Partially defined predicates

Is cutting one head off a two-headed person a case of beheading?

Examples like this are normally used as illustrations of vagueness. It’s natural to think of cases like this as ones where we have a predicate defined over a domain and being applied outside it. Thus, “is being beheaded” is defined over n-headed animals that are being deprived of all heads or of no heads.

I don’t like vagueness. So let’s put aside the vagueness option. What else can we say?

First, we could say that somehow there are deep facts about the language and/or the world that determine the extension of the predicate outside of the domain where we thought we had defined it. Thus, perhaps, n-headed people are beheaded when all heads are cut off, or when one head is cut off, or when the number of heads cut off is sufficient to kill. But I would rather not suppose a slew of facts about what words mean that are rather mysterious.

Second, we could deny that sentences using predicates outside of their domain lack truth value. But that leads to a non-classical logic. Let’s put that aside.

I want to consider two other options. The first, and simplest, is to take the predicates to never apply outside of their domain of definition. Thus,

1. False: Cutting one head off Dikefalos (who is two headed) is a beheading.

2. True: Cutting one head off Dikefalos is not a beheading

3. False: Cutting one head off Dikefalos is a non-beheading.

4. True: Cutting one head off Dikefalos is not a non-beheading.

(Since non-beheading is defined over the same domain as beheading). If a pre-scientific English-speaking people never encountered whales, then in their language:

5. False: Whales are fish.

6. True: Whales are not fish.

7. False: Whales are non-fish.

8. True: Whales are not non-fish.

The second approach is a way modeled after Russell’s account of definite descriptors: A sentence using a predicate includes the claim that the predicate is being used in its domain of definition and, thus, all of the eight sentences exhibited above are false.

I don’t like the Russellian way, because it is difficult to see how to naturally extend it to cases where the predicate is applied to a variable in the scope of a quantifier. On the other hand, the approach of taking the undefined predicates to be false is very straightforward:

9. False: Every marine mammal is a fish.

10: False: Every marine mammal is a non-fish.

This leads to a “very strict and nitpicky” way of taking language. I kind of like it.

Disjunctive predicates

I have found myself thinking these two thoughts, on different occasions, without ever noticing that they appear contradictory:

1. Other things being equal, a disjunctive predicate is less natural than a conjunctive one.

2. A predicate is natural to the extent that its expression in terms of perfectly natural predicates is shorter. (David Lewis)

For by (2), the predicates “has spin or mass” and “has spin and mass” are equally natural, but by (1) the disjunctive one is less natural.

There is a way out of this. In (2), we can specify that the expression is supposed to be done in terms of perfectly natural predicates and perfectly natural logical symbols. And then we can hypothesize that disjunction is defined in terms of conjunction (p ∨ q iff ∼(∼p ∧ ∼q)). Then “has spin or mass” will have the naturalness of “doesn’t have both non-spin and non-mass”, which will indeed be less natural than “has spin and mass” by (2) with the suggested modification.

Interestingly, this doesn’t quite solve the problem. For any two predicates whose expression in terms of perfectly natural predicates and perfectly natural logical symbols is countably infinite will be equally natural by the modified version of (2). And thus a countably infinite disjunction of perfectly natural predicates will be equally natural as a countably infinite conjunction of perfectly natural predicates, thereby contradicting (1) (the De Morgan expansion of the disjunctions will not change the kind of infinity we have).

Perhaps, though, we shouldn’t worry about infinite predicates too much. Maybe the real problem with the above is the question of how we are to figure out which logical symbols are perfectly natural. In truth-functional logic, is it conjunction and negation, is it negation and material conditional, is it nand, is it nor, or is it some weird 7-ary connective? My intuition goes with conjunction and negation, but I think my grounds for that are weak.

Some weird languages

Platonism would allow one to reduce the number of predicates to a single multigrade predicate Instantiates(x₁, ..., x_n, p), by introducing a name p for every property. The resulting language could have one fundamental quantifier ∃, one fundamental predicate Instantiates(x₁, ..., x_n, p), and lots of names. One could then introduce a “for a, which exists” existential quantifier ∃_a in place of every name a, and get a language with one fundamental multigrade predicate, Instantiates(x₁, ..., x_n, p), and lots of fundamental quantifiers. In this language, we could say that Jim is tall as follows: ∃_Jimx Instantiates(x, tallness).

On the other hand, once we allow for a large plurality of quantifiers we could reduce the number of predicates to one in a different way by introducing a new n-ary existential quantifier ∃_F(x₁, …, x_n) (with the corresponding ∀_P defined by De Morgan duality) in place of each n-ary predicate F other than identity. The remaining fundamental predicate is identity. Then instead of saying F(a), one would say ∃_Fx(x = a). One could then remove names from the language by introducing quantifiers for them as before. The resulting language would have many fundamental quantifiers, but only only one fundamental binary predicate, identity. In this language we would say that Jim is tall as follows: ∃_Jimx∃_Tally(x = y).

We have two languages, in each of which there is one fundamental predicate and many quantifiers. In the Platonic language, the fundamental predicate is multigrade but the quantifiers are all unary. In the identity language, the fundamental predicate is binary but the quantifiers have many arities.

And of course we have standard First Order Logic: one fundamental quantifier (say, ∃), many predicates and many names. We can then get rid of names by introducing an IsX(x) unary predicate for each name X. The resulting language has one quantifier and many predicates.

So in our search for fundamental parsimony in our language we have a choice:

• one quantifier and many predicates
• one predicate and many quantifiers.

Are these more parsimonious than many quantifiers and many predicates? I think so: for if there is only one quantifier or only one predicate, then we can collapse levels—to be a (fundamental) quantifier just is to be ∃ and to be a (fundamental) predicate just is to be Instantiates or identity.

I wonder what metaphysical case one could make for some of these weird fundamental language proposals.

Quantifier and Predicate Variance

Suppose we decide to speak with a quantifier family (a quantifier family includes ∃ and ∀, but may also include things like "many" and "most" and "at least three", and maybe even two-place quantifiers, all ranging over the same domain) that makes arbitrary pluralities of things have a fusion, i.e., a mereologically universalist quantifier family. According to the Quantifier Variance thesis, this decision to extend quantifiers is a linguistic decision that needs merely pragmatic justification, a decision that introduces a quantifier perhaps different from the one most ordinarily used.

This sounds like a decision solely about quantifiers. Not so. For now we need to say something about the meaning of predicates applied to variables bound by these quantifiers. For instance, we need to be able to meaningfully say using the new "there is" whether there is something whose mass is 455 tons, i.e., whether ∃x(Mass(x, 455T)). (Using van Inwagen quantifiers which range over simples and organisms, unless there are alien organisms much larger than blue whales, there isn't anything of that mass.) We can give a semantics for the universalist quantifiers in terms of plural quantification, but we need to account for how much a plurality masses. Intuitively, the mass of a plurality is the sum total of the masses of the simples in the plurality (some technical problems: what if there is gunk? do we count the mass-equivalent of the energy of the bonds between the simples?), and so ∃x(Mass(x, 455T)) provided that there are ys which plurally mass 455T. I suppose extending mass in this way once one has extended the quantifiers is pretty obvious.

But not all predicates extend in an obvious way. For instance, consider the predicate that says that something is spatially extended. Does that predicate apply to the fusion of the number seven with the Empire State Building? Here we have a decision to make, roughly about what it is to be plurally extended: Do we say that for the ys to be plurally extended, each one of them must be extended, or is it enough that one of them is extended? The former decision will fit better with the intuition that extended objects are material. The latter with the intuition that occupying space is sufficient for extension. And either way, we will have some technicalities (can't a plurality of unextended points make up something extended?). Or take the causal relation. Did the people who built the Empire State Building cause the fusion of the number seven with the Empire State Building? It's hard to say. And aesthetic properties will be particularly hairy (is the fusion of Beethoven's Ninth with Michelangelo's David beautiful? is it a work of art?).

Thinking about such examples makes it clear that the linguistic decision to speak with a new quantifier family needs to come along with a correlate decision about the semantics of predicates extended to work with this new quantifier family (a decision that could, sometimes, be simply to leave a predicate underdefined or vague). This means that it is a bit misleading to talk of "Quantifier Variance". The relevant thesis is "Quantifier and Predicate Variance". (And we may also need to have "name variance", unless we consider names to be a kind of quantifier.)

Note that none of this is an issue if one creates a new quantifier merely by domain restriction. It's domain expansion that generates the problems.

An argument against heavy-weight Platonism

Heavy-weight Platonism explains (or grounds) something's being green by its instantiating greenness. Light-weight Platonism refrains form making such an explanatory claim, restricting itself to saying that something is green if and only if it instantiates greenness. Let's think about a suggestive argument against heavy-weight Platonism.

It would be ad hoc to hold the explanatory thesis for properties but not for relations. The unrestricted heavy-weight Platonist will thus hold that for all n>0:

1. For any any n-ary predicate F, if x₁,...,x_n are F, this is because x₁,...,x_n instantiate Fness.

(One might want to build in an ad hoc exception for the predicate "instantiates" to avoid regress.) But just as it was unlikely that the initial n=1 case would hold without the relation cases, i.e., the n>1 cases, so too:

2. If (1) holds for each n>0, then it also holds for n=0.

What is the n=0 case? Well, a 0-ary predicate is just a sentence, a 0-ary property is a proposition, the "-ness" operator when applied to a sentence yields the proposition expressed by the sentence, and instantiation in the 0-ary case is just truth. Thus:

3. If (1) holds for each n>0, then for any sentence s, if s, then this is because because of the truth of the proposition that s.

(The quantification is substitutional.) For any sentence s, let <s> be the proposition that s. The following is very plausible:

4. For any sentence s, if s, then <s> is true because s.

But (4) conflicts with (3) (assuming some sentence is true). In fact, to generate a problem for (3), we don't even need (4) for all s just for some, and surely the proposition <The sky is blue> is true because the sky is blue, rather than the other way around: the facts about the physical world explain the relevant truth facts about propositions. Thus:

5. It is false that (1) holds for each n>0.

The above argument is compatible, however, with a restricted heavy-weight Platonism on which sometimes instantiation facts explain the possession of attributes. Perhaps, for instance, if "is green" is a fundamental predicate, then Sam is green because Sam instantiates greenness, but this is not so for non-fundamental predicates. And maybe there are no fundamental sentences (a fundamental sentence would perhaps need to be grammatically unstructured in a language that cuts nature at the joints, and maybe a language that cuts nature at the joints will require all sentences to include predication or quantification or both, and hence not to be unstructured). If so, that would give a non-arbitrary distinction between the n>0 cases and the n=0 case. There is some independent reason, after all, to think that (1) fails for complex predicates. For instance, it doesn't seem right to say that Sam is green-and-round because he instantiates greenandroundness. Rather, Sam is green-and-round because Sam is green and Sam is round.

Deflation of predicates

Some deflationary theories take some predicate, such as "is necessary" or "is true", and claim that there is really nothing in the predicate for philosophical investigation—the predicate is not in any way natural, but just attributes some messy, perhaps even infinite, combination of more natural properties.

But I know only three candidates for a way that we could come to grasp a meaningful predicate. One way is by ostension to a natural property. Here's a rough idea. The predicate "is circular" might be introduced as follows. We are shown a bunch of objects, A₁,...,A_n, and told that each "is circular", and a bunch of other objects, B₁,...,B_m, and told that each "is not circular." The predicate "is circular" is then grasped to indicate some property that all or almost all of the As have and all or almost all of the Bs lack. But there may be many (abundant) properties like that (for instance, being one of A₁,...,A_n). Which one do we mean by "is circular"? Answer: The most natural of the bunch.

The second way depends on a non-natural view of mind. It could be that our minds, unlike language, can directly be in contact with some properties. And it may be that a predicate tends to be used in circumstances in which both speaker and listener are directly contemplating a particular property, and that makes the predicate mean that property.

The third way is by stipulation. I just say: "Say that x is frozzly if and only if x is frozen and green."

The predicates, like "is true" and "is necessary", that are the subjects of these deflationary theories are not introduced in the first way if the theories are correct to hold that the predicates do not correspond to a natural property. Are they introduced in the third way? That is very unlikely. I doubt there was a first user of "is necessary" who stood up and said: "I say that p is necessary if and only if...." That leaves the second way, the non-naturalistic way. Therefore:

1. If these deflationary theories are correct, naturalism is false.

Which is interesting since the motivation for the theories is sometimes naturalistic (e.g., Hartry Field in the case of "is true"[note 1]).

But in any case, the following is very plausible. Any properties we are in direct non-natural cognitive contact with are either innately known or natural. So, the deflated predicates must refer to innately known predicates. I doubt, however, that we innately know any entirely non-natural predicates. And that leaves little room for these theories.

More generally, the above considerations make it difficult to see how we could have any genuine non-natural, non-stipulative predicates. Thus, if we have good reason to think that P does not indicate a natural property, and is not stipulative, we have good reason to have an error theory about P.

Concepts of artifacts appear to be a counterexample. "Is a chair" is neither natural nor stipulated. My inclination is to say that it is not really a predicate ("Bob is chair" expresses some sentence about Bobbish reality being chairwise arranged, or something like that), which makes for a kind of error theory.

Length and other predicates

The length of a pencil is measured in a straight line from tip to end. This is equal to the length of the region of space occupied by the pencil. The length of a rope is measured along the rope, so that the length does not change much when the rope is coiled or uncoiled, and so unless the rope is straightened out, the length of the rope is not equal to any dimension of the region of space occupied by the rope. The length of a bow is (typically) the length of the string plus three inches. On the other hand, the length of a computer program is something quite different, not measured in length-dimensions, but in units like lines or lines-of-code or characters.

Similar points apply to almost all other predicates. These are a matter of decision rather than discovery. When we extend our language to start talking about pencils, ropes, bows and programs, we also need to decide how all the many predicates that could apply to them are to be extended. Quantifier pluralism requires predicate pluralism.