Despite some pioneering work in the late 19C, I think it’s fair to say that only in the 20C was logic given a sufficiently rigorous formalization for straightforward proofs to become possible. Before that, logic was largely in the condition of (the rest of?) philosophy. Imprecision in the definitions of terms and concepts made it hard to distinguish substantial disagreements from merely terminological quarrels. Logicians could go a long way simply by having better command over their language than their rivals.
F.H. Bradley, though hardly an insubstantial thinker, drew enormous benefit from his mastery over the use of language. But his achievements were largely negative. He was very good at showing that his opponents often used terms in confused or meaningless ways. He was not always so good at explaining what the terms, properly used, actually did mean.
A case in point is the treatment of predication in his Principles of Logic (1883). Bradley opens that work by criticising the view that had been part of ‘traditional logic’ since the Port Royal Logic (1662), that judging and predicating are one and the same logical operation. To judge, as the Port Royal Logic put it (ch.2, §3), is to either affirm or deny something of a subject. This was generally glossed as meaning that to judge that S is P is the same thing as to predicate P of S.
To this Bradley replied, as Frege was doing at roughly the same time, that it is perfectly possible to predicate P of S without judging that S is P, as for instance where ‘S is P’ occurs as part of an if-then clause. When I make the judgement ‘If the shirt fits, you will buy it’ I predicate ‘fits’ of the shirt, but I do not judge that the shirt fits; I judge only that if it does then you will buy it.
Bradley correctly identified this confusion between predication and judgement. But having admirably achieved this negative task, he quickly became lost. It is not enough to say that in predicating we do not always judge; the next job should be to say what it is that we do when we predicate. Here Bradley had nothing to offer besides more negativity. He dispatched with three prominent theories of predication and then left it at that.
The first theory of which he dispatched was the view that in predicating P of S we state that one class, specified by S, belongs within another class, specified by P. Thus ‘the shirt fits’ should be analysed as meaning that the unit class consisting of the shirt falls within the class of things that fit. Bradley makes the sensible reply that when one makes such a judgement, one is rarely thinking, even implicitly, about classes. If (Bradley’s example) I judge that ‘this is my best coat’, it is very unlikely that I am thinking at all about the (unit) class of my best coats. More than this, as Bradley argues elsewhere, it leads towards nonsense to suppose that in judging that some men are taller than John I am including the class of some men in the class of things taller than John. The very idea of there being a class corresponding to every use of the term ‘some men’ is mistaken. If I say something like ‘If you leave the table outside, some men might come along and steal it’, the question, ‘Which men?’ does not always admit of a sensible answer.
The next view of predication rejected by Bradley is the view that predicating P of S is finding that the concept of P is included in that of S. Bradley rejects this view on the (I believe misguided) grounds that not every judgement occurs in subject-predicate form. But he is certainly right to reject it.
Finally, Bradley attacks the theory that predicating P of S is asserting some sort of identity between S and P. If I judge that iron is metal, I am not judging that ‘iron’ and ‘metal’ are simply two names for the same substance. Bradley examines the view put forward by Lewis Carroll (though Bradley doesn’t name him), that ‘iron is metal’ stands for ‘iron is iron-metal’. The latter could be construed as a sort of identity statement. But Bradley traps this view in a dilemma. If ‘iron-metal’ means just the same as ‘iron’, then the judgement ‘iron is iron-metal’ is an uninformative tautology disguised as an informative statement. Yet if ‘iron-metal’ does not mean the same as ‘iron’, then what right do we have to say that ‘iron is iron-metal’ is a statement of identity? Armed with a sense/reference distinction, we could have a ready answer for Bradley’s rhetorical question: ‘iron’ differs in sense from ‘iron-metal’, though both have the same reference. But Bradley is nevertheless right to reject Carroll’s theory of predication, since it involves a circular explanation; if we ask what ‘iron-metal’ means, the answer is ‘metal that is iron’, and the attempt to explain what that means brings us right back around to the original question: what does it means to say that S is P? This was pointed out by Peter Geach.
Having given us these three examples of what predication can’t be, Bradley does not go on to say what predication is. This becomes a problem when he goes on to give his own definition of judgement: ‘the reference of an ideal content to reality’. What he seems to mean by this is that in judging that iron is metal, I predicate the ‘ideal content’ expressed by ‘iron is metal’ of reality.
In other words, having successfully resolved the confusion of predication with judgement, he falls right back into the same confusion. Bradley ought to have asked himself why, if making a judgement is the same thing as predicating some ideal content of reality, we cannot embed such a predication within an if-then clause and thus make the predication without thereby making the judgement. I can’t help thinking that Bradley runs into the mistake for which he has just criticised others because of his failure to offer a theory of predication. This is the danger of having a primarily negative talent.
The same failure leads Bradley towards some of his more outlandish views – his view, for instance, that all judgements are hypothetical. The reasoning here is something like the following. If I judge that ‘this tree is green’, I am referring some ‘ideal content’ to a real tree that I perceive. I form a fairly abstract idea of a green tree and judge that it applies to this object of my perception. But thus far, Bradley argues, the judgement is false. ‘There are’, he argues, ‘more ways than one of saying the thing that is not true. It is not always necessary to go beyond the facts. It is often more than enough to fall short of them.’
Why should we agree with that? If the tree is green, then isn’t saying so telling the truth, though not the whole truth? Bradley’s reason for denying this is very difficult to follow, but it’s worth looking at how he expresses himself. As he puts it, if X=a, b, c, … , where X is something like the object of my perception and a, b, c, etc. are the various ideas that would be correctly applied to it, then it is false to say that X=a and simply leave it at that. The use of the ‘=’ sign is very telling here. Bradley seems to be operating with the implicit belief that in predicating a certain idea of an object of perception one is asserting an identity between the object and the content of the idea. Thus if I say that this object, O, is a green tree, I am asserting an identity between it and whatever is the content of my idea, ‘green tree’. But I would equally well assert that some distinct object, O’, is also a green tree. From my assertion that O and O’ are each identical with the content of ‘green tree’, it follows by the transitivity of identity that O and O’ are identical, and they are not.
The only way to avoid this sort of falsehood is to apply to the object not only some but all of the ideas that apply to it, all at once. All the ideas that apply to this object (including, say, ‘tree that is located in such-and-such a place’) will not, as a whole, apply to any other object. Thus, where X=a, b, c, …, the judgement that X=a is false except in conjunction with the judgements that X=b, that X=c, etc. (I think Bradley would take the conjunction of judgements, X=a ∧ X=b ∧ X=c ∧ …, to be logically equivalent to the judgement that X=a, b, c, …, and the latter is the only true judgement that can be made about X). And so the judgement that X=a can be true only as a hypothetical judgement X = a if X=b ∧ X=c ∧ ….
This tortured analysis only follows from the determination to see the ascription of an ideal content to an object as an assertion of identity. On a proper theory of predication this would not be the result. If I judge that this object is a green tree, I do not judge that a green tree is all that it is, and this is precisely because the ‘is’ here is one of predication rather than of identity. Bradley could not avail himself of this distinction, however, again because of his purely negative strategy. Although he rejected the view that predication is identity, he offered no alternative theory of predication. And so, when he ended up needing the notion of predication to explain how ‘ideal content’ is ‘referred’ to an object, he found that he did not have one. When you reject a lot of bad options and provide no better one, you make it inevitable that you’ll end up inadvertently with one of the bad options you’d already rejected. This is one thing we can learn from Bradley.