The workshop took place in Leuven, Belgium, and was hosted by the KU Leuven's Centre for Logic and Analytic Philosophy. The workshop’s theme was the syntactical treatment of (alethic, epistemic, etc.) modalities. The standard view on modalities nowadays is that they are operators. Syntactic theories, however, treat modalities as predicates, and thus have to assume a background theory which is sufficiently strong to encode its own formulas (usually, one works with some system of arithmetic and Gödel (...) coding). As a consequence, such theories suffer from paradoxes of self-referentiality. For example, just as the liar sentence states of itself that it is false, the knower sentence states of itself that it is unknown. Kaplan and Montague (1960: 'A paradox regained', Notre Dame J. Formal Logic, vol. 1, pp. 79–90) famously showed that any sufficiently strong theory that contains the knower is inconsistent. (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)

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)

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 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)

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)

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)

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)

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)

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)

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 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)

Ethical theorists often assume that the verb ‘ought’ means roughly ‘has an obligation’; however, this assumption is belied by the diversity of ‘flavours’ of ought-sentences in English. A natural response is that ‘ought’ is ambiguous. However, this response is incompatible with the standard treatment of ‘ought’ by theoretical semanticists, who classify ‘ought’ as a member of the family of modal verbs, which are treated uniformly as operators. To many ethical theorists, however, this popular treatment in linguistics seems to elide an (...) important distinction between agential and non-agential ought-statements. The thought is that ‘ought’ must have at least two senses, one implicating agency and connected to obligations, and another covering other uses. In this paper, I pursue some resolution of this tension between semantic theory and ethical theory with respect to the meaning of ‘ought’. To this end, I consider what I believe to be the most linguistically sophisticated argument for the view that the word ‘ought’ is ambiguous between agential and non-agential senses. This argument, due to Mark Schroeder, is instructive but based on a false claim about the syntax of agential ought-sentences—or so I attempt to show by first situating Schroeder's argument in its proper linguistic background and then discussing some syntactic evidence that he fails to appreciate. Then, I use the failure of this argument to motivate some more general reflections on how the standard treatment of ‘ought’ by theoretical semanticists might be refined in the light of the distinction important to ethical theory between agential and non-agential ought-statements, but also on how ethical theory might benefit from more careful study of the dominant treatment of modals as operators in theoretical semantics. (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 the recent literature of the philosophy of science, much space has been given to the problem of analyzing theories of the deductive and natural sciences in a way which makes explicit some of the syntactic and semantic features which seem to be implicitly present in their structures. This pa- per is concerned with the same problem; however, some other problems of syntax and semantics are touched upon along the way. After some prelim- inaries, a very general method of constructing (...) symbolic languages is intro- duced. The resulting languages are interpreted in either empty or non- empty sets and so in a way which permits them to contain terms which stand for nothing at all and an existence predicate which is not applicable to all terms. Also, certain variable binders are interpreted in a way which leaves open the possibility of interpreting them modally and a semantic operation of degree of truth for arbitrary formulas is introduced. An axiomless logic commensurate with a semantics of this kind and so one which contains a new treatment of definite descriptions is then constructed and shown to be both sound and semantically complete. Finally, theories and various concepts pertaining to theories are defined on the basis of this syntax and semantics. Among the concepts are operations which assign degrees of confirmation, explanatory powers, degrees of deductive simplic- ity, and degrees of adequacy to theories, relations and concreteness and abstractness for expressions, a relation of significance for theories, and operations which assign denotations and meanings to expressions. (shrink)

Kaplan, Stalnaker and Wettstein all urge a two-stage theory of language whereon the propositions expressed by sentences are generated prior to being evaluated. A new ambiguity for sentences emerges, propositional rather syntactic or semantic. Kaplan and Wettstein then propose to explain Donnellan's referential/attributive ambiguity as simply being two-stage propositional ambiguity. This is tacitly seen as further confirmation for two-stage theory. Modal ambiguities are prime motivators for two-stage theory which distinguishes local from exotic evaluation to explain them. But if sentences can (...) be found which exhibit both modal and referential/attribute ambiguity, an apparent paradox arises for a two-stage account. The theory recognizes both singular and general propositions, in Kaplan's senses. But reflecting one sense of such a doubly ambiguous sentence, two-stage theory would seem to need a proposition both singular and general with respect to a definite description attributively used. Since modal operators will come into rendering the problem sentences, an obvious idea is to let scope distinctions rescue two-stage theory from the apparent paradox. But while a rescue based on multiple renderings is proposed, it is not strictly a scope rescue, though different scopes are involved. Readers are asked to trust the author on missing formalities of an intuitively transparent two-sorted modal language that is employed. Two-stage theorists explicitly oppose scope treatments of modal ambiguities seeing them as rivals. Stalnaker, in particular, argues against them. But his arguments are shown not to count against the proposed rescue, on which the anticipated rivalry proves to be minimal. (shrink)

Perhaps the most pressing challenge for singularism—the predominant view that definite plurals like ‘the students’ singularly refer to a collective entity, such as a mereological sum or set—is that it threatens paradox. Indeed, this serves as a primary motivation for pluralism—the opposing view that definite plurals refer to multiple individuals simultaneously through the primitive relation of plural reference. Groups represent one domain in which this threat is immediate. After all, groups resemble sets in having a kind of membership-relation and iterating: (...) we can have groups of groups, groups of groups of groups, etc. Yet there cannot be a group of all non-self-membered groups. In response, we develop a potentialist theory of groups according to which we always can, but do not have to, form a group from any sum. Modalizing group-formation makes it a species of potential, as opposed to actual or completed, infinity. This allows for a consistent, plausible, and empirically adequate treatment of natural language plurals, one which is motivated by the iterative nature of syntactic and semantic processes more generally. (shrink)

We started from the fact that type theory, in the way it was implemented in IL, makes it costly to deal with nominalization processes. We have also argued that the type hierarchy as such doesn't play any real role in a grammar; the classification it provides for different semantic objects is already contained, in some sense, in the categorial structure of the grammar itself. So, on the basis of a theory of properties (Cocchiarella's HST*) we have tried to build a (...) language (IL*) whose syntax does not contain any explicit typing of expressions. Some of the consequences that this move brings about in the overall organization of the grammar can be summarized as follows:it allows for a simple treatment of infinitives, gerunds, factives and, in general, all those phenomena which might be analyzed as cases of nominalization;it provides a simpler and more constrained semantics than IL, since IL* doesn't go beyond second order, and its non-modal basis is axiomatizable;it suggests that the role of logical form in a theory of grammar could be that of a family of theories of semantic objects;it eliminates the extrinsic limitations of a type hierarchy on the choice of the system of syntactic categories.I think that it is interesting to notice how having an explicit semantic framework helps to provide a sense in which it is legitimate to regard the syntax of a language as ‘autonomous’. (shrink)

Human communication is multi-modal. It is an empirical fact that many of our acts of communication exploit a variety of means to make our communicative intentions recognisable. Scholars readily distinguish between verbal and non-verbal means of communication, and very often they deal with them separately. So it is that a great number of semanticists and pragmaticists give verbal communication preferential treatment. The non-verbal aspects of an act of communication are treated as if they were not underlain by communicative intentions. They (...) are “relegated” to mere aspects of the context. However, several schools of thoughts have a different take on the issue. Thus psychologists or semioticians of gesture have shown how intricately gestures and speech are related in utterances. And, in a different area of the theoretical landscape, so-called “Relevance Theorists” have made the same point. Thus, Robyn Carston writes that “the domain of pragmatics is a natural class of environmental phenomena, that of ostensive stimuli; verbal utterances are the central case, but not the only one, and they themselves are frequently accompanied by other ostensive gestures of the face, hands, voice etc, all of which have to be interpreted together if one is to correctly infer what is being communicated”. This position rests on the assumption that there is a single “pragmatic system” or module at work in the interpretation of “ostensive stimuli”. When it comes to interpreting verbal stimuli, the same mechanisms and resources are used as when it comes to processing non-verbal ones. If there is no distinct “linguistic pragmatic system”, then the scholar who studies communication should not favour the verbal at the expense of the non-verbal. In this paper, I want to make a contribution to the study of multi-modal messages by considering a type of utterances that mix the verbal with the non-verbal in such a way that a piece of non-linguistic communication seems to stand in for a linguistic constituent which remains unrealised. Here is a real-life example : I didn't see the [IMITATION OF FRIGHTENING GRUMPINESS] woman today; will she be back this week? The square-bracketed string in small capitals is meant to capture the facial expressions and gestures performed in the conversational setting. What is intriguing here is that this instance of ostensive mimicry does not come as a mere complement to some linguistic stimulus; it appears to take the place of that stimulus. I shall try to show that a linguistic analysis can indeed be offered for cases like – though, I believe, without succumbing to the pro-linguistic bias that Carston warns against. I will, however, argue against an ellipsis-based account : the structure of sentence does not contain an unrealised adjective phrase. Instead, I shall defend a ‘syntactic-recruitment' account. (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)

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)

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...

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 62.1 (2001) 121-140 [Access article in PDF] The Rhetoric of Modal Equivocacy in Cartesian Transubstantiation Julian Bourg Everyday language, in which words are not defined, is a medium in which nobody can express himself unequivocally. Robert Musil 1René Descartes's attempt to explain Eucharistic transubstantiation has long been understood as a dramatically significant moment in his tightrope walk across the medieval-to-modern divide. 2 Modeled (...) on the Last Supper in the New Testament, the Eucharistic rite had been explained in the Catholic tradition after the twelfth century in the scholastic philosophical language of substance and accident. 3 As the Council of Trent reaffirmed in 1551, this transubstantiation was a conversion "of the whole substance of the bread into the substance of the Body and of the whole substance of the wine into the substance of His blood... the appearances [speciebus] only of the bread and wine remaining." 4 Christ was [End Page 121] held to be "truly, really, and substantially contained under the species of those sensible things," under what still appeared to be bread and wine. 5Descartes's dilemma was that on one hand he was dead set against the hylomorphic theory of substance and accident, and on the other hand he found himself obliged by some of his interlocutors to comment on transubstantiation. It was a balancing act that pleased no one for very long, since despite his sometimes eager discussions of the Eucharist, he was not, as he admitted, "a theologian by profession." 6 The putative heterodoxy of Cartesian transubstantiation was principally responsible for his works being placed on the Index of Prohibited Books in 1663 and for his thought being banned from schools under Louis XIV in 1671. 7 His treatment of the Eucharistic question also distressed others in the seventeenth and early eighteenth centuries, such as Thomas Hobbes and Isaac Newton, who were unhappy that Descartes had bothered with such a theological non-issue. 8 For the many post-seventeenth-century advocates of a proto-secular Descartes as the godfather of modern philosophy, Cartesian transubstantiation has proved similarly inconvenient.The indisputable connections between Descartes and his scholastic forerunners, exemplified by the Eucharistic question, service a basic question of contemporary Cartesian scholarship. As Émile Boutroux put it, how was new wine poured from old bottles? 9 Accepting as axiomatic that there was something both radical and original in the Cartesian project but without falling back on pieties about his turn toward the modern subject, we must ask how Descartes put forth novel claims in conceptual language regarded by his contemporaries as resonant, if not consonant, with the scholastic intellectual culture of Catholic Reform France. More precisely, setting aside the not-very-defensible position that such language was merely incidental to Cartesian thought (a position that dehistoricizes Descartes as a precocious Enlightener), the dilemma then becomes explaining how original positions were elaborated within scholastic language. One important task is accounting for what Descartes did with that vocabulary--how he pointed words and concepts in new and surprising directions. The Eucharist becomes crucial in this perspective. [End Page 122]Though the question of Cartesian transubstantiation has been discussed for some time, the topic is worth revisiting again in light of two themes, rhetoric and the Cartesian theory of modes. My argument is that Descartes used one particular scholastic term, "mode," in equivocal ways and that this equivocation was in multiple senses rhetorical. This rhetoric coalesced and came to the fore around the Eucharist because, given the intellectual-political climate of the Catholic Reformation, the stakes of transubstantiation were so high. That Descartes used certain scholastic terms in equivocal or ambiguous ways should come as no surprise. Of interest, though, is the fact that by definition "mode" meant many things. This ambiguous term facilitated ambiguous intellectual positions about the Eucharist that allowed Descartes more generally (no matter what he "really" intended) to skirt between an orthodoxy that represented the past and an originality, not without dangers in his time, that seem to gesture toward our day. Foregrounded with regard to... (shrink)

By Solovay’s celebrated completeness result [31] on formal provability we know that the provability logic ${\textbf {GL}}$ describes exactly all provable structural properties for any sound and strong enough arithmetical theory with a decidable axiomatisation. Japaridze generalised this result in [22] by considering a polymodal version ${\mathsf {GLP}}$ of ${\textbf {GL}}$ with modalities $[n]$ for each natural number n referring to ever increasing notions of provability. Modern treatments of ${\mathsf {GLP}}$ tend to interpret the $[n]$ provability notion as (...) “provable in a base theory T together with all true $\Pi ^0_n$ formulas as oracles.” In this paper we generalise this interpretation into the transfinite. In order to do so, a main difficulty to overcome is to generalise the syntactical characterisations of the oracle formulas of complexity $\Pi ^0_n$ to the hyper-arithmetical hierarchy. The paper exploits the fact that provability is $\Sigma ^0_1$ complete and that similar results hold for stronger provability notions. As such, the oracle sentences to define provability at level $\alpha $ will recursively be taken to be consistency statements at lower levels: provability through provability whence the name of the paper. The paper proves soundness and completeness for the proposed interpretation for a wide class of theories, namely for any theory that can formalise the recursion described above and that has some further very natural properties. Some remarks are provided on how the recursion can be formalised into second order arithmetic and on lowering the proof-theoretical strength of these systems of second order arithmetic. (shrink)

Decidability of monadic first-order classical logic was established by Löwenheim in 1915. The proof made use of a semantic argument and 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)

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)

In his original semantics for counterfactuals, David Lewis presupposed that the ordering of worlds relevant to the evaluation of a counterfactual admitted no incomparability between worlds. He later came to abandon this assumption. But the approach to incomparability he endorsed makes counterintuitive predictions about a class of examples circumscribed in this paper. The same underlying problem is present in the theories of modals and conditionals developed by Bas van Fraassen, Frank Veltman, and Angelika Kratzer. I show how to reformulate all (...) these theories in terms of lower bounds on partial preorders, conceived of as maximal antichains, and I show that treating lower bounds as cutsets does strictly better at capturing our intuitions about the semantics of modals, counterfactuals, and deontic conditionals. (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)

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)

A general characterization of logical opposition is given in the present paper, where oppositions are defined by specific answers in an algebraic question-answer game. It is shown that opposition is essentially a semantic relation of truth values between syntactic opposites, before generalizing the theory of opposition from the initial Apuleian square to a variety of alter- native geometrical representations. In the light of this generalization, the famous problem of existential import is traced back to an ambiguous interpretation of assertoric sentences (...) in Aristotle's traditional logic. Following Abelard’s distinction between two alternative readings of the O-vertex: Non omnis and Quidam non, a logical difference is made between negation and denial by means of a more fine- grained modal analysis. A consistent treatment of assertoric oppositions is thus made possible by an underlying abstract theory of logical opposition, where the central concept is negation. A parallel is finally drawn between opposition and consequence, laying the ground for future works on an abstract operator of opposition that would characterize logical negation just as does Tarski’s operator of consequence for logical truth. (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 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)

This volume explores the linguistic expression of modality in natural language from a cross-linguistic perspective. Modal expressions provide the basic tools that allow us to dissociate what we say from what is actually going on, allowing us to talk about what might happen or might have happened, as well as what is required, desirable, or permitted. Chapters in the book demonstrate that modality involves many more syntactic categories and levels of syntactic structure than traditionally assumed. The volume distinguishes between three (...) types of modality: 'low modality', which concerns modal interpretations associated with the verbal and nominal cartographies in syntax; 'middle modality', or modal interpretation associated to the syntactic cartography internal to the clause; and 'high modality', relating to the left periphery. It combines cross-linguistic discussions of the more widely-studied sources of modality with analyses of novel or unexpected sources, and shows how the meanings associated with the three types of modality are realized across a wide range of languages. (shrink)

The success of deep brain stimulation (DBS) for movement disorders and the improved understanding of the neurobiologic and neuroanatomic bases of psychiatric diseases have led to proposals to expand current DBS applications. Recent preclinical and clinical work with Alzheimer's disease and obsessive-compulsive disorder, for example, supports the safety of stimulating regions in the hypothalamus and nucleus accumbens in humans. These regions are known to be involved in addiction and overeating associated with obesity. However, the use of DBS targeting these areas (...) as a treatment modality raises common ethical considerations, which include informed consent, coercion, enhancement, threat to personhood, and manipulation of the reward center. Pilot studies for both of these conditions are currently investigational. If these studies show promise, then there is a need to address the ethical concerns related to the initiation of clinical trials including the reliability of preclinical evidence, patient selection, study design, compensation for participation and injury, cost-effectiveness, and the need for long-term follow-up. Multidisciplinary teams are necessary for the ethical execution of such studies. In addition to establishing safety and efficacy, the consideration of these ethical issues is vital to the adoption of DBS as a treatment for these conditions. We offer suggestions about the pursuit of future clinical trials of DBS for the treatment of addiction and overeating associated with obesity and provide a framework for addressing ethical concerns related to treatment. (shrink)