Parsons, Geach, Distribution

tumblr_mawkconknz1qfoty5o1_500Terrence Parsons has a very interesting article defending the doctrine of distribution in traditional logic against the famous attacks made by Peter Geach in Reference and Generality (and elsewhere).

I like Parsons’ defence, especially since it draws upon the Port Royal Logic (which I’ve been studying for a while). But I’m wondering if Parsons misses one of Geach’s most formidable attacks.

I’m fairly convinced, I should say by the way, by Parsons’ reply to Geach’s historical claim that the doctrine of distribution has its origin in the medieval notion of supposition. Parsons proposes that it more likely arose out of studies of Aristotle’s On Interpretation – particularly the discussion of particular and universal terms and what it means to ‘take a term universally’. That all sounds right to me, but then this isn’t my period. Corrections very welcome.

Geach’s first criticism of the doctrine of distribution comes in the form of a rhetorical question:

When I say ‘Some men are P’, does the subject term refer to just such men as the predicate is true of? But then which men will the subject-term refer to if a predication of this sort is false?

Parsons deals with this by finding an answer to the rhetorical question in the Port Royal Logic and then developing it into a more modern idiom. Here it is in the modern idiom:

If the proposition is false, on either account the term will refer to no men. This is because there are not any men that the predicate is true of. So, when the extension of the subject is limited by that of the predicate, nothing is left. (p.72)

Parsons notes that Geach gives reasons for rejecting the idea that the reference of a term can depend upon whether the proposition in which it occurs is true or false. Parsons has an interesting reply to this that seems right to me, though I admit I haven’t thought about it much.

But Parsons doesn’t seem to deal with a problem Geach raises with the view that when ‘Some men are P’ is true, ‘some men’ refers to just those men of which P is true. This is fine as a doctrine of reference on its own, but it causes problems when we come to employ propositions in syllogisms.

Geach gives the clearest explanation of the problem in his inaugural lecture at Leeds, entitled ‘History of the Corruptions of Logic’. (Here he informs us, by the way, that ‘between [traditional] logic and genuine logic there can only be war; what fellowship has light with darkness?’)

Here is the problem. If we say that a term like ‘some men’ in a proposition of the form ‘Some men are P’ refers just to those men that are P (so long as the proposition is true), then we cannot hold that ‘some men’ refers to the same men (or class of men) throughout the whole course of a syllogistic argument. But if we cannot hold that the reference of the term stays fixed throughout the course of a syllogistic argument, then the following argument is not valid:

  1. Some men are philosophers.
  2. All philosophers can control their temper.
  3. Ergo: Some men can control their temper.

Geach writes:

For without this rule [viz., the rule that the reference of ‘some men’ stays fixed throughout the argument] ‘some men’ in the conclusion would possibly refer to a different class of men from ‘some men’ in the premise; and then the syllogism would be invalid, just as a syllogism containing a proper name would be invalid if the name meant a different person in the premise and the conclusion. (Logic Matters, p.57)

Geach next shows that if we embrace the rule then we end up validating invalid syllogisms. He concludes that the problem is not simply to find the right rule for the reference of ‘some men’; the problem lies with treating such a term as a referring term at all.

At any rate, the Port Royal Logic version of the doctrine of distribution could not allow us to embrace the rule that the reference of ‘some men’ is fixed throughout the course of an argument. It holds that: ‘the extension-in-a-proposition of the subject of an affirmative proposition consists of everything in its original extension which is also in the original extension of the predicate.’ (Parsons, p.70) Thus ‘some men’ must hold of whatever men are philosophers in Premise 1 of Geach’s syllogism and of whatever men can control their temper in Premise 2. There is no guarantee that these will be the same.

This certainly does make the syllogism invalid, as we can see by considering the following case. Suppose a universe of three men: A, B, and C. Suppose that A and B are philosophers and that all three men can control their tempers. Premise 1 and Premise 2 exclusively discuss A and B. In the Conclusion, the extension of ‘some men’ is A, B, and C. No amount of knowledge simply of A and B should validate any conclusion about A, B, and C. Thus Geach’s syllogism must indeed be invalid.

Is this a major worry? It might appear that all that is required to make the syllogism valid is to rewrite the Conclusion: ‘Some men, who are philosophers, can control their temper’. Since this is the only thing we’re entitled to conclude from the syllogism anyway, we can’t rankle too much about having to restrict the concluding proposition in this way.

What we have then done is to add what the Port Royal Logic calls a ‘determinative’ to the concluding proposition. There hasn’t been much discussion of the role of determinatives in traditional logic. I think they’re very important, but the PRL doesn’t have all that much to say about them.

The only discussion is in Chapter 7 of Part 2, where we are told that pronominal phrases of the form ‘who is F’, or ‘that is F’, come in two sorts: ‘explicative’ and ‘determinative’. Explicatives don’t change the reference of the subject to which they are applied; they just give extra information about the referent (e.g., ‘Alexander, who is the Son of Philip’). Determinatives (e.g., ‘men who are pious’, ‘kings who love their people’) do change the reference of the subject to which they are applied. But we aren’t told precisely how.

What we are told is that a determinative, when applied to a subject in a proposition, creates a ‘tacit or virtual’ proposition within the larger explicit proposition. We might then suppose that a participle applied to a particular subject in a ‘virtual proposition’ restricts the extension of that subject within that of the participle, just as a predicate does in a proposition. But what is the virtual proposition implied by a participial phrase?

Clearly the virtual proposition cannot affirm the participle of the subject. When I say that ‘Politicians who are honest would be nice to have’ I don’t mean to affirm honesty of any actual politicians. The PRL settles for saying that the virtual proposition affirms the possibility of the participle’s holding of the subject. Thus ‘Politicians who are honest would be nice to have’ contains as a virtual proposition something like: ‘Politicians could be honest’. Meanwhile ‘Squares that are round are excitable’ contains the virtual proposition ‘Squares can be round’. This virtual proposition is false, says the PRL (I suppose the authors are thinking about de dicto modality), and so the whole proposition is false.

We then end up with the view that in the modified conclusion of Geach’s syllogism, ‘Some men, who are philosophers, can control their temper’, we have the virtual proposition: ‘Some men can be philosophers’. Now we can take the extension of ‘some men’ to be restricted within the predicate ‘can be philosophers’.

This restriction, however, might be looser than that imposed on ‘some men’ by the predicate ‘are philosophers’ in Premise 1. In our model universe, it might be that A, B, and C all can be philosophers, even while only A and B are. Thus the syllogism still appears to be invalid. Something more than the PRL theory of participles is needed to rescue the traditional doctrine of distribution from this criticism by Geach, if it can be rescued.

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3 thoughts on “Parsons, Geach, Distribution

  1. Nathanael

    There’s a reason this stuff was generally abandoned in favor of formal logic, which makes all this crystal clear. Rather than “Some men are P”, we write “There exists an X, member of the set Men, for which proposition P(X) is true”.

    From a formal logic point of view, this entire discussion is one of language: of the semantics of language. Rather than arguing about what a particular sentence ought to mean (since arguably it means whatever the speaker thinks it means, or whatever the listener thinks it means, and it is certainly ambiguous) the mathematical logician asks that the statement be rephrased in a clear, unambiguous manner using well-defined semantics. But you know this… I don’t know if you know that there are a lot of specialized mathematical logics designed to formalize specific types of concepts.

    Ordinary language is quite deliberately ambiguous. This ambiguity is considered valuable by, for example, politicians — it is extremely valuable when it comes to rhetoric, a subject everyone should study. However, the project of philosophy has historically been one which is often *opposed* to the project of rhetoric: a project of clarity, rather than of ambiguity. Mathematics is the extreme example of a demand for clarity.

    I think Parsons hasn’t worked out why logic was formalized, and why natural language can’t be formalized — it’s because natural language ambiguity is a *feature* for rhetoric even though it’s a *bug* for philosophy and for science.

    Parsons is trying to advocate for “natural language philosophy”, but his program is fundamentally misguided, and only of value for the historical elements (studying the history of philosophy).

    If one is going to spend time on the semantics of natural language, one should move over to the linguistics or psychology department (where there is no “right meaning” of a statement, only observational study — with interviews and surveys — of the interpretations which actual people take from a statement). If one is trying to study logic, do it mathematically (where the formalization makes it clear what meaning is meant).

    There’s a fundamental job-security problem happening (over the course of the last several hundred years) in philosophy departments.

    This is demonstrated by the huge number of departments which issue a “Ph.D.” — every time a subsection of philosophy starts really achieving intellectual accomplishment, it gets hived off into its own department. Every subject which issues a Ph.D. was originally part of philosophy. “Science” was “natural philosophy” when it first split off. Even subject which philosophers still claim to work on have mostly split off, with nearly all the good work being done in other departments: aesthetics and “philosophy of mind” and part of epistemology went straight into the psychology department; logic and most of epistemology into the math department; social and political philosophy went into sociology, anthropology, poli-sci, women’s studies, economics, and lots of other departments; textual analysis and other bits of philosophy went into literature departments; other parts of philosophy went into linguistics departments. Et cetera.)

    The philosophy department is often left with the dregs: the subjects where nobody is making any progress (like ethics, where no progress seems to have been made in 5000 years, and although the biologists and psychologists may be getting somewhere nobody wants to accept that, and metaphysics, which appears to be a subject with a lot of words but void of content), and the cranks who refuse to learn from the progress made in their subjects since the other department split off.

    Reply
    1. axdouglas Post author

      Geach himself makes the point that quantifier-and-variable notation solves these problems. Parsons isn’t disputing that here; his interest in historical, as is mine.

      Geach would construe ‘Some men are P’ as ‘∃x:Man(x)[P(x)]’, restricting quantification to the *domain* Men (*not* the set), with the variable bound by the quantifier. Others might use unrestricted quantification and just have ∃x: Man(x) & P(x). Some won’t have the variable bound at all, and deal with open sentences that are true or false in various models. Some, like Quine, aspire to get rid of variables altogether:

      http://cas.uchicago.edu/workshops/wittgenstein/files/2008/05/quine-variables-explained-away.pdf

      I doubt, however, that many logicians would like your idea of involving set theory in the construal of statements involving quantifiers. First order logic is demonstrably sound and complete; to force set theory into it would ruin that.

      Reply
  2. Billikin

    I thought that you might enjoy this little tidbit. One of my favorite songs is “Across the Great Divide” ( https://www.youtube.com/watch?v=8v2xjsozr6Y ), the Great Divide being the continental divide of the US and Canada. Part of the chorus goes, “I find myself on the mountainside where the rivers change direction across the great divide.” Now it is true that the rivers change direction across the great divide, but no river changes direction across the great divide. 😉

    Reply

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