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In logic, an argument is said to be valid if and only if the supposed truth of its premises guarantee the truth of its conclusion. There are other ways to formulate this basic definition: the premises entail the conclusion; it cannot be the case both that the premises are true and the conclusion false; the falsehood of the conclusion entails the falsehood of at least one premise; etc.
A formula (also sentence or statement), as opposed to an argument, is said to be valid (or tautologous) if and only if it has all Ts (the truth-value true) in its truth table (refer to the truth table article). The rest of the article is about the validity of arguments, not formulas.
The definition of validity does not say that the premises are in fact (or actually) true. An argument is valid iff, supposing the premises are true, the conclusion follows. In actuality the premises might be false. Consider an example of a valid argument with actually false premises:
- All dogs have eight legs.
- The President is a dog.
- Therefore, the President has eight legs.
The above argument is perfectly valid because we are assuming the truth of the premises.
Valid arguments needn't be sound. A sound argument is valid and its premises are actually true. The above argument, while valid, is clearly not sound (i.e. dogs do not have eight legs and the president is not a dog). What makes the argument valid is its form. Consider the following argument form.
- All P are Q
- A is P
- Therefore, A is Q
The above is an argument form because the letters P, Q, A represent unanalyzed or uninterpreted sentences. When we substitute P, Q, A for actual sentences like those above about dogs and presidents, then we have an actual argument instead of merely an argument form. We can determine the validity of an actual argument by translating it into an argument form, and then analyzing the argument form for validity. To determine the validity of an argument form, usually we derive a proof of it. This rests on well-established theorems proving that only valid formulas are provable and only provable formulas are valid. Thus, if there is a proof of an argument form, then we know it is valid. The proofs of these theorems establishing, respectively, completeness and soundness are omitted here.
See also: validity (psychometric).
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