Definition:Contingent Statement

Definition
A contingent statement is a statement form which is neither a tautology nor a contradiction, but whose truth value depends upon the truth value of its component substatements.

Formal Definition
Let $\mathfrak M$ be a collection of models for a particular formal language $\mathcal L$.

A well-formed word $\mathbf A$ of $\mathcal L$ is said to be contingent (for $\mathfrak M$) :


 * $\exists \mathcal M_1, \mathcal M_2 \in \mathfrak M: \mathcal M_1 \models \mathbf A \land \mathcal M_2 \not \models \mathbf A$

that is, some models in $\mathfrak M$ approve, and others disapprove of $\mathbf A$.

Also known as
In the context of propositional formulas the term satisfiable is usually used:

A propositional formula is satisfiable if its value is True in at least one boolean interpretation.

A propositional formula is not-valid or falsifiable if its value is False in at least one boolean interpretation.

Set of Logical Formulas
Let $U = \left\{{P_1, P_2, \ldots, P_n}\right\}$ be a set of propositional formulas.

Let $U' = \left\{{p_1, p_2, \ldots, p_m}\right\}$ be the set of all the atoms of all the propositional 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 (mutually) satisfiable if there exists a boolean interpretation $v$ for all the atoms in $U'$ such that $v \left({P_1}\right) = v \left({P_2}\right) = \cdots = v \left({P_n}\right) = T$.

Also see

 * Definition:Tautology
 * Definition:Contradiction