A New Semantics for Systems of Logic of Essence

The purpose of the present paper is to provide a way of understanding systems of logic of essence by introducing a new semantic framework for them. Three central results are achieved: first, the now standard Fitting semantics for the propositional logic of evidence is adapted in order to provide a new, simplified semantics for the propositional logic of essence; secondly, we show how it is possible to construe the concept of necessary truth explicitly by using the concept of essential truth; finally, Fitting semantics is adapted in order to present a simplified semantics for the quantified logic of essence.     

Modelling simultaneous games in dynamic logic

We make a proposal for formalizing simultaneous games at the abstraction level of player’s powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of ‘concurrent game logic’ CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic.     

Agreement Theorems in Dynamic-Epistemic Logic

This paper introduces Agreement Theorems to dynamic-epistemic logic. We show first that common belief of posteriors is sufficient for agreement in epistemic-plausibility models, under common and well-founded priors. We do not restrict ourselves to the finite case, showing that in countable structures the results hold if and only if the underlying plausibility ordering is well-founded. We then show that neither well-foundedness nor common priors are expressible in the language commonly used to describe and reason about epistemic-plausibility models. The static agreement result is, however, finitely derivable in an extended modal logic. We provide the full derivation. We finally consider dynamic agreement results. We show they have a counterpart in epistemic-plausibility models, and provide a new form of agreements via public announcements.     

Kripke Models,Distributive Lattices,and Medvedev Degrees

We define a variant of the standard Kripke semantics for intuitionistic logic, motivated by the connection between constructive logic and the Medvedev lattice. We show that while the new semantics is still complete, it gives a simple and direct correspondence between Kripke models and algebraic structures such as factors of the Medvedev lattice. Presented by Daniele Mundici     

From IF to BI

We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges. We show how Hodges’ semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O’Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural rôle, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural rôle.     

Refutations,Proofs, and Models in the Modal Logic K4

In this paper we study the method of refutation rules in the modal logic K4. We introduce refutation rules with certain normal forms that provide a new syntactic decision procedure for this logic. As corollaries we obtain such results for the following important extensions: S4, the provability logic G, and Grzegorczyk's logic. We also show that tree-type models can be constructed from syntactic refutations of this kind.     

A logic for diffusion in social networks

This paper introduces a general logical framework for reasoning about diffusion processes within social networks. The new “Logic for Diffusion in Social Networks” is a dynamic extension of standard hybrid logic, allowing to model complex phenomena involving several properties of agents. We provide a complete axiomatization and a terminating and complete tableau system for this logic and show how to apply the framework to diffusion phenomena documented in social networks analysis.     

The Surprise Examination in Dynamic Epistemic Logic

We examine the paradox of the surprise examination using dynamic epistemic logic. This logic contains means of expressing epistemic facts as well as the effects of learning new facts, and is therefore a natural framework for representing the puzzle. We discuss a number of different interpretations of the puzzle in this context, and show how the failure of principle of success, that states that sentences, when learned, remain to be true and come to be believed, plays a central role in understanding the puzzle.     

Grishin Algebras and Cover Systems for Classical Bilinear Logic

Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (??propositions??) of cover systems that use an orthogonality relation to interpret the negations. The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover system, by a map that preserves all existing joins and meets. This representation is then used to give a cover system semantics for a version of classical bilinear logic that has first-order quantifiers and infinitary conjunctions and disjunctions.     

Inexact Knowledge with Introspection

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 iteration-sensitive up to degree n and insensitive beyond n. The multi-agent version of the semantics yields a resource-sensitive logic with implications for the representation of common knowledge in situations of bounded rationality.     

A Note on Bisimulation and Modal Equivalence in Provability Logic and Interpretability Logic

Provability logic is a modal logic for studying properties of provability predicates, and Interpretability logic for studying interpretability between logical theories. Their natural models are GL-models and Veltman models, for which the accessibility relation is well-founded. That’s why the usual counterexample showing the necessity of finite image property in Hennessy-Milner theorem (see [1]) doesn’t exist for them. However, we show that the analogous condition must still hold, by constructing two GL-models with worlds in them that are modally equivalent but not bisimilar, and showing how these GL-models can be converted to Veltman models with the same properties. In the process we develop some useful constructions: games on Veltman models, chains, and general method of transformation from GL-models/frames to Veltman ones.     

Reasoning About Social Choice Functions

We introduce a logic specifically designed to support reasoning about social choice functions. The logic includes operators to capture strategic ability, and operators to capture agent preferences. We establish a correspondence between formulae in the logic and properties of social choice functions, and show that the logic is expressively complete with respect to social choice functions, i.e., that every social choice function can be characterised as a formula of the logic. We prove that the logic is decidable, and give a complete axiomatization. To demonstrate the value of the logic, we show in particular how it can be applied to the problem of determining whether a social choice function is strategy-proof.     

Dynamical Models of Sentence Processing

We suggest that the theory of dynamical systems provides a revealing general framework for modeling the representations and mechanism underlying syntactic processing. We show how a particular dynamical model, the Visitation Set Gravitation model of Tabor, Juliano, and Tanenhaus (1997), develops syntactic representations and models a set of contingent frequency effects in parsing that are problematic for other models. We also present new simulations showing how the model accounts for semantic effects in parsing, and propose a new account of the distinction between syntactic and semantic incongruity. The results show how symbolic structures useful in parsing arise as emergent properties of connectionist dynamical systems.     

Bisimulation for Conditional Modalities

We give a definition of bisimulation for conditional modalities interpreted on selection functions and prove the correspondence between bisimilarity and modal equivalence, generalizing the Hennessy–Milner Theorem to a wide class of conditional operators. We further investigate the operators and semantics to which these results apply. First, we show how to derive a solid notion of bisimulation for conditional belief, behaving as desired both on plausibility models and on evidence models. These novel definitions of bisimulations are exploited in a series of undefinability results. Second, we treat relativized common knowledge, underlining how the same results still hold for a different modality in a different semantics. Third, we show the flexibility of the approach by generalizing it to multi-agent systems, encompassing the case of multi-agent plausibility models.     

Weak Rejection

Linguistic evidence supports the claim that certain, weak rejections are less specific than assertions. On the basis of this evidence, it has been argued that rejected sentences cannot be premisses and conclusions in inferences. We give examples of inferences with weakly rejected sentences as premisses and conclusions. We then propose a logic of weak rejection which accounts for the relevant phenomena and is motivated by principles of coherence in dialogue. We give a semantics for which this logic is sound and complete, show that it axiomatizes the modal logic KD45 and prove that it still derives classical logic on its asserted fragment. Finally, we defend previous logics of strong rejection as being about the linguistically preferred interpretations of weak rejections.     

Reactive intuitionistic tableaux

We introduce reactive Kripke models for intuitionistic logic and show that the reactive semantics is stronger than the ordinary semantics. We develop Beth tableaux for the reactive semantics.     

Does opposition logic provide evidence for conscious and unconscious processes in artificial grammar learning?

The question of whether studies of human learning provide evidence for distinct conscious and unconscious influences remains as controversial today as ever. Much of this controversy arises from the use of the logic of dissociation. The controversy has prompted the use of an alternative approach that places conscious and unconscious influences on memory retrieval in opposition. Here we ask whether evidence acquired via the logic of opposition requires a dual-process account or whether it can be accommodated within a single similarity-based account. We report simulations using a simple neural network model of two artificial grammar learning experiments reported by that dissociated conscious and unconscious influences on classification. The simulations demonstrate that opposition logic is insufficient to distinguish between single- and multiple-system models.     

Subjective Situations and Logical Omniscience

The beliefs of the agents in a multi-agent system have been formally modelled in the last decades using doxastic logics. The possible worlds model and its associated Kripke semantics provide an intuitive semantics for these logics, but they commit us to model agents that are logically omniscient. We propose a way of avoiding this problem, using a new kind of entities called subjective situations. We define a new doxastic logic based on these entities and we show how the belief operators have some desirable properties, while avoiding logical omniscience. A comparison with two well-known proposals (Levesque's logic of explicit and implicit beliefs and Thijsse's hybrid sieve systems) is also provided.     

Negationless intuitionism

The present paper deals with natural intuitionistic semantics for intuitionistic logic within an intuitionistic metamathematics. We show how strong completeness of full first order logic fails. We then consider a negationless semantics à la Henkin for second order intuitionistic logic. By using the theory of lawless sequences we prove that, for such semantics, strong completeness is restorable. We argue that lawless negationless semantics is a suitable framework for a constructive structuralist interpretation of any second order formalizable theory (classical or intuitionistic, contradictory or not).