Satisfiability { Philosophy Index } # Philosophy Index

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# Satisfiability

In logic, satisfiability with respect to a formula refers to whether or not that formula can ever be considered true. A formula that is true in at least one valuation of its terms is satisfiable, otherwise it is unsatisfiable.

A set of formulas is similarly said to be satisfiable when at least one valuation of its terms renders every formula in the set true. Otherwise, the set is unsatisfiable, though some individual formulas in it may be satisfiable themselves.

## Examples

Consider the following set of formulas:

Γ = { (A ∨ B), (P → Q), P }

This set is satisfiable, because there are valuations of the terms A, B, P and Q that make all of the formulas in this set true. For instance, if A, B, P and Q are each true themselves, then every formula in this set is true, thereby satisfying it.

By contrast, consider the following set of formulas:

Γ = { (A ∨ B), B, (A ∧ ¬A) }

This set is unsatisfiable because it contains a contradictory formula, (A ∧ ¬A). Because this formula is false on every valuation of its terms, the set can never be satisfied (having all formulas true).

## Contradictions

Individual formulas that are unsatisfiable (never true under any valuation) are said to be contradictory.