Validity (logic)
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In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, specifically in
deductive reasoning Deductive reasoning is the process of drawing valid inferences. An inference is valid if its conclusion follows logically from its premises, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, t ...
, an
argument An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
is valid
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it takes a form that makes it impossible for the
premise A premise or premiss is a proposition—a true or false declarative statement—used in an argument to prove the truth of another proposition called the conclusion. Arguments consist of a set of premises and a conclusion. An argument is meaningf ...
s to be
true True most commonly refers to truth, the state of being in congruence with fact or reality. True may also refer to: Places * True, West Virginia, an unincorporated community in the United States * True, Wisconsin, a town in the United States * ...
and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, 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 In logic, the logical form of a statement is a precisely specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unamb ...
.


Arguments

In logic, an
argument An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
is a set of related statements expressing the ''premises'' (which may consists of non-empirical evidence, empirical evidence or may contain some axiomatic truths) and a ''necessary conclusion based on the relationship of the premises.'' 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. Validity does not require the truth of the premises, instead it merely necessitates that conclusion follows from the premises without violating the correctness of the
logical form In logic, the logical form of a statement is a precisely specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unamb ...
. If also the premises of a valid argument are proven true, this is said to be
sound In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the br ...
. The corresponding conditional of a valid argument is a
logical truth Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement whic ...
and the negation of its corresponding conditional is a
contradiction In traditional logic, a contradiction involves a proposition conflicting either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
. The conclusion is a necessary consequence of its premises. An argument that is not valid is said to be "invalid". An example of a valid (and
sound In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the br ...
) argument is given by the following well-known
syllogism A syllogism (, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. In its earliest form (defin ...
: : 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. Validity is about the tie in relationship between the two premises the necessity of the conclusion. There needs to be a relationship established between the premises i.e., a middle term between the premises. If you just have two unrelated premises there is no argument. Notice some of the terms repeat: men is a variation man in premises one and two, Socrates and the term mortal repeats in the conclusion. The argument would be just as valid if both premises and conclusion were false. The following argument is of the same
logical form In logic, the logical form of a statement is a precisely specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unamb ...
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. Formal arguments that are invalid are often associated with at least one fallacy which should be verifiable. 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's 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 In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...
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, in some systems of logic like in Modal logic if the statement is true in all interpretations. In Aristotelian logic statements are not valid per se. Validity refers to entire arguments. The same is true in propositional logic (statements can be true or false but not called valid or invalid).


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.


Satisfiability

Model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
analyzes formulae with respect to particular classes of interpretation in suitable mathematical structures. On this reading, a 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''.


Preservation

In ''truth-preserving'' validity, the interpretation under which all variables are assigned a
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in ...
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'.Robert Cogan, ''Critical Thinking: Step by Step'', University Press of America, 1998
p. 48
:


See also

*
Logical consequence Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more stat ...
*
Reductio ad absurdum In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical argument'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absur ...
*
Mathematical fallacy In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple ''mistake'' and a ''mathematical fallacy'' in a proof ...
*
Soundness In logic and deductive reasoning, an argument is sound if it is both Validity (logic), valid in form and has no false premises. Soundness has a related meaning in mathematical logic, wherein a Formal system, formal system of logic is sound if and o ...
* Ω-validity


References


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. {{Mathematical logic Arguments Concepts in logic Deductive reasoning Logical truth