Logic Dynamics for Deductive Inference — Its Stability and Neural Basis

Abstract

We propose a dynamical model that represents a process of deductive inference. We discuss the stability of logic dynamics and a neural basis for the dynamics. We propose a new concept of descriptive stability, thereby enabling a structure of stable descriptions of mathematical models concerning dynamic phenomena to be clarified. The present theory is based on the wider and deeper thoughts of John S. Nicolis. In particular, it is based on our joint paper on the chaos theory of human short-term memories with a magic number of seven plus or minus two.

Genomic Intelligence as Über Bio-Cybersecurity: The Gödel Sentence in Immuno-Cognitive Systems

“…This warrants the moniker of Self-Rep (Self Representation) of Hofstader [ 40 ]. The Self-Rep scheme in Equation (5) highlights the two-step procedure mentioned in [ 46 ] on the underlying operation of self-reference, Diag ( g ) on the RHS of (5) and the copy of the same on the LHS of (5). Meta-analyses of software based alterations to basal codes, as will be seen, involve virtual recursive function operations on σ ( g , g ).…”

“…Except for Markose [ 44 , 45 ] and Tsuda [ 46 ], there has not been an explicit acknowledgement of the necessity of a precise Gödel meta-mirror system that is parallel to codes from online machine executions by self in a model of cognition. The epochal discovery of the Mirror Neuron System (MNS) in the brain by the Parma Group [ 47 , 48 , 49 ] in the 1990s led to the characterization of MNS in the facilitation of social cognition of other’s actions by self-referential reuse of codes from the neuronal firings from the agent’s own motor and sensory activity (see [ 50 , 51 ]).…”

“…The latter have been called canonical neurons [ 52 ]. Tsuda [ 46 ] has stated how neural systems which need to process a self-referential description use the recursive mappings of a mirror system as in the mathematics of the Gödel’s incompleteness theorem: “When neural systems process a self-referential description, they may first have to make a copy (italics added) of the object of self-reference and then refer to this copy. This two-stage formulation can be realized mathematically in the proof of Gödel’s incompleteness theorem through the processes of projecting mathematical statements to natural numbers and of referring to meta-mathematical statements by providing mathematical statements about such numbers.…”

“…However, [ 46 ] did not extend the necessity of the Gödel framework for the identification of hostile viral software. This has been variously referred to as “Viruses in Turing’s Garden” (see [ 42 , 43 , 53 ]), the Liar strategy [ 44 , 45 ] and by [ 54 ] on the necessity of self-referential structures involving inverter/negator machines for the generation of the undecidable syntactic objects associated with recursive novelty production.…”

Complex Type 4 Structure Changing Dynamics of Digital Agents: Nash Equilibria of a Game with Arms Race in Innovations

“…There is now a widely held view that operations on digital information, also referred to as recursive or computable functions 2 , contribute to the key building blocks for innovation in wide ranging areas. These include evolution and operations of the genome (Holland [36], Ben-Jacob [11], Mirowski [49], Pellionisz [57], LiebermanAiden [40]); protean behaviours associated with the growth of the neocortex, complex neural dynamics in cognition (Markose [44], Corballis [20], Tsuda [70]); human language (Chomsky [19,18], Hauser et. al [32]); and, human technological progress (Arthur [6], Beinhocker [10], Youn et.…”