Propositional logic · Definition
Contradiction
logicpropositional
Before you start you need to know
A contradiction is a formula that is false under every assignment of its components. It is the opposite of a tautology, which is true under every assignment.
Definition
A compound proposition \(P\) is a contradiction if it is false under every assignment of its components.
\[P\land\neg P\equiv\bot\]
A contradiction is a compound formula that turns out to be false under every possible assignment.
The form \(P\land\neg P\) is the basic example: a proposition and its negation cannot be true simultaneously in classical logic.
Contradictions help detect incompatibilities and appear in proofs by contradiction, where a statement is assumed and an impossibility is derived.
Think about it
What appears in the final column of a contradiction?
False in all rows of the table.
Exercises 11
- A contradiction is false...
- Which one is a contradiction?
- In the table of a contradiction, the final column contains...
- If P is true, P ∧ ¬P turns out...
- A contradiction cannot be true...
- Under which assignments is a contradiction false?
- Which of these formulas is a contradiction?
- The final column of a contradiction contains...
- Which of these formulas is contradictory?
- A formula and its negation, taken in conjunction, form...
- Let \(Q=(P\Rightarrow R)\land P\land\neg R\). Is \(Q\) a contradiction?
References
- Frederic P. Schuller, Lectures on the Geometrical Anatomy of Theoretical Physics · Lecture 1, Introduction / Logic of Propositions and Predicates, YouTube, 2015.