### The Logical Implication Table in Binary Propositional Calculus: Justification, Proof Automatability, and Effect on Scientific Reasoning

#### Abstract

Logic is the discipline concerned with providing valid general rules on which scientific reasoning and the resulting propositions are based. To evaluate the validity of sentences in propositional calculus, we, typically, perform a complete case analysis of all the possible truth-values assigned to the sentence’s propositional variables. Truth tables provide a systematic method for performing such analysis in order to determine whether the sentence is valid, satisfiable, contradictory, consistent, etc. However, in order to validate logical statements, we have to use valid truth tables, i.e., truth tables that are provably consistent and justifiable by some natural criteria. The justification of the truth table of some logical connectives is straightforward, due to the support of the table in everyday applications. Nevertheless, the justification of one of the logical connectives, namely, the implication operator, has always been difficult to build and understand. Though, the logical implication is arguably the most important operator because of its applications as an inference engine for reasoning in science in general and control engineering in particular.

In this paper, the author presents this problem introducing a non-exhaustive proof, which justifies the logical implication’s truth table in one phase. The author then proposes another optimal proof, discussing the points of optimization and the effects of the resulting linguistic and philosophical interpretation on the scientific reasoning processes. Finally, the paper envisions possible extension of the proposed methodology to solve similar problems in various types of logic.

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