Which errors of reasoning come purely from an argument's form, regardless of what it is about?
identify and explain formal fallacies, including affirming the consequent, denying the antecedent and the undistributed middle
A focused QCE Unit 3 answer on formal fallacies. Covers what makes a fallacy formal rather than informal, the propositional fallacies of affirming the consequent and denying the antecedent, the categorical fallacy of the undistributed middle, and how to expose each by counterexample.
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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.
Related dot points
- distinguish validity from soundness, and evaluate deductive arguments for both, using premises and conclusions
A focused QCE Unit 3 answer on validity and soundness. Covers the structure of deductive arguments, the difference between truth and validity, what soundness adds, common valid forms such as modus ponens and modus tollens, and how to test arguments by counterexample.
- translate and symbolise propositions using logical operators, and use truth tables to test propositional arguments for validity
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- identify and explain common informal fallacies in arguments, including fallacies of relevance, ambiguity and presumption
A focused QCE Unit 3 answer on informal fallacies. Covers the difference between formal and informal fallacies and explains the major fallacies of relevance, presumption and ambiguity, including ad hominem, straw man, appeal to authority, false dilemma, begging the question and equivocation, with examples.