Abductive reasoning in neural-symbolic systems

Abduction is or subsumes a process of inference. It entertains possible hypotheses and it chooses hypotheses for further scrutiny. There is a large literature on various aspects of non-symbolic, subconscious abduction. There is also a very active research community working on the symbolic (logical) characterisation of abduction, which typically treats it as a form of hypothetico-deductive reasoning. In this paper we start to bridge the gap between the symbolic and sub-symbolic approaches to abduction. We are interested in benefiting from developments made by each community. In particular, we are interested in the ability of non-symbolic systems (neural networks) to learn from experience using efficient algorithms and to perform massively parallel computations of alternative abductive explanations. At the same time, we would like to benefit from the rigour and semantic clarity of symbolic logic. We present two approaches to dealing with abduction in neural networks. One of them uses Connectionist Modal Logic and a translation of Horn clauses into modal clauses to come up with a neural network ensemble that computes abductive explanations in a top-down fashion. The other combines neural-symbolic systems and abductive logic programming and proposes a neural architecture which performs a more systematic, bottom-up computation of alternative abductive explanations. Both approaches employ standard neural network architectures which are already known to be highly effective in practical learning applications. Differently from previous work in the area, our aim is to promote the integration of reasoning and learning in a way that the neural network provides the machinery for cognitive computation, inductive learning and hypothetical reasoning, while logic provides the rigour and explanation capability to the systems, facilitating the interaction with the outside world. Although it is left as future work to determine whether the structure of one of the proposed approaches is more amenable to learning than the other, we hope to have contributed to the development of the area by approaching it from the perspective of symbolic and sub-symbolic integration.

A Note on Binary Inductive Logic

We consider the problem of induction over languages containing binary relations and outline a way of interpreting and constructing a class of probability functions on the sentences of such a language. Some principles of inductive reasoning satisfied by these probability functions are discussed, leading in turn to a representation theorem for a more general class of probability functions satisfying these principles.     

Pandora: A Reasoning Toolbox using Natural Deduction Style

Pandora is a tool for supporting the learning of first ordernatural deduction. It includes a help window, an interactivecontext sensitive tutorial known as the "e-tutor" and facilitiesto save, reload and export to LATEX. Every attempt to applya natural deduction rule is met with either success or a helpfulerror message, providing the student with instant feedback.Detailed electronic logs of student usage are recorded for evaluationpurposes. This paper describes the basic functionality, thee-tutor, our experiences of using the tool in teaching and ourfuture plans.     

How fish do geometry in large and in small spaces

It has been shown that children and non-human animals seem to integrate geometric and featural information to different extents in order to reorient themselves in environments of different spatial scales. We trained fish (redtail splitfins, Xenotoca eiseni) to reorient to find a corner in a rectangular tank with a distinctive featural cue (a blue wall). Then we tested fish after displacement of the feature on another adjacent wall. In the large enclosure, fish chose the two corners with the feature, and also tended to choose among them the one that maintained the correct arrangement of the featural cue with respect to geometric sense (i.e. left-right position). In contrast, in the small enclosure, fish chose both the two corners with the features and the corner, without any feature, that maintained the correct metric arrangement of the walls with respect to geometric sense. Possible reasons for species differences in the use of geometric and non-geometric information are discussed.     

On the Nature of Continuous Physical Quantities in Classical and Quantum Mechanics

Journal of Philosophical Logic - Within the traditional Hilbert space formalism of quantum mechanics, it is not possible to describe a particle as possessing, simultaneously, a sharp position value...     

First-order Expressivity for S5-models: Modal vs. Two-sorted Languages

Standard models for model predicate logic consist of a Kripke frame whose worlds come equipped with relational structures. Both modal and two-sorted predicate logic are natural languages for speaking about such models. In this paper we compare their expressivity. We determine a fragment of the two-sorted language for which the modal language is expressively complete on S5-models. Decidable criteria for modal definability are presented.     

Term-Modal Logics

Many powerful logics exist today for reasoning about multi-agent systems, but in most of these it is hard to reason about an infinite or indeterminate number of agents. Also the naming schemes used in the logics often lack expressiveness to name agents in an intuitive way.To obtain a more expressive language for multi-agent reasoning and a better naming scheme for agents, we introduce a family of logics called term-modal logics. A main feature of our logics is the use of modal operators indexed by the terms of the logics. Thus, one can quantify over variables occurring in modal operators. In term-modal logics agents can be represented by terms, and knowledge of agents is expressed with formulas within the scope of modal operators.This gives us a flexible and uniform language for reasoning about the agents themselves and their knowledge. This article gives examples of the expressiveness of the languages and provides sequent-style and tableau-based proof systems for the logics. Furthermore we give proofs of soundness and completeness with respect to the possible world semantics.     

On Different Intuitionistic Calculi and Embeddings from Int to S4

In this paper, we compare several cut-free sequent systems for propositional intuitionistic logic Intwith respect to polynomial simulations. Such calculi can be divided into two classes, namely single-succedent calculi (like Gentzen's LJ) and multi-succedent calculi. We show that the latter allow for more compact proofs than the former. Moreover, for some classes of formulae, the same is true if proofs in single-succedent calculi are directed acyclic graphs (dags) instead of trees. Additionally, we investigate the effect of weakening rules on the structure and length of dag proofs.The second topic of this paper is the effect of different embeddings from Int to S4. We select two different embeddings from the literature and show that translated (propositional) intuitionistic formulae have sometimes exponentially shorter minimal proofs in a cut-free Gentzen system for S4than the original formula in a cut-free single-succedent Gentzen system for Int. Moreover, the length and the structure of proofs of translated formulae crucially depend on the chosen embedding.     

Searching for the Roots of the Circumstantial Ad Hominem

This paper looks into the known evidence on the origins of the type of argument called the circumstantial ad hominemargument in modern logic textbooks, and introduces some new evidence. This new evidence comes primarily from recent historical work by Jaap Mansfeld and Jonathan Barnes citing many cases where philosophers in the ancient world were attacked on the grounds that their personal actions failed to be consistent with their philosophical teachings. On the total body of evidence, two hypotheses about the roots of the circumstantial ad hominem are considered. One is that it came from Aristotle through Locke. The other is that it may have had separate roots in these ancient philosophical writings that criticized philosophers for not practicing what they preached.