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Firstorder classical modal logic
 Studia Logica 84 (2006), 171
"... Abstract. The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer a series of new completeness results for salient classical syste ..."
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Abstract. The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K) in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics. We conclude by introducing general first order neighborhood frames and we offer a general completeness result in terms of them which circumvents some wellknown problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of neighborhood frames). We argue that the semantical program that thus arises surpasses both in expressivity and adequacy the standard Kripkean approach, even when it comes to the study of first order normal systems.
FirstOrder Classical Modal Logic: Applications in logics of knowledge and probability
"... The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we o#er a series of new completeness results for salient classical systems of fi ..."
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The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we o#er a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL+K) in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. We conclude by o#ering a general completeness result for the entire family of first order classical modal logics (encompassing both normal and nonnormal systems).
Inexact knowledge with introspection
 Journal of Philosophical Logic
, 2009
"... Standard Kripke models are inadequate to model situations of inexact knowledge with introspection, since positive and negative introspection force the relation of epistemic indiscernibility to be transitive and euclidean. Correlatively, Williamson’s margin for error semantics for inexact knowledge i ..."
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Standard Kripke models are inadequate to model situations of inexact knowledge with introspection, since positive and negative introspection force the relation of epistemic indiscernibility to be transitive and euclidean. Correlatively, Williamson’s margin for error semantics for inexact knowledge invalidates axioms 4 and 5. We present a new semantics for modal logic which is shown to be complete for K45, without constraining the accessibility relation to be transitive or euclidean. The semantics corresponds to a system of modular knowledge, in which iterated modalities and simple modalities are not on a par. We show how the semantics helps to solve Williamson’s luminosity paradox, and argue that it corresponds to an integrated model of perceptual and introspective knowledge that is psychologically more plausible than the one defended by Williamson. We formulate a generalized version of the semantics, called token semantics, in which modalities are iterationsensitive up to degree n and insensitive beyond n. The multiagent version of the semantics yields a resourcesensitive logic with implications for the representation of common knowledge in situations of bounded rationality.
Similarity, Approximations and Vagueness
 PROC. RSFDGRC’05 (E. A. SLEZAK D., ED.), LNAI, NUMBER 3641 IN LNAI
, 2005
"... The relation of similarity is essential in understanding and developing frameworks for reasoning with vague and approximate concepts. There is a wide spectrum of choice as to what properties we associate with similarity and such choices determine the nature of vague and approximate concepts define ..."
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The relation of similarity is essential in understanding and developing frameworks for reasoning with vague and approximate concepts. There is a wide spectrum of choice as to what properties we associate with similarity and such choices determine the nature of vague and approximate concepts defined in terms of these relations. Additionally, robotic systems naturally have to deal with vague and approximate concepts due to the limitations in reasoning and sensor capabilities. Halpern [1] introduces the use of subjective and objective states in a modal logic formalizing vagueness and distinctions in transitivity when an agent reasons in the context of sensory and other limitations. He also relates these ideas to a solution to the Sorities and other paradoxes. In this paper, we generalize and apply the idea of similarity and tolerance spaces [2,3,4,5], a means of constructing approximate and vague concepts from such spaces and an explicit way to distinguish between an agent’s objective and subjective states. We also show how some of the intuitions from Halpern can be used with similarity spaces to formalize the abovementioned Sorities and other paradoxes.
Inexact Knowledge, Margin for Error and Positive Introspection
"... Williamson (2000a) has argued that positive introspection is incompatible with inexact knowledge. His argument relies on a marginforerror requirement for inexact knowledge based on a intuitive safety principle for knowledge, but leads to the counterintuitive conclusion that no possible creature co ..."
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Williamson (2000a) has argued that positive introspection is incompatible with inexact knowledge. His argument relies on a marginforerror requirement for inexact knowledge based on a intuitive safety principle for knowledge, but leads to the counterintuitive conclusion that no possible creature could have both inexact knowledge and positive introspection. Following Halpern (2004) I put forward an alternative marginforerror requirement that preserves the safety requirement while blocking Williamson’s argument. I argue that the infallibilist conception of knowledge that underlies the new requirement
Ambiguous Language and Differences in Beliefs
"... Standard models of multiagent modal logic do not capture the fact that information is often ambiguous, and may be interpreted in different ways by different agents. We propose a framework that can model this, and consider different semantics that capture different assumptions about the agents ’ bel ..."
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Standard models of multiagent modal logic do not capture the fact that information is often ambiguous, and may be interpreted in different ways by different agents. We propose a framework that can model this, and consider different semantics that capture different assumptions about the agents ’ beliefs regarding whether or not there is ambiguity. We consider the impact of ambiguity on a seminal result in economics: Aumann’s result saying that agents with a common prior cannot agree to disagree. This result is known not to hold if agents do not have a common prior; we show that it also does not hold in the presence of ambiguity. We then consider the tradeoff between assuming a common interpretation (i.e., no ambiguity) and a common prior (i.e., shared initial beliefs). 1
Possible Worlds and Possible Meanings: a Semantics for the Interpretation of Vague Languages
"... The paper develops a formal model for interpreting vague languages based on a variant of supervaluation semantics. Two modes of semantic variability are modelled, corresponding to different aspects of vagueness: one mode arises where there can be multiple definitions of a term; and the other relates ..."
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The paper develops a formal model for interpreting vague languages based on a variant of supervaluation semantics. Two modes of semantic variability are modelled, corresponding to different aspects of vagueness: one mode arises where there can be multiple definitions of a term; and the other relates to the threshold of applicability of a vague term with respect to the magnitude of relevant observable values. The truth of a proposition depends on both the possible world and the precisification with respect to which it is evaluated. Structures representing possible worlds and precisifications are both specified in terms of primitive functions representing observable measurements, so that the semantics is grounded upon an underlying theory of physical reality. On the basis of this semantics, the acceptability of a proposition to an agent is characterised in terms of a combination of agent’s beliefs about the world and their attitude to admissible interpretations of vague predicates. 1
Margins for Error in Context
"... According to the epistemic theory of vagueness defended in particular by Sorensen (2001) and Williamson (1994: 237), vagueness is due to our limited powers of discrimination: looking at a particular shade of red fabric, I may not be able to recognize that it is red, as a result of the specific granu ..."
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According to the epistemic theory of vagueness defended in particular by Sorensen (2001) and Williamson (1994: 237), vagueness is due to our limited powers of discrimination: looking at a particular shade of red fabric, I may not be able to recognize that it is red, as a result of the specific granularity of my perceptual apparatus, which for instance makes the shade look somewhere between orange and red to me. Conversely, whenever I am confident that a particular shade of color is red, then this means that a slight variation in color should leave intact the fact that the shade is red. In Williamson’s account of vagueness, this idea is expressed in terms of what Williamson calls margin for error principles for knowledge: whenever my knowledge is inexact in the sense of being approximative, it requires a sufficient margin for error in order to hold. More abstractly, the margin for error principle says that in order for me to know that some property P holds of an object d, then a slight modification of some relevant parameter in the object d should leave it in the extension of P. Expressed in terms of propositions and contexts, this means that in order to know that some proposition p holds in a context w, then p should still hold in a context w ′ that is only slightly different from w. An important consequence of Williamson’s margin for error semantics for inexact knowledge
Proceedings of the Thirteenth International Conference on Principles of Knowledge Representation and Reasoning Ambiguous Language and Differences in Beliefs
"... Standard models of multiagent modal logic do not capture the fact that information is often ambiguous, and may be interpreted in different ways by different agents. We propose a framework that can model this, and consider different semantics that capture different assumptions about the agents ’ bel ..."
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Standard models of multiagent modal logic do not capture the fact that information is often ambiguous, and may be interpreted in different ways by different agents. We propose a framework that can model this, and consider different semantics that capture different assumptions about the agents ’ beliefs regarding whether or not there is ambiguity. We consider the impact of ambiguity on a seminal result in economics: Aumann’s result saying that agents with a common prior cannot agree to disagree. This result is known not to hold if agents do not have a common prior; we show that it also does not hold in the presence of ambiguity. We then consider the tradeoff between assuming a common interpretation (i.e., no ambiguity) and a common prior (i.e., shared initial beliefs). 1