# Validity: Wikis

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# Encyclopedia

## Contents

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 sound. 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 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:

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]

## n-Validity

A formula A of a first order language $\mathcal{Q}$ is n-valid iff it is true for every interpretation of $\mathcal{Q}$ 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.

## References

1. ^ L. T. F. Gamut, Logic, Language, and Meaning: Introduction to logic, p. 115
2. ^ Robert Cogan,"Critical thinking: step by step", University Press of America, 1998, p48 [1]
• Beer, Francis A. "Validities: A Political Science Perspective", Social Epistemology 7, 1 (1993): 85-105.

# Study guide

Up to date as of January 14, 2010

## Contents

Test validity refers to the extent to which a psychological test measures what it is purported to measure. Validity is commonly discussed in terms of there being three main types: content validity, construct validity, and criterion validity. In classical test theory, validity is limited by reliability.

## Content validity

Content validity refers to the degree to which the content of the test matches the content domain of the construct. For example, a test of ability to add two-digit numbers should cover the full range of combinations of digits. A test with only one-digit numbers, or only even numbers, would not have good coverage of the content domain.

Content validity evidence typically involves subject matter experts evaluating test items against the test specifications. Expert-based testing of content validity is distinguished from face validity which refers to prima facie estimates of whether a test appears to measure a certain criterion. Note that face validity does not guarantee that the test actually measures phenomena in that domain. Indeed, when a test is subject to faking (malingering), low face validity might make the test more valid.

## Construct validity

Construct validity refers to the totality of evidence about whether a particular operationalization of a construct adequately represents what is intended by theoretical account of the construct being measured.

Construct validity involves empirical and theoretical support for the interpretation of the construct. Lines of evidence include statistical analyses of the internal structure of the test including the relationships between responses to different test items (see internal consistency). It also includes relationships between the test and measures of other constructs. Construct validity is not distinct from the support for the substantive theory of the construct that the test is designed to measure.

There are two main approaches to construct validity:

1. Convergent validity: refers to the degree to which a measure is correlated with other measures that it is theoretically predicted to correlate with.
2. Discriminant validity: describes the degree to which the operationalization does not correlate with other operationalizations that it theoretically should not correlated with.

## Criterion validity

Criterion validity involves the correlation between the test and a criterion variable (or variables) taken as representative of the construct.

There are two types of criterion validity:

1. Concurrent validity: The test data and criterion data are collected at the same time, e.g.,
• Self-report and observer measures collected at the same time
2. Predictive validity: The test data is collected first in order to predict criterion data collected at a later point in time, e.g.,
• employee selection tests are often validated against measures of job performance.
• measures of risk of recidivism among those convicted of a crime can be validated against measures of recidivism.

1. Cronbach, L. J. & Meehl, P. E. (1955). Construct validity in psychological tests. Psychological Bulletin, 52, 281-302.
2. Validity (statistics) (Wikipedia)

# Genealogy

Up to date as of February 01, 2010

### From Familypedia

The term validity as it occurs in logic refers generally to a property of deductive arguments, although many logic texts apply the term to statements as well (a statement is a sentence that “has a truth value,” i.e., that is either true or false). For the purposes of this article, an argument is a set of statements, one of which is the conclusion and the rest of which are premises. The premises are reasons intended to show that the conclusion is, or is probably, true.

When an argument is set forth to show that its conclusion is true (as opposed to probably true), then the argument is intended to be deductive. An argument set forth to show that its conclusion is probably true may be regarded as inductive. To say that an argument is valid is to say that the conclusion really does follow from the premises. That is, an argument is valid precisely when it cannot possibly lead from true premises to a false conclusion. The following definition is fairly typical:

• An argument is deductively valid if it cannot possibly have all true premises and a false conclusion.

An argument that is not valid is said to be ‘’invalid’’.

All men are mortal
Socrates is a man
Therefore, Socrates is mortal.

What makes this a valid argument is not the mere fact that it has true premises and a true conclusion, but the fact of the logical impossibility of things being otherwise. No matter how the universe might be constructed, it could never be the case that this argument should turn out to have simultaneously true premises but a false conclusion. The above argument may be contrasted with the following invalid one:

All men are mortal
Socrates is mortal
Therefore, Socrates is a man

In this case, there is no impossibility of true premises but false conclusion: it is easily imagined that there is a woman named ‘Socrates’, so that in fact the above premises would be true but the conclusion false—hence it is possible that the argument has true premises and a false conclusion. This possibility is what constitutes invalidity. (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 for which no interpretation exists under which the premises are all true but the conclusion false.

As already seen, the interpretation given above does cause the second argument form to have true premises and false conclusion, hence demonstrating its invalidity.