Logical Homology

     In homology theory we can associate a sequence of algebraic objects such as B-modules (boolean modules) to other mathematical objects such as ProLT’s. This paper investigates cycles found in exact sequences of logical expressions by way of their associated unique B-modules and ProLT’s. By identifying cycles in various chains of logical expressions we may be able to develop an idea of logical homology which may give insight to logical reasoning. For example, given the same premises and rules, can we come to the same conclusion using the ‘Snake Lemma’ on a logical complex. We may even be able to algebraically measure how different the conclusions of one statement be from another given different premises and a set of logical rules. There are various tools such as the Ext functor which may help understand to what extent some chain of logical expressions is not exact and what this means for the validity of the statement.

     Work by Psychoanalyst Ignacio Matte-Blanco focused heavily on how structures and their relations were interpreted in the unconscious vs. the conscious. As we try to ‘topologize’ the mind, we look for objects of the mind in which we can represent concepts such as Matte-Blanco described. This paper focuses on the use of simplicial homology so as to build logical simplical complexes. It is envisioned one could create simplexes by conjoining (gluing) logical expressions together where their sub-expressions (sections) overlap. Using these logical simplexes we can build a notion of ‘concepts’ (future) by conjoining simplices.

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