Classical propositional logic—introduction

Propositional logic as a clarification of ambiguity, truth tables, propositional worlds, logical entailment, validity, tautologies

A Panorama of Logic

Welcome to this first installment from my new book, A Panorama of Logic, which I shall be serializing here on my substack over the coming months.

I should like to show you the breadth of the subject of logic, a subject that is broader and deeper than many beginners expect. I shall paint a panorama of logic, stretching across the subject from mathematical logic to philosophical logic, emphasizing the enormity of it while also indicating how each part can be pursued more deeply. We'll cover propositional logic, multi-valued logic, relational logic, the theory of orders, predicate logic, definability, interpretability, model theory, the logic of games, proof theory, modal logic, and much more. I shall present the examples and results that frame my own understanding of the subject and the ideas that I find illuminating. My further goal will be to help you develop your ability to consider technical matters with clarity and precision.

The text is suitable as an introduction to logic for mathematicians, philosophers, or computer scientists, at either the advanced undergraduate or introductory graduate level.

Let us begin very simply at the beginning, with an elementary introduction to classical propositional logic.

Classical propositional logic

Propositional logic is the logic of propositions, the logic of how to reason with sentential assertions and the analysis of the conditions that make them true or false. We aim to understand the elementary logical structure of propositional assertions and their semantics.

Classical propositional logic lies within the elementary core of a broad contemporary investigation of diverse topics in logic. It exhibits in elementary form many of the themes that will become prominent later in more sophisticated contexts with more powerful logics, themes such as the concept of formal language, the concept of truth in a model, the accompanying interplay between syntax and semantics, the notions of satisfiability, consistency, completeness, and the connections of these notions with computability and complexity.

I tend to view propositional logic consequently as a kind of logic playground, filled with enticing logic toys—let's play! As with all good toys, we shall have an enjoyable time, while also beginning to grasp the ideas whose extent will be more fully realized later in more refined contexts.

Propositional logic as a clarification of ambiguity

Natural language is excellent for conveying shades of nuance—we communicate profound, subtle meanings by our utterances, telling one another our passions, joys, hopes, fears, and dreams. What I should like to discuss, however, is the shocking degree of logical ambiguity that we often allow unnoticed into our natural language assertions. A statement may seem at first to be completely clear and distinct, but upon reflection gigantic holes of uncertainty open up concerning exactly it means and what the truth conditions of the statement might be.

Suppose, for example, that you are a boat builder contracted to deliver a yacht to your customer's berth at the marina on a certain day, unless an imminent hurricane is predicted.

I shall deliver the yacht to your berth at the marina on Tuesday, unless an imminent hurricane is predicted.

You diligently prepare the delivery and successfully deliver the yacht as scheduled, although it turns out there is a hurricane predicted, which you knew about, and your delivery arrives just as the storm is beginning to surge. Your customer is furious, however, as the boat is subsequently destroyed in the raging waves. He vows to sue. But have you violated the agreement?