# Logic: The Propositional Calculus

By Stanley Chang,
Professor of Mathematics, Wellesley College, MA

In many college mathematics curricula students are challenged to
bridge the mental gap between heavily computational courses such as
multivariable calculus and proof-based courses such as linear algebra and real
analysis. While a number of institutions offer semester-long classes on
set theory and logic, most mathematics departments consign the pedagogy of
such topics to a few lectures in their intermediate curricular offerings,
where they receive only perfunctory and incomplete treatment at best.

This minicourse remedies this lacuna in the university mathematics
curriculum by offering to students a concise yet thorough description
of mathematical reasoning and logic. With exercises, self-correcting
problem sets and a host of examples, the syllabus provides an ideal
platform for self-guided study that students can tackle at any point
during the academic year.

The Formal Language

Introduction

Mathematical Sentences

Well-formed Formulas 1 2 3

Interpreting Well-formed Formulas

Decomposing Well-formed Formulas

Introduction to Truth Tables

The Truth Table for Conjunction

Building Truth Tables: Negation and Conjunction

The Truth Tables for Disjunction and Implication

Problem Set LM1.1

Tautologies and Contradictions

Equivalent Well-formed Formulas

Problem Set LM1.2

Converse and Contrapositive

Equivalence and Tautologies

Negating formulas

Wffs, Meta-wffs, and Instances

Truth Table Templates

The Tautology Principle

Problem Set LM1.3

Introducing Demonstrations

Modus Ponens

Applying Modus Ponens

Demonstrations and Program Outlines 1 2

What is a Demonstration?

Problem Set LM1.4

Conjunctive Inference

Conjunctive Simplification

Problem Set LM1.5

Unnecessary Hypotheses

Contrapositive Inference and Modus Tollens

Applying Modus Tollens

Problem Set LM1.6

Syllogistic Inference

Inference by Cases

Problem Set LM1.7

Modus Ponens for the Biconditional

Contrapositive Inference and Modus Tollens for the Biconditional

The Substitution Principle 1 2

Problem Set LM1.8

Tautologies and Demonstrations 1 2

The Substitution Principle and Equivalence

Consequences of the Substitution Principle

Rules of Inference: A Summary

Problem Set LM1.9

The estimated completion time is 20 hours.
With a clear understanding of deduction and rules of inference,
students in intermediate-level mathematics courses are better prepared
to follow and to create arguments based on these logical principles.