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!

Literature Survey: The Empty Domain

The state of play in the empty domain, pun-intended, seems to be this: nobody has really gotten a generally satisfactory proof theory and semantics for (non-free) inclusive logic—logic inclusive of the empty domain. I think that getting a satisfactory semantics for it stands to benefit discussions of metaphysical nihilism—the view that, possibly, there are no concrete entities—a priori justification for contingent claims, and of proof-theoretic and model-theoretic semantics for logic.

So in a kind of literature survey, I want to lay out the currently-available approaches to inclusive logic. The terminology ‘inclusive logic’ comes from Quine’s 1954. My treatment here largely follows the quasi-literature survey in (Williamson, 1999). Since it has been twenty years, I thought that the state of play was worth reviewing again in an accessible, public-domain format.

  • Mostowski’s 1951 Trick: Interpret the value of every formula containing free variables as the empty set, and the value of all formulas without free variables as either true or false. In this case, all universally-closed formulas are true, and all existentially-closed formulas are false.
    • Motivation: Universally-closed formulas should come out vacuously true, whereas existentially-closed formulas should come out false.
      • We might also think that formulas containing free variables should come out true, since the meaning of these is seemingly odd if the empty domain is allowed. In suhc cases, we have a kind of vacuous quantification—quantification where the bound variable does not occur in the formula occurring within the scope of the quantifier, as in (∀x)(Fb). As vacuous quantifiers are redundant, they should not weigh on whether the formula is true or not. As such, these formulas should be counted as false in the empty domain, just as Fb is.
    • Problem: The rule of modus ponens is not truth-preserving in the empty domain (as Mostowski himself pointed out). The formulas (φ∨¬φ) and (φ∨¬φ)→(∃x)(φ∨¬φ) come out true, but (∃x)(φ∨¬φ) does not. We only get modus ponens in a restricted form (it is truth-preserving whenever φ and ψ share all the free variables—then we do have that φ and φ→ψ imply ψ).
  • Quine’s 1954 Trick: Mark every formula with an intial universal quantifier (∀x)(φ) as true, and mark every formula with an initial existential quantifier (∃x)(φ) as false. Apply truth-functional considerations elsewhere.
    • Motivation: for this is that universally-closed formulas should come out vacuously true, whereas existentially-closed formulas should come out false.
      • Also, we want to preserve extensionality: vacuous quantifiers (∀x)(Fb) are justified by the equivalence (∀x)(Fb)↔(∀x)(Fb∧(Fx→Fx)). And this sytem preserves extensionality (in that it preserves the vacuity of vacuous quantifiers).
    • Problem: Universally-closed contradictions (∀x)(φ∧¬φ) come out true and existentially-closed tautologies (∃x)(φ∨¬φ) come out false. But it seems that a closed tautology should be true, and a closed contradiction should be false.
  • Schneider’s 1958 Trick: First, we define satisfaction under limited interpretations—interpretations of free predicate and individual variables in a given formula φ only.  Second we consider unlimited interpretations—interpretations of all predicate and individual variables in the language, not just those occurring in a given formula.
    • Explanation of Limited Interpretations: In limited interpretations over the empty domain, any open formula comes out true because there are no assignments of their free variables to anything—there are no functions from variables to elements of the domain. As such, all open formulas are vacuously satisfied. Limited interpretations of closed formulas, in contrast, still operate in the truth-functional way, as do ones of universally-closed formulas (which are true) and of existentially-closed formulas (which are false).
    • Motivation for Limited Interpretations: In limited-interpretation satisfaction in the empty domain, we have satisfaction in the empty domain roughly as on Quine’s trick. Open formulas are all valid, and we can treat quantifiers truth-functionally as on Quine’s trick.
    • Problem for Limited Interpretations: The odd formulas that should not be true, like universally-closed contradictions (∀x)(φ∧¬φ), come out true, while the odd formulas that should not be false, like existentially-closed tautologies (∃x)(φ∨¬φ), come out false.
    • Explanation of Unlimited Interpretations: In unlimited intepretations, we interpret all predicate and individual variables. But in the empty domain, there is no assignment for such formulas. As such, any formula is vacuously true in the empty domain under an unlimited interpretation.
    • Motivation for Unlimited Interpretations: Since logical truth is usually characterized as true in every non-empty domain, including the empty domain does not change what counts as logically true: for every formula that is true in all non-empty domains is also true in the empty domain under unlimited interpretations. So we get as universally valid, in all domains including empty ones, all formulas that are valid in non-empty domains. In this approach to empty-domain logic, we keep as logical truths formulas like (∃x)(φ∨¬φ) and (∀x)(Fx)→(∃x)(Fx), while rejecting as false formulas like (∀x)(φ∧¬φ).
    • Problem for Unlimited Interpretations: Under unlimited interpretations, formulas like φ∧¬φ and (∃x)(Fx) come out true in the empty domain (though not as true in all domains). But the first example seems like it should come out false in every domain, and the second seemingly should come out false in the empty domain (on the intended meaning of “(∃x)(Fx)”). So we preserve what we might want from non-empty domain logic, but at the cost of undermining practically any intended interpretation of truth in the empty domian (besides the deeply implausible one on which everything is true because everything is vacuous).

These are the approaches that do not rely on inner-domain and outer-domain semantics that are typical in free logic (and in semantics for quantified modal logic). These are also the alternatives that preserve compositionality—the truth-conditions of complex formulas is given recursively through the truth-conditions of their constituent formulas (so that we do not get, for instance, that φ∨¬φ is true while φ and ¬φ are both false).

I believe that covers the available approaches. I am, of course, fallible. So if you know of others that should be added to the list, let me know!

When was the first commutative diagram?

Thanks to @LogicalAnalysis, I saw an interesting post from Tom Leinster at The n-Category Café. The post considers whether Russell’s 1919 Introduction to Mathematical Philosophy contains the first printed occurrence of a commutative diagram. I have two remarks on this suggestion: (1) pretty much the same diagram occurs in Principia some eight years earlier; (2) while it is controversial whether this diagram really should count as a commutative diagram in the modern sense of the phrase “commutative diagram”, I will argue that it is one.

The diagram in question occurs on page 54 in the public domain text hosted on Archive.org. Here is a screenschot:

Russell IMP Commutative Diagram.png

Now for remark (1): this diagram is not the first such to occur in print. Whitehead and Russell’s Principia Mathematica contains an earlier occurrence. It is given on page 321 in the PDF (page 295 in the text) in the public domain text of Volume II hosted on Archive.org. Here is a screenshot:

Russell PM Commutative Diagram.png

The upshot of this is that it is not Russell, but Whitehead and Russell, who printed this commutative diagram. I do not know of an earlier source for it. No such diagram occurs in Volume 1 of Principia. Readers: do you have an earlier text in which such a diagram occurs?

There is a kind of anticipation of this diagram in Whitehead’s 1911 An Introduction to Mathematics. It is given on page 94 in this public domain text hosted on Archive.org. Here is a screenshot:

Whitehead IM Commutative Diagram.png

This diagram is purposefully coordinatized, as the context and reference to Descartes’ Discourse makes clear. But as the context, especially the subsequent discussion on page 96, also makes clear, Whitehead is abstracting from this diagram the commutativity and associativity of addition. Given that this diagram is also published in 1911, like the diagram above, we can just focus on the Principia diagram, which Whitehead of course also had a hand in.

(Similar uses of coordinatized diagrams to represent mathematical properties occur in Whitehead’s 1898 A Treatise on Universal Algebra, but nothing so distinctly used to represent algebraic properties like commutativity and associativity: most of these earlier diagrams occur as force diagrams or as representing geometric properties. I am open to correction on that score from someone who is more knowledgeable about Whitehead.)

Now for remark (2): the phrase “commutative diagram” is often used specifically to maps between categories. Categories are a specific kind of mathematical object, and if a map must be between such objects to count as a commutative diagram, then Whitehead and Russell most definitely did not print the first commutative diagram. However, it certainly seems to be a commutative diagram if we merely want the following feature, namely, that is commutes in the usual mathematical sense of “commutes”:

Generally, a diagram is said to commute if whenever there are two paths from an object X to an object Y, the map from X to Y obtained by composing along one path is equal to the map obtained by composing along the other. (Leinster, 2014, Basic Category Theory, page 11, link)

This is exemplified by the notation in the above screenshot from Principia,


which is clearly the analogue of gfjih in the screenshot from Leinster’s text (ibid.):

Leinster BCT Commutative Diagram.png

So I am comfortable claiming that Whitehead and Russell did indeed print the first commutative diagram in the usual sense in their 1911 Principia Mathematica: Volume II. I am always open to correction on this from someone who knows of an earlier occurrence.