This paper proposes a new model of graded modal judgment. It begins by problematizing the phenomenon: given plausible constraints on the logic of epistemic modality, it is impossible to model graded attitudes toward modal claims as judgments of probability targeting epistemically modal propositions. This paper considers two alternative models, on which modal operators are non-proposition-forming: (1) Moss (2015), in which graded attitudes toward modal claims are represented as judgments of probability targeting a “proxy” proposition, belief in which (...) would underwrite belief in the modal claim. (2) A model on which graded attitudes toward modal claims are represented as judgments of credence taking as their objects (non-propositional) modal representations (rather than proxy propositions). The second model, like Moss’ model, is shown to be semantically and mathematically tractable. The second model, however, can be straightforwardly integrated into a plausible model of the role of graded attitudes toward modal claims in cognition and normative epistemology. (shrink)

This work intends to be a generalization and a simplification of the techniques employed in [2], by the proposal of a general strategy to prove satisfiability theorems for NLGM-s (= normal logics with graded modalities), analogously to the well known technique of the canonical models by Lemmon and Scott for classical modal logics.

We study a modal system ¯T, that extends the classical (prepositional) modal system T and whose language is provided with modal operators M inn (nN) to be interpreted, in the usual kripkean semantics, as there are more than n accessible worlds such that.... We find reasonable axioms for ¯T and we prove for it completeness, compactness and decidability theorems.

We go on along the trend of [2] and [1], giving an axiomatization of S4 0 and proving its completeness and compactness with respect to the usual reflexive and transitive Kripke models. To reach this results, we use techniques from [1], with suitable adaptations to our specific case.

We prove a completeness theorem for Kmath image, the infinitary extension of the graded version K0 of the minimal normal logic K, allowing conjunctions and disjunctions of countable sets of formulas. This goal is achieved using both the usual tools of the normal logics with graded modalities and the machinery of the predicate infinitary logics in a version adapted to modal logic.

We introduce a notion of bisimulation for graded modal logic. Using this notion, the model theory of graded modal logic can be developed in a uniform manner. We illustrate this by establishing the finite model property and proving invariance and definability results.

We extend the languages of both basic and graded modal logic with the infinity diamond, a modality that expresses the existence of infinitely many successors having a certain property. In both cases we define a natural notion of bisimilarity for the resulting formalisms, that we dub $\mathtt {ML}^{\infty }$ and $\mathtt {GML}^{\infty }$, respectively. We then characterise these logics as the bisimulation-invariant fragments of the naturally corresponding predicate logic, viz., the extension of first-order logic with the infinity quantifier. Furthermore, (...) for both $\mathtt {ML}^{\infty }$ and $\mathtt {GML}^{\infty }$ we provide a sound and complete axiomatisation for the set of formulas that are valid in every Kripke frame, we prove a small model property with respect to a widened class of weighted models, and we establish decidability of the satisfiability problem. (shrink)

We introduce a class of neighbourhood frames for graded modal logic embedding Kripke frames into neighbourhood frames. This class of neighbourhood frames is shown to be first-order definable but not modally definable. We also obtain a new definition of graded bisimulation with respect to Kripke frames by modifying the definition of monotonic bisimulation.

In this work we extend results from [4], [3] and [2] about propositional calculi with graded modalities to the predicative level. Our semantic is based on Kripke models with a single domain of interpretation for all the worlds. Therefore the axiomatic system will need a suitable generalization of the Barcan formula. We haven't considered semantics with world-relative domains because they don't present any new difficulties with respect to classical case. Our language will have, as in [1], constant and (...) function symbols, but they will have a rigid interpretation. In this instance the terms will be considered rigid designators, that is, their interpretations will be the same in all possible worlds (cfr. [5]). Nevertheless we will not consider languages with identity: in this way we avoid the problems of identity in modal contexts. On the other hand, we think that in that context would arise essentially the same problems which occur in classical predicative modal logic. Finally we will consider only denumerable languages because the extension to non denumerable ones isn't conceptually problematic. (shrink)

We prove the canonical models introduced in [D] do not exist for some graded normal logics with symmetric models, namelyKB°, KBD°, KBT°, so that we define a new kind of canonical models, the general ones, and show they exist and work well in every case.

The paper presents an infinite hierarchy PR m [ m = 1, 2, . . . ] of sound and complete axiomatic systems for modal logic with graded probabilistic modalities , which are to reflect what I have elsewhere called the Bolding-Ekelöf degrees of evidential strength as applied to the establishment of matters of fact in law-courts. Our present approach is seen to differ from earlier work by the author in that it treats the logic of these (...) class='Hi'>graded modalities not only from a semantical or model-theoretic viewpoint but from a prooftheoretical and axiomatic stance as well. A paramount feature of the approach is its use of so-called systematic frame constants as labels of diverse grades of probability. Apart from this novel feature our approach can be seen to go back to pioneering work by Lou Goble in 1970. (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.

Reprinted in Quine, W. V. O. 1966. The Ways of Paradox. (New York: Random House.).

Psychophysics and neuroscience demonstrate that different sensory systems interact and influence each other. Perceiving involves extensive cooperation and coordination among systems associated with sight, hearing, touch, smell, and taste. Nonetheless, it remains unclear in what respects conscious perceptual awareness is multisensory. This paper distinguishes six differing varieties of multisensory awareness, explicates their consequences, and thereby elucidates the multisensory nature of perception. It argues on these grounds that perceptual awareness need not be exhausted by that which is associated with each of (...) the respective sensory modalities along with whatever accrues thanks to simple co-consciousness. (shrink)

Though it has become a commonplace that probabilistic contexts are intentional, the precise sense in which this is true has never, to my knowledge, been stated. By making use of a relatively non-controversial set of distinctions regarding the grades of modal involvement, I am able to state more exactly than has been done previously the grade of intensionality which probability statements have prima facie. The distinctions I employ are, with certain qualifications, those introduced by Quine in his wellknown paper, Three (...) Grades of Modal Involvement. By means of paraphrases, I am able to show that probability statements exemplify what Quine calls the third grade of modal involvement. Consequently, they are intensional in a way not acceptable to the majority of extensionalists. (shrink)

Biological modalities (e.g., biological possibility, necessity and counterfactuality) play an important explanatory role in biological practice. However, biological modalities lack truth conditions and the inferential relationship between biological and other modalities is unclear. This thesis addresses these problems, first, by improving upon Daniel Dennett's Library of Mendel. Second, a family of modal logics is introduced. In the simplest model, states are interpreted as codons, the binary relation is interpreted as single substitution mutation and the valuation induces a (...) partition of blocks of codons that code for some amino acid. In addition, the proposed framework allows for a grading of biological possibility and captures various competing notions of biological possibility. Applications include a model checker and a lawless Lewis-style semantics for biological counterfactuals. In short, this thesis lays the groundwork for a theory of biological modalities. (shrink)

Graphs are among the most frequently used structures in Computer Science. Some of the properties that must be checked in many applications are connectivity, acyclicity and the Eulerian and Hamiltonian properties. In this work, we analyze how we can express these four properties with modal logics. This involves two issues: whether each of the modal languages under consideration has enough expressive power to describe these properties and how complex it is to use these logics to actually test whether a given (...) graph has some desired property. First, we show that these properties are not definable in a basic modal logic or in any bisimulation-invariant extension of it, like the modal μ-calculus. We then show that it is possible to express some of the above properties in a basic hybrid logic. Unfortunately, the Hamiltonian and Eulerian properties still cannot be efficiently checked. In a second attempt, we propose an extension of CTL* with nominals and show that the Hamiltonian property can be more efficiently checked in this logic than in the previous one. In a third attempt, we extend the basic hybrid logic with the ↓ operator and show that we can check the Hamiltonian property with optimal complexity in this logic. Finally, we tackle the Eulerian property in two different ways. First, we develop a generic method to express edge-related properties in hybrid logics and use it to express the Eulerian property. Second, we express a necessary and sufficient condition for the Eulerian property to hold using a graded modal logic. (shrink)

In the paper (Braüner, 2001) we gave a minimal condition for the existence of a homophonic theory of truth for a modal or tense logic. In the present paper we generalise this result to arbitrary modal logics and we also show that a modal logic permits the existence of a homophonic theory of truth if and only if it permits the definition of a socalled master modality. Moreover, we explore a connection between the master modality and hybrid logic: We show (...) that if attention is restricted to bidirectional frames, then the expressive power of the master modality is exactly what is needed to translate the bounded fragment of first-order logic into hybrid logic in a truth preserving way. We believe that this throws new light on Arthur Prior's fourth grade tense logic. (shrink)

I propose an account of indicative conditionals that combines features of minimal change semantics and information semantics. As in information semantics, conditionals are interpreted relative to an information state in accordance with the Ramsey test idea: “if p then q” is supported at a state s iff q is supported at the hypothetical state s[p] obtained by restricting s to the p-worlds. However, information states are not modeled as simple sets of worlds, but by means of a Lewisian system of (...) spheres. Worlds in the inner sphere are considered possible; worlds outside of it are ruled out, but to different degrees. In this way, even when a state supports “not p”, it is still possible to suppose p consistently. I argue that this account does better than its predecessors with respect to a set of desiderata concerning inferences with conditionals. In particular, it captures three important facts: that a conditional is logically independent from its antecedent; that a sequence of antecedents behaves like a single conjunctive antecedent ; and that conditionals restrict the quantification domain of epistemic modals. I also discuss two ways to construe the role of a premise, and propose a generalized notion of entailment that keeps the two apart. (shrink)

Adding certain cardinality quantifiers into first-order language will give substantially more expressive languages. Thus, many mathematical concepts beyond first-order logic can be handled. Since basic modal logic can be seen as the bisimular invariant fragment of first-order logic on the level of models, it has no ability to handle modally these mathematical concepts beyond first-order logic. By adding modalities regarding the cardinalities of successor states, we can, in principle, investigate modal logics of all cardinalities. Thus ways of exploring model-theoretic (...) logics can be transferred to modal logics. (shrink)

Adding certain cardinality quantifiers into first-order language will give substantially more expressive languages. Thus, many mathematical concepts beyond first-order logic can be handled. Since basic modal logic can be seen as the bisimular invariant fragment of first-order logic on the level of models, it has no ability to handle modally these mathematical concepts beyond first-order logic. By adding modalities regarding the cardinalities of successor states, we can, in principle, investigate modal logics of all cardinalities. Thus ways of exploring model-theoretic (...) logics can be transferred to modal logics. (shrink)

We prove strong completeness results for some modal logics with the universal modality, with respect to their topological semantics over 0-dimensional dense-in-themselves metric spaces. We also use failure of compactness to show that, for some languages and spaces, no standard modal deductive system is strongly complete.

The debate about the nature of the representational format of concepts seems to have reached an impasse. The debate faces two fundamental problems. Firstly, amodalists (i.e., those who argue that concepts are represented by amodal symbols) and modalists (i.e., those who see concepts as involving crucially representations including sensorimotor information) claim that the same empirical evidence is compatible with their views. Secondly, there is no shared understanding of what a modal or amodal format amounts to. Both camps recognize that the (...) two formats play essential roles in higher cognition, leading to an increasing number of hybrid proposals. In this paper, I argue that the existence of those fundamental problems should make us suspicious about a modal/amodal dichotomy. Also, I suggest that hybrid approaches, as they currently stand, do not provide suitable solutions to the impasse. Instead, we should overcome the dichotomy and treat the modal/amodal distinction as a graded phenomenon. I illustrate this hypothesis with an example of a cognitive-computational model of concepts based on the Predictive Processing framework. (shrink)

The paper presents a survey of author's results on definable fixed points in modal, temporal, and intuitionistic propositional logics. The well-known Fixed Point Theorem considers the modalized case, but here we investigate the positive case. We give a classification of fixed point theorems, describe some classes of models with definable least fixed points of positive operators, special positive operators, and give some examples of undefinable least fixed points. Some other interesting phenomena are discovered – definability by formulas that do not (...) preserve positivity of parameters and definability by finite sets of formulas. We also consider negative operators, graded modalities, construct undefinable inflationary fixed points, and put some problems. (shrink)

Of late, evidentiality has received great attention in formal semantics. In this paper I develop ‘evidentiality-informed’ truth conditions for modal operators such as must and may . With language data drawn from Luoping Nase (a Tibeto-Burman language spoken in the P.R. of China and belonging to the Yi Nationality), I illustrate that epistemic modals clash with clauses articulating first-hand information. I then demonstrate that existing models such as Kratzer’s graded possible-worlds semantics fail to provide accurate truth conditions for modals (...) tagging clauses with first-hand information. As a remedy I propose a fuzzy version of possible-worlds semantics with various grades of belief and knowledge. In addition to preserving the expressive power of graded possible-worlds semantics, the fuzzy model will be shown to supply appropriate truth conditions for epistemic modals appended to evidential clauses (i.e. clauses expressing first-hand information). (shrink)

The current work investigated whether differences in phonological overlap between the past- and present-tense forms of regular and irregular verbs can account for the graded neurophysiological effects of verb regularity observed in past-tense priming designs. Event-related potentials were recorded from 16 healthy participants who performed a lexical-decision task in which past-tense primes immediately preceded present-tense targets. To minimize intra-modal phonological priming effects, cross-modal presentation between auditory primes and visual targets was employed, and results were compared to a companion intra-modal (...) auditory study (Justus, T., Larsen, J., de Mornay Davies, P., Swick, D. (2008). Interpreting dissociations between regular and irregular past-tense morphology: evidence from event-related potentials. Cognitive, Affective, Behavioral Neuroscience, 8, 178–194.). For both regular and irregular verbs, faster response times and reduced N400 components were observed for present-tense forms when primed by the corresponding past-tense forms. Although behavioral facilitation was observed with a pseudopast phonological control condition, neither this condition nor an orthographic-phonological control produced significant N400 priming effects. Instead, these two types of priming were associated with a post-lexical anterior negativity (PLAN). Results are discussed with regard to dual- and single-system theories of inflectional morphology, as well as intra- and cross-modal prelexical priming. (shrink)

This paper deals with the system of modal logicGL, in particular with a formulation of it in terms of sequents. We prove some proof theoretical properties ofGL that allow to get the cut-elimination theorem according to Gentzen's procedure, that is, by double induction on grade and rank.

Possibilistic logic and modal logic are knowledge representation frameworks sharing some common features, such as the duality between possibility and necessity, and the decomposability of necessity for conjunctions, as well as some obvious differences since possibility theory is graded. At the semantic level, possibilistic logic relies on possibility distributions and modal logic on accessibility relations. In the last 30 years, there have been a series of attempts for bridging the two frameworks in one way or another. In this paper, (...) we compare the relational semantics of epistemic logics with simpler possibilistic semantics of a fragment of such logics that only uses modal formulas of depth 1. This minimal epistemic logic handles both all-or-nothing beliefs and explicitly ignored facts. We also contrast epistemic logic with the S5-based rough set logic. Finally, this paper presents extensions of generalized possibilistic logic with objective and non-nested multimodal formulas, in the style of modal logics KD45 and S5. (shrink)

We present a natural standard translation of inquisitive modal logic $\mathrm{InqML}$ into first-order logic over the natural two-sorted relational representations of the intended models, which captures the built-in higher-order features of $\mathrm{InqML}$. This translation is based on a graded notion of flatness that ties the inherent second-order, team-semantic features of $\mathrm{InqML}$ over information states to subsets or tuples of bounded size. A natural notion of pseudo-models, which relaxes the non-elementary constraints on the intended models, gives rise to an elementary, (...) purely model-theoretic proof of the compactness property for $\mathrm{InqML}$. Moreover, we prove a Hennessy-Milner theorem for $\mathrm{InqML}$, which crucially uses $\omega $ -saturated pseudo-models and the new standard translation. As corollaries we also obtain van Benthem style characterisation theorems. (shrink)

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)

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

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