Exclusive Premises

The categorical logic fallacies are called “formal” fallacies, because they are all violations of proper syllogistic form. The categorical syllogism is central to Aristotelian logic, which strongly influenced later logical traditions. These fallacies, however, have only been termed “fallacies” for a couple hundred years. When Aristotle wrote his description of the categorical syllogism in the Prior Analytics, he pointed out that in only certain combinations of terms, no syllogism is possible, no syllogism is produced, or no syllogism can be made. Alexander of Aphrodisias (1991) referred to these combinations as “non‐syllogistic” in his commentary on Aristotle’s Prior Analytics. These were combinations of propositions that simply didn’t lead anywhere.

Proper fallacies, at the time, were rather errors in reasoning based on ambiguities, such as those Aristotle speaks of in Sophistical Refutations and on which Galen comments in De Captionibus (On Fallacies). In this account of the Aristotelian tradition, arguments are described in terms of matter and form, but the combinations now called “formal” fallacies do not supply a valid syllogistic form. William of Sherwood, a medieval thinker, whose Introduction to Logic explains the rules for the proper formation of a syllogism, called them “useless combinations.” The tradition of calling them “fallacies” in modern logic really begins with Richard Whately’s Elements of Logic, the 1826 book that is an expansion of his earlier article in Encyclopaedia Metropolitana. The book to which Whately claims he is most indebted, Henry Aldrich’s Artis Logicæ Compendium of 1691, talks about such problems as illicit processes of terms and undistributed middles, but they are not included in Aldrich’s appendix on formal fallacies.

Formal fallacies, according to Aldrich, include any reasoning that violates the law of identity, law of non‐contradiction, or law of excluded middle. It is only since Whately’s Elements of Logic that a standard table of fallacies, that is, reasons for the invalidity of certain combinations of premises, includes an illicit major or minor (see, for instance, Thomas Solly’s A Syllabus of Logic from 1839 or Samuel Neil’s The Art of Reasoning from 1853).

Each categorical fallacy is a violation of a rule for the formation of valid syllogisms. Aristotle defines the syllogism in Prior Analytics at 24a19–22:

A syllogism is a discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so. I mean by the last phrase that it follows because of them, and by this, that no further term is required from without in order to make the consequence necessary.

Alexander of Aphrodisias (1991) praises the force of syllogistic deduction:

When a part is justified from the whole, such a justification is called a syllogism; and this is the most compelling type of justification. For anything which applies to or holds of a universal and a totality, by necessity also holds of what is within it and is included in it. (104)

A valid syllogism guarantees a true conclusion when its premises are true. Aristotle also describes combinations of premises that do not yield a syllogism. The exclusive premises fallacy (EP) is one such combination. Related fallacies include four terms (Chapter 5), the illicit major and minor terms fallacies (Chapter 6), and the fallacy of the undistributed middle term (Chapter 7).

The exclusive premises fallacy occurs when a syllogism has two negative premises—that is, propositions that deny a connection between the subject and predicate. For example, the proposition, “No monkeys are cats,” makes the claim that there is no connection between the set of monkeys and the set of cats. Likewise, the proposition, “Some monkeys are not plants,” says that at least one monkey lies outside the set of plants. In all, negative premises usually (not always!) contain the word “not” or “no” in them. A standard form categorical syllogism commits this fallacy if both premises are negative in quality, whether in the universal form “no X are Y” or the particular form “some X are not Y,” or any combination thereof.

The quality or quantity of the premises of a syllogism can make them exclusive to one another. By quality, I mean whether the premise is affirmative or negative, and by quantity, I mean whether the premise is particular or universal. The terminology is comparatively recent, but the rule against two negative premises has a long history. Aristotle notes the problem in Prior Analytics at 26a9–13.

Nor again can a deduction be formed when neither the first term belongs to any of the middle, nor the middle to any of the last. As an example of a positive relation between the extremes take the terms science, line, medicine: of a negative relation science, line, unit.

We can see this illustrated in one of Aristotle’s examples of a non‐syllogism at 27a21–23:

Nor is a syllogism possible when M is predicated neither of any N nor of any O. Terms to illustrate a positive relation are line, animal, man: a negative relation, line, animal, stone.

If we fill in the blanks for a negative relation:

1. No animals are lines.
2. No stones are lines.
3. Therefore, …?

Those premises do not establish a categorical relation between animals and stones. Alexander of Aphrodisias (1991) quotes the former passage in his commentary, explaining the reasoning behind it:

The reason why nothing is deduced syllogistically in this combination is that the middle bears no relation to either of the extremes (it is as if the middle had not been taken at all – and syllogisms depend on the middle term).

The middle term excludes each extreme term without establishing a categorical relation between them. For example, if I say:

1. No monkeys are cats.
2. No monkeys are plants.

Together, these statements still do not establish a categorical relation between cats and plants. It’s as if the premises have nothing to do with each other besides the fact that both of them mention monkeys. They are exclusive. These two premises, taken together, tell me nothing of the relation between the sets of cats and plants. Fallacies like this are especially dangerous because the conclusion can still seem plausible. Changing the terms while keeping the same exclusive form often reveals the error clearly: what appears to be good reasoning with one set of terms collapses into obvious nonsense with another, yet the logical mistake is identical. This was a well‐known rule for the formation of any and all syllogisms, right from Aristotle, through the medievals, and into the modern era. William of Sherwood (1966) includes it in his rules for syllogisms: “Note that nothing follows from two negatives or from two particulars” (68). So does John Buridan (2015): “Second Conclusion: No syllogism can be validly drawn from two negatives” (119). And so does Walter Burley (2000): “Therefore, I say that there are two general rules for every syllogism, in no matter what figure or mood it occurs, namely that it have (a) one universal proposition and (b) one affirmative one. For nothing follows syllogistically from negatives or from particulars” (26).

Neil (1853) uses the “agreement/disagreement” terminology to describe assertions and negations, which helps explain why two negative categorical premises do not establish the required relation between the extreme terms.