Category Archives: History of Logic

(Part 2) Collingwood’s Revolution in Logic

In my last post I looked at Collingwood’s attempted revolution in logic and its anti-realist implications. Sam Lebens raised the interesting question of how the proposed revolution in the Autobiography connects with the more radical line pushed in the Essay on Metaphysics.

The theory in the Autobiography is, if I interpreted it rightly, relatively simple. Truth-bearers become question-answer complexes rather than propositions. Their truth-values depend on the answer’s being the right answer to the question, defined in terms of justifiability. The Essay on Metaphysics presents a much more complex structure, involving propositions, questions to which propositions are answers, and presuppositions giving rise to questions. “Logicians”, Collingwood opines, “have paid a great deal of attention to some kinds of connexion between thoughts, but to other kinds not so much.” (23)

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Collingwood’s Revolution in Logic

“In logic”, writes R.G. Collingwood in his autobiography, “I am a revolutionary; and like other revolutionaries I can thank God for the reactionaries. They clarify the issue.” (52)

The revolution in logic he hoped to bring about does not seem to have come about. The matter is not helped by Collingwood’s vague and confusing way of presenting his alternative logic.

The reactionaries in his story are those who subscribe to what he calls “propositional logic”. His use of this name is apt to be somewhat confusing, since he does not mean, as most people mean by that term, the study of true and false propositions and sentential connectives. Rather, “propositional logic”, for Collingwood, is any logic that holds that propositions are truth-bearers:

According to propositional logic (under which denomination I include the so-called ‘traditional’ logic, the ‘idealistic’ logic of the eighteenth and nineteenth centuries, and the ‘symbolic’ logic of the nineteenth and twentieth), truth or falsehood, which are what logic is chiefly concerned with, belongs to propositions as such. This doctrine was often expressed by calling the proposition the ‘unit of thought’, meaning that if you divide it up into parts such as subject, copula, predicate, any of these parts taken singly is not a complete thought, that is, not capable of being true or false. (34)

Collingwood’s objection to this doctrine is not that the components of a proposition are units of thought (or what Frege simply called “Thoughts” – those items for which the question of truth arises). Rather propositions themselves have the status that the “reactionary” logicians would ascribe to subjects, predicates, etc.: they can be parts of a larger whole to which truth and falsity can be ascribed, but truth and falsity cannot be ascribed to them directly. Truth-bearers, for Collingwood, are complexes of which propositions form only a part:

It seemed to me that truth, if that meant the kind of thing which I was accustomed to pursue in my ordinary work as a philosopher or historian – truth in the sense in which a philosophical theory or an historical narrative is called true, which seemed to me the proper sense of the word – was something that belonged not to any single proposition, nor even, as the coherence-theorists maintained, to a complex of propositions taken together; but to a complex consisting of questions and answers. (37)

Collingwood implies that propositions in his logic have the same status as subject- or predicate-terms in standard logic. They can thus be presented as arguments in truth-functions, the questions being the truth-functions. For instance, the question “Who ate the eggs?” can be treated as a function that yields the value true if the proposition “Ms. Jones ate the eggs” is taken as an argument. In this way any proposition will be part of both true and false complexes. Who-ate-the-eggs?(Ms.-Jones-ate-the-eggs) will come out true, as will What-did-Ms.Jones-eat?(Ms.-Jones-ate-the-eggs), whereas Why-is-the-sky-blue?(Ms.-Jones-ate-the-eggs) will come out false. The form “Q?(x)” is meant to represent a question-function, where “Q?” specifies the question and “x” represents the argument-place into which various propositions can be inserted to yield truth or falsity to the whole complex.

To retain various benefits regarding quantification, etc., Collingwood’s logic could be made to include Frege’s functional analysis of names and predicates. Thus “x ate the eggs” can be taken as a function that yields different values when different names are given as arguments for x. Unlike in the Fregean analysis, however, the values yielded will not be true and false; rather, they will be items of a logically intermediate status that yield truth and falsity when taken as the arguments in question-functions.

It is interesting to consider the anti-realist implications of this. Collingwood is hard on what he calls “realism” and connects with propositional logic in the Autobiography; in Essay on Metaphysics he claims that realism “has the grandest foundation a philosophy can have, namely, human stupidity” (34). Here, however, I mean “realism” in Michael Dummett’s sense: one is a realist about a class of propositions, roughly, if one believes that classical logic, especially bivalence, holds for them. It is not entirely clear how close realism in Dummett’s sense is to realism in Collingwood’s sense, but certainly there is a connection.

Is Collingwood’s logic anti-realist in Dummett’s sense?

One failure of classical logic for propositions is suggested by Collingwood when he implies that his logic is paraconsistent. But his choice of example to illustrate this, “The contents of this box are both one thing and many things”, is extremely unfortunate, since the contradiction turns out to be only apparent; context reveals that the first part of the statement counts sets of chessmen as things whereas the second counts individual chessmen as things. The resolution of the contradiction does not require the adoption of a non-classical logic; it requires one to recognise a point that Frege often made: a number assignment attaches to a concept or kind of thing (a Begriff, in Frege’s terminology), not to a thing directly. Collingwood provides no other examples to show that his logic can permit true contradictions in a way that classical logic cannot.

On the other hand, it is clear that one and the same proposition can contribute to both true and false question-answer complexes. Who-is-Caesar?(Caesar-is-Emperor-of-Rome) could be a true complex while Why-is-there-something-rather-than-nothing?(Caesar-is-Emperor-of-Rome) is obviously a false complex. Since the answer part of the complex is usually the only part that is explicitly articulated – according to Collingwood we need to pay attention to wider context to work out the question part – this can give the appearance of allowing for a single proposition to be both true and false at the same time. But it is only an appearance; no proposition is either true or false, and the fact that a proposition can yield both truth and falsity when taken as the argument for different functions entails no more a rejection of classical rules than the fact that a single name can have both true and false predications made of it.

There is a similar appearance of a failure of bivalence. For instance, the proposition “I have exactly 10,003 hairs on my head” would usually be thought to be either true or false. Collingwood’s logic of question and answer entails that the proposition independently of its being offered as the answer to any question has no truth-value at all. In this sense, Collingwood could be said to be an anti-realist concerning a large class of ordinary propositions.

We need to remember, however, that in ordinary usage, according to Collingwood, the utterance, “I have exactly 10,003 hairs on my head”, expresses not a proposition on its own but rather a question-answer complex (with the question expressed through the context of utterance). And then it might well be, for all Collingwood says, that the complex How-many-hairs-do-I-have-on-my-head?(I-have-exactly-10,003-hairs-on-my-head) is decisively either true or false; it may be true or false even if nobody actually asks the question. Once we accept that thoughts, in the Fregean sense, are now question-answer complexes rather than propositions, classical logic can continue to apply to thoughts, though not to propositions.

Anti-realist implications begin to creep in, however, where Collingwood’s account requires an explanation of how a proposition yields truth or falsity in a question-function. It seems that a question-and-answer complex is true or false depending on whether the answer to the question is right or wrong. But this leaves rightness and wrongness unexplained. They can’t be explained in terms of the truth or falsity of the answer, since truth and falsity only belong to the question-and-answer complexes.

One possible explanation of rightness might be as follows: a right answer is a justifiable answer. We need no recourse to the concept of truth to explain justifiability: an answer is justifiable if it meets with certain standards embodied in our social and linguistic practices. Thus for Collingwood a proposition can contribute truth to a question-function by being a justifiable answer to the question; it can be (though it may not in fact be) shown to be worthy of acceptance according to some standard. Meanwhile wrongness can be explained in terms of an answer’s being capable of being shown worthy of rejection. This is not said explicitly by Collingwood, so far as I know, but it seems a plausible option; at least I can’t think of a better explanation of rightness and wrongness in answers to questions.

If this is right, then Collingwood’s theory of meaning is a justificationist theory – the classic recipe for anti-realism in Dummett’s sense. Question-answer complexes will be true in virtue of the answer’s being right, meaning (roughly) it can be shown to be worthy of acceptance; they will be false in virtue of the answer’s being wrong, meaning (roughly) it can be shown to be worthy of rejection. A truth-value gap will appear if an answer cannot be decidedly shown worthy of either acceptance or rejection. Negation will function non-classically where there is a difference between being able to show that an answer is not worthy of rejection (~~Q?(a)) and being able to show that it is worthy of acceptance (Q?(a)). Obviously this all needs a lot more working out.

Descartes on the Syllogism

2000px-square_of_opposition_set_diagrams-svgDescartes made several criticisms of the syllogism. In the Discourse on Method, he remarks that “syllogisms … are of less use for learning things than for explaining to others the things one already knows”. This might lead us to think that Descartes’s main criticism is that syllogisms are non-ampliative. This is the general line pushed by Stephen Gaukroger in his Cartesian Logic. But arguably it presents Descartes as falling into ignoratio elenchi (“of all the fallacies, that which has the widest range”, as De Morgan claimed – Formal Logic, p.260).

No doubt the role of the syllogism was conceived variously by philosophers of Descartes’s time. Many regarded it as a purely didactic device. But it does not follow from the fact that it is non-ampliative that it must be constrained to that role. The power of non-ampliative knowledge can also be harnessed in a decision method. And Descartes, after all, was happy to use such knowledge for such a purpose. His own method of drawing out the consequences of innate ideas by intellectual intuition, in order to decide what is known for certain, seems a paradigm case of such an application. Nothing appears in the consequent that is not contained in the innate idea serving as antecedent. We might try to soften the non-ampliativity by saying that the consequent is only implicitly contained in the antecedent, but I don’t see why we can’t place the same qualification onto the claim that the consequent of a syllogism is contained in its antecedent.

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