Denying the Antecedent | Examples & Definition

Denying the antecedent is the error of assuming that if the initial condition (P) is not met, the expected result (Q) won’t occur either.

This logical fallacy is expressed as follows:

Denying the antecedent fallacy example
  • If it is snowing, then it is cold outside.
  • It is not snowing.
  • Therefore, it is not cold outside.

The logical fallacy of denying the antecedent is typically found in domains that involve formal logical reasoning, such as math, science, and law.

What is denying the antecedent?

Denying the antecedent occurs when someone mistakenly concludes that if the initial condition (P) of a situation isn’t fulfilled, then the expected outcome (Q) can’t occur either.

It is a formal logical fallacy that occurs in deductive reasoning using hypothetical syllogisms. The fallacy is defined by the following formula:

  • If P, then Q.
  • Not P.
  • Therefore, not Q.

This logical structure is recognized as invalid because the truth of Q is not solely dependent on the condition P being true. Other conditions could still make Q true. Thus, the conclusion does not follow logically from the premises.

Denying the antecedent examples

Examples of the denying the antecedent fallacy are often found in contexts such as math, where deductive reasoning is essential.

Denying the antecedent fallacy example in math
  • If a function is linear, then its graph is a straight line.
  • The function is not linear.
  • Therefore, its graph is not a straight line.

The graph could be a constant function, which is a horizontal straight line.

They are also commonly found in scientific contexts.

Denying the antecedent fallacy example in science
  • If an animal is a mammal, then it is warm-blooded.
  • This animal is not a mammal.
  • Therefore, it is not warm-blooded.

The animal could be a bird, which is also warm-blooded.

Note
The denying the antecedent fallacy, like the affirming the consequent fallacy, is attractive because it sometimes does lead to a correct conclusion. However, it is important to remember that this is a matter of chance, not logic. This line of reasoning cannot be used to prove anything logically.

Denying the antecedent vs modus tollens

The valid way to negate hypothetical syllogisms is called modus tollens (or “denying the consequent”). It involves proving that an initial condition (P) didn’t happen because a necessary consequence (Q) didn’t happen. It takes the following form:

Modus tollens (valid):

  • If P, then Q.
  • Not Q.
  • Therefore, not P.
Modus tollens (denying the consequent) example
  • If twins are identical, then they are the same sex.
  • These twins are not the same sex.
  • Therefore, they are not identical.

In contrast, denying the antecedent involves making the mistake of erroneously trying to prove that an effect didn’t happen just because one possible (but nonexclusive) cause didn’t happen.

Denying the antecedent (invalid):

  • If P, then Q.
  • Not P.
  • Therefore, not Q.
Denying the antecedent fallacy example
  • If twins are identical, then they are the same sex.
  • These twins are not identical.
  • Therefore, they are not the same sex.

The conclusion is invalid. Non-identical (fraternal) twins could be the same sex.

Frequently asked questions about denying the antecedent

Why is denying the antecedent a fallacy?

Denying the antecedent is a logical fallacy because the absence of one potential cause doesn’t mean that no other causes exist.

Consider the following example:

  • If it’s raining (antecedent), then the ground is wet (consequent).
  • It’s not raining.
  • Therefore, the ground is not wet.

This argument is clearly faulty because the ground could be wet for many reasons other than rain (e.g., lawn sprinklers). In other words, the conclusion is not solely dependent on the premise.

Is denying the antecedent valid or invalid?

Denying the antecedent is an invalid argument form. In other words, it is a formal logical fallacy.

In logic, the term “invalid” describes a type of argument in which the premises do not guarantee the truth of the conclusion, even if all the premises are true. In the fallacy of denying the antecedent, it is possible that the expected outcome could occur without one specific cause being true.

Consider the following example:

  • If an animal is a bird, then it lays eggs.
  • This animal is not a bird.
  • Therefore, it does not lay eggs.

It is clear that this argument is invalid. The animal could be an insect or a reptile or many other animals. The conclusion is not guaranteed by the premises.

What is a real-life example of denying the antecedent?

A real-life example of denying the antecedent is the following argument:

  • If someone is a professor, then they have a PhD.
  • Maria is not a professor.
  • Therefore, Maria does not have a PhD.

This is an invalid argument because the fact that Maria is not a professor does not necessarily mean she does not have a PhD. Maria might be someone who has a PhD but chose a non-academic career path.

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Magedah Shabo

Magedah is the author of Rhetoric, Logic, & Argumentation and Techniques of Propaganda and Persuasion. She began her career in the educational publishing industry and has over 15 years of experience as a writer and editor. Her books have been used in high school and university classrooms across the US, including courses at Harvard and Johns Hopkins. She has taught ESL from elementary through college levels.