Symmetry vs. Duality in Logic: An Interpretation of Bi-Logic to Model Cognitive Processes Beyond Inference

Symmetry vs. Duality in Logic: An Interpretation of Bi-Logic to Model Cognitive Processes Beyond Inference

Giulia Battilotti (Department of Mathematics, University of Padua, Padua, Italy)
DOI: 10.4018/ijcini.2014100105
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Abstract

The author discusses the problem of symmetry, namely of the orientation of the logical consequence. The author shows that the problem is surprisingly entangled with the problem of “being infinite”. The author presents a model based on quantum states and shows that it features satisfy the requirements of the symmetric mode of Bi-logic, a logic introduced in the '70s by the psychoanalyst I. Matte Blanco to describe the logic of the unconscious. The author discusess symmetry, in the model, to include correlations, in order to obtain a possible approach to displacement. In this setting, the author finds a possible reading of the structural rules of sequent calculus, whose role in computation, on one side, and in the representation of human reasoning, on the other, has been debated for a long time.
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Introduction

Quoting Wang (2012): “The logical model of the brain and the abstract intelligence theory of the natural intelligence will enable the development of cognitive computers that perceive, think and learn. The functional and theoretical difference between cognitive computers and classic computers are that the latter are data processors based on Boolean algebra and its logical counterparts; while the former are knowledge processors based on contemporary denotational mathematics.”

The quotation makes it clear the need to distinguish “cognitive computers” from the usual ones in order to develop cognitive informatics (Wang et al., 2010; Wang et al., 2011) moreover it makes clear that “knowledge processors” should escape the strict limitations of classical logic in favour of the much broader means offered by mathematics.

Then the question: “Which logic should knowledge processors follow?” The question has very intriguing consequences both from a theoretical and practical point of view. By the limitative results proved in the '30s, first order logic cannot derive all mathematical truth. This implies that the logic of knowledge processors should include also different logical tools. In Wang (2007), the author describes the cognitive processes of formal inferences such as deduction, induction, abduction and analogy. However, even if inferences different from deduction are considered, their logical formalization exploits the usual framework of a deductive theory.

Since the best example of knowledge processors at our disposal are human beings, in the present paper, we tackle the problem from the point of view of Bi-logic, the logic proposed by the Chilean psychoanalyst Ignacio Matte Blanco to grasp all the features of the processes of judgements performed by human beings, including those due to the Unconscious, see Matte Blanco (1975) and Matte Blanco (1988).

Bi-logic has two modes:

  • 1.

    The asymmetric mode: Proper of the conscious reasoning, which deals with non-symmetric relations, can separate objects, and permits sound logic, where two distinct truth values are present;

  • 2.

    The symmetric mode: That is the mode of the unconscious. The symmetric mode has symmetric relations only, it gathers, identifies objects, creates links between judgements different from those considered at the conscious level and has an unsound logical behaviour.

Moreover, since, by symmetry, any part is treated as the whole thing, any subset and the whole set are idempotent, and then, following Matte Blanco, the unconscious deals with infinite sets.

Symmetry has the following logical and computational consequences:

  • 1.

    Condensation: The opposites coexist - no mutual contradiction - no negation;

  • 2.

    No temporal processes: (Matte Blanco suggests to speak about “manifestations” rather than “processes” of the unconscious) and hence no algorithmic/step-by-step processes - no logical consequence;

  • 3.

    Displacement: Different hidden symmetric links between judgements.

Total symmetrization characterizes the “indivisible mode”, where “the endless number of things tend to become, mysteriously, only one thing” (Matte Blanco, 1988).

The present work finds a possible logical approach to the symmetric mode, considering a model proposed in the framework of quantum computational logics. Since in quantum physics one has the uncertainty principle, adopting our quantum model, in particular, we explain why the symmetric mode should be so hidden to our usual logical mode, and so, why logic could not supply enough means to cognitive informatics, in order to perform satisfactory results, from the human point of view.

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