Propositional logic · Definition
Tautology
Before you start you need to know
A tautology is a formula that is true no matter which truth values its components take. Here you will learn to recognize it using truth tables.
Definition
A compound proposition \(P\) is a tautology if it is true under every assignment of its components.
Properties: a tautology is true in every row of its truth table, so its final column contains only T. Its truth does not depend on the concrete values of its atoms.
Guiding example: in \(P\lor\neg P\), if \(P\) is T then \(P\) is T; if \(P\) is F then \(\neg P\) is T. So the final column is T, T.
Typical error: confusing a tautology with a formula true in only some cases, such as \(P\land Q\), or thinking one true row is enough.
Think about it
How is a tautology recognized in its table?
The final column contains true in all its rows.
Exercises 10
- A tautology is true...
- Which one is a tautology?
- In the table of a tautology, the final column contains...
- If P is false, P ∨ ¬P turns out...
- A tautology is a property of...
- Under which assignments is a tautology true?
- Which one is a classic tautology?
- A tautology has in its final table...
- Select the tautology.
- If a formula has no counterexamples, then it is...
References
- Frederic P. Schuller, Lectures on the Geometrical Anatomy of Theoretical Physics · Lecture 1, Introduction / Logic of Propositions and Predicates, YouTube, 2015.