identify and explain formal fallacies, including affirming the consequent, denying the antecedent and the undistributed middle

What this dot point is asking

QCAA distinguishes two families of fallacy. Informal fallacies fail because of content or context; formal fallacies fail because of structure alone. This dot point is about the formal ones: invalid forms that can be identified without knowing what the argument is about. You need to name the main formal fallacies, show why each is invalid, and expose them by counterexample.

The answer

What makes a fallacy formal

A formal fallacy is an error in the logical form of an argument: the form is invalid no matter what content you plug in. Because validity is purely structural, you can detect a formal fallacy by symbolising the argument and checking the form, or by finding a counterexample (a substitution with true premises and a false conclusion). This contrasts with informal fallacies, which require attention to meaning and context.

Affirming the consequent

The valid form modus ponens is: if P then Q; P; therefore Q. The formal fallacy of affirming the consequent reverses it:

• If P then Q.
• Q.
• Therefore P. (Invalid.)

Example: "If it is raining, the ground is wet; the ground is wet; therefore it is raining." Counterexample: a sprinkler wets the ground with no rain. The conditional does not say Q happens only when P does, so affirming Q tells us nothing about P.

Denying the antecedent

The valid form modus tollens is: if P then Q; not-Q; therefore not-P. The formal fallacy of denying the antecedent mistakes it:

• If P then Q.
• Not-P.
• Therefore not-Q. (Invalid.)

Example: "If you study law, you will be employable; she did not study law; therefore she is not employable." Counterexample: she studied medicine and is employable. The conditional gives one route to Q, not the only route.

The undistributed middle

In categorical syllogisms, the middle term (the one appearing in both premises but not the conclusion) must be distributed (referring to all members of its class) at least once. The fallacy of the undistributed middle breaks this rule:

• All cats are mammals.
• All dogs are mammals.
• Therefore all cats are dogs. (Invalid.)

The middle term "mammals" is never distributed, so sharing it does not link cats and dogs. Counterexample is built in: the premises are true and the conclusion plainly false.

Other formal fallacies

• Illicit major / illicit minor: a term distributed in the conclusion but not in the premise where it appears.
• Affirming a disjunct: P or Q; P; therefore not-Q. Invalid where "or" is inclusive, since both could be true.
• Fallacy of the inverse and converse: treating "if P then Q" as equivalent to "if not-P then not-Q" (inverse) or "if Q then P" (converse), neither of which follows.

Exposing a formal fallacy

To show an argument commits a formal fallacy: (1) symbolise it to reveal the bare form; (2) name the invalid form; (3) give a counterexample with the same form, true premises and a false conclusion. The counterexample is the clinching move, because it demonstrates the form does not preserve truth, regardless of the original content.

Try this

Q1. Identify the fallacy and explain why it is invalid: "If she is guilty, she will be nervous; she is nervous; so she is guilty." [3 marks]

• Cue. Affirming the consequent; nervousness has other causes, so a counterexample has true premises and a false conclusion.

Q2. Explain the fallacy of the undistributed middle. [3 marks]

• Cue. The middle term is never distributed, so sharing it fails to connect the two end terms.

Q3. Distinguish denying the antecedent from modus tollens. [2 marks]

• Cue. Modus tollens denies the consequent (not-Q, so not-P) and is valid; denying the antecedent (not-P, so not-Q) is invalid.