

In this paper the PA-completeness of modal logic is studied by syntactical and constructive methods. The main results are theorems on the structure of the PA-proofs of suitable arithmetical interpretationsS of a modal sequentS, which allow the transformation of PA-proofs ofS into proof-trees similar to modal proof-trees. As an application of such theorems, a proof of Solovay's theorem on arithmetical completeness of the modal system G is presented for the class of modal sequents of Boolean combinations of formulas of (...) the form p i,m i=0, 1, 2, ... The paper is the preliminary step for a forthcoming global syntactical resolution of the PA-completeness problem for modal logic. (shrink)

The purpose of this paper is to explore the issue of how the validity of the parallel inference is possible in view of its deep semantic-syntactic structure. I fi...

We introduce and study hierarchies of extensions of the propositional modal and temporal languages with pairs of new syntactic devices: point of reference-reference pointer which enable semantic references to be made within a formula. We propose three different but equivalent semantics for the extended languages, discuss and compare their expressiveness. The languages with reference pointers are shown to have great expressive power (especially when their frugal syntax is taken into account), perspicuous semantics, and simple deductive systems. For instance, Kamp's and (...) Stavi's temporal operators, as well as nominals (names, clock variables), are definable in them. Universal validity in these languages is proved undecidable. The basic modal and temporal logics with reference pointers are uniformly axiomatized and a strong completeness theorem is proved for them and extended to some classes of their extensions. (shrink)

Certain religious texts are deemed part of legal texts that are characterised by high sensitivity and sacredness. Amongst such religious texts are Islamic legal texts that are replete with Islamic legal terms that designate particular legal concepts peculiar to Islamic legal system and legal culture. However, from the syntactic perspective, Islamic legal texts prove lengthy and condensed, with an extensive use of coordinated, subordinate and relative clauses, which separate the main verb from the subject, and which, of course, carry a (...) heavy load of legal detail. The present paper seeks to examine the syntactic features of Islamic legal texts and the syntactic translation implications involved through studying three Islamic legal Arabic excerpts and their English translations. The paper argues that amongst the syntactic features of Islamic legal texts are nominalisation, participles, modals and complex structures. It also claims that the syntactic translation implications are indeed syntactic features of legal English, which are sentence combining versus sentence break, nominalisation, wh-deletion, passivisation, modals and multiple negations. Moreover, nominalisation, modality and complex structures are features of both Islamic legal texts and legal English, albeit with varying degrees. (shrink)

Crosslinguistically, the same modal words can be used to express a wide range of interpretations. This crosslinguistic trend supports a Kratzerian analysis, where each modal has a core lexical entry and where the difference between an epistemic and a root interpretation is contextually determined. A long-standing problem for such a unified account is the equally robust crosslinguistic correlation between a modal’s interpretation and its syntactic behavior: epistemics scope high (in particular higher than tense and aspect) and roots low, a fact (...) which has led to proposals that hardwire different syntactic positions for epistemics and roots (cf. Cinque’s hierarchy). This paper argues that the range of interpretations a modal receives is even more restricted: a modal must be keyed to certain time-individual pairs, but not others. I show that this can be captured straightforwardly by minimally modifying the Kratzerian account: modals are relative to an event—rather than a world—of evaluation, which readily provides a time (the event’s running time) and (an) individual(s) (the event’s participants). I propose that this event relativity of modals can in turn explain the correlation between type of interpretation and syntactic position, without stipulation of an interpretation-specific height for modals. (shrink)

In this paper, I examine phenomenologically the structure of the normative noema, which I call the “nomothetic noema.” I distinguish the meaning content, its normative characters, which I call “ductive forces”, and its modes of givenness. Next, I introduce the traditional difference between modalities de re and de dicto. I argue that the current tendency, in deontic logic, to treat deontic expressions as operators over sentences induces, at least on the syntactic surface, a de dicto reading. I then discuss (...) some of the theoretical decisions that underlie deontic logic, starting from two basic asymmetries between modal alethic logic and deontic logic. The decisive point is to recover a strong sense of permission that I equate with the factor of free agency, which is difficult to formalize. Finally, I return to the analysis of the nomothetic noema and try to show how a de re reading is better suited to account for normative intentionality. I simultaneously introduce the concept of status-imposing norms, which must be formulated with a normative use of the copulas “is” and “has.” I conclude that the treatment of normative expressions is more appropriate when they are understood as complex copulas, irreducible to assertive sentences, rather than as operators over whole sentences. I emphasize on this occasion that, from a phenomenological point of view and within the framework of genetic analysis, the primitive basic datum is the freedom of agency, on which the normative domain of assigning statuses to agents, obligations, and prohibitions is built. (shrink)

Decidability of monadic first-order classical logic was established by Löwenheim in 1915. The proof made use of a semantic argument, but a purely syntactic proof has never been provided. In the present paper we introduce a syntactic proof of decidability of monadic first-order logic in innex normal form which exploits G3-style sequent calculi. In particular, we introduce a cut- and contraction-free calculus having a (complexity optimal) terminating proof-search procedure. We also show that this logic can be faithfully embedded in the (...) modal logic T. (shrink)

This thesis concerns the source of modal truth. I aim to answer the question: what is it in virtue of which there are truths concerning what must have been the case as a matter of necessity, or could have been the case but isn't. I begin by looking at a dilemma put forward by Simon Blackburn which attempts to show that any realist answer to this question must fail, and I conclude that either horn of his dilemma can be resisted. (...) I then move on to clarify the nature of the propositions whose truth I am aiming to find the source of. I distinguish necessity de re from necessity de dicto, and argue for a counterpart theoretic treatment of necessity de re. As a result, I argue that there is no special problem concerning the source of de re modal facts. The problem is simply to account for what it is in virtue of which there are qualitative ways the world could have been, and qualitative ways it couldn't have been. I look at two ways to answer this question: by appealing to truthmakers in the actual world, or by appealing to non-actual ontology. I develop a theory of truthmakers, but argue that it is unlikely that there are truthmakers for modal truths among the ontology of the actual. I look at the main possibilist ontology, David Lewis' modal realism, but argue that warrant for that ontology is unobtainable, and that we shouldn't admit non-actual possibilia into our ontology. I end by sketching a quasi-conventionalist approach to modality which denies that there are modal facts, but nevertheless allows that we can speak truly when we use modal language. (shrink)

Aristotle studies syllogistic argumentation in Sophistical Refutations and Prior Analytics. In the latter he focuses on the formal and syntactic character of arguments and treats the sullogismoi and non-sullogismoi as argument patterns with valid or invalid instances. In the former Aristotle focuses on semantics and rhetoric to study apparent sullogismoi as object language arguments. Interpreters usually take Sophistical Refutations as considerably less mature than Prior Analytics. Our interpretation holds that the two works are more of a piece than previously believed (...) and, indeed, that Aristotle's treatment of fallacious reasoning presupposes the results of the formal theory. (shrink)

For some modal fixed point logics, there are deductive systems that enjoy syntactic cut-elimination. An early example is the system in Pliuskevicius [15] for LTL. More recent examples are the systems by the authors of this paper for the logic of common knowledge [5] and by Hill and Poggiolesi for PDL[8], which are based on a form of deep inference. These logics can be seen as fragments of the modal mu-calculus. Here we are interested in how far this approach can (...) be pushed in general. To this end, we introduce a nested sequent system with syntactic cut-elimination which is incomplete for the modal mu-calculus, but complete with respect to a restricted language that allows only fixed points of a certain form. This restricted language includes the logic of common knowledge and PDL. There is also a traditional sequent system for the modal mu-calculus by Jäger et al. [9], without a syntactic cut-elimination procedure. We embed that system into ours and vice versa, thus establishing cut-elimination also for the shallow system, when only the restricted language is considered. (shrink)

Review of the paper mentioned in the title.

In their searching contributions to this Symposium, Gideon Rosen, Timothy Williamson and Crispin Wright identify a set of issues crucial for assessing the principle-based treatment of modality I presented in Chapter Four of Being Known. I thank them for such focused and thoughtful discussions. This response is organized as a series of questions and proposed answers that aim to address the issues they raise. I hope their contributions will be as helpful to the reader as they have been to me (...) in understanding what is, and what is not, involved in the principle-based conception. My response will not consist only in defense and elaboration. On some issues, I will argue that the position formulated in Being Known needs supplementation. On others, especially those concerned with actualism and possibilia, it needs revision and a new positive theory. (shrink)

Punctuation has so far attracted attention within the linguistics community mostly from a syntactic perspective. In this paper, we give a preliminary account of the information-based aspects of punctuation, drawing our points from assorted, naturally occurring sentences. We present our formal models of these sentences and the semantic contributions of punctuation marks. Our formalism is a simplified analogue of an extension --- due to Nicholas Asher --- of Discourse Representation Theory.

In the propositional modal treatment of two-variable first-order logic equality is modelled by a ‘diagonal’ constant, interpreted in square products of universal frames as the identity relation. Here we study the decision problem of products of two arbitrary modal logics equipped with such a diagonal. As the presence or absence of equality in two-variable first-order logic does not influence the complexity of its satisfiability problem, one might expect that adding a diagonal to product logics in general is similarly harmless. We (...) show that this is far from being the case, and there can be quite a big jump in complexity, even from decidable to the highly undecidable. Our undecidable logics can also be viewed as new fragments of first-order logic where adding equality changes a decidable fragment to undecidable. We prove our results by a novel application of counter machine problems. While our formalism apparently cannot force reliable counter machine computations directly, the presence of a unique diagonal in the models makes it possible to encode both lossy and insertion-error computations, for the same sequence of instructions. We show that, given such a pair of faulty computations, it is then possible to reconstruct a reliable run from them. (shrink)