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Validity (logic)

August 25, 2023

For other uses, see Validity.

In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.[1] It is not required for a valid argument to have premises that are actually true,[2] 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 formulas (also called wffs or simply formulas).

The validity of an argument can be tested, proved or disproved, and depends on its logical form.[3]

Arguments

 
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.[3] Validity does not require the truth of the premises, instead it merely 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 sound.[3]

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 (and sound) argument is given by the following well-known syllogism:

All men are mortal. (True)
Socrates is a man. (True)
Therefore, Socrates is mortal. (True)

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. (False)
Socrates is a cup. (False)
Therefore, Socrates is green. (False)

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. (False)
Socrates is a man. (True)
Therefore, Socrates is mortal. (True)

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 third argument becomes:

No P's are Q.
S is a P.
Therefore, S is not 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 of the language. In propositional logic, they are tautologies.

Statements

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

Soundness

Main article: 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. (False)
All humans are animals. (True)
Therefore, all humans live on Mars. (False)

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.[3]

Satisfiability

Main article: 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.[4]

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'.[5]

Preservation properties Logical connective sentences
True and false preserving: Proposition  • Logical conjunction (AND,   )  • Logical disjunction (OR,   )
True preserving only: Tautology (   )  • Biconditional (XNOR,   )  • Implication (   )  • Converse implication (   )
False preserving only: Contradiction (   ) • Exclusive disjunction (XOR,   )  • Nonimplication (   )  • Converse nonimplication (   )
Non-preserving: Negation (   )  • Alternative denial (NAND,   ) • Joint denial (NOR,   )

See also

References

  1. ^ Validity and Soundness – Internet Encyclopedia of Philosophy
  2. ^ Jc Beall and Greg Restall, "Logical Consequence", The Stanford Encyclopedia of Philosophy (Fall 2014 Edition).
  3. ^ a b c d Gensler, Harry J. (January 6, 2017). Introduction to logic (Third ed.). New York: Routledge. ISBN 978-1-138-91058-4. OCLC 957680480.
  4. ^ L. T. F. Gamut, Logic, Language, and Meaning: Introduction to Logic, University of Chicago Press, 1991, p. 115.
  5. ^ Robert Cogan, Critical Thinking: Step by Step, University Press of America, 1998, p. 48.

Further reading

  • 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.

August 25, 2023
validity, logic, other, uses, validity, logic, specifically, deductive, reasoning, argument, valid, only, takes, form, that, makes, impossible, premises, true, conclusion, nevertheless, false, required, valid, argument, have, premises, that, actually, true, ha. For other uses see Validity In logic specifically in deductive reasoning an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false 1 It is not required for a valid argument to have premises that are actually true 2 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 formulas also called wffs or simply formulas The validity of an argument can be tested proved or disproved and depends on its logical form 3 Contents 1 Arguments 2 Valid formula 3 Statements 4 Soundness 5 Satisfiability 6 Preservation 7 See also 8 References 9 Further readingArguments Edit Argument terminology used in logicIn 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 3 Validity does not require the truth of the premises instead it merely 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 sound 3 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 and sound argument is given by the following well known syllogism All men are mortal True Socrates is a man True Therefore Socrates is mortal True 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 False Socrates is a cup False Therefore Socrates is green False 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 False Socrates is a man True Therefore Socrates is mortal True 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 third argument becomes No P s are Q S is a P Therefore S is not 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 EditA formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language In propositional logic they are tautologies Statements EditA statement can be called valid i e logical truth if it is true in all interpretations Soundness EditMain article 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 False All humans are animals True Therefore all humans live on Mars False 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 3 Satisfiability EditMain article 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 4 Preservation EditIn 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 5 Preservation properties Logical connective sentencesTrue and false preserving Proposition 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 Negation displaystyle neg Alternative denial NAND displaystyle uparrow Joint denial NOR displaystyle downarrow See also Edit Philosophy portalLogical consequence Reductio ad absurdum Mathematical fallacy Soundness W validityReferences Edit Validity and Soundness Internet Encyclopedia of Philosophy Jc Beall and Greg Restall Logical Consequence The Stanford Encyclopedia of Philosophy Fall 2014 Edition a b c d Gensler Harry J January 6 2017 Introduction to logic Third ed New York Routledge ISBN 978 1 138 91058 4 OCLC 957680480 L T F Gamut Logic Language and Meaning Introduction to Logic University of Chicago Press 1991 p 115 Robert Cogan Critical Thinking Step by Step University Press of America 1998 p 48 Further reading EditBarwise Jon Etchemendy John Language Proof and Logic 1999 42 Beer Francis A Validities A Political Science Perspective Social Epistemology 7 1 1993 85 105 Wiktionary has definitions related to Validity Retrieved from https en wikipedia org w index php title Validity logic amp oldid 1170722969, wikipedia, wiki, book, books, library,

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