A New Formal Framework Resolves a Previously Unrecognized Logical Contradiction Stemming from Standard Variable Usage and Yields Cross Disciplinary Insights

Variables and constants lack axiomatic definitions. Instead, existing accounts rely on natural language and intuition to describe what they areand how they differ. A possible manifestation of this is the fact a logicalcontradiction stemming from standard variable usage, which is rigorouslydemonstrated and resolved in this paper, has never been formally acknowledged, let alone resolved. Although informal notions reflecting thiscontradiction have been expressed and non-rigorous explanations of whatvariables and constants are, some explicitly describing their differences,have been presented as resolutions to such notions, I demonstrate it arisesindependently of such informal perceptions, which are also notably heterogeneous. In resolving the contradiction, I axiomatically define variablesand constants. To enable this, I establish a new formal framework in firstorder logic built over set-theoretic foundations: Instantiation StructuredMathematics (ISM). ISM, in turn, enables parameters, which also lack anaxiomatic definition, to be given one, paves the way for further formalizations, and yields insights across fields of study, including in mathematics,metaphysics and quantum mechanics (QM). A novel axiom that defines therelation between a symbol and its interpretation— two notions conflatedin all accounts discussed in this paper that use terms such as "symbol" toexplain what variables or constants are— is also presented.