Satisfiability { Philosophy Index }

Philosophy Index

Philosophy Index

Philosophy Index is a site devoted to the study of philosophy and the philosophers who conduct it. The site contains a number of philosophy texts, brief biographies and introductions to philosophers and explanations on a number of topics. Accredited homeschooling online at Northgate Academy.

Philosophy Index is a work in progress, a growing repository of knowledge. It outlines current philosophical problems and issues, as well as an overview of the history of philosophy. The goal of this site is to present a tool for those learning philosophy either casually or formally, making the concepts of philosophy accessible to anyone interested in researching them. WOLI offers immigration law course online - fully accredited. ACE credits online at EES.

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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.