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.

In the Rules for the Direction of the Mind there is, I believe, a more substantive criticism of the syllogism as a decision method. A decision method is a formal test of validity. It is a reasonable requirement of such a method that applying it should not require one to know ahead of time whether the argument being tested is in fact valid. I see Descartes’s criticisms in the Rules as an attempt to show that the syllogistic fails to meet this requirement.

There are two steps in applying a formal decision method:

  1. to present the argument under discussion in a way that makes its logical form perspicuous, and
  2. to work out whether this form falls into the class of valid forms specified by the method.

One common criticism of the syllogistic is that it fails to provide an exhaustive class of valid forms. But this is arguably another ignoratio elenchi; the significant question is whether the syllogistic provides a legitimate way of determining whether an argument is valid-under-syllogism, in the same way that first order logic provides a decision method for determining whether an argument is valid-under-FOL. I propose that Descartes wanted to show that the traditional syllogistic does not even yield a way of deciding whether an argument is a valid syllogism.

The problems, in other words, are all with step 1 above. Here is what Descartes says in the Rules:

Some will perhaps be surprised that in this context, where we are searching for ways of making ourselves more skilful at deducing some truths on the basis of others, we make no mention of any of the precepts with which dialecticians suppose they govern human reason. They prescribe certain forms of reasoning in which the conclusions follow with such irresistible necessity that if our reason relies on them, even though it takes, as it were, a rest from considering a particular inference clearly and attentively, it can nevertheless draw a conclusion which is certain simply in virtue of the form. But, as we have noticed, truth often slips through these fetters, while those who employ them are left entrapped in them. (Rule 10)

I think most readers of Descartes have interpreted this as a distinct criticism from the one he makes in the Discourse, and which he appears to repeat further on in Rule 10: “dialecticians are unable to formulate a syllogism with a true conclusion unless they are already in possession of the substance of the conclusion, i.e. unless they have previous knowledge of the very truth deduced in the syllogism.” I propose that they are the same criticism. If we take the second criticism to be, again, a complaint that the syllogism is non-ampliative, we are overlooking the fact that Descartes has earlier been quite clear that the stated purpose of the syllogistic is not to make discoveries but rather to test for the validity of arguments purely on the basis of their form.

The truth deduced in the syllogism is not the consequence of the syllogism. As Łukasiewicz pointed out, the traditional Aristotelian syllogism is in the form of a true hypothetical proposition, not an inference-pattern. A first figure syllogism, for instance, can be: if MaP and SaM then SaP. It cannot be: MaP, SaM ⊢ SaP. The syllogistic is a method for determining the truth of hypothetical statements, not the truth of the consequents of such hypothetical statements. And so when Descartes says that the substance of the conclusion of the syllogism must be known ahead of time before it is revealed by the syllogistic, I take him to mean that the truth of the hypothetical proposition that is the syllogism – the validity of the argument – must be known ahead of time. The syllogistic fails as a test of validity, not as a method for discovering truths.

But why would the validity of an argument need to be known ahead of time for the syllogistic to be successfully applied? Here Descartes’s comments about truth slipping through the fetters while dialecticians remain trapped in them must be borne in mind. Descartes could have spared us a lot of misunderstanding by giving examples, but examples are given in that great pioneering work of Cartesian logic, the Port-Royal Logic (Part 3, Chapter 9). Here is one:

The Divine Law commands that we honour kings.

Louis XIV is a king.

Therefore the divine law commands that we honour Louis XIV.

This argument is clearly valid. But does the syllogistic help us to see that it is? Remember that the first step in applying the decision method is to make the logical form perspicuous. The PRL imagines a poor logic student identifying the terms of the syllogism as follows:

S = The Divine Law commands that we honour

M = king(s)

P = Louis XIV

If we identify the terms in that way, then the syllogistic rules the argument invalid. The form appears to be: SiM / PiM | SiP, which is not a valid second-figure form, since it contains only affirmative propositions.

Now because we know the argument is valid, we know that something must have gone wrong in identifying the terms. But to identify them correctly, we need to rewrite the original argument in natural language into a logically equivalent form:

Kings must be honoured (according to the Divine Law).

Louis XIV is a king.

Therefore Louis XIV must be honoured (according to the Divine Law).

Now we can assign the terms as follows:

M = king(s)

P = must be honoured

S = Louis XIV

Now the form of the syllogism is MaP / SiM | SiP. That is a valid first figure syllogism.

This is an example of “truth slipping through the fetters” of the syllogistic. What was required to make the argument fit into the form of a valid syllogism was a rearrangement in natural language, so that the terms could be properly assigned. But it wasn’t the syllogistic that told us such a rearrangement was necessary, nor that the proposed rearrangement was logically equivalent to the original argument. If we hadn’t known those things, we might have accepted the verdict of the syllogistic in the first instance and ruled the argument invalid. And that would be an example of dialecticians being trapped in the fetters of the syllogistic.

Now in this case we could reply that the mistake in formalising the syllogism above was that we got the quantity assignment in the wrong place. If “the Divine Law commands that we honour kings” means “the Divine Law commands that we honour all kings”, and if we assign “kings” as the middle term, then we must put the syllogism into a form where M is in the subject place of at least one proposition (so it can’t be a second-figure syllogism). But the PRL gives other examples that are even more troubling, for instance the invalid argument: We should follow Scripture; tradition is not Scripture; thus we should not follow tradition. (It is, of course, no coincidence that the PRL chooses examples of reasoning that have enormous and contentious political and religious implications – what they are concerned to build is a logic that can help us to think when it matters the most.) In this case, if we wrongly assign “we” to S, “should believe Scripture” / “Scripture” to M, and “tradition” to P, we can appear to get a valid second-figure form: SaM / PeM | SeP.

Obviously the error here is to treat “Scripture” and “should believe Scripture” as synonymous and thus to identify both as a single middle term. But the syllogistic itself provides no rules of synonymy. We need knowledge from outside the syllogistic to know how to apply the syllogistic.

One might find it hard to imagine that we could know how to properly assign terms, so as to apply syllogistic analysis to an argument, without already knowing whether that argument is valid. That, I believe, is what Descartes thought, and that is what he meant when he claimed that dialecticians are unable to formulate a syllogism with a true conclusion without already possessing the substance of the conclusion.


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