Kripke's standard meter

Back when there was a standard meter, Kripke claimed that it was contingent a priori that the standard meter is a meter in length.

This seems wrong. For anything narrowly logically entailed by something that’s a priori is also a priori. But that the standard meter is a meter in length entails that there is an extended object. And that there is an extended object is clearly a posteriori.

Kripke’s reasoning is that to know that the standard meter is a meter in length all you need to know is how “meter” is stipulated, namely as the actual length of the standard meterstick, and anything you can know From knowing how the terms are stipulated is known a priori.

There is something fishy here. We don’t know a priori that the stipulation was successful (it might have failed if, for instance, the “standard meter” never existed but with a conspiracy to pretend it exists). In fact, we don’t know a priori that any stipulations were ever made—that, too, is clearly a posteriori.

Maybe what we need here is some concept of “stipulational content”, and the idea is that something is a priori if you can derive it a priori from the stipulational content of the terms. But the stipulational content of a term needs to be defined in such a way that it’s neutral on whether the stipulation happened or succeeded. If so, then Kripke should have said that it’s a priori that if there is a standard meterstick, it is a meter long.

Elegance and stipulation

Depending on metaphysics, wholes depend on their parts or the parts depend on the wholes. But nothing depends on itself: that would be a vicious circularity. So nothing is a part of itself. On my own preferred story about parts, they are modes of wholes. But perhaps apart from God, nothing is a mode of itself. So, again, nothing is a part of itself (we shouldn't say that God is a part of himself, except trivially if everything is a part of itself).

Yet contemporary usage in mereology makes each thing a part of itself. One is free to stipulate how one wishes. If "part" is the ordinary notion, the contemporary mereologist can stipulate that parthood* is a disjunction of parthood and identity, i.e.,

1. x is a part* of y if and only if x is a part of y or x=y.

However, while one can stipulate how one wishes, one wouldn't expect a disjunctive stipulation to cut reality at its joints.

Why does this matter? Well, one of the interesting questions about parts is what axioms of mereology are true. We have several criteria for what makes a plausible axiom. It's supposed to be intuitive in itself, it's supposed to not lead to paradox, but it's also supposed to be elegant. It seems, however, that one can always ensure the elegance of any axioms with stipulation (just stipulate a zero-place predicate that says that the conjunction of the axioms is true). So it seems we want axioms to be elegant when expressed in terms that cut nature at the joints. In mereology, this would mean that we want axioms to be elegant when expressed in terms of proper parthood* (since proper parthood* is just parthood, the joint-carving natural concept) rather than in terms of parthood*.

This is a bit problematic. For it seems that the standard axioms of mereology get some of their prettiness by using the overlap relation:

2. Oxy iff x and y have a part* in common.

But the overlap relation is a nasty disjunctive mess when expressed in terms of proper parthood*:

3. Oxy iff x=y or x is a proper part* of y or y is a proper part* of x or x and y have a proper part* in common.

This suggests that much of the apparent elegance of the axioms of classical mereology may be spurious. They end up being a mess when you rewrite them in terms of parthood rather than parthood*.

I think the above negative conclusion about the elegance of the axioms of classical mereology is premature, and buys into a mistaken way to measure the elegance of the axioms of a theory. The mistake is to think that one rewrites all the axioms in what Lewis calls "perfectly natural" terms, and then looks at how brief the result is. Mathematicians frequently think that some set of axioms--say, group axioms--are quite "elegant and natural" even when rewriting the axioms in terms of the set membership relation ∈ produces a mess, as it generally does. (Just think of what a mess is produced when anything using the ordered pair (x,y) is rewritten using the set {{x},{x,y}}, and how just about everything in mathematics uses functions and hence ordered pairs.)

One can indeed make any set of axioms brief by careful choice of stipulations. But in some cases the stipulation will itself be very messy (the extreme case is where one replaces all the axioms with a single zero-place predicate) and in other cases there will be many stipulations. But if one can make a set of axioms brief by making a small number of relatively simple stipulations, that is impressive.

A theory can, thus, be elegant even if it is messy and long when all the axioms are written out in perfectly natural terms to the extent that the theory can be elegantly generated from an elegantly small set of elegant stipulations. Classical mereology can satisfy this elegance condition on theories even if I am right that the natural concept of parthood does not allow for proper parts. One just makes the fairly elegant (it's just a disjunction of two natural conditions) disjunctive stipulation (1), and then uses this stipulated notion of parthood* to elegantly stipulate a notion of overlap by means of (2), and then elegantly formulates the rest of the theory in terms of these.

The suggestion I am making is that we measure the complexity of a theory in terms of the brevity of expression in a language that has significant higher-order generative resources that, nonetheless, start with perfectly natural terms. These generative resources allow, in particular, for multiple levels of stipulation. We philosophers have a tendency to simply ignore stipulative definitions. But they do matter. If one takes classical mereology and rewrites the axioms in terms of (proper) parthood, one gets a mess; but the hierarchical stipulative structure of the classical theory is a part of the theory. Furthermore, the generative resources should also allow one to see an axiom schema as simpler and better unified than the sum total of the individual axioms falling under the schema. An axiom schema is not just the sum of the axioms falling under it.

This approach would also let one compare the complexity two different higher-level scientific theories, say in geology or organic chemistry, and say that one is simpler than the other even if both are equally intractable messes when fully expanded out in the vocabulary of fundamental physics. And one can do this even if one does not know how to make the needed stipulations--nobody knows how to define "tectonic plate" in the terminology of fundamental physics, but we can suppose the stipulation to have been made and proceed onward. All this makes it easier to be a reductionist about higher-level theories (I'm not happy about this, mais c'est la vie).

None of this should be news at all to those who are enamoured of computational notions of complexity.

One deep question here is just what generative resources the language should have.

And another deep question should be asked. When we formulate axioms by careful use of stipulation or axiom schemata, what we are really doing is describing the axioms in higher level terms: we are describing a set of sentences formulated in lower level terms. Patterns in reality are sometimes most aptly described not by first-order sentences in fundamental terms, but by describing how to generate those first-order sentences (say, as instances of a schema, or as the result of filling out a sequence of stipulations). We should then ask: How can such patterns be explanatory? I think that if such patterns are explanatory, if they are not mere coincidence, then in an important way reality is suffused with logos, in both of the main sense of the word (language and rationality).

There are, I think, three main options here. One is that we create this reality with our language. Realism forbids that. The second is that we are living in a computer simulation. But this explains the linguistic-type patterns only in contingent reality. But the axioms of mereology or of set theory are not merely contingent. The third is a supernaturalist story like theism, panentheism or pantheism.

Ineffability

Consider this argument against divine ineffability: Let p be the conjunction of all fundamental truths intrinsically about God (I'm thinking here of something like the Jon Jacobs account of ineffability, but the point should work on other similar accounts). Stipulate that the sentence "It divines" (a feature-placing sentence or zero-place predicate, like in "It rains") expresses p. It divines. It seems I have just said the conjunction of all fundamental truths intrinsically about God. Hence God is not ineffable.

But this argument cannot be sound, since God is in fact ineffable--divine ineffability is, for instance, part of the creed of the Fourth Lateran Council. So what goes wrong with the argument?

First, one might have technical worries about infinite conjunctions or arbitrary linguistic stipulations. I'll put those to one side, though they are worth thinking about.

More deeply, one might worry whether there are any fundamental truths intrinsically about God. Truths are true propositions. Perhaps the fundamental reality of God not only cannot be expressed in language, but cannot even be given propositional form. I am not sure about this, though it is a promising response to the argument. But, plausibly, propositions are divine thoughts. And God surely does express his fundamental reality in his thought (indeed, this is central to Augustine's Trinitarianism).

I want to try out a different response to the argument: question the last step in the argument, the inference "Hence God is not ineffable." This response allows that we can stipulate and assert a sentence that means the conjunction of all fundamental truths intrinsically about God, but denies that this is a problem for ineffability. Ineffability isn't a denial of the possibility of asserting a sentence whose semantic content is such-and-such truths about the divine nature. Rather, it is the denial of the possibility of linguistically communicating these truths. For me to linguistically communicate a truth to you it is required that my sentence give rise to your thinking that truth. But the truth expressed by "It divines" isn't a truth you can think. On this understanding, divine ineffability is an immediate consequence of divine incomprehensibility, and rather than being a doctrine about semantics, it's a doctrine about communication.

If this is right, then stipulation allows the semantics of our language to outrun communication and thought. You can think some deep philosophical truth that I don't know, and I can stipulate that "It xyzzes" means that truth, and I can sincerely assert "It xyzzes." But I don't thereby think that truth. I can, of course, think the second order thought that "It xyzzes" is true, but to do that is not the same as to think that it xyzzes. Similarly, I can think that "It divines" is true, but that's a thought about a piece of stipulated language rather than a thought about God. Indeed, it divines, but I don't understand the sentence "It divines" as I can't grasp the proposition it expresses.

Sometimes people are accused of a certain kind of insincerity like this: "You're just saying the words but don't really understand." This is a different kind of insincerity than when people are lying. A person who is "just saying the words" may believe that the sentence composed of the words is really true, and if so, then she isn't lying. (Corollary: One can say something one doesn't believe and yet not be a liar, as long as one believes that what one is saying is true.) The reason that there may be insincerity in "just saying the words" is that normally one implicates that one believes (and hence has a minimal understanding of) the content of what one says. But that's an implicature that can be canceled to avoid even this kind of insincerity: "I don't exactly know what 'God loves you' means, but I believe that it is true. God loves you." And when people are talking of a topic neither is close to being an expert on, the implicature of understanding one's words may be contextually canceled.

Whatever content can be conveyed metaphorically can be said literally

Suppose you say something metaphorical, and by means of that you convey to me a content p. I now stipulate that "It's zinging" expresses precisely the content you conveyed. Technically, "It's zinging" is a zero-place predicate, like "It's raining." And now I say: "It's zinging." The literal content expressed by "It's zinging" is now equal to the metaphorical content conveyed by what you said. A third party can then pick up the phrase "It's zinging" from me without having heard the original metaphor, get a vague idea of its literal content from observing my use of it, and now a literal statement which has the same content as was conveyed by the metaphorical statement can start roaming the linguistic community.

Thus: If you cannot say something literally, you cannot whistle it either. For if you could convey it by whistling, you could stipulate a zero-place predicate to mean that which the whistling conveys.

Objection 1: My grasp of "It's zinging" is parasitic on your metaphor, while the third-party doesn't have any understanding.

Response: Yes, and so what what? I wasn't arguing that you can usefully get rid of metaphor. It may well be essential to understanding the content in question. My point was simply that there can be a statement whose literal semantic content is the same as the content conveyed by your metaphor. Understanding is something further. This is very familiar in cases of semantic deference. (I hear physicists talking about a new property of particles. I don't really understand what they're saying, but I make the suggestion that they call that property: "Zinginess." My suggestion catches on. I can say: "There are zingy particles", and what I say has the same content as the scientists' attribution of that property. But while the scientists understand what they're saying, I have very little understanding.) The third party who hasn't heard the original metaphor may not understand much of what he's saying with "It's zinging." But what he's saying nonetheless has the literal semantic content it does by deference to my use of the sentence, and my use of the sentence has the literal semantic content it does by stipulation. All this is quite compatible with the claim that any decent understanding of "It's zinging" will require getting back to the metaphor. But, nonetheless, "It's zinging" literally means what the metaphor metaphorically conveyed.

Objection 2: The stipulation does not succeed. (This is due to Mike Rea.)

Response: Why not? If I can refer to an entity, I can stipulate a name for it, no matter how little I know about it. I may have no idea who killed certain people, but I can stipulate "Jack the Ripper" names that individual. My stipulation will succeed if and only if exactly one individual killed those people. Similarly, if I can refer to a property, I can stipulate a one-place predicate that expresses that property. (If a certain kind of Platonism is true, this just follows from the name case: I name the property "Bob", and then I have the predicate "instantiates Bob".) In cases without vagueness, contents seem to be propositions, and zero-place predicates express propositions, so just as I can stipulate a one-place predicate to express a property, I should be able to stipulate a zero-place predicate to express a propositions. And in cases of vagueness, where maybe a set of propositions (or, better, a weighted set of propositions) is a content, I should be able to stipulate a similarly vague literal zero-place predicate as having as its content the same set of propositions.

There are many ways of introducing a new term into our language. One way is by stipulating it in terms of literal language. That's common in mathematics and the sciences, but rare in other cases. Another way is by ostension. Another is just by talking-around, hoping you'll get it. One way of doing this talking-around is to engage in metaphor: "I think we need a new word in English, 'shmet'. You know that butterflies in the stomach feeling? That's what I mean." We all understand what's going on when people do this kind of stipulation. For all we know, significant parts of our language came about this way.

A puzzle about stipulation

I don't know whether Mars ever had life.

But it seems I can very easily get to know.  Let P be the property of having once had life.  Now, stipulate the predicate "is xyzzy" as follows: if Mars once had life, the predicate expresses P, and otherwise, it expresses ~P.  Use "<s>" to abbreviate "the proposition that s".  Then, plausibly:

1. (Premise) I know the proposition <Mars is xyzzy>.  (For I know that "is xyzzy" expresses P if and only if Mars has P, and I can do all the logic needed to yield the claim that Mars is xyzzy.)
2. (Premise) The proposition <Mars is xyzzy> is the proposition <Mars once had life> if "is xyzzy" expresses P.
3. (Premise) The proposition <Mars is xyzzy> is the proposition <Mars never had life> if "is xyzzy" expresses ~P.
4. (Premise) "Is xyzzy" either expresses P or it expresses ~P.
5. So, either I know <Mars once had life> or I know <Mars never had life>.  (By 1-4)
6. So, either I know the proposition that Mars once had life or I know the proposition that Mars never had life. (Expanding abbreviations)
7. (Premise schema) If I know the proposition that s, then I know that s.
8. So, either I know that Mars once had life or I know that Mars never had life. (By 6 and 7)

And so whichever disjunct is true, I know whether Mars once had life!  So, I didn't know it, but once I stipulated "xyzzy" and came to know that Mars is xyzzy, I got to know it.

The argument is valid, but the conclusion is surely false.  So what should we deny?

My inclination is either to deny that it is possible to stipulate predicates in the way I stipulated "is xyzzy" (and hence (1) is nonsense, since it uses a bit of nonsense, namely "is xyzzy", as if it were a predicate) or to allow such stipulation but not allow that sentences using the stipulated predicate express precisely the propositions that (2) and (3) claim they do.  I also feel some pull to denying (7).

An interesting move would be to deny (1) on the grounds that I once know that it is true that Mars is xyzzy, but I do not know that Mars is xyzzy. That sounds odd.

I think Jon Kvanvig may have once used something in the vicinity of this puzzle, but I could be wrong.