Denying the antecedent
Encyclopedia
Denying the antecedent, sometimes also called inverse error, is a formal fallacy
, committed by reasoning in the form:
Argument
s of this form are invalid
. Informally, this means that arguments of this form do not give good reason to establish their conclusions, even if their premises are true.
The name denying the antecedent derives from the premise "not P", which denies the "if" clause
of the conditional
premise.
One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example:
That argument is obviously bad, but arguments of the same form can sometimes seem superficially convincing, as in the following example imagined by Alan Turing
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 everyday discourse, however, such cases are rare, typically only occurring when the "if-then" premise is actually an "if and only if
" claim (i.e., a biconditional
). For example:
The above argument is not valid, but would be if the first premise ended thus: "...and if I can veto Congress, then I am the U.S. President" (as is in fact true). More to the point, the validity of the new argument stems not from denying the antecedent, but denying the consequent.
Formal fallacy
In philosophy, a formal fallacy is a pattern of reasoning that is always wrong. This is due to a flaw in the logical structure of the argument which renders the argument invalid...
, committed by reasoning in the form:
- If P, then Q.
- Not P.
- Therefore, not Q.
Argument
Argument
In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...
s of this form are invalid
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
. Informally, this means that arguments of this form do not give good reason to establish their conclusions, even if their premises are true.
The name denying the antecedent derives from the premise "not P", which denies the "if" clause
Antecedent
An antecedent is a preceding event, condition, cause, phrase, or word. It may refer to:* Antecedent moisture, a hydrologic term describing the relative wetness condition of a sewershed.* Antecedent , the first half of a hypothetical proposition....
of the conditional
Indicative conditional
In natural languages, an indicative conditional is the logical operation given by statements of the form "If A then B". Unlike the material conditional, an indicative conditional does not have a stipulated definition...
premise.
One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example:
- If Queen ElizabethElizabeth II of the United KingdomElizabeth II is the constitutional monarch of 16 sovereign states known as the Commonwealth realms: the United Kingdom, Canada, Australia, New Zealand, Jamaica, Barbados, the Bahamas, Grenada, Papua New Guinea, the Solomon Islands, Tuvalu, Saint Lucia, Saint Vincent and the Grenadines, Belize,...
is an American citizen, then she is a human being. - Queen Elizabeth is not an American citizen.
- Therefore, Queen Elizabeth is not a human being.
That argument is obviously bad, but arguments of the same form can sometimes seem superficially convincing, as in the following example imagined by Alan Turing
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
in the article "Computing Machinery and Intelligence
Computing machinery and intelligence
Computing Machinery and Intelligence, written by Alan Turing and published in 1950 in Mind, is a seminal paper on the topic of artificial intelligence in which the concept of what is now known as the Turing test was introduced to a wide audience....
":
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
Begging the question
Begging the question is a type of logical fallacy in which the proposition to be proven is assumed implicitly or explicitly in the premise....
. In everyday discourse, however, such cases are rare, typically only occurring when the "if-then" premise is actually an "if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
" claim (i.e., a biconditional
Logical biconditional
In logic and mathematics, the logical biconditional is the logical connective of two statements asserting "p if and only if q", where q is a hypothesis and p is a conclusion...
). For example:
- If I am President of the United StatesPresident of the United StatesThe President of the United States of America is the head of state and head of government of the United States. The president leads the executive branch of the federal government and is the commander-in-chief of the United States Armed Forces....
, then I can veto Congress. - I am not President.
- Therefore, I cannot veto Congress.
The above argument is not valid, but would be if the first premise ended thus: "...and if I can veto Congress, then I am the U.S. President" (as is in fact true). More to the point, the validity of the new argument stems not from denying the antecedent, but denying the consequent.
See also
- Affirming the consequentAffirming the consequentAffirming the consequent, sometimes called converse error, is a formal fallacy, committed by reasoning in the form:#If P, then Q.#Q.#Therefore, P....
- Modus ponensModus ponensIn classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...
- Modus tollensModus tollensIn classical logic, modus tollens has the following argument form:- Formal notation :...
- Necessary and sufficient conditionsNecessary and sufficient conditionsIn logic, the words necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.-Definitions:A necessary condition...
- To His Coy MistressTo His Coy MistressTo His Coy Mistress is a metaphysical poem written by the British author and statesman Andrew Marvell either during or just before the Interregnum....