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.

Deflation of the foundations of probability

I don't really want to commit to the following, but it has some attraction.

Question 1: What is probability?

Answer: Any assignment of values that satisfies the Kolmogorov axioms or an appropriate analogue of them (say, a propositional one).

Question 2: Are probabilities to be interpreted along frequentist, propensity or epistemic/logical lines?

Answer: Frequency-based, propensity-based and epistemically-based assignments of weights are all probabilities when the assignments satisfy the axioms or an appropriate analogue of them. In particular, improved frequentist probabilities are genuine probabilities when they can be defined, but so are propensity-based objective probabilities if they satisfy the axioms, and likewise logical probabilities. Each of these may have a place in the world.

Question 3: But what about the big metaphysical and epistemological questions, say about the grounds of objective tendencies and epistemic probabilities?

Answer: Those questions are intact. But they are not questions about the interpretation of probability as such. They are questions about the grounds of objective propensity or about the grounds of epistemic assignments. Thus, the former question belongs to the philosophy of science and the metaphysics of causation and the latter to epistemology.

Question 4: But surely one of the interpretations of probability is fundamental.

Answer: Maybe, but do we need to think so? Take the axioms of group theory. There are many kinds of structures that satisfy these axioms. Why think one kind of structure satisfying the axioms of group theory is fundamental?

Question 5: Still, couldn't there be connections, such as that logical probabilities ultimately derive from propensities via some version of the Principal Principle, or the other way around?

Answer: Maybe. But even if so, that doesn't affect the deflationary theory. There are plenty more structures that satisfy the probability calculus that do not derive from propensities.

Question 6: But shouldn't we think there is a focal Aristotelian sense of probability from which the others derive?

Answer: Maybe, but unlikely given the wide variety of things that instantiate the axioms. Maybe instead of an Aristotelian pros hen analogy, all we have is structural resemblance.

Reflections on Horwich's minimalism

Horwich's minimalism is a theory of truth generated by the axiom:

1. If p is true, then p is a proposition

and the axiom schema obtained by taking all sentences s of extensions of English and substituting them into:

2. <s> is true if and only if s

(with this understood in the same extension of English as s was), with the exception of those sentences that lead to paradox (e.g., "This sentence is false"). Here, <s> denotes the proposition that s.

Here are some issues. None of them are fatal. But they all mean that minimalism isn't quite as simple as it initially seems.

Issue 1: What is an extension of English? We need to include sentences of extensions of English because no doubt there are propositions that no sentence of English can express. Now, an extension of English had better not change the meanings of "is true" or "if and only if"—for if that is allowed to change, then some instances of (2) will become false. Presumably, then, what makes L an extension of English is that for any linguistic element e of English, e is also a linguistic element of L, and it has the same meaning (semantic value, etc.) in L as it does in English. Thus, Horwich's minimalism in its description of the axiom schema presupposes the concept of meaning (semantic value, etc.). To avoid circularity, the concept of meaning had better not depend on that of truth.

Issue 2: Nitpicky stuff. Strictly speaking, (2) generates bad orthography. Suppose s is "Snow is white." Then we are told that <Snow is white.> is true if and only if Snow is white. But the last "Snow" should not be capitalized. This can be easily handled—we specify that when substituting s in, we adjust the first letter's case as needed. There is also that odd looking period after the first "white"; again, we can specify that it is to be omitted. A slightly less easy case is where s is "This sentence is short." Suppose s is true. But now consider:

3. <This sentence is short> is true if and only if this sentence is short.

But the second occurrence of "this sentence" refers not to s but to (3), and that sentence is not short, so (3) is false. Presumably, we handle this by not allowing sentences with indexicals or demonstratives. This requires the substantive assumption that any proposition that can be expressed with indexicals/demonstratives can be expressed without them. Next, let s be "u if and only if v" (for some u and v). Then (2) yields:

4. <u if and only if v> if and only if u if and only if v.

But this is wrong or badly ambiguous. Maybe then we're supposed to use an extension of English that has grouping parentheses, and then replace (2) with:

5. <s> is true if and only if (s).

Issue 3: Contingent liar. Axioms normally are supposed to not vary between worlds. But there are contingent liar sentences, like "The sentence on Alex's board is false", which is paradoxical when it is the unique sentence on Alex's board but need not be paradoxical when written on Jon's board (unless we have something like "The sentence on Jon's board is false" as the only sentence on Alex's board). This means that dropping those instances of (2) (or of (5)) that generates the paradox requires dropping different instances in different worlds, thereby making the axioms of truth differ from world to world.

There are two ways for axioms to differ between worlds. In the weak sense, whether p is an axiom varies between worlds, but p is true at all worlds. In the strong sense, the truth value varies between worlds, too. This is the kind of variation that we'd need to get out of the contingent liar. And this just doesn't fit with what we understand by "axiom", I think.