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,

P=S|Q|Š,

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.

Being Corrected: The Best Part of Philosophy Conferences

Reflecting on the Central APA in Denver, I have to admit that one of the best parts of philosophy conferences is being corrected. This came up as follows: I indirectly pointed out to a speaker that their interpretation of an author’s notion of diagram should also be extended to natural deduction proofs. They thanked me after the talk for pointing out “a blind spot in their paper” that they wanted to think more about.

That got me thinking about the many times that I had been corrected at conferences. I confess that constructive criticism that improves my argument, or my presentation of it, is one of the most pleasurable experiences to come out of conferences (or the peer–review process). It really makes attending conferences entirely worthwhile.

The pleasant feeling of being corrected is perhaps best encapsulated in Socrates’ words to Theaetetus at the end of Plato’s Theaetetus (link). Their search for a definition of knowledge has apparently come up empty-handed. But Socrates makes clear that there has been some definite philosophical improvement by the elimination of ignorance:

SOCRATES: …And so, Theaetetus, knowledge is neither sensation nor true opinion, nor yet definition and explanation accompanying and added to true opinion?

THEAETETUS: I suppose not.

SOCRATES: And are you still in labour and travail, my dear friend, or have you brought all that you have to say about knowledge to the birth?

THEAETETUS: I am sure, Socrates, that you have elicited from me a good deal more than ever was in me.

SOCRATES: And does not my art show that you have brought forth wind, and that the offspring of your brain are not worth bringing up?

THEAETETUS: Very true.

SOCRATES: But if, Theaetetus, you should ever conceive afresh, you will be all the better for the present investigation, and if not, you will be soberer and humbler and gentler to other men, and will be too modest to fancy that you know what you do not know.

Much the same could be said of the experience of being corrected at conferences. We are all like Socratic midwives to one another, helping each other to “birth” philosophical ideas.

This makes Socrates’ goodbye to Theaetetus all the more applicable to my colleagues in academic philosophy:

SOCRATES: To-morrow morning, Theodorus, I shall hope to see you again at this place.

And indeed, I will hope to see you, my fellow philosopher, again at the next conference.

Philosophy Paper Guide

Over break, I did some work: I wrote a one-page guide for introductory philosophy students writing term papers. Feedback and criticism is welcome!


The goal in a philosophy paper is to give a good argument for your thesis. It is a stellar idea in making an argument outline to put your argument in one of the argument forms like we use in class. This format does not need to appear in your actual paper, but it can.

Giving a good argument for a philosophical thesis usually involves related tasks, like:

  • formulating and choosing a thesis that seems plausible and defensible in a term paper
  • clarifying what the issue is and bracketing other tangential issues
  • researching data and evidence to see if your premises and thesis are plausible
  • making your premises and methods clear and explicit
  • defending the premises that are used in your argument for your thesis
  • fairly presenting and critically discussing alternative theses
  • considering and critically replying to objections and implications of your thesis

In doing the above, you may find, as I often have, that a given thesis is not plausible and should be rejected. This—being wrong—is part of philosophizing, just as it is part of any scientic inquiry. As such, your paper may end with a devastating objection to your thesis or premises: it is perfectly acceptable to end your paper by rejecting your thesis with which you began. What I want to see, ultimately, is you critically arguing well.

Here are some further suggestions as to what a good philosophy paper does:

  1. Thesis Thesis is clear, concise, and at the beginning, perhaps after a brief introduction. The thesis is not trivial, like puppies are cute is. It is also defensible in a term paper.
  2. Issue The main issue is explicitly explained, perhaps by connecting it to the course content. No filler or bullshit is used. Why we should care about the issue is also explained.
  3. Premises The premises are clearly and explicitly stated, along with some basis for accepting them. The premises are first-blush plausible or well-supported by research.
  4. Clarification The thesis and premises, and the main issue, are claried with examples and explanations of key philosophical terms. Your non-philosopher friend should understand your thesis, premises, and the issue. Do not write as if I am the audience.
  5. Argument The thesis is cogently argued for using the premises. The argument for the thesis is persuasive to anyone who accepts your premises and methods.
  6. Alternatives Alternatives to your thesis or view on the main issue are fairly presented. Present them even better than their advocates do. Then critically discuss them. You might object to consequences of them, or agree with them to the extent that you do.
  7. Objections You fairly present and critically discuss concerns with your thesis and premises.
  8. Research Your sources and data are documented in a bibliography, and are acceptable in academic, scientic contexts wherein getting the facts right is the first, foremost goal.

Book Published!

I am happy to report that I have published my first book, The Philosophy of Logical Atomism: A Centenary Reappraisal! I co-edited this volume with Gregory Landini (University of Iowa), and contributed a chapter. You can see previews, buy chapters, or buy the whole thing at this link.

The volume collects fourteen original essays on Russell’s 1918 logical atomism lectures. The book is published with Palgrave Macmillan in their History of Analytic Philosophy series edited by Michael Beaney (Humboldt-Universität zu Berlin / King’s College–London). The other books in this series are also quite good.

Palgrave collection 2018 cover.png