Transfers between Logics and their Applications

In this paper, logics are conceived as two-sorted first-order structures, and we argue that this broad definition encompasses a wide class of logics with theoretical interest as well as interest from the point of view of applications. The language, concepts and methods of model theory can thus be used to describe the relationship between logics through morphisms of structures called transfers. This leads to a formal framework for studying several properties of abstract logics and their attributes such as consequence operator, syntactical structure, and internal transformations. In particular, we treat Belief Revision Systems (BRS) as our main example, defining the Wide Belief Revision Systems (WBRS's). This generalization allows us to define BRS's in an abstract setting for classical and non-standard logics. We also show how the concept of translation between logics can be obtained as a particular case of transfers.     

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.     

Closure Operators and Complete Embeddings of Residuated Lattices

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follow, including various types of completeness theorems of substructural logics.     

The Undecidability of Grisin's Set Theory

We investigate a contractionless naive set theory, due to Grisin [11]. We prove that the theory is undecidable.     

Algorithmic Proof Methods and Cut Elimination for Implicational Logics Part I: Modal Implication

In this work we develop goal-directed deduction methods for the implicational fragment of several modal logics. We give sound and complete procedures for strict implication of K, T, K4, S4, K5, K45, KB, KTB, S5, G and for some intuitionistic variants. In order to achieve a uniform and concise presentation, we first develop our methods in the framework of Labelled Deductive Systems [Gabbay 96]. The proof systems we present are strongly analytical and satisfy a basic property of cut admissibility. We then show that for most of the systems under consideration the labelling mechanism can be avoided by choosing an appropriate way of structuring theories. One peculiar feature of our proof systems is the use of restart rules which allow to re-ask the original goal of a deduction. In case of K, K4, S4 and G, we can eliminate such a rule, without loosing completeness. In all the other cases, by dropping such a rule, we get an intuitionistic variant of each system. The present results are part of a larger project of a goal directed proof theory for non-classical logics; the purpose of this project is to show that most implicational logics stem from slight variations of a unique deduction method, and from different ways of structuring theories. Moreover, the proof systems we present follow the logic programming style of deduction and seem promising for proof search [Gabbay and Reyle 84, Miller et al. 91].     

Many-Valued Logics and Suszko's Thesis Revisited

Suszko's Thesis maintains that many-valued logics do not exist at all. In order to support it, R. Suszko offered a method for providing any structural abstract logic with a complete set of bivaluations. G. Malinowski challenged Suszko's Thesis by constructing a new class of logics (called q-logics by him) for which Suszko's method fails. He argued that the key for logical two-valuedness was the "bivalent" partition of the Lindenbaum bundle associated with all structural abstract logics, while his q-logics were generated by "trivalent" matrices. This paper will show that contrary to these intuitions, logical two-valuedness has more to do with the geometrical properties of the deduction relation of a logical structure than with the algebraic properties embedded on it.     

Ranking Theory and Conditional Reasoning

Ranking theory is a formal epistemology that has been developed in over 600 pages in Spohn's recent book The Laws of Belief, which aims to provide a normative account of the dynamics of beliefs that presents an alternative to current probabilistic approaches. It has long been received in the AI community, but it has not yet found application in experimental psychology. The purpose of this paper is to derive clear, quantitative predictions by exploiting a parallel between ranking theory and a statistical model called logistic regression. This approach is illustrated by the development of a model for the conditional inference task using Spohn's (2013) ranking theoretic approach to conditionals.     

Leaping to Conclusions: Why Premise Relevance Affects Argument Strength

Everyday reasoning requires more evidence than raw data alone can provide. We explore the idea that people can go beyond this data by reasoning about how the data was sampled. This idea is investigated through an examination of premise non‐monotonicity, in which adding premises to a category‐based argument weakens rather than strengthens it. Relevance theories explain this phenomenon in terms of people's sensitivity to the relationships among premise items. We show that a Bayesian model of category‐based induction taking premise sampling assumptions and category similarity into account complements such theories and yields two important predictions: First, that sensitivity to premise relationships can be violated by inducing a weak sampling assumption; and second, that premise monotonicity should be restored as a result. We test these predictions with an experiment that manipulates people's assumptions in this regard, showing that people draw qualitatively different conclusions in each case.     

Relevance drives attention: Attentional bias for gain- and loss-related stimuli is driven by delayed disengagement

Attentional bias to gain- and loss-related stimuli was investigated in a dot-probe task. We used coloured stimuli that had acquired their valence during the experiment by signalling the chance to either win or lose points in a game task. Replicating previous findings with the additional singleton paradigm, we found attentional bias effects for both gain- and loss-related colours. The effects were due to delayed disengagement from valent stimuli, especially if they were positive, and could not be explained by nonattentional processes like behavioural freezing. Our findings suggest that stimuli signalling opportunities and dangers hold attention, supporting a general motivational relevance principle of the orienting of attention.