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Mathematics

Propositional Logic

Connectives, Truth Tables, and the Logic of Valid Arguments — A TLDR Primer

Logic shows up without warning — in a discrete mathematics course, a philosophy class, a computer science prerequisite, or buried inside a proof-based math sequence. When it does, most students hit the same wall: the symbols look foreign, truth tables feel mechanical, and it's not obvious what any of it has to do with actual reasoning. This guide cuts through that.

**TLDR Propositional Logic** is a focused, concise guide covering exactly what a high school or early college student needs to get oriented fast. It starts with what a proposition is and why formal logic strips language down to true/false atoms. From there it builds through the five connectives (negation, conjunction, disjunction, conditional, biconditional) and how to translate messy English sentences into clean symbols. You'll learn to build truth tables systematically, classify statements as tautologies or contradictions, and apply equivalence laws — including De Morgan's and the contrapositive — to simplify formulas without tables. The final section covers valid arguments and the core inference rules: modus ponens, modus tollens, and hypothetical syllogism, plus the fallacies students most often confuse for valid reasoning.

This is a symbolic logic practice and reference guide, not a textbook. Every term is defined the first time it appears. Every concept comes with a worked example. If you're prepping for a discrete math unit, a logic and reasoning module, or just trying to help a student who's stuck, this is the shortest path to confident understanding.

Pick it up and work through it in a single sitting.

What you'll learn
  • Translate everyday English statements into propositional logic using the five standard connectives.
  • Build and read truth tables to test whether a compound statement is a tautology, contradiction, or contingency.
  • Recognize tautological equivalences (De Morgan's laws, contrapositive, distribution) and use them to simplify formulas.
  • Distinguish valid from invalid arguments and identify common fallacies like affirming the consequent.
  • Apply basic inference rules (modus ponens, modus tollens, hypothetical syllogism) to construct short proofs.
What's inside
  1. 1. What Propositional Logic Is
    Introduces propositions, the goal of formal logic, and why we strip statements down to true/false atoms.
  2. 2. The Five Connectives and How to Translate English
    Defines negation, conjunction, disjunction, conditional, and biconditional, with translation practice from English to symbols.
  3. 3. Truth Tables and Classifying Statements
    Shows how to build truth tables systematically and use them to identify tautologies, contradictions, and contingencies.
  4. 4. Logical Equivalences and Simplification
    Covers the key equivalence laws — De Morgan's, contrapositive, distribution, double negation — and how to use them to rewrite formulas without truth tables.
  5. 5. Valid Arguments and Inference Rules
    Defines validity, introduces modus ponens, modus tollens, and hypothetical syllogism, and shows how to spot common fallacies.
  6. 6. Where This Shows Up Next
    Connects propositional logic to circuits, programming conditionals, mathematical proof, and previews predicate logic.
Published by Solid State Press
Propositional Logic cover
TLDR STUDY GUIDES

Propositional Logic

Connectives, Truth Tables, and the Logic of Valid Arguments — A TLDR Primer
Solid State Press

Propositional Logic

Connectives, Truth Tables, and the Logic of Valid Arguments — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 What Propositional Logic Is
  2. 2 The Five Connectives and How to Translate English
  3. 3 Truth Tables and Classifying Statements
  4. 4 Logical Equivalences and Simplification
  5. 5 Valid Arguments and Inference Rules
  6. 6 Where This Shows Up Next
Chapter 1

What Propositional Logic Is

Every argument, proof, and decision you make rests on statements that are either true or false. Propositional logic is the system mathematicians and computer scientists use to study exactly those statements — and to figure out, with precision, when a conclusion genuinely follows from a set of premises.

The building block is a proposition: a declarative sentence that has exactly one truth value — either true or false, never both, never neither. "The sum of two even numbers is even" is a proposition (it's true). "7 is greater than 10" is a proposition (it's false). "Close the door" is not a proposition — it's a command, and commands aren't true or false. "This statement is false" is famously not a proposition either, because it can't consistently be assigned either truth value.

Getting comfortable with identifying propositions is the first skill, and the test is simple: ask yourself whether the sentence makes a claim that could, in principle, be checked and declared true or false. If yes, it's a proposition.

Example. Which of the following are propositions?

  1. Washington, D.C. is the capital of the United States.
  2. Is math hard?
  3. $3 + 4 = 8$
  4. Every even integer greater than 2 is the sum of two primes.

Solution.

  1. Yes — it makes a checkable claim. (True.)
  2. No — it's a question.
  3. Yes — it makes a checkable claim. (False: $3 + 4 = 7$.)
  4. Yes — this is the famous Goldbach conjecture. Nobody has proved or disproved it yet, but it is still a proposition, because it either is true or it isn't. Our ignorance of which doesn't change that.

That last case is worth sitting with. A proposition doesn't have to be known to be true or false — it just has to be one or the other. Propositional logic is about the structure of reasoning, not about our current state of knowledge.

Atomic and Compound Statements

About This Book

If you're looking for an intro to propositional logic for high school or you're a college freshman hitting formal logic for the first time in a discrete math or philosophy course, this guide was written for you. It's also useful for anyone preparing for the SAT, the LSAT, or a computer science course where logic gates and circuit design appear early in the syllabus.

This book covers the five logical connectives, how to build and read truth tables, and how to classify statements as tautologies or contradictions. It doubles as a truth tables and logic gates study guide and a discrete math logic beginner study guide — moving from translation of English sentences into symbols, through logical equivalences, all the way to modus ponens and inference rules explained clearly with worked examples. A concise overview with no filler.

Read straight through once, work every example as you go, then use the symbolic logic practice problems at the end to test yourself. This math reasoning and formal logic primer is built around logic connectives and proofs for beginners — meaning you need no prior background beyond basic algebra.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon