George Campbell on the Syllogism
Mill on Enumeration
Joseph on Enumeration
Britannica 1911 article on the syllogism
Ben-Yami on Plural Referring Expressions


This is a collection of pages on the (mostly) nineteenth century dispute about the relation between the universal judgment (every A is a B), and the so-called hypothetical judgment (Any X that is A, is B) and the enumerative judgment (all the A's are B's). The dispute was about whether they are really distinct forms of judgment, or whether some of them could be reduced to one or more of the others.

It seemed to many (in particular, to Frege) as though the universal judgment is not equivalent to an enumerative judgment, otherwise every syllogism containing a universal premiss would involve a petitio principii. The enumerative judgment is equivalent to a collection of singular judgments, but if 'All men are mortal', is equivalent to such a collection of singular judgments about every man who has ever lived, it also includes the judgment that Socrates is mortal. But then the inference 'All men are mortal, therefore Socrates is mortal' is valid, and the minor premiss 'Socrates is a man' is redundant. Mill endorsed such a view. The proposition 'Socrates is mortal' is presupposed in the more general assumption that all men are mortal, and thus every syllogism contains a petitio. (This is the basis for his theory of induction. 'No reasoning from generals to particulars can, as such, prove anything, since from a general principle we cannot infer any particulars, but those which the principle itself assumes as known' [reference]). However, this seems implausible. As Frege later argued, the universal judgment 'all whales are mammals' does not contain the judgment that this object in front of me is a mammal, without the additional (singular) judgment that it is a whale, as to which the universal judgment says nothing. If I utter a sentence with the grammatical subject 'all men', I do not wish to say something about 'some Central African chief' wholly unknown to me.

Moreover, the universal judgment (as the Oxford logician H.W.B. Joseph argues) expresses a necessary or essential connection between subject and predicate, whereas the enumerative judgment, being merely a list of singular judgments predicating B of each A on the list, does not express any essential connection between these A's, and being B.

On the other hand, it seemed as though a universal judgment cannot be equivalent to a hypothetical judgment of the form 'If any X is A, X is B', for the hypothetical judgment does not imply the existence of any subject A, whereas the universal judgment, at least in the traditional theory of the proposition, does imply or presuppose or assert the existence of a subject.

The dispute was finally settled by (or at least with) the arrival of modern logic. In modern predicate calculus, the universal judgment is really hypothetical: 'All A's are B's' must be interpreted as 'for every x, if x is A, x is B', and thus is not existential. Enumerative judgments were forgotten about, and modern logic leaves us with singular judgments, and universals expressed as hypotheticals.

The page is under development, but begins with a passage from H.W.B. Joseph's An Introduction to Logic. Later I shall include Mill's discussion of the petitio problem, and discussions by Bosanquet and Bradley on the nature of the universal judgment. I also include a link to an excellent paper by Hanoch Ben-Yami whom I met at the Square of Opposition conference at Montreux in June 2007. Hanach argues persuasively that we shouldn't forget about enumerative judgments (which in common with many modern revisionist logicians, he characterises in terms of 'plural referring expressions). He comments (quite correctly, in my view) on the astonishing weakness of Frege’s arguments for the universal judgment being hypothetical, and that Frege failed to justify his claim that common nouns in the subject position in quantified subject-predicate sentences are not logical subject-terms, designating a plurality of particulars.

Edward Buckner
London, June 2007

THE LOGIC MUSEUM Copyright (c) E.D.Buckner 2007.