The term validity in logic (also logical validity) is largely synonymous with logical truth. However, the term is used in different contexts. Validity is a property of formulae, statements and arguments. A logically valid argument is one where the conclusion follows from the premises. An invalid argument is where the conclusion does not follow from the premises. A deductive argument may be valid but not sound. In other words, validity is a necessary condition for truth of a deductive syllogism but is not a sufficient condition.
A formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language.
Validity of arguments
An argument is valid if and only if the truth of its premises entails the truth of its conclusion. It would be self-contradictory to affirm the premises and deny the conclusion. The corresponding conditional of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction. The conclusion is a logical consequence of its premises.
An argument that is not valid is said to be “invalid”.
An example of a valid argument is given by the following well-known syllogism:
What makes this a valid argument is not that it has true premises and a true conclusion, but the logical necessity of the conclusion, given the two premises. The argument would be just as valid were the premises and conclusion false. The following argument is of the same logical form but with false premises and a false conclusion, and it is equally valid:
No matter how the universe might be constructed, it could never be the case that these arguments should turn out to have simultaneously true premises but a false conclusion. The above arguments may be contrasted with the following invalid one:
In this case, the conclusion does not follow inescapably from the premises. All men are mortal, but not all mortals are men. Every living creature is mortal; therefore, even though both premises are true and the conclusion happens to be true in this instance, the argument is invalid.
A standard view is that whether an argument is valid is a matter of the argument’s logical form. Many techniques are employed by logicians to represent an argument’s logical form. A simple example, applied to two of the above illustrations, is the following: Let the letters ‘P’, ‘Q’, and ‘S’ stand, respectively, for the set of men, the set of mortals, and Socrates. Using these symbols, the first argument may be abbreviated as:
Similarly, the third argument becomes:
An argument is formally valid if its form is one such that for each interpretation under which the premises are all true, the conclusion is also true. As already seen, the interpretation given above does cause the second argument form to have true premises and false conclusion, hence demonstrating its invalidity.
Validity of statements
A statement can be called valid, i.e. logical truth, if it is true in all interpretations.
For example: If no god is mortal, then no mortal is a god.
In logical form, this is:
A given statement may be entailed by other statements, i.e. the given statement must be true if the other statements are true. This means that an argument with the given statement as its conclusion and the other statements as its premises is a valid argument. The corresponding conditional of a valid argument is a logical truth.
Validity and soundness
Validity of deduction is not affected by the truth of the premise or the truth of the conclusion. The following deduction is perfectly valid:
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