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 formulas, 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 true. In other words, validity is a necessary condition for truth of a deductive syllogism but is not a sufficient condition.

Valid formula

A formula of a formal language is a valid formula iff 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:

All men are mortal.

Socrates is a man.

Therefore, Socrates is mortal.

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:

All cups are green.

Socrates is a cup.

Therefore, Socrates is green.

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:

All men are mortal.

Socrates is mortal.

Therefore, Socrates is a man.

In this case, the conclusion does not follow inescapably from the premises: a universe is easily imagined in which ‘Socrates’ is not a man but a woman, so that in fact the above premises would be true but the conclusion false. This possibility makes the argument invalid. (Although, whether or not an argument is valid does not depend on what anyone could actually imagine to be the case, this approach helps us evaluate some arguments.)

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 the above two 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:

All P are Q.

S is a P.

Therefore, S is a Q.

Similarly, the second argument becomes:

All P are Q.

S is a Q.

Therefore, S is a P.

These abbreviations make plain the logical form of each respective argument. At this level, notice that we can talk about any arguments that may take on one or the other of the above two configurations, by replacing the letters P, Q and S by appropriate expressions. Of particular interest is the fact that we may exploit an argument's form to help discover whether or not the argument from which it has been obtained is or is not valid. To do this, we define an “interpretation” of the argument as an assignment of sets of objects to the upper-case letters in the argument form, and the assignment of a single individual member of a set to the lower-case letters of the argument form. Thus, letting P stand for the set of men, Q stand for the set of mortals, and S stand for Socrates is an interpretation of each of the above arguments. Using this terminology, we may give a formal analogue of the definition of deductive validity:

An argument is formally valid if its form is one such that for each interpretation under which the premises are all true also the conclusion is 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:

If (No P is a Q), then (No Q is a P).

A given statement may be entailed by other statements, i.e. if 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

One thing we should note is that the validity of deduction is not at all affected by the truth of the premise or the truth of the conclusion. The following deduction is perfectly valid:

All fire-breathing rabbits live on Mars

All humans are fire-breathing rabbits

Therefore all humans live on Mars

The problem with the argument is that it is not sound. In order for a deductive argument to be sound, the deduction must be valid and the premise must be true.

Satisfiability and validity

Model theory analyses formulae with respect to particular classes of interpretation in suitable mathematical structures. On this reading, formula is valid if all such interpretations make it true. An inference is valid if all interpretations that validate the premisses, validate the conclusion. This is known as semantic validity^[1].

Preservation

In truth-preserving validity, the interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true'.

In a false-preserving validity, the interpretation under which all variables are assigned a truth value of ‘false’ produces a truth value of ‘false'. ^[2]

Preservation properties	Logical connective sentences
True and false preserving:	Logical conjunction (AND, ${\displaystyle \land }$ )  • Logical disjunction (OR, ${\displaystyle \lor }$ )
True preserving only:	Tautology ( ${\displaystyle \top }$ )  • Biconditional (XNOR, ${\displaystyle \leftrightarrow }$ )  • Implication ( ${\displaystyle \rightarrow }$ )  • Converse implication ( ${\displaystyle \leftarrow }$ )
False preserving only:	Contradiction ( ${\displaystyle \bot }$ ) • Exclusive disjunction (XOR, ${\displaystyle \oplus }$ )  • Nonimplication ( ${\displaystyle \nrightarrow }$ )  • Converse nonimplication ( ${\displaystyle \nleftarrow }$ )
Non-preserving:	Proposition  • Negation ( ${\displaystyle \neg }$ )  • Alternative denial (NAND, ${\displaystyle \uparrow }$ ) • Joint denial (NOR, ${\displaystyle \downarrow }$ )

n-Validity

A formula A of a first order language ${\displaystyle {\mathcal {Q))}$ is n-valid iff it is true for every interpretation of ${\displaystyle {\mathcal {Q))}$ that has a domain of exactly n members.

ω-Validity

A formula of a first order language is ω-valid iff it is true for every interpretation of the language and it has a domain with an infinite number of members.