Definitions

In the previous post, I offered a criticism of defining logical consequence by means of proofs. A more precise way to put my criticism would be:

1. Logical consequence is equally well defined by (i) tree-proofs or by (ii) Fitch-proofs.

2. If (1), then logical consequence is either correctly defined by (i) and correctly defined by (ii) or it is not correctly defined by either.

3. If logical consequence is correctly defined by one of (i) and (ii), it is not correctly defined by the other.

4. Logical consequence is not both correctly defined by (i) and and correctly defined by (ii). (By 3)

5. Logical consequence is neither correctly defined by (i) nor by (ii). (By 1, 2, and 4)

When writing the post I had a disquiet about the argument, which I think amounts to a worry that there are parallel arguments that are bad. Consider the parallel argument against the standard definition of a bachelor:

6. A bachelor is equally well defined as (iii) an unmarried individual that is a man or as (iv) a man that is unmarried.

7. If (6), then a bachelor is either correctly defined by (iii) and correctly defined by (iv) or it is not correctly defined by either.

8. If logical consequence is correctly defined by one of (iii) and (iv), it is not correctly defined by the other.

9. A bachelor is not both correctly defined by (iii) and correctly defined by (iv). (By 9)

10. A bachelor is neither correctly defined by (iii) nor by (iv). (By 6, 7, and 10)

Whatever the problems of the standard definition of a bachelor (is a pope or a widower a bachelor?), this argument is not a problem. Premise (9) is false: there is no problem with saying that both (iii) and (iv) are good definitions, given that they are equivalent as definitions.

But now can’t the inferentialist say the same thing about premise (3) of my original argument?

No. Here’s why. That ψ has a tree-proof from ϕ is a different fact from the fact that ψ has a Fitch-proof from ϕ. It’s a different fact because it depends on the existence of a different entity—a tree-proof versus a Fitch-proof. We can put the point here in terms of grounding or truth-making: the grounds of one involve one entity and the grounds of the other involve a different entity. On the other hand, that Bob is an unmarried individual who is a bachelor and that Bob is a bachelor who is unmarried are the same fact, and have the same grounds: Bob’s being unmarried and Bob’s being a man.

Suppose one polytheist believes in two necessarily existing and essentially omniscient gods, A and B, and defines truth as what A believes, while her coreligionist defines truth as what B believes. The two thinkers genuinely disagree as to what truth is, since for the first thinker the grounds of a proposition’s being true are beliefs by A while for the second the grounds are beliefs by B. That necessarily each definition picks out the same truth facts does not save the definition. A good definition has to be hyperintensionally correct.

Laws of nature are hyperintensional

Are the laws of nature hyperintensional? I.e., if p and q are logically equivalent, could it be that one of them is a law of nature and the other is not?

I am inclined to think so.

Argument 1: The laws of nature in our world do not make reference to particular substances. But if p is a law of nature, then let q be the proposition that p is true and either Biden is president or Biden is not presiden. Then p and q are logically equivalent, but q is not a law as it makes reference to a particular substance.

Argument 2: The laws of nature in our world are first-order. But any first-order proposition p is logically equivalent to the second-order proposition that p is true.

Argument 3: Plausibly, the values of fundamental constants like the fine-structure constant α are a part of the laws of nature. But now imagine that it turns out that the infinitely many significant digits of α express the infinite list of all arithmetical propositions and their truth values in some specific simple encoding scheme. There are two possibilities. Supposing that it is a law of nature that the digits of α have this curious property, then after verifying this property for a sufficiently large number of digits, we could know which of the remaining arithmetical propositions are true simply by measuring α to a high degree of precision. But if the law of nature is simply the brute fact that the digits are 0.007297352569…, and it just happens that these digits encode arithmetical truths in that encoding scheme, then we wouldn’t know truths by just measuring α. (Compare: Imagine a machine where you input an arithmetical proposition, and the machine flips a coin to yield an output of “True” and “False”. Even if we are so lucky that the machine always gives the right answer, that answer wouldn’t be knowledge. It would be just luck.) This means that there is a difference between having a law that says that the digits of α are determined by the arithmetical truths according to that encoding scheme and having an infinite law that simply states the digits, even though the two laws are logically equivalent (assuming the truths of arithmetic are logically necessary; if not, replace the truths of arithmetic by any sequence of hard to know logically necessary truths).

Argument 4: Laws of nature figure in explanations, but explanation is hyperintensional. The correct explanation of why the apple fell down is not that F = Gm₁m₂/r² and either Biden is president or Biden is not president, but simply that F = Gm₁m₂/r².

Argument 5: One of our best accounts of laws of nature is the Lewis-Ramsey best-systems model. But on that model it is very natural to identify the laws of nature with the axioms of the best system, and not just with propositions equivalent to the axioms of the best system.

Final note: I wonder, though, whether there is a unique proposition that expresses any given law of nature. Is there really a fact of the matter whether the law is F = Gm₁m₂/r² or F = m₁m₂(G/r²)?

Hyperintensional vagueness

“Water” and “H₂O” don’t mean the same thing in ordinary English: it is not a priori that water is H₂O. But I suspect that when a chemist uses the word “water” in the right kind of professional context, they use it synonymously with “H₂O”. Suppose this is right. But what if the chemist uses the word with fellow chemists in an “ordinary” way, telling a colleague that the tea water has boiled?

Here is a possibility: we then have a case of merely hyperintensional vagueness. In cases of merely hyperintensional vagueness, there is vagueness as to what an utterance means, but this vagueness has no effect on truth value.

I suspect that hyperintensional vagueness is a common phenomenon. Likely some people use “triangle” to mean a polygon with three angles (as the etymology indicates) and some use it to mean a polygon with three sides. (We can capture the difference by noting that to the latter group it is trivial that triangles have three sides while for the former it is a not entirely trivial theorem.) But consider a child who inherits the word “triangle” from two parents, one of whom uses it in the angle way and the other uses it in a side way. This is surely not an unusual phenomenon: much of the semantics of our language is inherited from users around us, and these users often have hyperintensional (or worse!) differences in meaning.

Hyperintensional vagueness

The typical examples of vagueness in the literature are ones where it is vague whether a subject has a property (e.g., vagueness) or whether a statement is true. But there is another kind of vagueness which we might call “hyperintensional vagueness”, which looks like it should be quite widespread. The easiest way to introduce this is in a supervaluationist context: a term has vagueness provided it has more than one precisification. But one possibility here is that all the precisifications of the term are intensionally the same. In that case, we can say that the term is merely hyperintensionally vague.

For instance, the English word “triangle” looks like it’s only hyperintensionally vague. It has two precisifications: a three-sided polygon and a three-angled polygon (the etymology favors the latter, but we cannot rely on etymology for semantics). Since necessarily all and only three-sided polygons are three-angled polygons, the two precisifications are intensionally the same.

Hyeprintensional vagueness doesn’t affect first-order logic or even modal logic, so it doesn’t get much talked about. But it does seem to be an interesting phenomenon that is even harder to get rid of than extensional or even intensional vagueness. Consider the vagueness in “bachelor”: it is extensionally vague whether a man who had his marriage annulled or the Pope is a bachelor. But even after we settle all the intensional vagueness by giving precise truth conditions for “x is a bachelor” such as “x is a never validly married, marriageable man”, there will still be hyperintensionally differing precisifications of “bachelor” such as:

• a marriageable man none of whose past marriages was valid

• a marriageable man none of whose past valid statuses was a marriage

• a human being none of whose past marriages was valid and who is a man.

This makes things even harder for epistemicists who have to uphold a fact of the matter as to the hyperintensionally correct precisification. Moreover, at this point epistemicists cannnot make use of the standard classical logic argument for epistemicism. For while that argument has much force against extensional vagueness, it has no force against hyperintensional vagueness. One could hold that there is no extensional or intensional vagueness but there is hyperintensional vagueness, but that sounds bad to me.

"John and John"

I just sent out an email to two philosophers whose first name was "John" and the email's first line said "Dear John and John". After I sent the email, I wondered to myself: Is there a fact of the matter as to which token of "John" referred to whom?

Normally, if I write an email to two people, I think about the issue of which order to list their names in, and I typically proceed alphabetically. But in this case, I didn't think about the order of names I was writing down. It is possible that I thought about the one while typing the first "John" and then about the other while typing the second "John". Would that be enough to determine which token refers to whom? Maybe. But I don't know if I did anything like that, and we may suppose I didn't.

Now:

1. John and John are philosophers.

This is true. But I didn't think of a particular one of the two while typing a particular "John" token. It seems unlikely that there be a fact of the matter as to which "John" refers to whom. But the sentence is, nonetheless, true, and hence meaningful.

Is the sentence ambiguous in its speaker meaning? If so, that's a hyperintensional ambiguity, because necessarily "x and y are Fs" and "y and x are Fs" have the same truth value. I am hesitant to say that (1) is ambiguous in its speaker meaning. (I will leave its lexical meaning alone, not to complicate things.)

Suppose that there is no ambiguity in speaker meaning, or at least none arising from the issue of which token refers to whom (maybe "philosopher" is ambiguous). Then this rather complicates compositional semantics on which the content of a whole arises from the content of the parts. For if either token of "John" in (1) has a content, the other token has the same content, since they are on par. But if the content is the same, we're not going to get out of this a sentence that means the same thing as (1) does. Suppose, for instance, the content of each token of "John" is the same as that of of "x or y", where "x" and "y" are unambiguous names for the two philosophers. Then we would have to say that (1) is equivalent to:

2. (x or y) and (x or y) are philosophers,

but in fact (1) and (2) are not equivalent—all that (2) needs for its truth is that one of x and y be a philosopher.

Maybe the solution is this. Neither "John" in (1) refers. But "John and John" is the name of a plurality. I think not, though. Here's why. Suppose instead I said: "John and the most productive member of my Department and John are all philosophers." Well, "John and the most productive member of my Department and John" is not a name, as it does not refer rigidly.

I am just a dilettante on semantics, and it would not surprise me if this was exhaustively discussed in the literature.

Two fun counterfactuals

1. If I were a better football player than everybody else, I would be very strong.
2. If everyone else were a worse football player than I, nobody would be very strong.

Both of these conditionals are true. But their antecedents are logically equivalent. This shows[note 1] that one cannot substitute logical equivalents for logical equivalents in the antecedents of counterfactuals while preserving truth value, even when one restricts one's consideration to counterfactuals with possible antecedents—i.e., counterfactuals are hyyperintensional. And this, in turn, shows that possible worlds and probabilistic accounts of counterfactuals fail.

I am not happy with this argument. I want to say that the antecedents of (1) and (2) describe families of possible worlds. So we need a interpretation of the antecedents of (1) and (2) on which, although seeming logically equivalent, these antecedents rigidify different features. Thus, the antecedent of (1) rigidifies the range of others' abilities, while the antecedent of (2) rigidifies my abilities.

It is tempting to do this with the overused distinction between semantics and pragmatics: the antecedents of (1) and (2) implicate non-equivalent things, though their propositional content is the same. But if we did that, then either we need to depart from the possible worlds or probabilistic analysis (since that analysis is in terms of truth, not implicature), or we would have to say that although (1) is true and (2) is false, or (1) is false and (2) is true, the real communication goes on at the level of implicature. But the view that (1) is true and (2) is false is implausible, as is the view that (1) is false and (2) is true. (Lewis's closeness account forces one to keep everyone else's abilities constant, so I guess he has to say that (2) is false, but surely (2) is true—not just something that implicates truly.)

Indicative and material conditionals

I will use "p→q" for the indicative conditional "if p, then q". I will use "p⊃q" for the material conditional "(not p) or q". I will say that "indicatives are material" providing that p→q and p⊃q are logically equivalent for all p and q, where a and b are logically equivalent if and only if it is necessary that (a if and only if b). I will say that p entails q provided that it is necessary that p⊃q.

Almost no philosopher thinks indicatives are material. There are very plausible counterexamples. For instance, suppose it is lightly raining in Seattle and Seattle is not having a drought. Let p be "Seattle is having heavy rain" and let q be "Seattle is having a drought". Then p⊃q, since p is false. But it seems quite wrong to say that if Seattle is having heavy rain, then Seattle is having a drought, so p→q doesn't seem to be true.

I am going to offer some arguments that indicatives are material. Say that → is non-hyperintensional provided p→q and p*→q* are logically equivalent whenever p and p* are logically equivalent and q and q* are logically equivalent. Consider the following two theses:

1. For any possible world w: (p at w) → (q at w) if and only if (p→q at w).
2. For any predicates F and G, from "Every F is a G" (where "x is an F" is more euphonious way of saying that x satisfies F) together with the assumption that c exists, it logically follows that if c is an F, then c is a G.

I will argue in S5, and using some fairly uncontroversial further premises:

3. If (1) is true and → is non-hyperintensional, then indicatives are material.
4. If (2) is true and → is non-hyperintensional, then indicatives are material.

Moreover, I will try to make plausible:

5. If (2) is true, then one has to assign the same truth value as the material conditional does to a number of paradoxical-sounding examples of indicative conditional sentences that are relevantly just like the standard alleged counterexamples to the thesis that all indicatives are material.

I think (1) and (2) are quite plausible. I don't know, however, how plausible it is that → is non-hyperintensional. My argument for (3) is very similar to more general arguments in Williamson. However, given (5), even if we don't assume that → is hyperintensional, there seems to be little advantage to denying the elegant and simple view that indicatives are material.

Argument for (5): Take my heavy rain and drought in Seattle case. Suppose that as it happens, there is no place where there presently is heavy rain. Let Fx say that x is having heavy rain. Let Gx say that x is having drought. Then all Fs are Gs. (If you think, with Aristotle, that "All Fs are Gs" requires there to be an F, then add the premise that on Venus somewhere right now there is a drought but a very, very brief heavy rain is currently occurring. I will leave out such modifications in the future.) Then by (2), we have to say that if F(Seattle), then G(Seattle):

6. If Seattle is having heavy rain, then Seattle is having drought.

And that is pretty much a standard alleged counterexample to the view that indicatives are material, of the false-antecedent sort. The case divides into two: we might suppose that Seattle is having neither drought nor heavy rain, in which case (6) is false-antecedent, false-consequent, or we might suppose that Seattle is having drought and (unsurprisingly) no heavy rain, in which case we have false-antecedent, true-consequent.

We can also use (2) to manufacture a true-antecedent, true-consequent case. Suppose that it is raining in both Seattle and the Sahara. Then the following is a standard alleged counterexample of the true-antecedent, true-consequent sort:

7. If it's raining in Seattle, then it's raining in the Sahara.

Let Fx say that x is a planet on which it is raining in Seattle, and let Gx say that x is a planet on which it is raining in the Sahara. Then every F is a G, since the only F is earth. By (2):

8. If earth is a planet on which it is raining in Seattle, then earth is a planet on which it is raining in the Sahara.

And that sounds about as paradoxical as (7). That completes my argument for (5).

Argument for (3): First we need a special case:

9. If p and q are non-contingent and → is non-hyperintensional, then p→q is logically equivalent to p⊃q.

To argue for (9), consider the following four sentences:

10. 2+2=4→2+3=5. (necessary, necessary)
11. 2+2=5→2+3=6. (impossible, impossible)
12. 2+2=5→ (2+2=5 or 1+1=2 or both). (impossible, necessary)
13. 2+2=4→2+2=5. (necessary, impossible)

If p and q are non-contingent, then they are respectively logically equivalent to the antecedent and consequent of exactly one of (10)-(13). By non-hyperintensionality of →, it follows p→q must have the same truth value as the conditional in that line. But the truth values of (10)-(13) are just as the material conditional says they are: thus, clearly, (10)-(12) are (necessarily) true and (13) is (necessarily) false. So, p→q must have the same truth value as p⊃q, assuming p and q are non-contingent.

The argument for (3) is now easy. Observe that (p at w) and (q at w) are non-contingent, even if p and q are contingent. So,

14. (p at w) → (q at w) is logically equivalent to (p at w) ⊃ (q at w).

But, plainly:

15. (p⊃q at w) is logically equivalent to (p at w) ⊃ (q at w).

From (1), (14) and (15) we conclude that:

16. (p→q at w) is logically equivalent to (p⊃q at w)

and hence indicatives are material.

Argument for (4): The most intuitive form of the argument is to assume theism, and let Fx say that x is an omniscient being that knows that p, and let Gx say that x knows that q. Then as long as p⊃q, it will be the case that every F is a G (just think about the four possible truth-value combinations). Hence:

17. If God is an omniscient being that knows that p, then God knows that q.

Since God's existence and omniscience are necessary, the antecedent and consequent are logically equivalent to p and q, respectively, and so we get p→q by non-hyperintensionality.

If we don't want to suppose there is a God, let's suppose that numbers and sets exist necessarily. Let P be the singleton set whose only member is p. Let Q be the singleton set whose only members is q. Then, let Fx say that x is greater than zero and x equals the number of truths in P. Let Gx say that x is greater than zero and x equals the number of truths in Q. Then, if p⊃q, it is easy to see that all Fs are Gs, so:

18. If one is greater than zero and one equals the number of truths in P, then one is greater than zero and one equals the number of truths in Q.

But the antecedent and consequent are logically equivalent to p and q respectively, so by non-hyperintensionality we get p→q, once again.