### Logical and mathematical foundations of truth value patterns on statement constants columns of truth tables

#### Abstract

An investigation of the logical and mathematical foundations of the pattern (order) of truth values on truth tables‟ statement constants columns was carried out. The methods applied in the investigation were those of historical approach, where previous attempts at achieving the set objective were reviewed, as well as logical and mathematical analysis of concepts and formulas. It was discovered through analysis that the algebraic function, C_{y} = ^{μn}/_{μy} (α_{0}...α_{μ})^{μy/μ}, provides grounds for all modes of orderings of values on any truth table whatsoever; ranging from zero valued to infinitary valued logic. The function was interpreted for two valued logic as, C_{y} = ^{2n}/_{2y} (T,F) ^{2y/2}. Truth values' permutations with C_{y} were obtained for three valued logic. Further analysis of the function C_{y} = ^{2n}/_{2y} (T,F) ^{2y/2} and the concept of a table as a system of row-column coordinates resulted in a two valued logic truth table permutation's coordinate α^{x}_{y} = ^{x}/_{y}.y was shown to be equal to 2(^{2n}/_{2y}). Hence, a^{x}_{y} = ^{x}/_{y} is given as a^{x}_{y} = x/_{2(2n/2y)}. Results from mathematical analysis showed that and a set of five rules generated values similar to those of the truth table statement constants columns for two valued logic. It was concluded therefore that the function and the coordinate provide logical and mathematical justifications for the truth value patterns on the statement constants columns of the truth table. Novel concepts introduced in the paper are: C_{y} = ^{μn}/_{μy} (α_{0}...α_{μ})^{μy/μ} and its interpretation in three valued logic.

**Keywords:** Truth Table; Truth Values; Mathematical Foundations Statement Constants, Logic