Logic programs, iterated function systems, and recurrent radial basis function networks

Graphs of the single-step operator for first-order logic programs—displayed in the real plane—exhibit self-similar structures known from topological dynamics, i.e., they appear to be fractals, or more precisely, attractors of iterated function systems. We show that this observation can be made mathematically precise. In particular, we give conditions which ensure that those graphs coincide with attractors of suitably chosen iterated function systems, and conditions which allow the approximation of such graphs by iterated function systems or by fractal interpolation. Since iterated function systems can easily be encoded using recurrent radial basis function networks, we eventually obtain connectionist systems which approximate logic programs in the presence of function symbols.     

Conflicting imperatives and dyadic deontic logic

Often a set of imperatives or norms seems satisfiable from the outset, but conflicts arise when ways to fulfill all are ruled out by unfortunate circumstances. Semantic methods to handle normative conflicts were devised by B. van Fraassen and J.F. Horty, but these are not sensitive to circumstances. The present paper extends these resolution mechanisms to circumstantial inputs, defines dyadic deontic operators accordingly, and provides a sound and (weakly) complete axiomatic system for such deontic semantics.     

Reduction and emergence in the physical sciences: Reply to Rueger

I analyse Rueger’s application of Kim’s model of functional reduction to the relation between the thermal conductivities of metal bars at macroscopic and atomic scales. 1) I show that it is a misunderstanding to accuse the functional reduction model of not accounting for the fact that there are causal powers at the micro-level which have no equivalent at the macro-level. The model not only allows but requires that the causal powers by virtue of which a functional predicate is defined, are only a subset of the causal powers of the properties filling the functional specification. 2) The fact that the micro-equation does not converge to the macro-equation in general but only under the constraint of a “solvability condition” does not show that reduction is impossible, as Rueger claims, but only that reduction requires inter-level constraints. 3) Rueger tries to analyse inter-level reduction with the conceptual means of intra-level reduction. This threatens the coherence of his analysis, given that it makes no sense to ascribe macroproperties such as thermal conductivity to entities at the atomic level. Ignoring the distinction between theses two senses of “reduction” is especially confusing because they have opposite directions: in intra-level reduction, the more detailed account reduces to the less detailed one, whereas in inter-level reduction, the less detailed theory is reduced to the more detailed one. 4) Finally I criticize Rueger’s way of using Wimsatt’s criteria for emergence in terms of non-aggregativity, to construct a concept of synchronic emergence. It is wrong to require, over and above non-aggregativity, irreducibility as a criterion for emergence.     

Hybrid Probabilistic Logic Programs as Residuated Logic Programs

In this paper we show the embedding of Hybrid Probabilistic Logic Programs into the rather general framework of Residuated Logic Programs, where the main results of (definite) logic programming are validly extrapolated, namely the extension of the immediate consequences operator of van Emden and Kowalski. The importance of this result is that for the first time a framework encompassing several quite distinct logic programming semantics is described, namely Generalized Annotated Logic Programs, Fuzzy Logic Programming, Hybrid Probabilistic Logic Programs, and Possibilistic Logic Programming. Moreover, the embedding provides a more general semantical structure paving the way for defining paraconsistent probabilistic reasoning with a logic programming semantics.     

Towards the logic of projective identification

This paper is devoted to the ‘logic of the unconscious’ and its application to the analysis of projective intentionality in psychoanalysis.Subjective assumptions concerning the existence and identity of intentional objects are often unconscious. They result from personal experience through its assimilation and transformation in further psychological (e.g. defensive) processes. Formal aspects of these subjective assumptions and their influence on our judgment and action have been studied by a number of psychoanalytic authors, in particular by Silvano Arieti, Ignacio Matte-Blanco and their followers who tried to develop ‘logic of the unconscious’. My project consists in the reformulation, clarification and elaboration of the logic of the unconscious using contemporary modal and relevant logics, in particular Graham Priest's logic of intentionality. An important advantage of this logic is that it allows for truth indeterminacy and paraconsistency of the propositional content of intentional states. In this article I explore the logic of projective identification, which I assume plays the central role in the logic of the unconscious.Special attention is given to the logical analysis of the notion of an internal object, and to a logical reconstruction of the fantasy of projective identification.     

Learning Łukasiewicz logic

The integration between connectionist learning and logic-based reasoning is a longstanding foundational question in artificial intelligence, cognitive systems, and computer science in general. Research into neural-symbolic integration aims to tackle this challenge, developing approaches bridging the gap between sub-symbolic and symbolic representation and computation. In this line of work the core method has been suggested as a way of translating logic programs into a multilayer perceptron computing least models of the programs. In particular, a variant of the core method for three valued Łukasiewicz logic has proven to be applicable to cognitive modelling among others in the context of Byrne’s suppression task. Building on the underlying formal results and the corresponding computational framework, the present article provides a modified core method suitable for the supervised learning of Łukasiewicz logic (and of a closely-related variant thereof), implements and executes the corresponding supervised learning with the backpropagation algorithm and, finally, constructs a rule extraction method in order to close the neural-symbolic cycle. The resulting system is then evaluated in several empirical test cases, and recommendations for future developments are derived.