Formal Fallacies (Math Lair)

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A formal fallacy is a fallacy in the form, or structure of an argument. Formal fallacies can be contrasted with informal fallacies, which are fallacies that do not involve errors in the form of the argument. Here is a list of some formal fallacies:

Non Sequitur

A non sequitur, Latin for "it does not follow", is an argument where the conclusion is drawn from premises not logically connected with it. All formal fallacies are specific types of the non sequitur fallacy.

Affirming a disjunct

This fallacy is an argument of the form "A or B; A; therefore not B". Because, logically, or is always used in an inclusive sense, it could still be possible that B is true if A is true.

Affirming the Consequent

This fallacy is an argument of the form "A implies B, B is true, therefore A is true". For example, "If someone owns a large mansion, then they are rich. Richard is rich. Therefore Richard owns a large mansion".

Denying the Antecedent

This fallacy is an argument of the form "A implies B, A is false, therefore B is false".

Illicit major

This fallacy occurs when the major term of a categorical syllogism is undistributed in the major premise but is distributed in the conclusion.

Illicit minor

This fallacy occurs when the minor term of a categorical syllogism is undistributed in the minor premise but is distributed in the conclusion.

Fallacy of the undistributed middle

This fallacy occurs where the middle term in a categorical syllogism is not distributed. For example: "All dogs are mammals. All cats are mammals. Therefore all dogs are cats."

Fallacy of exclusive premises:

A categorical syllogism that draws a conclusion from two negative premises commits this fallacy.

Affirmative conclusion from a negative premise

A categorical syllogism that draws a positive conclusion but has at least one negative premise commits this fallacy.

Fallacy of the four terms

A syllogism should have only three terms: a subject, a predicate, and a middle term that relates the subject and predicate. If the "middle term" represents something different in each premise, then the argument is invalid. Sometimes this may be a case of equivocation (see linguistic fallacies). This fallacy can also sometimes occur subtly. For example:

Nothing is better than butter.
Margarine is better than nothing.
Margarine is better than butter.

While at first glance this syllogism may appear to have only three terms, the middle term in the first premise ("nothing") is different from the middle term in the second premise ("better than nothing"). Also note that it is possible, in theory, to have a "fallacy of the five terms" or "fallacy of the six terms", but these are rarely seen as it is quite difficult to create a reasonable-sounding syllogism using five or six terms.

Converting a conditional

Also known as illicit conversion, this fallacy is related to affirming the consequent, above. This fallacy is an argument of the form "If A then B; therefore, if B then A." For example:

If it's snowing, then it's cold outside.
If it's cold outside, then it's snowing.

Existential Fallacy

In the traditional formal logic that was developed by Aristotle and later logicians in the Middle Ages, all propositions have existential import; in other words, all statements are implicitly taken to assert that the classes of things referred to by the subject and predicate terms are non-empty.

When first-order logic was developed in the nineteenth century, logicians dropped this assumption of non-empty terms, and adopted the Boolean interpretation of universal quantifiers. Under this interpretation, universal propositions (A-type and E-type) do not have existential import, while particular (I-type and O-type) propositions have existential import (incidentally, other interpretations are also possible; for example, under another interpretation, affirmative premises have existential import, while negative premises lack it). Under this interpretation, an otherwise valid syllogism with universal premises and a particular conclusion could be false if some of the terms referred to empty classes. For example:

All space aliens are lifeforms.
All space aliens come from other planets.
Some lifeforms come from other planets. (AAI-3)
If no space aliens exist, the two premisses are true but the conclusion is false.