“Why is it, according to Russell, not the case that every proposition is of the subject-predicate form?”

This very nice question was recently asked on the Reddit forum r/askphilosophy. Here is the (reposted) answer.


Russell’s reasoning is given in the passages immediately preceding that quote. Russell wants to argue that, in the case of symmetrical relations, we can understand them as one-place predicates if we wish. This is because we can, using one-place relations, preserve the usual inferences involving symmetrical relations. For example:

  1. A and B both have property G. [premise]

  2. B and C both have property G. [premise]

  3. Thus, A and C both have property G. [from 1-2]

On a theory involving only one-place relations, this inference can be preserved. This is because A and B, and C, too, all have the same property. So there is no problem of relating A’s color to B’s color, and then to C’s color. They all have the same color. So one could at least see how the one-place relations story could go for symmetric relations.

The case of asymmetric relations is entirely different. Russell’s view is that, if we try to eliminate all relations in favor of one-place relations, then we cannot preserve ordinary inferences involving asymmetric relations. Take Russell’s example:

  1. A is greater than B. [premise]

  2. B is greater than C. [premise]

  3. So, A is greater than C. [from 1-2]

Russell says about this sort of example:

Take for example “A is greater than B”. It is obvious that “A is greater than B” does not consist in A and B having a common predicate, for if it did it would require that B should also be greater than A. It is also obvious that it does not consist merely in their having different predicates, because if A has a different predicate from B, B has a different predicate from A, so that in either case, whether of sameness or difference of predicate, you get a symmetrical relation.

Russell is outlining two strategies for accounting for this clearly valid inference, assuming we only have one-place relations. We can say that A being greater than B consists in them having (a) the same property, or (b) different one-place properties.

Strategy (a) fails. A having the property being greater than B does not imply that B has this same property (indeed, B cannot have this property). So we have to introduce distinct properties for A and B. That is, we must pursue strategy (b). But then we have the problem of relating these one-place relations together.

This problem cannot be overcome. There is no way to get from the first two claims to the third (absent, perhaps, some modal logical principles about tallness). Just consider how that argument would look if we only have one-place relations:

  1. A has the property being greater than B.

  2. B has the property being greater than C.

  3. So, A has the property being greater than C.

This argument gets the form:

  1. F(A)

  2. G(B)

  3. H(A)?

And it won’t help to add one-place relations either, so that A being taller than B consists in their having different predicates. Just consider:

  1. A has the property being greater than B and B has the property being smaller than A.

  2. B has the property being greater than C and C has the property being smaller than B.

  3. So, A has the property being greater than C.

But (3) doesn’t follow still. The argument has the form:

  1. F(A) and P(B)

  2. G(B) and Q(C)

  3. H(A)?

So, in short, if we eliminate all relations in favor of one-place relations, then there is no practicable way to make inferences involving the asymmetric or transitive properties of relations.

C. I. Lewis and the Orthodox and Heterodox Views of Logic

Clarence Irving Lewis (1883-1964) was an American logician who, in 1918, published an influential book about logic, A Survey of Symbolic Logic. I was reading it as part of the Journal of the History of Philosophy‘s Master Class on Frege with Juliet Floyd. I was struck by Lewis how distinguishes two views of logic, the “orthodox” and “heterodox” (pg 354):

The differences between the “orthodox” and this “heterodox” view have to do principally with two questions: (1) What is the nature of the fundamental operations in mathematics; are they essentially of the nature of logical inference and the like, or are they fundamentally arbitrary and extra-logical? (2) Is logistic ideally to be stated so that all its assertions are metaphysically true, or is its principal business the exhibition of logical types of order without reference to any interpretation or application? The two questions are related.

A fascinating thing about this passage is how Lewis sees the issue of whether theses of symbolic logic are synthetic a priori truths under an intended interpretation (as they would need to be to count as “metaphysically true”) as bound up with the issue of whether logicism is true. Putting aside my casting of the distinction Lewis is drawing, Lewis is claiming that there is an inverse relationship between the substantial truth of symbolic logic’s principles and mathematics lying outside logic: logic lacks substantially true principles if and only if mathematics lies outside logic.

That is striking to me because it is consonant with my view that logic being synthetic a priori and a genuine science is crucial for Russell. In 1914, Russell wants to make logic the essence of philosophy. He can hardly do that if logic lacks informative and non-tautologous principles such as would count as “metaphysically true” in Lewis’ sense (that is, as synthetic a priori in my casting of Lewis’ point).

There is much more to say, especially about what Lewis proceeds to say about taking logic as a symbolic apparatus amenable to various interpretations, and how doing so purges logic and mathematics of all meaning and reference. (Frege would be furious.) Lewis takes the upshot of this to be that one removes what is going on in the mind of a mathematician from mathematics. That is a point for another day. What I want to say is: this is all just to show that it is always philosophically fruitful to return to the classics.

A Weird Passage in Stebbing’s MIL

I found a very strange passage in L. S. Stebbing’s A Modern Introduction to Logic. The passage is on page 155 of the third edition (I adjusted the logical notation for universal quantification because subscripts are a pain in WordPress):

Thus [Russell] uses (ιx)(Φx) is two ways. The two uses are:

  1. E!(ιx)(Φx)
  2. f[(ιx)(Φx)]

[Russell] evidently supposes…(i) that an incomplete symbol has ‘no meaning in isolation’ but only ‘a definition in use’; (ii) that in the analysed statement the incomplete symbol disappears. But the definition Mr. Russell gives of one is

E!(ιx)(Φx) .=: (∃c)(∀x):Φx.≡. x=c. Df.

Here both E! and (ιx)(Φx) have disappeared in the analysis, so that, according to Russell’s account, both E! and (ιx)(Φx), or the set of symbols E!(ιx)(Φx) are incomplete symbols having only a definition in use. Together they mean what is given in the right-hand side expression. But his definition of (2) is

fx)(Φx) .=: (∃c)(∀x):Φx.≡. x=c : fc. Df.

But the first part of the right-hand side expression is the analysis of E!(ιx)(Φx); hence, it is not true in this case, to say that (ιx)(Φx) has no meaning in isolation. Nor has “f” disappeared. Consequently, Russell does not seem to have been clear in what sense exactly he can assert that ‘(ιx)(Φx) is always an incomplete symbol’.

Stebbing seems to be arguing that (ιx)(Φx) does not always lack meaning in isolation because, if (1) is defined as above, then (ιx)(Φx) has meaning in isolation in (2). Evidence: the “(ιx)(Φx)” part in (2) has the same symbolic expression as (1) itself.

But that is a flatly mistaken inference, or so it seems to me. The symbols occurring in the definitional expansion of (2) contain as a sub-formula the definitional expansion of (1): this does not imply that any sub-expression of (1)’s or (2)’s definiendum is to be identified with any sub-expression of (2)’s definiens. So Stebbing’s move to identify the meaning or definition of “(ιx)(Φx)” in (1) or in (2) with “(∃c)(∀x):Φx.≡. x=c” seems ill-motivated

If anything, the only thing that could conceivably be justified, by the definition in (1), is the following substitution into (2):

(3) E!(ιx)(Φx) : fc.

But even that move is illicit, because now we have changed the scope of the quantifier in (2) from a primary occurrence to a secondary occurrence, so that c is now occurring free in (3), whereas it was not occurring free in (2).

So I am lost: all I see in this argument is further confirmation that Russell was right to warn against supposing that “(ιx)(Φx)” has meaning in isolation. But Stebbing is sharp, so perhaps I am missing a crucial part of the argument. Comments welcome!

Updated Projects Page!

I am happy to say that my projects page has been updated with some sweet pictures! The gallery of these photos is below. The four current projects listed there are:

  • Coding the Proofs of Principia in Coq (with a photo of the first two proofs of Principia and the corresponding implementation in Coq)
  • Digitizing and Publishing Bertrand Russell’s Pocket Diaries (with a photo of Russell’s 20-22 January 1918 pocket diary, which includes his first logical atomism lecture)
  • A Public-Domain Prototractatus (with a photo of the University of Iowa Tractatus Map)
  • The Search for Logical Forms: In Defense of Logical Atomism

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This last project was my thesis, and I am developing a number of papers that build on my new reading of logical atomism—including papers that I am presenting at the British Society for the History of Philosophy, and for the Society for the Study of the History of Analytical Philosophy this 2019!

Treat Yourself to Mid-Semester Feedback

Student evaluations, since they are anonymous and revealed only after courses end, do not always help the students that are currently enrolled. So I like to ask for anonymous mid-semester feedback. This gives students a chance to suggest minor improvements or make requests, or to report on material that they liked.

Besides the usual and helpful suggestions, and also showing students that you actually are responsive to their needs, an unexpected benefit was that this gets you compliments! My students said so many nice things about my course. So if you want to do right by your students and treat yourself to some positive reinforcement, do mid-semester feedback.

As motivation for you, here are some of the nice things that students said about my course:

  • I like the interesting and different topics discussed in class. It is a thought provoking environment.
  • I enjoy the discussions. Debating theories is fun.
  • I enjoy the discussion based format.
  • I like the in class discussion. It helps me understand the material better.
  • The teaching style [is] very open. I enjoy being comfortable and saying “I don’t know” and not feeling like it’s a bad thing.
  • I like the chill atmosphere and that you are lenient but not too lenient. Also, we have great discussions.
  • I like that the class is structured based on daily teaching objectives and class is discussion based.
  • I really enjoy how we engage in class, it helps me understand the context better.
  • I like how it forces me to think deeper and form questions that I normally wouldn’t think about.
  • I really like the fact that I have to think hard about the answer.

Besides this shamelessly self-promotional material, I also got thoughtful remarks about:

  • assessment aspects of the course, like a suggestion that there be more spacing between discussing a reading and submitting an assignment,
  • the comparative difficulty of various readings, which is always a useful opportunity to reflect on the course content and making adjustments to the syllabus, and
  • the in-class presentation and discussion of the material, like a request for more historical background to philosophers and their ideas—which, as a historian of philosophy, I am only too happy to oblige!

So you get concrete suggestions for improvement and a confidene boost to boot. In summary, the takeaway regarding mid-semester feedback is: do it!