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{{other uses|Validity (disambiguation)}}
In [[logic]], an [[argument]] is '''valid''' if and only if its conclusion is logically [[entailment|entailed]] by its premises. A [[well-formed formula|formula]] is valid [[if and only if]] it is true under every [[interpretation (logic)|interpretation]], and an argument form (or schema) is valid if and only if every argument of that [[logical form]] is valid.

==Validity of arguments==

An  [[Argument (logic)|argument]] is valid if and only if the truth of its premises [[entailment|entails]] the truth of its conclusion and each step, sub-argument, or logical operation in the argument is valid.  Under such conditions 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. 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 because it depends on an incorrect operation of implication. Such fallacious arguments have much in common with what are known as [[Mathematical fallacy#Howlers|howlers]] in mathematics.

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:

: All P are Q.
: S is a P.
: Therefore, S is a Q.

Similarly, the third argument becomes:

: All P are Q.
: S is a Q.
: Therefore, S is a P.

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 (for the third argument) does cause the second argument form to have true premises and false conclusion (if P is a not human creature), hence demonstrating its invalidity.

==Valid formula==
{{Main|Well-formed formula}}

A formula of a [[formal language]] is a valid formula if and only if it is true under every possible [[Interpretation (logic)|interpretation]] of the language. In propositional logic, they are [[tautology (logic)|tautologies]].

==Validity of statements==
A statement can be called valid, i.e. logical truth, if it is true in all interpretations.

==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:

: All animals live on Mars.
: All humans are animals.
: Therefore, all humans live on Mars.

The problem with the argument is that it is not ''[[soundness|sound]]''.  In order for a deductive argument to be sound, the deduction must be valid and '''all'''  the premises true.

==Satisfiability and validity==
{{Main|Satisfiability and validity}}

[[Model theory]] analyzes 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 premises validate the conclusion.  This is known as ''semantic validity''.<ref>L. T. F. Gamut, ''Logic, Language, and Meaning: Introduction to logic'', p. 115</ref>

==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'.<ref>Robert Cogan,"Critical thinking: step by step", University Press of America, 1998, p48 [http://books.google.com/books?id=Fit9rjXiLBUC&pg=PA48&lpg=PA48&dq=truth+preserving&source=bl&ots=79aWhBY_p6&sig=4MgG6kOzky8riS9e4oR853qgIjg&hl=en&ei=zOjsSp6gOYn2sgO1tPXhBw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAsQ6AEwADge#v=onepage&q=truth%20preserving&f=false]</ref>

:{| class="wikitable" style="text-align:left"
! '''Preservation properties''' !! '''[[Logical connective]] sentences'''
|-
|'''True and false preserving:''' || [[Logical conjunction| Logical conjunction (AND, <math>\and</math> )]] {{*}} [[Logical disjunction| Logical disjunction (OR, <math>\or</math> )]]
|-
|'''True preserving only:''' ||  [[Tautology (logic)|Tautology ( <math>\top</math> )]] {{*}} [[Logical biconditional|Biconditional (XNOR, <math>\leftrightarrow</math> )]] {{*}} [[Material conditional|Implication ( <math>\rightarrow</math> )]] {{*}} [[Converse implication|Converse implication ( <math>\leftarrow</math> )]]
|-
|'''False preserving only:''' ||  [[Contradiction|Contradiction ( <math>\bot</math> )]]{{*}} [[Exclusive or|Exclusive disjunction (XOR, <math>\oplus</math> )]] {{*}} [[Material nonimplication|Nonimplication ( <math>\nrightarrow</math> )]] {{*}} [[Converse nonimplication|Converse nonimplication ( <math>\nleftarrow</math> )]]
|-
|'''Non-preserving:''' ||  [[Proposition (philosophy)|Proposition]] {{*}} [[negation|Negation ( <math>\neg</math> )]] {{*}} [[Sheffer stroke|Alternative denial (NAND, <math>\uparrow</math> )]]{{*}}[[Logical NOR|Joint denial (NOR, <math>\downarrow</math> )]]
|}

==n-Validity==

A formula '''A''' of a first order language <math>\mathcal{Q}</math> is [[n-valid]] [[If and only if|iff]] it is true for every interpretation of <math>\mathcal{Q}</math> that has a [[domain (mathematics)|domain]] of exactly n members.

===ω-Validity===

A formula of a first order language is [[ω-valid]] [[If and only if|iff]] it is true for every interpretation of the language and it has a [[domain (mathematics)|domain]] with an [[countable set|infinite]] number of members.

==See also==
{{Portal|Logic}}
*[[Entailment]]
*[[Philosophy of science#Grounds of validity of scientific reasoning|Grounds of validity of scientific reasoning]]
*[[Mathematical fallacies]]
*[[Soundness]]

==References==
{{Reflist}}
*Barwise, Jon; Etchemendy, John. ''Language, Proof and Logic'' (1999): 42.
*Beer, Francis A. "Validities: A Political Science Perspective", ''Social Epistemology'' 7, 1 (1993): 85-105.

==External links==
{{Sisterprojectlinks|wikt=validity|b=no|n=no|commons=no|q=no|s=no|v=no|d=no|species=no|voy=no}}

{{Logic}}

[[Category:Arguments]]
[[Category:Concepts in logic]]
[[Category:Deductive reasoning]]
[[Category:Logical truth]]