In this paper we study the expressive power and definability for modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt–Thomason definability theorem in terms of the well-established first-order topological language.

Several extensions of the basic modal language are characterized in terms of interpolation. Our main results are of the following form: Language ℒ' is the least expressive extension of ℒ with interpolation. For instance, let ℳ be the extension of the basic modal language with a difference operator [7]. First-order logic is the least expressive extension of ℳ with interpolation. These characterizations are subsequently used to derive new results about hybrid logic, relation algebra and the (...) guarded fragment. (shrink)

Does general validity or real world validity better represent the intuitive notion of logical truth for sentential modal languages with an actuality connective? In (Philosophical Studies 130:436–459, 2006) I argued in favor of general validity, and I criticized the arguments of Zalta (Journal of Philosophy 85:57–74, 1988) for real world validity. But in Nelson and Zalta (Philosophical Studies 157:153–162, 2012) Michael Nelson and Edward Zalta criticize my arguments and claim to have established the superiority of real world validity. Section (...) 1 of the present paper introduces the problem and sets out the basic issues. In Sect. 2 I consider three of Nelson and Zalta’s arguments and find all of them deficient. In Sect. 3 I note that Nelson and Zalta direct much of their criticism at a phrase (‘true at a world from the point of view of some distinct world as actual’) I used only inessentially in Hanson (Philosophical Studies 130:436–459, 2006), and that their account of the philosophical foundations of modal semantics leaves them ill equipped to account for the plausibility of modal logics weaker than S5. Along the way I make several general suggestions for ways in which philosophical discussions of logical matters–especially, but not limited to, discussions of truth and logical truth for languages containing modal and indexical terms–might be facilitated and made more productive. (shrink)

A system of modal logic with the operator is proposed, and proved complete. In contrast with a previous one by Stalnaker and Thomason, this system does not require two categories of singular terms.

Many powerful logics exist today for reasoning about multi-agent systems, but in most of these it is hard to reason about an infinite or indeterminate number of agents. Also the naming schemes used in the logics often lack expressiveness to name agents in an intuitive way.To obtain a more expressive language for multi-agent reasoning and a better naming scheme for agents, we introduce a family of logics called term-modal logics. A main feature of our logics is the (...) use of modal operators indexed by the terms of the logics. Thus, one can quantify over variables occurring in modal operators. In term-modal logics agents can be represented by terms, and knowledge of agents is expressed with formulas within the scope of modal operators. (shrink)

All languages, both spoken and signed, make a formal distinction between two types of terms in a proposition – terms that identify what is to be talked about (nominals) and terms that say something about this topic (predicates). Here we explore conditions that could lead to this property by charting its development in a newly emerging language – Nicaraguan Sign Language (NSL). We examine how handshape is used in nominals vs. predicates in three Nicaraguan groups: (1) homesigners who (...) are not part of the Deaf com- munity and use their own gestures, called homesigns, to communicate; (2) NSL cohort 1 signers who fashioned the first stage of NSL; (3) NSL cohort 2 signers who learned NSL from cohort 1. We compare these three groups to a fourth: (4) native signers of American Sign Language (ASL), an established sign language. We focus on handshape in predicates that are part of a productive classifier system in ASL; handshape in these predicates varies sys- tematically across agent vs. no-agent contexts, unlike handshape in the nominals we study, which does not vary across these contexts. We found that all four groups, including home- signers, used handshape differently in nominals vs. predicates – they displayed variability in handshape form across agent vs. no-agent contexts in predicates, but not in nominals. Variability thus differed in predicates and nominals: (1) In predicates, the variability across grammatical contexts (agent vs. no-agent) was systematic in all four groups, suggesting that handshape functioned as a productive morphological marker on predicate signs, even in homesign. This grammatical use of handshape can thus appear in the earliest stages of an emerging language. (2) In nominals, there was no variability across grammatical con- texts (agent vs. no-agent), but there was variability within- and across-individuals in the handshape used in the nominal for a particular object. This variability was striking in homesigners (an individual homesigner did not necessarily use the same handshape in every nominal he produced for a particular object), but decreased in the first cohort of NSL and remained relatively constant in the second cohort. Stability in the lexical use of handshape in nominals thus does not seem to emerge unless there is pressure from a peer linguistic community. Taken together, our findings argue that a community of users is essential to arrive at a stable nominal lexicon, but not to establish a productive morpholog- ical marker in predicates. Examining the steps a manual communication system takes as it moves toward becoming a fully-fledged language offers a unique window onto factors that have made human language what it is. (shrink)

abstract. A first-order dynamic epistemic logic is developed where the names of the agents are also terms in the sense of first-order logic. Consequently one can quantify over epistemic modalities. Us- ing constructs from dynamic logic one can express many interesting concepts. First-order update models are developed and added to the language as modalities.

Sign language interpreting is a cognitively challenging task performed mostly by second language learners. SLI students must first gain language fluency in a new visuospatial modality and then move between spoken and signed modalities as they interpret. As a result, many students plateau before reaching working fluency, and SLI training program drop-out rates are high. However, we know little about the requisite skills to become a successful interpreter: the few existing studies investigating SLI aptitude in terms of (...) linguistic and cognitive skills lack baseline measures. Here we report a 3-year exploratory longitudinal skills assessments study with British Sign Language -English SLI students at two universities. Our aims were two-fold: first, to better understand the prerequisite skills that lead to successful SLI outcomes; second, to better understand how signing and interpreting skills impact other aspects of cognition. A battery of tasks was completed at four time points to assess skills, including but not limited to: multimodal and unimodal working memory, 2-dimensional and 3-dimensional mental rotation, and English comprehension. Dependent measures were BSL and SLI course grades, BSL reproduction tests, and consecutive SLI tasks. Results reveal that initial BSL proficiency and 2D-MR were associated with selection for the degree program, while visuospatial working memory was linked to continuing with the program. 3D-MR improved throughout the degree, alongside some limited gains in auditory, visuospatial, and multimodal working memory tasks. Visuospatial working memory and MR were the skills closest associated with BSL and SLI outcomes, particularly those tasks involving sign language production, thus, highlighting the importance of cognition related to the visuospatial modality. These preliminary data will inform SLI training programs, from applicant selection to curriculum design. (shrink)

It is widely assumed in the developmental literature that certain classes of modal expression appear later in language acquisition than others; specifically, epistemic interpretations lag behind non-epistemic interpretations. An explanation for these findings is proposed in terms of the child’s developing theory of mind, i.e. the ability to attribute to oneself and others mental representations, and to reason inferentially about them. It is hypothesized that epistemic modality crucially implicates theory-of-mind abilities and is therefore expected to depend on prior (...) developments in the child's ability to handle representations of mental representations. In support of this hypothesis, it is shown that autistic individuals have difficulty with epistemics. (shrink)

Taboo terms offer a playground for linguistic creativity in language after language, and sign languages form no exception. The present paper offers the first investigation of taboo terms in sign languages from a cognitive linguistic perspective. We analyze the linguistic mechanisms that introduce offense, focusing on the combined effects of cognitive metonymy and iconicity. Using the Think Aloud Protocol, we elicited offensive or crass signs and dysphemisms from nine signers. We find that German Sign Language uses a (...) variety of linguistic means to introduce and enhance offense, many of which rely on iconic properties of the taboo sign. In conjunction with cross-linguistically common metonymic word-formation strategies, the degree of visual explicitness of a sign increases its potential to offend. Semantically similar taboo signs based on the same metonymic anchor but differing in their degree of iconicity also differ in offensiveness. This allows for creating dysphemisms and euphemisms via phonological changes to a sign. We further show that embodiment creates modality-enhanced ‘vicarious embarrassment’ in the viewer that results in the respective signs being judged obscene or offensive. Further, lexical blending and non-manual enhancement play a role in the creation of dysphemisms in DGS. Lastly, we propose that iconicity as a cognitive structuring principle of linguistic expressions constrains the possible semantic extensions of iconic taboo terms. (shrink)

A concept language with role intersection and number restriction is defined and its modal equivalent is provided. The main reasoning tasks of satisfiability and subsumption checking are formulated in terms of modal logic and an algorithm for their solution is provided. An axiomatization for a restricted graded modal language with intersection of modalities (the modal counterpart of the concept language we examine)is given and used in the proposed algorithm.

Consider two standard quantified modal languages A and P whose vocabularies comprise the identity predicate and the existence predicate, each endowed with a standard S5 Kripke semantics where the models have a distinguished actual world, which differ only in that the quantifiers of A are actualist while those of P are possibilist. Is it possible to enrich these languages in the same manner, in a non-trivial way, so that the two resulting languages are equally expressive-i.e., so that for each (...) sentence of one language there is a sentence of the other language such that given any model, the former sentence is true at the actual world of the model iff the latter is? Forbes (1989) shows that this can be done by adding to both languages a pair of sentential operators called Vlach-operators, and imposing a syntactic restriction on their occurrences in formulas. As Forbes himself recognizes, this restriction is somewhat artificial. The first result I establish in this paper is that one gets sameness of expressivity by introducing infinitely many distinct pairs of indexed Vlach-operators. I then study the effect of adding to our enriched modal languages a rigid actuality operator. Finally, I discuss another means of enriching both languages which makes them expressively equivalent, one that exploits devices introduced in Peacocke (1978). Forbes himself mentions that option but does not prove that the resulting languages are equally expressive. I do, and I also compare the Peacockian and the Vlachian methods. In due course, I introduce an alternative notion of expressivity and I compare the Peacockian and the Vlachian languages in terms of that other notion. (shrink)

This article introduces a special issue on mechanisms in language evolution research. It describes processes relevant for the emergence of protolanguage and the transition thereof to modern language. Protolanguage is one of the key terms in the field of language evolution, used to designate a hypothesised intermediate stage in the emergence of language present in extinct hominins: qualitatively different from non-human primate communication in possessing some, but not all, of the features that characterise modern language. (...) Much debate in language evolution focuses on the exact delineation of these features, as well as the means whereby the transitions occurred: first from non-human primate communication systems to protolanguage, and then from protolanguage to modern language. In what follows, we first propose a comprehensive typology of protolanguage debates, taking into account the postulated structural organisation of protolanguage, its functions, and its communicative modality. This makes it possible to show how a specific focus on mechanisms and processes deemed relevant for the emergence of these features allows us to assess the explanatory scope of the existing theories of protolanguage. (shrink)

Over the last few years the diplomatic language of UN resolutions has repeatedly been questioned for the excessive presence of vagueness. The use of vague terms could be connected to the genre of diplomatic texts, as resolutions should be applicable to every international contingency and used to mitigate tensions between different legal cultures. However, excessive vagueness could also lead to biased or even strategically-motivated interpretations of resolutions, undermining their legal impact and triggering conflicts instead of diplomatic solutions. This study (...) aims at investigating intentional vagueness in Security Council resolutions, by focussing on the analysis of the resolutions relating to the second Gulf war. Using the qualitative Discourse-Historical approach (Wodak in Rhetorics of racism and antisemitism, Taylor & Francis Ltd., London [2000]) and quantitative analysis tools (Antconc and Sketch Engine), special attention is given to the historical/political consequences of the vagueness and indeterminacy used in that framework and to the study of vague ‘weasel words’ (Mellinkoff in The language of the law, Little, Brown & Company, Boston [1963]), modals, and adjectives contained in the corpus. The hypothesis of intentional vagueness is further reinforced through an analysis of the US legislation related to the outbreak of the war, to reveal how the US has legally interpreted UN legislation and to understand the purposes and consequences of vague language contained in it. The findings indicate that vagueness in resolutions has triggered the Iraqi conflict instead of diplomatic solutions with the overall legislative intent of using intentional vagueness as a political strategy. (shrink)

Recent research on algebraic models of _quasi-Nelson logic_ has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a _nucleus_. Among these various algebraic structures, for which we employ the umbrella term _intuitionistic modal algebras_, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for (...) their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of _weak implicative semilattices_, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus. (shrink)

The essay discusses a recurrent criticism of the isomorphism-invariance criterion for logical terms, according to which the criterion pertains only to the extension of logical terms, and neglects the meaning, or the way the extension is fixed. A term, so claim the critics, can be invariant under isomorphisms and yet involve a contingent or a posteriori component in its meaning, thus compromising the necessity or apriority of logical truth and logical consequence. This essay shows that the arguments underlying the (...) criticism are flawed since they rely on an invalid inference from the modal or epistemic status of statements in the metalanguage to that of statements in the object-language. The essay focuses on McCarthy’s version of the argument, but refers to Hanson and McGee’s versions as well. (shrink)

Metaphor abounds in both sign and spoken languages. However, in sign languages, languages in the visual-manual modality, metaphors work a bit differently than they do in spoken languages. In this paper we explore some of the ways in which metaphors in sign languages differ from metaphors in spoken languages. We address three differences: (a) Some metaphors are very common in spoken languages yet are infelicitous in sign languages; (b) Body-part terms are possible in very specific types of metaphors in sign (...) languages, but are not so restricted in spoken languages; (c) Similes in some sign languages are dispreferred in predicative positions in which metaphors are fine, in contrast to spoken languages where both can appear in these environments. We argue that these differences can be explained by two seemingly unrelated principles: the Double Mapping Constraint (Meir 2010), which accounts for the interaction between metaphor and iconicity in languages, and Croft's (2003) constraint regarding the autonomy and dependency of elements in metaphorical constructions. We further argue that the study of metaphor in the signed modality offers novel insights concerning the nature of metaphor in general, and the role of figurative speech in language. (shrink)

Modal logic has been applied in many different areas, as reasoning about time, knowledge and belief, necessity and possibility, to mention only some examples. In the present paper, an attempt is made to use modal logic to account for the semantics of theoretical sentences in scientific language. Theoretical sentences have been studied extensively since the work of Ramsey and Carnap. The present attempt at a modal analysis is motivated by there being several intended interpretations of the (...) theoretical terms once these terms are introduced through the axioms of a theory. (shrink)

The Continuum Companion to Philosophy of Language offers the definitive guide to contemporary philosophy of language. The book covers all the fundamental questions asked by the philosophy of language - areas that have continued to attract interest historically as well as topics that have emerged more recently as active areas of research. Ten specially commissioned essays from an international team of experts reveal where important work continues to be done in the area and, most valuably, the exciting (...) new directions the field is taking. The Companion explores issues pertaining to the nature of language, form semantics, theories of meaning, reference, intensional contexts, context-dependence, pragmatics, the normativity of language, analyticity, a priority and modality. Featuring a series of indispensable research tools, including an A to Z of key terms and concepts, a detailed list of resources and a fully annotated bibliography, this is the essential reference tool for anyone working in the philosophy of language. (shrink)

Volume I. Foundational articles -- volume II. Semantics -- volume III. Singular terms, propositional attitudes, and modality -- volume IV. Pragmatics, thought, and some contemporary issues.

In spoken languages, children acquire locative terms in a cross‐linguistically stable order. Terms similar in meaning to in and on emerge earlier than those similar to front and behind, followed by left and right. This order has been attributed to the complexity of the relations expressed by different locative terms. An additional possibility is that children may be delayed in expressing certain spatial meanings partly due to difficulties in discovering the mappings between locative terms in speech and spatial relation they (...) express. We investigate cognitive and mapping difficulties in the domain of spatial language by comparing how children map spatial meanings onto speech versus visually motivated forms in co‐speech gesture across different spatial relations. Twenty‐four 8‐year‐old and 23 adult native Turkish‐speakers described four‐picture displays where the target picture depicted in‐on, front‐behind, or left‐right relations between objects. As the complexity of spatial relations increased, children were more likely to rely on gestures as opposed to speech to informatively express the spatial relation. Adults overwhelmingly relied on speech to informatively express the spatial relation, and this did not change across the complexity of spatial relations. Nevertheless, even when spatial expressions in both speech and co‐speech gesture were considered, children lagged behind adults when expressing the most complex left‐right relations. These findings suggest that cognitive development and mapping difficulties introduced by the modality of expressions interact in shaping the development of spatial language. (shrink)

First-order modal logics, as traditionally formulated, are not expressive enough. It is this that is behind the difficulties in formulating a good analog of Herbrand’s Theorem, as well as the well-known problems with equality, non-rigid designators, definite descriptions, and nondesignating terms. We show how all these problems disappear when modal language is made more expressive in a simple, natural way. We present a semantic tableaux system for the enhanced logic, and (very) briefly discuss implementation issues.

Philosophy of language explores some of the fundamental yet most technical problems in philosophy, such as meaning and reference, semantics, and propositional attitudes. Some of its greatest exponents, including Gottlob Frege, Ludwig Wittgenstein and Bertrand Russell are amongst the major figures in the history of philosophy. In this clear and carefully structured introduction to the subject Gary Kemp explains the following key topics: the basic nature of philosophy of language and its historical development early arguments concerning the role (...) of meaning, including cognitive meaning vs expressivism, context and compositionality Frege’s arguments concerning sense and reference; non-existent objects Russell and the theory of definite descriptions modern theories including Kripke and Putnam; arguments concerning necessity, analyticity and natural kind terms indexicality, context and modality. What are indexicals? Davidson’s theory of language and the ‘principle of charity’ propositional attitudes Quine’s naturalism and its consequences for philosophy of language. Chapter summaries, annotated further reading and a glossary make this an indispensable introduction to those teaching philosophy of language and will be particularly useful for students coming to the subject for the first time. (shrink)

There is a class of expressions which are perceived as ‘ungrammatical’ not because they are syntactically ill-formed but because they have interpretations which are informationally trivial. Triviality-driven unacceptability constrains the distribution of determiners, modals, attitude verbs, exhaustifiers, approximatives, among many other classes of logical terms. At the same time, many superficial tautologies and contradictions—pre-theoretically, the clearest examples of trivial expressions—are judged to be perfectly acceptable. This paper discusses two promising yet fundamentally opposed attempts to model triviality-driven unacceptability without over-generating ‘ungrammaticality’ (...) judgments. One approach combines the ‘Logicality’ view that the language system includes a deductive-inferential system (DS) that automatically identifies and filters out expressions with trivial interpretations, with the hypothesis that the DS runs on ‘modulated logical forms’, i.e., structures where all content-based terms and variables are subject to meaning-modulation operations. The other approach downplays Logicality and tries to reduce triviality-driven unacceptability to familiar kinds of pragmatic infelicities. The Logicality-based approach, I argue, is superior to the most sophisticated attempts to treat triviality-driven unacceptability as a species of pragmatic infelicity. This result suggests that our purely linguistic and logical competences are deeply intertwined, and sheds new light on the division of labor between syntax, semantics and pragmatics. (shrink)

In the paper we introduce a wide range of Anderson-like variants of Gödel's theory and prove for each of them strong completeness theorem wrt. corresponding class of modal structures.These theories — all formulated in the 2nd order modal language with a 2nd order unary predicate of positiveness — differ among themselves with respect of: properties of the necessity operator and of the predicate of positiveness, axioms characterizing identity between 1st sort terms, definitions of identity between 2nd sort (...) terms, the treatment of Godlikeness as a term of the 2nd order, the treatment of necessary existence as a term of the 2nd order if there exists the necessity of its presence, the treatment of Ix (Ix(y) is read: terms x, y are identical), for every 1st sort term x, as a term of the 2nd order if there exists the necessity of its presence.Concerning to semantical commitments, the following are essential: (i) 1st order terms order receive only rigid extensions in the constant objectual 1st order domain; (ii) 2nd order terms receive non-rigid extensions in the constant objectual domain of 2nd order and rigid intensions in the constant conceptual 2nd order domain; (iii) The behavior of the identity between 1st sort terms shifts from possible worlds to possible worlds. (shrink)

In the tradition of quantified modal logic, it was assumed that significantly different linguistic systems underlie reference to individuals, to times and to ‘possible worlds’. Various results from recent research in formal semantics suggest that this is not so, and that there is in fact apervasive symmetrybetween the linguistic means with which we refer to these three domains. Reference to individuals, times and worlds is uniformly effected through generalized quantifiers, definite descriptions, and pronouns, and in each domain grammatical features (...) situate the reference of terms as near, far or ‘further’ from the actual or from a reported speech act. We outline various directions in which a program ofontological symmetrycould be developed, and we offer in theAppendixa symmetric fragment developed in a logic that can be seen as a compromise between an extensional and an intensional system. (shrink)

We generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all (...) inductive formulae are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames. (shrink)

The concepts of the innateness, universality, species-specificity, and autonomy of the human language capacity have had an extreme impact on the psycholinguistic debate for over thirty years. These concepts are evaluated from several neurobiological perspectives, with an emphasis on the emergence of language and its decay due to brain lesion and progressive brain disease.Evidence of perceptuomotor homologies and preadaptations for human language in nonhuman primates suggests a gradual emergence of language during hominid evolution. Regarding ontogeny, the (...) innate component of language capacity is likely to be polygenic and shared with other developmental domains. Dissociations between verbal and nonverbal development are probably rooted in the perceptuomotor specializations of neural substrates rather than the autonomy of a grammar module. Aphasiologicaldata often assumed to suggest modular linguistic subsystems can be accounted for in terms of a neurofunctional model incorporating perceptuomotor-based regional specializationsand distributivity of representations. Thus, dissociations between grammatical functors and content words are due to different conditions of acquisition and resulting differences in neural representation. Human brains are characterized by multifactorial interindividual variability, and strict universality of functional organization is biologically unrealistic.A theoretical alternative is proposed according to which (1) linguistic specialization of brain areas is due to epigenetic and probabilistic maturational events, not to genetic ”hard-wiring,” and (2) linguistic knowledge is neurally represented in distributed cell assemblies whose topography reflects the perceptuomotor modalities involved in the acquisition and use of a given item of knowledge. (shrink)

In this paper, we aim to explore connections between a Carnapian semantics of theoretical terms and an eliminative structuralist approach in the philosophy of mathematics. Specifically, we will interpret the language of Peano arithmetic by applying the modal semantics of theoretical terms introduced in Andreas (Synthese 174(3):367–383, 2010). We will thereby show that the application to Peano arithmetic yields a formal semantics of universal structuralism, i.e., the view that ordinary mathematical statements in arithmetic express general claims about all (...) admissible interpretations of the Peano axioms. Moreover, we compare this application with the modal structuralism by Hellman (Mathematics without numbers: towards a modal-structural interpretation. Oxford University Press: Oxford, 1989), arguing that it provides us with an easier epistemology of statements in arithmetic. (shrink)

Word learning is basic to foreign language acquisition, however time consuming and not always successful. Empirical studies have shown that traditional (visual) word learning can be enhanced by gestures. The gesture benefit has been attributed to depth of encoding. Gestures can lead to depth of encoding because they trigger semantic processing and sensorimotor enrichment of the novel word. However, the neural underpinning of depth of encoding is still unclear. Here, we combined an fMRI and a behavioral study to investigate (...) word encoding online. In the scanner, participants encoded 30 novel words of an artificial language created for experimental purposes and their translation into the subjects’ native language. Participants encoded the words three times visually, audiovisually, and by additionally observing semantically related gestures performed by an actress. Hemodynamic activity during word encoding revealed the recruitment of cortical areas involved in stimulus processing. In this study, depth of encoding can be spelt out in terms of sensorimotor brain networks that grow larger the more sensory modalities are linked to the novel word. Word retention outside the scanner documented a positive effect of gestures in a free recall test in the short-term. (shrink)

A consequence relation is strongly classical if it has all the theorems and entailments of classical logic as well as the usual meta-rules (such as Conditional Proof). A consequence relation is weakly classical if it has all the theorems and entailments of classical logic but lacks the usual meta-rules. The most familiar example of a weakly classical consequence relation comes from a simple supervaluational approach to modelling vague language. This approach is formally equivalent to an account of logical consequence (...) according to which α1, ..., αn entails β just in case □α1, ..., □αn entails □β in the modal logic S5. This raises a natural question: If we start with a different underlying modal logic, can we generate a strongly classical logic? This paper explores this question. In particular, it discusses four related technical issues: (1) Which base modal logics generate strongly classical logics and which generate weakly classical logics? (2) Which base logics generate themselves? (3) How can we directly characterize the logic generated from a given base logic? (4) Given a logic that can be generated, which base logics generate it? The answers to these questions have philosophical interest. They can help us to determine whether there is a plausible supervaluational approach to modelling vague language that yields the usual meta-rules. They can also help us to determine the feasibility of other philosophical projects that rely on an analogous formalism, such as the project of defining logical consequence in terms of the preservation of an epistemic status. (shrink)

In the past three decades there has been a rapid development of the formal machinery for modal logic. Quantified modal logic has developed along with a semantics and model theory that is appropriate to it. With this technical development there has been relatively little discussion of what modality is all about. There are two fundamental questions that have gone unanswered. First, to what does necessity amount? Is this a new logical notion, or is it something that can be (...) further analyzed in terms of other notions that we already have at our disposal? The second question is what makes truths involving necessity true? What is their ontological grounding in the world? This essay is directed at the first of these questions. ;There are three possible reductions of metaphysical modality. First, there is the reduction suggested by the semantics for modal logic: a reduction in terms of possible worlds. Second, there is the reduction in terms of cognition. What is possible is what is conceivable. Third, there is the reduction in terms of language. Necessities arise from the structures of language and/or language use. ;I argue that none of these is successful. All of them suffer either from presupposing some form of metaphysical necessity in the reductive base, rendering them circular and not truly reductive, or they do not do justice to the modality being reduced. ;Since necessity is not reducible to any non-modal features of the world, we need to determine whether we ought to reject it as unacceptable on this basis, or whether we ought to retain it, but treat is as a new primitive that is introduced into a philosophical or logical theory. I urge the latter course is the one that we ought to take because without it too many things will likewise be deemed unacceptable. Causal necessity and explanation, normally accepted theories or knowledge and perception, logic and logical validity, and one common manner of distinguishing sets from properties will be deemed unacceptable if de re necessity is unacceptable. All of these presuppose de re metaphysical necessity in some way. I conclude that de re metaphysical necessity ought to be retained in our theory as primitive, unanalyzable notion. (shrink)

The paper addresses the question how descriptive language is used to express legal norms. Sentences we find in legislative acts, i.e. statutes, constitutions and regulations, express legal norms. Linguistically speaking, there are various grammatical and lexical ways of expressing norms, such as imperative mood, modal verbs, deontic verbs, etc. However, norms may also be expressed by descriptive sentences, namely sentences in present or future tense and indicative (declarative) mood (i.e. _The minister determines the tax rate_). In many civil (...) law countries (including Poland), this is a very common, if not the default, form of expressing norms in legislative texts. Often presented as a legal peculiarity, this phenomenon has yet to draw much academic attention. The normative meaning of descriptive sentences is usually attributed to purely pragmatic factors stemming from our shared assumptions about the legal system. However, a closer look reveals that similar grammatical constructions are ubiquitous in everyday communication and in different languages. We tend to utter various sorts of directives using descriptive sentences (_Now we add a spoon of salt to the sauce_; _credit cards are not accepted_). This suggests the possibility for a linguistic (as opposed to exclusively legal) explanation. This paper aims to offer such an explanation. Rather than resorting to formal semantics, so prevalent in legal theory, it borrows from Cognitive Linguistics to reveal the cognitive underpinnings of our surprising tendency to express normativity in descriptive terms. This involves four different, yet complementary, theories. Firstly, the theory of conceptual metaphor by George Lakoff and Mark Johnson which explains the phenomenon in question in terms of the metaphor “ ought is is ”; with “ ought ” as the more abstract target domain and “ is ” as the more concrete, cognitively simpler source domain. Secondly, the theory of speech act metonymy by Panther and Thornburg which presents descriptive legal sentences as referring to various components of the underlying cognitive scenario of obligation. Thirdly, Ronald Langacker’s notion of the virtuality of language as the explanation for non-present, including future, perpetual and directive, uses of the present tense. Fourtly, the notion of normative generics which points to the nominal, as opposed to verbal, structure of descriptive legal sentences as the source of their normativity. (shrink)

(1) modal logic is not needed, Since there are alternative accounts of modality. (2) modal logic does not function as logic even in the thinking of its advocates, As is revealed, E.G., When the semantics of modal logic is presented in an extensional metalanguage. Furthermore, (3) when a wider view is taken, One sees that modal logic treats as logical constants expressions that belong to a large and open syntactic class, Unlike other logical constants. Finally, (4) (...) modal logic treats as sentential operators devices that function in natural language as underlying predicates. The last two points also indicate that a theory of modality making use of modal predicates is to be preferred to an account that explains away modality in terms of quantification over possible worlds. (shrink)

This paper introduces a multi-modal polymorphic type theory to model epistemic processes characterized by trust, defined as a second-order relation affecting the communication process between sources and a receiver. In this language, a set of senders is expressed by a modal prioritized context, whereas the receiver is formulated in terms of a contextually derived modal judgement. Introduction and elimination rules for modalities are based on the polymorphism of terms in the language. This leads to a (...) multi-modal non-homogeneous version of a type theory, in which we show the embedding of the modal operators into standard group knowledge operators. (shrink)

The dissertation gives a second-order-logic-based explanation of modal incompleteness. The leading concept is that modal incompleteness is to be explained in terms of the incompleteness of standard second-order logic, since modal language is basically a second-order language. The development of Kripke-style semantics for modal logic has been underpinned by the conjecture that all modal systems are characterizable by classes of frames defined by first-order conditions on a binary relation. However, the discovery of certain (...) incomplete modal systems has undermined the all-encompassing feature of Kripke-style approach. There are logics not determined by any class of Kripke frames at all. ;In the dissertation I investigate a normal incomplete sentential modal system due to J. F. A. K. Van Benthem, and I address both the formal and the philosophical facets of modal incompleteness from the vantage point that modal logic is essentially second-order in its nature. Modal systems can be analyzed in terms of structures with a domain of second-order individuals that are assigned under an interpretation to propositional variables within languages of sentential modal logic. Hence, the phenomenon of there being incomplete modal systems can be accounted for through the incompleteness of standard second-order logic. ;Since second-order logic plays a crucial role in the explanation, Quine's animadversions upon second-order logic and in particular his views that second-order logic is 'Set Theory in Sheep's Clothing' are examined. Against Quine's stance I will seek to show that a vindication of second-order logic can be gotten provided a proper due is given to the sharp distinction between the logical notion of set which is the concern of second-order logic and the iterative notion of set which lies within the realm of set theory. (shrink)

In this paper the propositional logic LTop is introduced, as an extension of classical propositional logic by adding a paraconsistent negation. This logic has a very natural interpretation in terms of topological models. The logic LTop is nothing more than an alternative presentation of modal logic S4, but in the language of a paraconsistent logic. Moreover, LTop is a logic of formal inconsistency in which the consistency and inconsistency operators have a nice topological interpretation. This constitutes a new (...) proof of S4 as being "the logic of topological spaces", but now under the perspective of paraconsistency. (shrink)

Despite the wide acceptance of standard modal logic, there has always been a temptation to think that ordinary modal discourse may be correctly analyzed and adequately represented in terms of predicates rather than in terms of operators. The aim of the formal model outlined in this paper is to capture what I take to be the only plausible sense in which ‘possible’ and ‘necessary’ can be treated as predicates. The model is built by enriching the language of (...) standard modal logic with a quantificational apparatus that is “substitutional” rather than “objectual”, and by obtaining from the language so enriched another language in which constants for such predicates apply to singular terms that stand for propositions. (shrink)

If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics (...) for predicates of sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate □, we tackle both problems. Given a frame (W, R) consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret □ at every world in such a way that □ $\ulcorner A \ulcorner$ holds at a world ᵆ ∊ W if and only if A holds at every world $\upsilon$ ∊ W such that ᵆR $\upsilon$ . The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of □ at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic. (shrink)

This chapter examines modality in the history of analytic philosophy. There were, in this history, two principal types of reductionism or eliminativism about modality, and two corresponding phases in the rejection of anti-modal stances. First, the founders of analytic philosophy, Frege, Moore, and Russell, took necessity and possibility to be reducible to more fundamental logical notions, where logic for these thinkers consists of truths about a mind- and language-independent reality extending beyond the empirical world. Against this reductionism, C. (...) I. Lewis and Wittgenstein argued that logic itself requires modal notions. Second, Carnap advanced a pragmatically motivated account of modal terms as expressing pseudo-object properties, which appear to be properties of objects but can be construed as properties of their designations. Quine criticized this account and proposed a thoroughgoing elimination of modality. Quine’s well-known claim that quantifying into modal contexts is unintelligible without resorting to essentialism cannot be fully understood independently of Carnap’s notion of pseudo-object property. Against Quine, Ruth Marcus and Saul Kripke argued for the coherence of modal concepts and their entrenchment in ordinary language and thought, leading to the now dominant view that modal properties are mind- and language-independent features of the world. (shrink)

Drawing an analogy between modal structuralism about mathematics and theism, I o er a structuralist account that implicitly de nes theism in terms of three basic relations: logical and metaphysical priority, and epis- temic superiority. On this view, statements like `God is omniscient' have a hypothetical and a categorical component. The hypothetical component provides a translation pattern according to which statements in theistic language are converted into statements of second-order modal logic. The categorical component asserts the logical (...) possibility of the theism struc- ture on the basis of uncontroversial facts about the physical world. This structuralist reading of theism preserves objective truth-values for theistic statements while remaining neutral on the question of ontology. Thus, it o ers a way of understanding theism to which a naturalist cannot object, and it accommodates the fact that religious belief, for many theists, is an essentially relational matter. (shrink)

The present paper studies formal properties of so-called modal information logics (MILs)—modal logics first proposed in (van Benthem 1996 ) as a way of using possible-worlds semantics to model a theory of information. They do so by extending the language of propositional logic with a binary modality defined in terms of being the supremum of two states. First proposed in 1996, MILs have been around for some time, yet not much is known: (van Benthem 2017, 2019 ) (...) pose two central open problems, namely (1) axiomatizing the two basic MILs of suprema on preorders and posets, respectively, and (2) proving (un)decidability. The main results of the first part of this paper are solving these two problems: (1) by providing an axiomatization [with a completeness proof entailing the two logics to be the same], and (2) by proving decidability. In the proof of the latter, an emphasis is put on the method applied as a heuristic for proving decidability ‘via completeness’ for semantically introduced logics; the logics lack the FMP w.r.t. their classes of definition, but not w.r.t. a generalized class. These results are build upon to axiomatize and prove decidable the MILs attained by endowing the language with an ‘informational implication’—in doing so a link is also made to the work of (Buszkowski 2021 ) on the Lambek Calculus. (shrink)

In this volume, the author investigates and argues for, a particular answer to the question: What is the right way to logically analyze modalities from natural language within formal languages? The answer is: by formalizing modal expressions in terms of predicates. But, as in the case of truth, the most intuitive modal principles lead to paradox once the modal notions are conceived as predicates. -/- The book discusses the philosophical interpretation of these modal paradoxes and (...) argues that any satisfactory approach to modality will have to face the paradoxes independently of the grammatical category of the modal notion. By systematizing modal principles with respect to their joint consistency and inconsistency, Stern provides an overview of the options and limitations of the predicate approach to modality that may serve as a useful starting point for future work on predicate approaches to modality. Stern also develops a general strategy for constructing philosophically attractive theories of modal notions conceived as predicates. The idea is to characterize the modal predicate by appeal to its interaction with the truth predicate. This strategy is put to use by developing the modal theories Modal Friedman-Sheard and Modal Kripke-Feferman. (shrink)

Over the last fifteen years, linguists and philosophers of language have reexamined the canonical, Kratzerian semantics for modal expressions, with special attention paid to their epistemic and deontic uses. This article is an overview of the literature on deontic modal expressions. Section 1 provides an overview of the canonical semantics, noting some of its main advantages. Section 2 introduces a set of desiderata that have achieved the status of fixed points in the debates about whether the canonical (...) semantics is correct. These include the observations that deontic modal sentences have both deliberative and evaluative readings and both information-sensitive and -insensitive readings. Adequate resolutions of certain puzzles in deontic logic and resolving the Frege-Geach problem for Expressivism have also achieved this status. The third section provides an opinionated overview of some of the main extant rivals to the canonical semantics, including Cariani, Kaufmann, and Kaufmann’s (2013) complex contextualism , Yalcin’s (2012) Expressivism, Willer’s (2014) dynamic semantics, and Starr’s (2016) dynamic Expressivism. Section 4 provides an assessment of each of the views discussed in terms of the desiderata introduced in section 2. Section 5 is an overview of remaining issues that require more attention in the literature. (shrink)

Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with the (...) so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness. (shrink)

One can add the machinery of relation symbols and terms to a propositional modal logic without adding quantifiers. Ordinarily this is no extension beyond the propositional. But if terms are allowed to be non-rigid, a scoping mechanism (usually written using lambda abstraction) must also be introduced to avoid ambiguity. Since quantifiers are not present, this is not really a first-order logic, but it is not exactly propositional either. For propositional logics such as K, T and D, adding such machinery (...) produces a decidable logic, but adding it to S5 produces an undecidable one. Further, if an equality symbol is in the language, and interpreted by the equality relation, logics from K4 to S5 yield undecidable versions. (Thus transitivity is the villain here.) The proof of undecidability consists in showing that classical first-order logic can be embedded. (shrink)

This paper articulates the structure of a two species of weakly aggregative necessity in a common idiom, neighbourhood semantics, using the notion of a k-filter of propositions. A k-filter on a non-empty set I is a collection of subsets of I which contains I, is closed under supersets on I, and contains ∪{Xi ≤ Xj : 0 ≤ i < j ≤ k} whenever it contains the subsets X0,…, Xk. The mathematical content of the proof that weakly aggregative modal (...) logic is complete relative to k-ary frame theory, the standard semantic idiom for weakly aggregative modal logic is presented in language-independent terms as a representation theorem for k-filters: every non-trivial k-filter is included in the union of ≤ k non-trivial filters. The elementary theory of k-filters is developed and then applied in the form of an ultrafilter extension result for k-ary frame theory. (shrink)

Let us sum up. We began with the question, “What is the interest of a model-theoretic definition of validity?” Model theoretic validity consists in truth under all reinterpretations of non-logical constants. In this paper, we have described for each necessity concept a corresponding modal invariance property. Exemplification of that property by the logical constants of a language leads to an explanation of the necessity, in the corresponding sense, of its valid sentences. I have fixed upon the epistemic modalities (...) in characterizing the logical constants: to be a logical constant in the language of a population is to be invariant over a modality describing complete possible epistemic states of that population (or an idealized analogue thereof). The grounds for this characterization are these: (1) It leads, I believe, to an extensionally reasonable demarcation of the logical constants, including clear cases and excluding clear non-cases. It gives a principled criterion for deciding unclear cases. (2) It provides an analysis of the topic-neutrality of logic. (3) It leads to an explanation of the epistemic necessity of the logical truths in terms of the topic-neutrality of the logical constants.All the same, it is reasonable to ask, even if the suggested demarcation of logic is extensionally correct, whether it can reasonably be expected to be fundamental. The epistemic invariance of an expression is a rather striking property, one which we should want to explain. What is missing, then, is an explanation of the distinguishing epistemic properties of the constants in terms of more fundamental properties involving their understanding and use. It would be these that properly define the nature, not just the extent, of logic. (shrink)

I propose and defend an account on which the semantic content of propositions expressed by utterances making use of modals epistemically is constant; i.e., invariant. Although such proposals are typically considered non-starters, I aim to show that combining such a semantics with a performative account in which such utterances perform two speech acts is quite promising. I argue that a performative account, when combined with an invariant semantic content theory, does a good job of accounting for ordinary intuitions in some (...) of the standard test cases discussed in the literature. I further show that the contextualist has a little noticed problem with how a single speaker is able to consider the same epistemic possibility proposition over time. An invariant position has no such problem, and I argue that this counts strongly in its favor. (shrink)

We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. (...) To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics. (shrink)