Author | Crumpler, Charles Wingate | |
Date Accessioned | 2023-06-05T13:54:30Z | |
Date Available | 2023-06-05T13:54:30Z | |
Date Created | 2023-05 | |
Date of Issue | 2023-05-03 | |
xmlui.metadata.dc.date.submitted | May 2023 | |
Identifier (URI) | http://hdl.handle.net/10342/12853 | |
Description | 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 $\F_4$ and describing the algebraic structure over $\F_4$. We then define the connectives $\wedge$, $\vee$, and $\neg$ over $\F_4$ by extending their standard definition over $\F_2$. We define the basic syntax and semantics of $\F_4$. We show that $\wedge$ and $\vee$ are dual over $\F_4$ with respect to $\neg$ and that $\F_4$ is functionally complete over $\{ \ \wedge, \vee, \neg \ \} \cup \F_4$. We develop a notion of inferences over $\F_4$ 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 $\F_4$ behaves in ways comparable to, but not equivalent to, those over a field of two values in characteristic $2$. | |
Mimetype | application/pdf | |
Language | en | |
Publisher | East Carolina University | |
Subject | logic, many-valued logic, soundness, completeness | |
Title | Inferences Over Fields: A Preliminary Investigation into the Deductive Capabilities of Field-Theoretically Defined Logical Connectives | |
Type | Master's Thesis | |
xmlui.metadata.dc.date.updated | 2023-06-02T15:41:01Z | |
Department | Mathematics | |
xmlui.metadata.dc.degree.name | M.A. | |
xmlui.metadata.dc.degree.level | Masters | |
xmlui.metadata.dc.degree.discipline | MA-Mathematics | |
xmlui.metadata.dc.degree.grantor | East Carolina University | |
xmlui.metadata.dc.degree.department | Mathematics | |
xmlui.metadata.dc.type.material | text | |