Definition:Contradiction
Contents |
Definition
A contradiction is a statement form which is always false, no matter what the truth value of its component substatements.
A contradiction is symbolised by $\bot$, and referred to as bottom.
This is also known as a logical falsehood or logical falsity.
See the Principle of Non-Contradiction, which states that any contradiction can be seen to arise from a statement in conjunction with its negation:
- $\perp \dashv \vdash \left({p \land \neg \, p}\right)$
Logical Formula
In the context of logical formulas the term unsatisfiable is sometimes used:
A logical formula $P$ is unsatisfiable if its value is False in all boolean interpretations.
Interpretation by Models
By definition of logical consequence:
- $\forall \mathcal M: \mathcal M \not \models \left({P \land \neg P}\right)$
Propositional Calculus
Let $\mathbf A$ be a propositional WFF.
Then $\mathbf A$ is a contradiction iff it is false in every model:
- $\forall \mathcal M: \mathcal M \not \models \mathbf A$
Set of Logical Formulas
Let $U = \left\{{P_1, P_2, \ldots, P_n}\right\}$ be a set of logical formulas.
Let $U' = \left\{{p_1, p_2, \ldots, p_m}\right\}$ be the set of all the atoms of all the logical formulas in $U$.
(Some of these atoms, and indeed this will most likely be the case, may be in more than one logical formula.)
Then $U$ is unsatisfiable if for every boolean interpretation $v$ there exists an $i$ such that $v \left({P_i}\right) = F$.
That is, if it is not possible to find a boolean interpretation such that $v \left({P_i}\right) = T$ for all the logical formulas in $U$.
Boolean Interpretation
There is only one boolean interpretation for $\bot$:
- $v \left({\bot}\right) = F$
Also see
Sources
- Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.2$: Definition $1.5 \ \text{(b)}$
- D.J. O'Connor and Betty Powell: Elementary Logic (1980): $\S 1.3$
- Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000): $\S 1.2.1$: Definition $1.19$
