Denying the antecedent, sometimes also called
inverse error, is a
formal fallacyIn philosophy, a formal fallacy or a logical fallacy is a pattern of reasoning which 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
formIn logic, the argument form or test form of an argument results from replacing the different words, or sentences, that make up the argument with letters, along the lines of algebra; the letters represent logical variables. This is of importance since the validity of an argument is determined solely...
:
- If P, then Q.
- Not P.
- Therefore, not Q.
Argument* In logic, an argument is a set of one or more meaningful declarative sentences known as the premises along with another meaningful declarative sentence known as the conclusion...
s of this form are
invalidThe term Validity in logic applies to arguments or statements.-Validity of arguments:An argument is valid if and only if the truth of its premises entails the truth of its conclusion. It would be self-contradictory to affirm the premises and deny the conclusion...
. 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" clauseAn 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
conditionalIn 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.
Denying the antecedent, sometimes also called
inverse error, is a
formal fallacyIn philosophy, a formal fallacy or a logical fallacy is a pattern of reasoning which 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
formIn logic, the argument form or test form of an argument results from replacing the different words, or sentences, that make up the argument with letters, along the lines of algebra; the letters represent logical variables. This is of importance since the validity of an argument is determined solely...
:
- If P, then Q.
- Not P.
- Therefore, not Q.
Argument* In logic, an argument is a set of one or more meaningful declarative sentences known as the premises along with another meaningful declarative sentence known as the conclusion...
s of this form are
invalidThe term Validity in logic applies to arguments or statements.-Validity of arguments:An argument is valid if and only if the truth of its premises entails the truth of its conclusion. It would be self-contradictory to affirm the premises and deny the conclusion...
. 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" clauseAn 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
conditionalIn 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 Elizabeth
Elizabeth II is the queen regnant of sixteen independent states known informally 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,...
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 TuringAlan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was influential in the development of computer science and provided an influential formalisation of the concept of the algorithm and computation with the Turing machine...
in the article "
Computing Machinery and IntelligenceComputing 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 notes) 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 claims
P and
Q express the same proposition, then the argument would be trivially valid, as it would
beg the questionBegging the question is a logical fallacy in which the proposition to be proved is assumed implicitly or explicitly in the premise. Begging the question is related to circular argument, circulus in probando or circular reasoning but they are considered absolutely different by Aristotle...
. In everyday discourse, however, such cases are rare, typically only occurring when the "if-then" premise is actually an "
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...
" claim (i.e., a
biconditionalIn logic and mathematics, the logical biconditional is a logical operator connecting two statements to assert "p if and only if q", where p is a hypothesis and q is a conclusion...
). For example:
- If I am President of the United States
The President of the United States is the head of state and head of government of the United States and is the highest political official in the United States by influence and recognition...
, 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 consequent
Affirming the consequent, sometimes called converse error, is a formal fallacy, committed by reasoning in the form:#If P, then Q.#Q.#Therefore, P....
- Modus ponens
In classical logic, modus ponendo ponens is a valid, simple argument form sometimes referred to as affirming the antecedent or the law of detachment...
- Modus tollens
In classical logic, modus tollens has the following argument form:...
- Necessary and sufficient conditions
In 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.* A necessary condition of a...
- To His Coy Mistress
To His Coy Mistress is a witty metaphysical poem written by the British author and statesman Andrew Marvell either during or just before the Interregnum. The poem is often considered an argument for the concept of 'carpe diem'...
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