Logic in a Dynamic Brain

Abstract

The ability of the human brain to carry out logical reasoning can be interpreted, in general, as a by-product of adaptive capacities of complex neural networks. Thus, we seek to base abstract logical operations in the general properties of neural networks designed as learning modules. We show that logical operations executable by McCulloch-Pitts binary networks can also be programmed in analog neural networks built with associative memory modules that process inputs as logical gates. These modules can interact among themselves to generate dynamical systems that extend the repertoire of logical operations. We demonstrate how the operations of the exclusive-OR or the implication appear as outputs of these interacting modules. In particular, we provide a model of the exclusive-OR that succeeds in evaluating an odd number of options (the exclusive-OR of classical logic fails in his case), thus paving the way for a more reasonable biological model of this important logical operator. We propose that a brain trained to compute can associate a complex logical operation to an orderly structured but temporary contingent episode by establishing a codified association among memory modules. This explanation offers an interpretation of complex logical processes (eventually learned) as associations of contingent events in memorized episodes. We suggest, as an example, a cognitive model that describes these "logical episodes".

On the consistency of Sherlock Holmes’ old maxim: A logical and neurocomputational approach

“…Recently, the investigation of the links between reasoning and neural models acquired a new impulse and new perspectives promoted in part from the advances in cognitive sciences and the interest in neural computations. Between these new approaches, we want to mention the panoramic contribution of Stenning and van Lambalgen (2008, in In the present work, we opted to use models based on Anderson-Kohonen matrix memories, where the logical computations are performed by networks of interconnected matrix modules (Mizraji and Lin 2011). In turn, each modular unit is composed by a large set of interconnected neurons represented by high dimensional matrices and can be programmed or instructed via a learning algorithm (e.g.…”

“…Using the neural-inspired logical formalism (Mizraji 2008 andMizraji andLin 2011), we showed in equation (29) how Holmes' maxim is represented using modal logical operators and how truth-evaluations are expressed with a matrix-vector formalism. As was previously mentioned, these logical operators are interpretable as memory modules that integrate a modular network (in fact a network of networks, because each one of the logi-cal modules is by itself a neural network).…”

“…Our formal approach is neutral in the debate about the "Nature versus Nurture" origin of our logical abilities: these neural logical modules can be the result of a genetically coded ontogenetic process, or, on the contrary, they can be the result of a learning process occurring in a particular cultural environment (for a discussion of this point, see Mizraji and Lin 2011). We argue that the spontaneous computation involved in the understanding of Holmes' maxim is only one example among many others; all of them emerging from a natural biological design that obviously includes our cognitive brains, together with our sensory and motor systems.…”

“…One of these approaches (Mizraji and Lin 2011) uses the Kronecker product to integrate inputs and contexts. In this framework, the matrix memory can be expressed as (13) where p j is the context associated with the input f i , and g ij is the output associated with the contextualized input.…”

“…This is the only assumption concerning probabilities. It implies a kind of conservation inside the judgment (conservation mapped in the complementarity of assigned probabilities for the two canonical truth-values s and n, and in turn supported by the complementarity of the associated conceptual sets (Mizraji and Lin 2011)). In this work we will leave as an open problem the link between these "cognitive probabilities" and the formal probabilities involved in the Bayesian treatment illustrated in Section 2.…”