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.

Informative characterizations

It is hard to characterize an “informative characterization”. Here is an instructive illustration.

Ned Markosian in his famous brutal composition paper says that an informative, or non-trivial, characterization of when the xs compose something is one that is not synonymous with the statement that the xs compose something. But by that definition, here is a non-trivial characterization of when the xs compose something:

• water is H₂O and the xs compose something.

This statement is not synonymous with the statement that the xs compose something. Nor are the two statements provably equivalent. Nor are they a priori equivalent. But they are metaphysically necessarily equivalent.

Van Inwagen in Material Beings proceeds seemingly more restrictively. He wants a characterization of when the xs compose something that doesn’t use mereological vocabulary. But here is such a characterization:

• the xs have the property expressed by the actual world’s English phrase “compose something”.

This characterization mentions mereological vocabulary, but doesn’t use it. And if we want, we can avoid mentioning mereological vocabulary as well:

• the xs have the property referred to in the second bulleted item in this post in the actual world.

Obviously, none of these characterizations of “compose something” are informative.

A quick heuristic for testing conjunctive accounts

Suppose someone proposes an account of some concept A in conjunctive form:

• x is a case of A if and only if x is a case of A₁ and of A₂ and ... of A_n.

It may seem initially plausible to you that anything that is a case of A is a case of A₁,...,A_n. There is a very quick and simple heuristic for whether you should be convinced. Ask yourself:

• Suppose we can come up with a case where it's merely a coincidence that x is a case of A₁,A₂,...,A_n. Am I confident that x is still a case of A then?

In most cases the answer will be negative, and this gives you good reason to doubt the initial account. And to produce a counterexample, likely all you need to do is to think up some case where it's merely a coincidence that A₁,A₂,...,A_n are satisfied. But even if you can't think of a counterexample, there is a good chance that you will no longer be convinced of the initial account as soon as you ask the coincidence question. In any case, if the answer to the coincidence question is negative, then the initial account is only good if there is no way for the conditions to hold coincidentally. And so now the proponent of the account owes us a reason to think that the conditions cannot hold coincidentally. The onus is on the proponent, because for any conditions the presumption is surely that they can hold coincidentally.

Consider for instance someone who offers a complicated account of knowledge:

• x knows p if and only if (i) x believes p; (ii) p is true; (iii) x is justified in believing p; (iv) some complicated further condition holds.

Without thinking through the details of the complicated further condition, ask the coincidence question. If there were a way for (i)-(iv) to hold merely coincidentally, would I have any confidence that this is a case of knowledge? I suspect that the answer is going to be negative, unless (iv) is something weaselly like "(i)-(iii) hold epistemically non-aberrantly". And once we have a negative answer to the coincidence question, then we conclude that the account of knowledge is only good if there is no way for the conditions to hold coincidentally. So now we can search for a counterexample by looking for cases of coincidental satisfaction, or we can turn the tables on the proponent of the account of knowledge by asking for a reason to think that (i)-(iv) cannot hold coincidentally.

Most proposed accounts crumble under this challenge. Just about the only account I know that doesn't is:

• x commits adultery with y if and only if (i) x or y is married; (ii) x is not married to y; (iii) x and y have sex.

Here I answer the coincidence question in the positive: even if (i)-(iii) are merely coincidentally true (e.g., x believes that he is married to y but due to mistaken identity is married to someone else), it's adultery.

A method for testing definitions

I have a new method for testing definitions. Read a definiens to someone, out of context, and ask her what she thinks the definiendum is. If she doesn't come up with something pretty close to the definiendum, you've got reason to think the definition is bad.

One can also do this as a thought experiment, though it's probably less effective that way. What does "justified true belief with no false lemmas" define? Answer: nothing other than justified true belief with no false lemmas. (Maybe you were trying to define knowledge?) What does "Sex between two people at least one of whom is married and who are not married to each other" define? Answer: adultery. (Right!)

Vagueness, definitions and translations

If we are to define a vague term, the definiens will need to be vague in exactly the same way as the definiendum is. But it is exceedingly improbable that the contextual profile of the vagueness of the definiens would exactly match the contextual profile of any complex definiendum that we could practically state, or maybe even that we could state in principle.

For instance, suppose we're trying to define "short". Now, "short" has a certain contextual vagueness profile which specifies, perhaps vaguely, in what context what lengths do and do not count as short and in what way, Either there is vagueness all the way up or at some level we get definiteness.

Suppose first that at some level we get definiteness. For simplicity, suppose it's after the first level of vagueness. Then for any context C, there will be precise lengths x₁ and x₂ such that anything shorter than x₁ is definitely short, anything of length between x₁ and x₂ is vaguely short, and anything longer than x₂ is definitely non-short. These precise lengths will be some exact real numbers determined by our actual linguistic practices—which things we've called "short" and which we haven't. It is exceedingly unlikely that we could construct a definiendum which will make the definitely/vaguely/definitely-not transitions in exactly the same spot. Suppose, for instance, we define "is short" as "has small length." Well, small will have its own vagueness profile, defined by a different set of social practices. It is exceedingly unlikely that this vagueness profile would exactly correspond to that of "is short", so that the exact point of transition between being definitely short and vaguely short should be the point of transition between being definitely of small length and being vaguely of small length.

Suppose now that we have vagueness all the way up. Then we're going to have arbitrarily long predications like "a is vaguely definitely vaguely vaguely vaguely definitely definitely vaguely definitely short." And which such predications apply to which objects will be determined by our complex linguistic practices surrounding "is short". It is, again, exceedingly unlikely that our complex linguistic practices surrounding some other term, like "has small length" would in every context match those of "is short".

For exactly the same reason, except when the users of one language self-consciously use a term as an exact translation of a term used by another language, it is exceedingly unlikely that we could find an exact simple translation of a vague term from one language to another, and for the same reasons as above, a complex translation is also unlikely. For we would have to exactly match the vagueness profile, and since the social practices underlying the different languages are subtly and unsubtly different, it is very unlikely we would succeed.

It may be worse than that. It may well be that no two people have the same vagueness profile in their homophonic terms, except when both defer in their usage to exactly the same community. And they rarely do.

In practice, when translating and giving dictionary definitions, we are satisfied with significant similarity between vagueness profiles.

Spinoza on truth and falsity

In Actuality, Possibility, and Worlds, I attribute to Spinoza the view that no belief is false (though I think i also emphasize that nothing rides on the accuracy of the historical claim).  Rather, there are more or less confused beliefs, and in the extreme case there are empty words--words that do not signify any proposition.

I was led to the attribution by a focus on passages, especially in Part II of the Ethics and in the Treatise on the Emendation of the Intellect, that insist that every idea has an ideatum, that of which it is the idea, and hence corresponds to something real.  The claim that every idea has an ideatum is central to Spinoza's work.  It is a consequence of the central 2 Prop. 7 (which is the most fecund claim outside Part I) which claims that the order and connection of ideas is the order and connection of things, and it is also a consequence of the correspondence of modes between attributes.

These passages stand in some tension, however, to other passages where Spinoza expressly talks of false ideas, which are basically ideas that are too confused to be adequate or to be knowledge (the details won't matter for this post).

I think it is easy to reconcile the two sets of passages when we recognize that Spinoza has an idiosyncratic sense of "true" and "false".  In Spinoza's sense, an idea is true if the individual having the idea is right to have it, and it is false if the individual having it is not right to have it (cf. Campbell's "action-based" view of truth, but of course Campbell will not go along with Spinoza's internalism), where the individual is right to have the idea provided that she knows the content, or knows it infallibly.  And Spinoza, rationalist that he is, has an internalist view of knowledge, where knowledge is a matter of clarity and distinctness and a grasp of the explaining cause of the known idea.

Hence, Spinoza uses the words "true" and "false" in an internalist sense.  But we do not.  "True" as used by us expresses a property for which correspondence to reality is sufficient, and "false" expresses a property incompatible with such correspondence.  Since every belief has an idea (in Spinoza's terminology) as its content, and according to Spinoza every idea corresponds to reality, namely to its ideatum, it follows that in our sense of the word, Spinoza holds that every belief is true and no belief is false.

The ordinary notion of truth includes ingredients such as that correspondence to reality is sufficient for truth and that truth is a good that our intellect aims at.  Spinoza insists on the second part of this notion, and finds it in tension with the first (cf. this argument).  But the first part is, in fact, the central one, which is why philosophers can agree on what truth is while disagreeing about whether belief is aimed at truth, knowledge, understanding or some other good.

So, we can say that in Spinoza's sense of "true", it is his view that some but not all beliefs are true.  And in our sense of "true", it is his view that all beliefs are true.  The sentence "Some beliefs are false" as used by Spinoza would express a proposition that Spinoza is committed to, while the sentence "Some beliefs are false" as used by us would express a proposition that Spinoza is committed to the denial of.

This move of distinguishing our sense of a seemingly ordinary word like "true" from that of a philosopher X is a risky exegetical move in general. Van Inwagen has argued libertarians should not hold that compatibilists have a different sense of the phrase "free will".  But I think there are times when the move is perfectly justified.  When the gap between how X uses some word and how we use it is too great, then we may simply have to concede that X uses the word in a different sense.  This is particularly appropriate in the case of Spinoza whose views are far from common sense, whose philosophical practice depends on giving definitions, and who expressly insists that many disagreements are merely apparent and are simply due to using the same words in diverse senses.  (Actually, I also wonder if van Inwagen's case of free will isn't also a case where the phrase is used in diverse senses.  Even if so, we should avoid making this move too often.)

Addendum: This reading is in some tension with 1 Axiom 6 which says that a true idea must agree with its ideatum. While strictly speaking, this sets out only a necessary condition for a true idea, and hence does not conflict with what I say above, it is not unusual for Spinoza to phrase biconditionals as mere conditionals. If we read 1 Axiom 6 as a biconditional, then maybe we should make a further distinction, that between the truth of an idea and truth of a believing. We take the truth of a believing to be the same as the truth of the idea (or proposition) that is the object of the believing. But Spinoza distinguishes, and takes more to be required for the truth of a believing. We then disambiguate various passages. The problem with this is that on Spinoza's view, the believing is identical with the idea. But nonetheless maybe we can distinguish between the idea qua believing and the idea qua idea?

Analyses: a hypothesis

In philosophy journals, one occasionally sees things like this:

Necessarily, x is an F if and only if x satisfies each of the following n conditions:
(i) ...
(ii) ...
(iii) ...
(iv) ...
...

I hypothesize that every philosophical claim of this form that has ever been made in print by a Western philosopher with the number of conditions n greater than or equal to 4 is:

1. false, and/or
2. stipulative, and/or
3. circular, and/or
4. redundant.

By "circular" I mean that Fness is implicitly or explicitly found in the conditions. By "redundant" I mean that one of the conditions is entailed by the others.

My evidence for the hypothesis is inductive. I have never seen a correct, non-stipulative, non-circular and non-redundant set of necessary and sufficient conditions for anything philosophical where there are more than three conditions.

It could be that the hypothesis is false. Is there a counterexample?