Inferences Over Fields: A Preliminary Investigation into the Deductive Capabilities of Field-Theoretically Defined Logical Connectives
Author
Crumpler, Charles Wingate
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 $\F_4$ 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 $2$.
Date
2023-05-03
Citation:
APA:
Crumpler, Charles Wingate.
(May 2023).
Inferences Over Fields: A Preliminary Investigation into the Deductive Capabilities of Field-Theoretically Defined Logical Connectives
(Master's Thesis, East Carolina University). Retrieved from the Scholarship.
(http://hdl.handle.net/10342/12853.)
MLA:
Crumpler, Charles Wingate.
Inferences Over Fields: A Preliminary Investigation into the Deductive Capabilities of Field-Theoretically Defined Logical Connectives.
Master's Thesis. East Carolina University,
May 2023. The Scholarship.
http://hdl.handle.net/10342/12853.
April 28, 2024.
Chicago:
Crumpler, Charles Wingate,
“Inferences Over Fields: A Preliminary Investigation into the Deductive Capabilities of Field-Theoretically Defined Logical Connectives”
(Master's Thesis., East Carolina University,
May 2023).
AMA:
Crumpler, Charles Wingate.
Inferences Over Fields: A Preliminary Investigation into the Deductive Capabilities of Field-Theoretically Defined Logical Connectives
[Master's Thesis]. Greenville, NC: East Carolina University;
May 2023.
Collections
Publisher
East Carolina University