Exclusive Premises

Charlene Elsby

From two negative premises no conclusion is deducible... e.g., though I assert Syrius is not a planet, Procyon is not a planet, these dicta prove nothing.

Samuel Neil, The Art of Reasoning (1853)

The categorical logic fallacies are called “formal” fallacies, because they are all violations of proper syllogistic form.

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 the Aristotelian tradition, arguments are describable in terms of their matter and form, but in the case of these modern “formal” fallacies, there is no form and, therefore, no argument. 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 Logics 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.

While a syllogism is by no means the only kind of argument, it is the best kind of argument, just as Alexander of Aphrodisias (1991) says it is:

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)

The reason a syllogism is so great is because if the syllogism is of the correct form, and as long as the premises are true, then the conclusion is guaranteed to be true - necessarily. Aristotle describes syllogisms that are not formed correctly and hence, fallacious. This chapter deals with the exclusive premises fallacy (EP). Also see the other chapters on the fallacy of four terms (Chapter 5), the illicit major and minor terms fallacies (Chapter 6), and the fallacy of the undistributed middle term (Chapter 7).

EP occurs when a syllogism has two negative premises, which are propositions that deny a connection between the subject and predicate.

Exclusive Premises 53

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. While nobody spoke of the EP until relatively recently, everyone has always known that no good syllogism has two negative premises. 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 at27a21-23:

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,...?

And there’s nothing we can conclude from that. 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).

Basically, we have a middle term and it is related to each of our major and minor terms in some way but in such a way that doesn’t allow us to make any connection between the major and minor terms in order to form a con­clusion.

(1) No monkeys are cats.

(2) No monkeys are plants.

then I still have nothing to conclude from these statements when I take them together. 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. 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 syllogis- tically from negatives or from particulars” (26).

In summary, you can’t have a good syllogism with two negative premises, or else your premises won’t be related in the right way, and you won’t be able to conclude anything at all.

Neil (1853) uses the “agreement/disagreement” terminology to describe assertions and negations, which helps to elucidate the fact that from two statements about what things aren’t, we can’t form any conclusion.

References

Alexander of Aphrodisias. 1991. On Aristotle’s Prior Analytics 1.1-7, translated by Jonathan Barnes, Susanne Bobzien, Kevin S.J. Flannery, and Katerina Ierodiakonou. London: Gerald Duckworth.

Aristotle. 1984. Prior Analytics. In The Complete Works of Aristotle: The Revised Oxford Translation, edited by Jonathan Barnes. Princeton: Princeton University Press.

Buridan, John. 2015. Treatise on Consequences, translated by Stephen Read. New York: Fordham University Press.

Burley, Walter. 2000. On the Purity of the Art of Logic, translated by P.V. Spade. New Haven, CT: Yale University Press.

Neil, Samuel. 1853. The Art of Reasoning. London: Walton and Maberly.

William of Sherwood. 1966. Introduction to Logic, translated by Norman Kretzmann. Minneapolis: University of Minnesota.