Keith’s Conundrums: Conventions

Keith Douglas

Last time, I discussed Leibniz’s Law (LL), stated as two conditionals in second-order logic. Most sophomore-level courses in logic, which discuss predicate logic, are restricted to so-called “first order.” And this is where the trickiness comes in at first. Reader Steve followed my hint and asked if location was a property. In my view, many discussions that surround properties in metaphysics and beyond are challenging to work through without quickly adopting a view about properties. 

For example, topics such as mental states, events, causation, laws, scientific realism, etc., all have properties playing a role. There is also a substantial literature on its own concerning properties. In fact, this topic in metaphysics was one of the first addressed by the pre-Socratics and most famously by Plato. In the early modern period, the Galileo-Boyle conception of primary properties (versus secondary properties) was important for the rise of semi-autonomous science.

There are, however, two contemporary philosophers whose work in general metaphysics is systematic enough to notice that theories of properties have substantial work to play in many of the other matters I mentioned. These are two philosophers whom regular readers of my column may remember — my own teacher, Mario Bunge, and his approximate contemporary, D. M. Armstrong. Both point out that logic deals with predicates and Bunge goes so far as to actually analyze and produce a calculus of properties. This task is only partially sketched by Armstrong. 

For our purposes this is extremely important, because one has to handle questions even more profound than Steve’s to attempt to solve our problem. Consider the predicate “is identical with x.” If that refers to a property (note the semantics here!) then trivialism looms. Alternatively, if the quantification is read as being over predicates, as is sometimes done, the same follows.

A topologist I was in correspondence with once inadvertently illustrated what Bunge and Armstrong point out: that in mathematics every predicate does correspond to a property, in the special domain here. My correspondent even called a relation just a special structured set of a given kind. This is interesting — set theoretic Platonism of the most stereotypical kind. But for many of us these are not properties. The use of the phrase is an equivocation. Bunge even suggested to me in class that “truth” as ordinarily understood is only an “honourific” and “used by tradition” in mathematics.

Moreover, Bunge and Armstrong balk at “disjunctive” properties and have trouble with what to say about “negative” properties. (Armstrong thinks they may exist; Bunge is categorically against the idea.) Oddly, these pose less of a problem for people like Max Black, who had an influential supposed counterexample to one direction of LL. Gary Wedeking, my MA thesis supervisor, also bought this argument, about two spheres. This allows us to finally return to Steve’s point. If we affix a coordinate system to one sphere, then one sphere is here (say with its center at the origin of coordinates) and the other over there. Black’s followers, including Wedeking, say that Leibniz’ law only applies to properties where that is meant to exclude relations. 

Further, by introducing a coordinate system, one has supposedly introduced “other things.” Interestingly, this is an argument that then reintroduces that relations make a difference. This looks inconsistent, on the face of it. Physics-literate readers will immediately notice I said “coordinate system,” not “reference frame.” In my conversations with Wedeking, I actually said the latter. Now I’m not sure that works — for a reason Bunge points out. 

In physics, a reference frame has several properties. One of them is velocity. Bunge suggests that this makes for a claim for reality for reference frames, since a formal object cannot have such a thing. I wonder now whether or not a reference frame can be regarded as one of his “controlled fictions” — like either idealizations (e.g., the perfectly frictionless surface) or calculational aids, like the “null object” in his mereology. Does this affect Steve’s question about relational properties? Why does this matter? Because Wedeking might hold that the frame is an additional item to Black’s “universe.”

I for one do think that monadic properties are not in any way ontologically special and that many hypothesized ones might have been guessed due to monadic predicates in ordinary language. So I would always want to state the two directions of LL in terms of properties, with an appropriate theory of properties that ruled out “disjunctive” and likely also “negative” ones, as well as ensure that it covered properties of all “airities” (valences, adicities, etc.). But I have no such theory to offer, other than those of Wedeking and Armstrong. 

Conventions

This month I would like to talk about conventions. It is said to be conventional that we put the stop signal on the top of a traffic light and the go signal on the bottom. It is also said to be conventional that hydrogen appears to the top and left of a periodic table, that the table itself is a convention. There’s something right about all of these claims, it seems to me, but they also irk me in a way — at least the latter two. What matters is that the conventions were adopted without thinking — or at least “deliberate” thinking.

So first conundrum this month: Is there a problem here? Is it purely verbal? Similarly, let us think about the conventions of the periodic table. First: There is no “the periodic table.” There are in fact many different table types, sometimes called “forms” of the periodic table. I would not want to claim that the most common form, or any other, is the only correct one. So calling the different tables “conventions” is correct up to a point. But it doesn’t strike me as being at all the same sort as either of the other two examples. 

Why? Because unlike the other two, the notion of law seems to play a role. Thus we should ask ourselves: What is an appropriate difference between the three cases? What is the source of my perplexity and discomfort? Do you share it? Why or why not? Does this shade into questions about realism and classification, or is there something else here? 

Final historical note. Dmitri Mendeleev himself regarded the invention of the periodic table to be an almost trivial accomplishment. He regarded the discovery of periodic law to be a much more worthy accomplishment, which the table attempts to represent. Mendeleev himself seems to have been an explicit scientific realist (like Einstein, Galileo, and Boyle). Does this matter?