View source for Module:Validity (logic)

{{Other uses|Validity (disambiguation){{!}}Validity}}
{{Use mdy dates|date = February 2019}}
{{Short description|Argument whose conclusion must be true if its premises are}}
{{Use American English|date = February 2019}}
In [[logic]], specifically in [[deductive reasoning]], an [[argument]] is '''valid''' if and only if it takes a form that makes it impossible for the [[premise]]s to be [[truth|true]] and the conclusion nevertheless to be [[False (logic)|false]].<ref>[http://www.iep.utm.edu/val-snd/ Validity and Soundness – Internet Encyclopedia of Philosophy]</ref> It is not required for a valid argument to have premises that are actually true,<ref>Jc Beall and Greg Restall, [http://plato.stanford.edu/archives/fall2014/entries/logical-consequence/ "Logical Consequence"], The Stanford Encyclopedia of Philosophy (Fall 2014 Edition).</ref> but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called [[well-formed formula|well-formed formulas]] (also called ''wffs'' or simply ''formulas''). 

The '''validity''' of an argument can be tested, proved or disproved, and depends on its [[logical form]].<ref name=":0" />

==Arguments==
[[File:Argument terminology used in logic (en).svg|thumb|Argument terminology used in logic]]
In logic, an [[argument]] is a set of statements expressing the ''premises'' (whatever consists of  empirical evidences and axiomatic truths) and an ''evidence-based conclusion.''

An argument is ''valid'' if and only if it would be contradictory for the conclusion to be false if all of the premises are true.<ref name=":0">{{Cite book|title=Introduction to logic|last=Gensler, Harry J., 1945-|isbn=978-1-138-91058-4|edition=Third|location=New York|oclc=957680480|date = January 6, 2017}}</ref> Validity doesn't require the truth of the premises, instead it merely [[Logical truth|necessitates]] that conclusion follows from the formers without violating the correctness of the [[logical form]]. If also the premises of a valid argument are proven true, this is said to be [[Soundness|sound]].<ref name=":0" />   

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 immortal.
: Socrates is a man.
: Therefore, Socrates is mortal.

In this case, the conclusion contradicts the deductive logic of the preceding premises, rather than deriving from it. Therefore, the argument is logically 'invalid', even though the conclusion could be considered 'true' in general terms. The premise 'All men are immortal' would likewise be deemed false outside of the framework of classical logic. However, within that system 'true' and 'false' essentially function more like mathematical states such as binary 1s and 0s than the philosophical concepts normally associated with those terms.

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 second argument becomes:

: All P are not Q.
: S is a P.
: Therefore, S is a Q.

An argument is termed formally valid if it has structural self-consistency, i.e. if when the operands between premises are all true, the derived conclusion is always also true. In the third example, the initial premises cannot logically result in the conclusion and is therefore categorized as an invalid argument.

==Valid 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]].

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

==Soundness==
{{main|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 ''sound''.  In order for a deductive argument to be sound, the argument must be valid and all the premises must be true.<ref name=":0" />

==Satisfiability==
{{Main|Satisfiability}}

[[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'', University of Chicago Press, 1991, 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, [https://archive.org/details/criticalthinking0000coga/page/48 p. 48].</ref>

:{| class="wikitable" style="text-align:left"
! Preservation properties !! [[Logical connective]] sentences
|-
|True and false preserving: || [[Proposition]] {{*}} [[Logical conjunction| Logical conjunction (AND, <math>\land</math> )]] {{*}} [[Logical disjunction| Logical disjunction (OR, <math>\lor</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: || [[negation|Negation ( <math>\neg</math> )]] {{*}} [[Sheffer stroke|Alternative denial (NAND, <math>\uparrow</math> )]]{{*}}[[Logical NOR|Joint denial (NOR, <math>\downarrow</math> )]]
|}

==See also==
{{Portal|Philosophy}}
*[[Logical consequence]]
*[[Reductio ad absurdum]]
*[[Mathematical fallacy]]
*[[Soundness]]
*[[Ω-logic#Analysis|Ω-validity]]

==References==
{{Reflist}}

==Further reading==
*[[Jon Barwise|Barwise, Jon]]; [[John Etchemendy|Etchemendy, John]]. ''Language, Proof and Logic'' (1999): 42.
*Beer, Francis A. "[https://www.tandfonline.com/doi/abs/10.1080/02691729308578683 Validities: A Political Science Perspective]", ''Social Epistemology'' 7, 1 (1993): 85-105.

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

{{Logic}}
{{Mathematical logic}}

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