Propositional Logic as a Boolean Algebra: A New Perspective (MWN Propositional Logic Book 1)

William S. Veatch
Propositional Logic as a Boolean Algebra – A New Perspective
Vol. 1

This Volume 1 considers the question of whether we can interpret Traditional Propositional Logic using the Logic Operations OR, AND, and NOT as a Boolean Algebra when viewed in the broader context of the Mathematics of Ideas as developed in the author’s book: “Math Without Numbers – The Mathematics of Ideas – Vol.1 Foundations.” The answer is “yes,” provided, that we make some changes to how OR, AND, and NOT are defined and implemented. Basically, we equate OR, AND, and NOT to Union, Intersection, and Complementation for purposes of combining Propositions to form sets, but we develop a new methodology for assigning Truth Values.

To implement our new style of Propositional Logic in Math Without Numbers, or MWN for short, the author creates three separate but related Universes of Discourse, each of which constitutes a Boolean Algebra using Union, Intersection, and Complementation: Ideas (Order 1), Propositions (Order 2), and Logic Formulas (Order 3). We see that the Truth Values of Propositions and Logic Formulas are inextricably linked to the set relationships of the Ideas comprising the subjects and predicates of the Propositions.

In the end, we see that we can view Traditional Propositional Logic as a subset of a larger system of MWN Propositional Logic. Traditional Propositional Logic is a special case concerning an Order 2 Domain with a single Atom, whereas MWN Propositional Logic goes on to examine Order 2 Domains with multiple Atoms. In developing this new theory of Propositional Logic, the author proposes a new methodology for assigning Truth Values.

The underlying premise is that every Idea is either an Atom or a Compound made up of Atoms, but only Atoms have a binary Truth/False Truth Value. Compounds, if homogeneous, may have a clear Truth Value, but unlike Atoms, Compounds may consist of a heterogeneous mix of True and False Atoms, such that there is no clear Truth Value for such “Mixed Sets” of Atoms. Depending upon the context, we may be able to create a rule for assigning a Truth Value to a Mixed Set, but it requires some exercise of discretion. This is consistent with the premise that mathematics can tell us how to think, but not what to think.

This book is intended for anyone interested in Logic.