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Denying the antecedent, sometimes also called inverse error or fallacy of the inverse, is a formal fallacy of inferring the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
from the original statement. It is committed by reasoning in the
form Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens. Form also refers to: *Form (document), a document (printed or electronic) with spaces in which to write or enter data * ...
: :If ''P'', then ''Q''. :Therefore, if not ''P'', then not ''Q''. which may also be phrased as :P \rightarrow Q (P implies Q) :\therefore \neg P \rightarrow \neg Q (therefore, not-P implies not-Q)
Argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
s of this form are
invalid Invalid may refer to: * Patient, a sick person * one who is confined to home or bed because of illness, disability or injury (sometimes considered a politically incorrect term) * .invalid, a top-level Internet domain not intended for real use As ...
. Informally, this means that arguments of this form do not give good reason to establish their conclusions, even if their premises are true. In this example, a valid conclusion would be: ~P or Q. The name ''denying the antecedent'' derives from the premise "not ''P''", which denies the "if" clause of the
conditional Conditional (if then) may refer to: *Causal conditional, if X then Y, where X is a cause of Y *Conditional probability, the probability of an event A given that another event B has occurred *Conditional proof, in logic: a proof that asserts a co ...
premise. One way to demonstrate the invalidity of this argument form is with an example that has true premises but an obviously false conclusion. For example: :If you are a ski instructor, then you have a job. :You are not a ski instructor. :Therefore, you have no job. That argument is intentionally bad, but arguments of the same form can sometimes seem superficially convincing, as in the following example offered by
Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical ...
in the article " Computing Machinery and Intelligence": However, men could still be machines that do not follow a definite set of rules. Thus, this argument (as Turing intends) is invalid. It is possible that an argument that denies the antecedent could be valid if the argument instantiates some other valid form. For example, if the claims ''P'' and ''Q'' express the same proposition, then the argument would be trivially valid, as it would
beg the question In classical rhetoric and logic, begging the question or assuming the conclusion (Latin: ') is an informal fallacy that occurs when an argument's premises assume the truth of the conclusion, instead of supporting it. For example: * "Green is t ...
. In everyday discourse, however, such cases are rare, typically only occurring when the "if-then" premise is actually an "
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
" claim (i.e., a biconditional/ equality). The following argument is not valid, but would be if the first premise was "If I can veto Congress, then I am the US President." This claim is now ''modus tollens,'' and thus valid. :If I am
President of the United States The president of the United States (POTUS) is the head of state and head of government of the United States of America. The president directs the Federal government of the United States#Executive branch, executive branch of the Federal gove ...
, then I can veto Congress. :I am not President. :Therefore, I cannot veto Congress.


See also

* Affirming the consequent * ''
Modus ponens In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference ...
'' * ''
Modus tollens In propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo tollens'' (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference. ''Modus tollens' ...
'' * Necessity and sufficiency


References


External links


FallacyFiles.org: Denying the Antecedent

safalra.com: Denying The Antecedent
{{Formal fallacy Propositional fallacies Dichotomies