logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic

, specifically in

deductive reasoning Deductive reasoning, also deductive logic, is the process of reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making ...

, an

argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

is valid if and only if it takes a form that makes it impossible for the

premise A premise or premiss is a true or false statement that helps form the body of an argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, appl ...

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 * Tr ...

true

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 In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbol (formal), symbols from a given alphabet (computer science), alphabet that is part of ...

(also called ''wffs'' or simply ''formulas''). The validity of an argument—its being valid—can be tested, proved or disproved, and depends on its

logical form In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents sta ...

Arguments

In logic, an

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. Validity doesn't require the truth of the premises, instead it merely necessitates that conclusion follows from the formers without violating the correctness of the

. If also the premises of a valid argument are proven true, this is said to be

sound In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

sound

. The

corresponding conditional Correspondence may refer to: *In general usage, non-concurrent, remote communication Communication (from Latin ''communicare'', meaning "to share") is the act of developing Semantics, meaning among Subject (philosophy), entities or Organization ...

of a valid argument is a

logical truth Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and arg ...

and the negation of its corresponding conditional is a

contradiction In traditional logicIn philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Phil ...

. The conclusion is a

logical consequence Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies ...

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 A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (bo ...

: : 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

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

. 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 In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...

is a valid formula if and only if it is true under every possible

interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event i ...

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

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.

Satisfiability

Model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...

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''.

Preservation

In ''truth-preserving'' validity, the interpretation under which all variables are assigned a

truth value In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

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
:

Arguments

Valid formula

Statements

Soundness

Satisfiability

Preservation

See also

References

Further reading