A Three-Valued Logic for Infinity-Dependent and Undecidable Statements

Motivation / 1. IntroductionIn everyday physical and mathematical practice, many naturally occurringdichotomiesare in fact trichotomies. A real quantity is not merely positive or negative, butpositive–zero–negative; motion is not just left or right, but left–rest–right; integerparity splits as even–odd–zero (where zero plays a special, structurally symmetricrole).In all these cases there are two opposed "poles" and one distinguished, neutralboundarystate in between. Classical two-valued logic implicitly forces us to collapse thisneutralmiddle point into one of the extremes (true or false), thereby breaking an underlyingsymmetry that our numerical and physical intuition treats as fundamental.The three-valued scheme proposed here, with truth values -1, 0, 1, is designedpreciselyto preserve that symmetry at the logical level. The values +1 and -1 correspond totwoopposed, stable truth assignments (analogous to positive/negative, left/right), while 0represents a boundary or neutral state: a statement whose status depends on furtherstructural assumptions (such as finiteness vs. infiniteness of the underlying universe)or on the choice of axioms and computational resources. In the same way that zero isnot "just another number" but the exact midpoint between positive and negative, thevalue 0 in this logic is not an arbitrary third option, but the logical analogue of thatmidpoint—capturing model-dependence and undecidability without artificiallyidentifyingthem with full truth or full falsity.