5.3 Formal Fallacies in Propositional Logic

Conditional Fallacies

A conditional statement has the form "If P, then Q," where P is the antecedent and Q is the consequent. Two of the most common formal fallacies come from drawing invalid inferences from these statements.

Invalid Inferences from Conditional Statements

Affirming the Consequent incorrectly concludes that the antecedent is true because the consequent is true. In symbolic form, it looks like this:

1. $P \rightarrow Q$
2. $Q$
3. $\therefore P$ (invalid)

This fallacy assumes the converse of a conditional is logically equivalent to the original. But a consequent can be true for reasons that have nothing to do with the antecedent.

Denying the Antecedent incorrectly concludes that the consequent is false because the antecedent is false. Symbolically:

1. $P \rightarrow Q$
2. $\neg P$
3. $\therefore \neg Q$ (invalid)

This fallacy assumes the inverse of a conditional is logically equivalent to the original. But the consequent might still be true even when the antecedent is false.

Compare these with the two valid inference patterns for conditionals: modus ponens (affirm the antecedent, conclude the consequent) and modus tollens (deny the consequent, conclude the negation of the antecedent). The fallacies are essentially modus ponens and modus tollens done backwards.

Misinterpreting Conditional Statements

These two fallacies are closely related to the ones above but focus on confusing a conditional with a different conditional derived from it.

Fallacy of the Converse occurs when you treat a conditional as interchangeable with its converse. The converse of $P \rightarrow Q$ is $Q \rightarrow P$. These are not logically equivalent.

Fallacy of the Inverse occurs when you treat a conditional as interchangeable with its inverse. The inverse of $P \rightarrow Q$ is $\neg P \rightarrow \neg Q$. Again, these are not logically equivalent.

Notice the pattern: affirming the consequent is essentially applying the converse in an argument, and denying the antecedent is essentially applying the inverse. These four errors are two pairs describing the same underlying mistakes.

Syllogistic Fallacies

Categorical syllogisms use terms like "All," "No," and "Some" to relate categories. Formal fallacies here arise from violating the structural rules that make syllogisms valid.

Fallacies Involving Categorical Syllogisms

Fallacy of Exclusive Premises occurs when a syllogism has two negative premises. A valid categorical syllogism requires at least one affirmative premise, because two negative premises fail to establish a positive link between the terms.

Existential Fallacy (from universal premises) occurs when a particular conclusion (one using "some") is drawn from two universal premises. If both premises are universal ("All..." or "No..."), the conclusion must also be universal.

Fallacies Involving Quantifiers and Existence

The existential fallacy in its most important form arises when an argument's premises are all universal statements and the conclusion asserts that something actually exists. Universal statements like "All S are P" don't guarantee that any S actually exists. They can be read as saying: if something is an S, then it is a P.

So when you draw a particular conclusion ("Some S are P"), you're claiming at least one S exists. If nothing guarantees that, the argument is invalid.

This matters most when the subject category is empty or potentially empty:

The key takeaway: in traditional (Aristotelian) logic, universal statements were assumed to be about non-empty categories, so this fallacy didn't arise. In modern (Boolean) logic, universal statements carry no existential commitment. You need to know which framework your course uses, because the existential fallacy only applies under the modern interpretation.