Inferences Over Fields: A Preliminary Investigation into the Deductive Capabilities of Field-Theoretically Defined Logical Connectives

Abstract

In this paper, we will be concerned with developing an inferential structure over the field with four elements in characteristic 2. We begin by discussing the historical context in which this research occurs. In subsequent sections, we will construct the field, called F₄ and describing the algebraic structure over F₄. We then define the connectives [wedge], [vee], and [negation] over F₄ by extending their standard definition over F₂. We define the basic syntax and semantics of F₄. We show that [wedge] and [vee] are dual over F₄ with respect to [negation] and that F₄ is functionally complete over {[wedge], [vee], [negation] } [union] F₄. We develop a notion of inferences over F₄ by imbuing it with a partial order, defining validity and the material implication, and defining a proof. Upon completing this, we prove the Deduction, Soundness, and Completeness Theorems, thereby showing that inferences over F₄ behaves in ways comparable to, but not equivalent to, those over a field of two values in characteristic 2.