Formal fallacy

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Not to be confused with non sequitur -- when an argument may be valid but the conclusion does not follow because of weak premises.
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 “”France is no longer France. —Donald Trump, apparently refuting the law of identity[1]

A formal fallacy is a logical fallacy that violates a particular rule of propositional calculus, such as modus ponens. These fallacies can be determined to be invalid simply by the inspection of the form or structure of the argument - at heart, a formal fallacy contains some sort of non sequitur. This doesn't necessarily mean that a conclusion is wrong, but it does mean that we will need a better reason or argument to derive the conclusion.

Formal fallacies are propositional, quantificational, syllogistical, modal, or fallacy fallacy.

Examples

For example, given the syllogism of A -> B, B -> C, A therefore C, it doesn't matter what A, B and C actually are. If the statement is incorrect because of the content of A, B or C, it would be an informal fallacy, not a formal fallacy because the logic works out. The difference can often be difficult to spot because we may have trouble splitting the content of an argument from its form.

Consider the following proposition:

1. Some men are doctors.
2. Some doctors are tall.
3. Therefore, some men are tall.

We immediately want to say "yes" in response to this, and take the proposition to be correct because this is what reality reflects. However, this is actually an inference beyond what the logical set up allows us to make. Saying that "some men are tall" given the first two premises is a formal logical fallacy. Why this is so can be more easily highlighted by replacing "tall" with a different property, such as "are women" to yield:

1. Some men are doctors.
2. Some doctors are women.
3. Therefore, some men are women.

This doesn't change the logical structure of the argument, but the fallacy is easier to spot because the content changes it to a more obvious absurdity.

More formally, this is not a valid rule of inference in predicate logic:

$\cfrac{\exists x (P(x) \implies Q(x)) \qquad \exists x (Q(x) \implies R(x))}{\exists x (P(x) \implies R(x))}$

because the first and second x are different bound variables.

Subfallacies

See the main article on this topic: Logical fallacy#Formal