The Method of Socratic Proofs for Modal Propositional Logics: K5, S4.2, S4.3, S4F,S4R,S4M and G

The aim of this paper is to present the method of Socratic proofs for seven modal propositional logics: K5, S4.2, S4.3, S4M, S4F, S4R and G. This work is an extension of [10] where the method was presented for the most common modal propositional logics: K, D, T, KB, K4, S4 and S5. Presented by Jacek Malinowski     

Type Logics and Pregroups

We discuss the logic of pregroups, introduced by Lambek [34], and its connections with other type logics and formal grammars. The paper contains some new ideas and results: the cut-elimination theorem and a normalization theorem for an extended system of this logic, its P-TIME decidability, its interpretation in L1, and a general construction of (preordered) bilinear algebras and pregroups whose universe is an arbitrary monoid. Special Issue Categorial Grammars and Pregroups Edited by Wojciech Buszkowski and Anne Preller     

LOGICAL PLURALISM AND SEMANTIC INFORMATION

Up to now theories of semantic information have implicitly relied on logical monism, or the view that there is one true logic. The latter position has been explicitly challenged by logical pluralists. Adopting an unbiased attitude in the philosophy of information, we take a suggestion from Beall and Restall at heart and exploit logical pluralism to recognise another kind of pluralism. The latter is called informational pluralism, a thesis whose implications for a theory of semantic information we explore. Research Assistant of the Fund for Scientific Research – Flanders (Belgium)     

Spatial and linguistic aspects of visual imagery in sentence comprehension

There is mounting evidence that language comprehension involves the activation of mental imagery of the content of utterances ( Barsalou, 1999 ; Bergen, Chang, & Narayan, 2004 ; Bergen, Narayan, & Feldman, 2003 ; Narayan, Bergen, & Weinberg, 2004 ; Richardson, Spivey, McRae, & Barsalou, 2003 ; Stanfield & Zwaan, 2001 ; Zwaan, Stanfield, & Yaxley, 2002 ). This imagery can have motor or perceptual content. Three main questions about the process remain under‐explored, however. First, are lexical associations with perception or motion sufficient to yield mental simulation, or is the integration of lexical semantics into larger structures, like sentences, necessary? Second, what linguistic elements (e.g., verbs, nouns, etc.) trigger mental simulations? Third, how detailed are the visual simulations that are performed? A series of behavioral experiments address these questions, using a visual object categorization task to investigate whether up‐ or down‐related language selectively interferes with visual processing in the same part of the visual field (following Richardson et al., 2003 ). The results demonstrate that either subject nouns or main verbs can trigger visual imagery, but only when used in literal sentences about real space—metaphorical language does not yield significant effects—which implies that it is the comprehension of the sentence as a whole and not simply lexical associations that yields imagery effects. These studies also show that the evoked imagery contains detail as to the part of the visual field where the described scene would take place.     

An Axiomatisation for the Multi-modal Logic of Knowledge and Linear Time LTK

The paper aims at providing the multi-modal propositional logicLTK with a sound and complete axiomatisation. This logic combinestemporal and epistemic operators and focuses on m odeling thebehaviour of a set of agents operating in a system on the backgroundof a temporal framework. Time is represented as linear and discrete,whereas knowledge is modeled as an S₅-like modality. A furthermodal operator intended to represent environment knowledge isadded to the system in order to achieve the expressive powersufficient to describe the piece of information available tothe agents at each moment in the flow of time.     

Line-based affine reasoning in Euclidean plane

We consider the binary relations of parallelism and convergence between lines in a 2-dimensional affine space. Associating with parallelism and convergence the binary predicates P and C and the modal connectives [P] and [C], we consider a first-order theory based on these predicates and a modal logic based on these modal connectives. We investigate the axiomatization/completeness and the decidability/complexity of this first-order theory and this modal logic.     

Cut-free common knowledge

Starting off from the infinitary system for common knowledge over multi-modal epistemic logic presented in [L. Alberucci, G. Jäger, About cut elimination for logics of common knowledge, Annals of Pure and Applied Logic 133 (2005) 73–99], we apply the finite model property to “finitize” this deductive system. The result is a cut-free, sound and complete sequent calculus for common knowledge.     

Hume’s naturalistic theory of representation

Hume is a naturalist in many different respects and about many different topics; this paper argues that he is also a naturalist about intentionality and representation. It does so in the course of answering four questions about his theory of mental representation: (1) Which perceptions represent? (2) What can perceptions represent? (3) Why do perceptions represent at all? (4) Howdo perceptions represent what they do? It appears that, for Hume, all perceptions except passions can represent; and they can represent bodies, minds, and persons, with their various qualities. In addition, ideas can represent impressions and other ideas. However, he explicitly rejects the view that ideas are inherently representational, and he implicitly adopts a view according to which things (whether mental or non-mental) represent in virtue of playing, through the production of mental effects and dispositions, a significant part of the causal and/or functional role of what they represent. It is in virtue of their particular functional roles that qualitatively identical ideas are capable of representing particulars or general kinds; substances or modes; relations; past, present, or future; and individuals or compounds.     

Definability and Interpolation in Non-Classical Logics

Algebraic approach to study of classical and non-classical logical calculi was developed and systematically presented by Helena Rasiowa in [48], [47]. It is very fruitful in investigation of non-classical logics because it makes possible to study large families of logics in an uniform way. In such research one can replace logics with suitable classes of algebras and apply powerful machinery of universal algebra. In this paper we present an overview of results on interpolation and definability in modal and positive logics,and also in extensions of Johansson's minimal logic. All these logics are strongly complete under algebraic semantics. It allows to combine syntactic methods with studying varieties of algebras and to flnd algebraic equivalents for interpolation and related properties. Moreover, we give exhaustive solution to interpolation and some related problems for many families of propositional logics and calculi. This paper is a version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005. Presented by Jacek Malinowski     

Beyond Rasiowa's Algebraic Approach to Non-classical Logics

This paper reviews the impact of Rasiowa's well-known book on the evolution of algebraic logic during the last thirty or forty years. It starts with some comments on the importance and influence of this book, highlighting some of the reasons for this influence, and some of its key points, mathematically speaking, concerning the general theory of algebraic logic, a theory nowadays called Abstract Algebraic Logic. Then, a consideration of the diverse ways in which these key points can be generalized allows us to survey some issues in the development of the field in the last twenty to thirty years. The last part of the paper reviews some recent lines of research that in some way transcend Rasiowa's approach. I hope in this way to give the reader a general view of Rasiowa's key position in the evolution of Algebraic Logic during the twentieth century. This paper is an extended version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005. Presented by Jacek Malinowski