Propositional logic · Definition
Logical equivalence
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Before you start you need to know
Logical equivalence compares two formulas by their truth value under each assignment, regardless of their form.
Definition
\(P\) and \(Q\) are logically equivalent if they have the same truth value under every assignment.
\[P\Leftrightarrow Q\]
Two formulas are logically equivalent when they have the same truth value under every possible assignment of their atomic propositions.
Equivalence does not require the formulas to have the same form or the same symbols. Two different expressions can describe exactly the same truth function.
Truth tables offer a direct method to check it: their final columns are compared row by row.
Think about it
What must coincide to prove \(P\equiv Q\) by means of tables?
The truth value of \(P\) and \(Q\) under every possible assignment.
Exercises 10
- Two propositions are logically equivalent if...
- Which symbol represents logical equivalence?
- P ⇔ P is...
- If two truth tables coincide row by row, the propositions are...
- A logical equivalence mainly compares...
- By De Morgan, ¬(P∧Q) is equivalent to...
- Logical equivalence is usually written...
- Which equivalence is correct?
- Select the logically equivalent pair.
- To prove P≡Q, a direct strategy is...
References
- Frederic P. Schuller, Lectures on the Geometrical Anatomy of Theoretical Physics · Lecture 1, Introduction / Logic of Propositions and Predicates, YouTube, 2015.