Pushing language too far

Quantum Mechanics has borne much fruit. Is this fruit poisonous? Probably not. But is the total weight of the fruit borne by Quantum Mechanics even or odd when rounded to the nearest pound? Unlike the question whether the fruit is poisonous, the question about whether the weight is even or odd is just silly—it pushes the metaphor too far, in the sense that there is no natural meaning in the metaphor to be assigned to any answer to it.

The formation rules for meaningful metaphorical discourse do not have unrestricted compositionality. While "The fruit borne by Quantum Mechanics is sometimes bitter" is probably meaningful, "The fruit borne by Quantum Mechanics is sometimes elongated" has no meaning unless we choose to assign one to it. The latter sentence takes the metaphor too far.

A sign of metaphor being pushed too far is that classical logic has the appearance of failing. It appears to be neither true that the fruit of Quantum Mechanics is sometimes elongated nor that the fruit of Quantum Mechanics is never elongated, contrary to bivalence. To ask which is the case is to be silly. The same apparent failure of classical logic can occur when we make too involved inferences from a metaphorical claim, for instance when we conclude that Quantum Mechanics is partly made of carbon atoms, because only plants bear fruit and all plants are partly made of carbon atoms.

Here is a different kind of metaphor (I heard this metaphor—though I don't remember if it was identified as a metaphor—in discussion at INPC): The average plumber has 2.3 children. Let's press on. Stipulate, no doubt contrary to fact, that exactly half of all plumbers are male and exactly half of all plumbers are female. So, is the average plumber male? No. Is the average plumber female? No. Is the average plumber human? Certainly (supposing there are no alien plumbers). So, the average plumber is a human who is neither male nor female. Now, maybe there are such rare humans (this is a difficult question about the metaphysics of sex), but since by stipulation none of them are plumbers, surely the average plumber isn't one of them. Wondering about this bit of weirdness is, however, silly. It is taking the metaphor of the average plumber too far. Once we start saying that the average plumber is a human who is neither male nor female, we take our metaphor beyond the narrow region of the space of sentences where it makes sense.

Now, some metaphysicians, including me, think that in an important sense there are no tables or chairs. There are only particles or fields arranged tablewise or chairwise. It is a tough problem for these metaphysicians to defend their own use of ordinary language about tables and chairs—their saying things like "There are ten chairs in the room."

I think our ordinary language about artifacts has some things in common with metaphorical language. Take something like the question of how much of the wood of the table you can replace, and in how large chunks, while maintaining the same table? I think one can have a sense of discomfort with the question. After enough fast replacement of wood, one is tempted both to deny that one has the same table and to deny that one has a distinct table. The question seems to be a matter for our decision—much as it is a question for our decision whether we count last year's average plumber, with his/her/its 2.29 children, as the same individual as this year's average plumber with his/her/its 2.30. In the plumber case, the decision is a decision what to understand identity across time in the metaphor as standing for (maybe by saying that the average plumber is the same last year as this year we want to metaphorically signal that there was no en masse replacement of plumbers). And, I think, the ordinary folk think there is something a bit humorous and unserious about pressing the question whether after the replacement we have the same table, just as they would in the plumber case.

These things suggest that when we ask whether we have the same table, we can be pushing language too far, just as we sometimes do in metaphorical cases. And this, in turn, suggests that we should not take "There is a table here" at face value.

I will stop short of saying that our ordinary language of tables and chairs is literally metaphorical, that "their existence is metaphorical". Instead I'd like to say that our ordinary language of tables and chairs behaves in certain important respects metaphorically. Among these respects is that we should not expect arbitrary compositions of such language to be meaningful, and we should not expect to have classical logic hold on the surface level.

I actually think classical logic holds even in cases of metaphor. But it holds not at the linguistic level, but at the level of the propositions expressed by the metaphorical claims. "The average plumber has 2.3 children" expresses the same proposition as some sentence like "The average of the numbers of children had by plumbers is 2.3", and the latter sentence better reflects the logical structure of the proposition.

Deviant logic

In the chapter on deviant logic in his philosophy of logic book, Quine makes the claim that:

1. The deviant logician changes the subject rather than disagreeing with classical logic.

Thus, the logician who denies excluded middle is not using the words "or" and "not" to indicate disjunction and negation. She is, perhaps with good reason (though Quine is sceptical of that), using some other connectives. Thus, when she denies "not not p entails p", she is not disagreeing with us when we assert "not not p entails p". The basic thought running behind this is that:

2. The rules of classical logic are grounded in the meanings of the logical connectives (using "connectives" very widely to include negation, quantifiers, etc.)

and so any departure from the rules is a change of subject.

There is a powerful kind of argument against deviant logic here. Claims (1) and (2) seem to tell us that it is not really possible to disagree with classical logic without self-contradiction. I am either using my words in the sense that they have in classical logic, in which case I had better not disagree with classical logic on pain of contradiction, or else I am using the words in a different sense and hence not disagreeing.

I now want to describe a class of apparently non-classical logics that do not change the subject. Thus, either a deviant logician doesn't always change the subject, or else these logics are not actually deviant. The idea is this. We have rules like:

• You can infer p from p.
• If r is a conjunction of p with q, then you can infer r from p and q, p from r and q from r (conjunction introduction and elimination).
• If r is a negation of a negation of p, then you can infer p from r.
• If you can infer p and a negation of p from r, and s is a negation of r, then you can infer s from r.

And so on. The interesting thing is the second rule does not tell us that p and q have a conjunction. And indeed that is how I am imagining the system deviating from classical logic. We simply disallow certain conjunctions, negations, etc.—there will be sentences that perhaps have no negation, and pairs of sentences that perhaps have no conjunction. If we represent the language along the lines of First Order Logic, there may be cases where "A" is a sentence and "B" is a sentence but "A and B" counts as malformed. The rules of disallowing combinations may take all sorts of forms. For instance, we might simply prohibit any sentences that contain a double negation. This would result in a severe intuitionist-type limitation on what can be proved.

The logic, thus, has standard classical rules in an important sense. The rules are correct whenever they can be applied—whenever there are output sentences that work. The subject is not changed—"or" means or, "and" means and, and "not" means not—but it can be a substantive claim whether for a pair of sentences A and B, there is a sentence that we might wish to denote "A or B".

This restriction does not count as a change of subject. Indeed, Quine himself notes that there can be languages which are incapable of translating all the English truthfunctionally and quantificationally connected sentences, and he seems to think that these languages do have connectives that mean the same thing as English ones. In fact, English itself has restrictions on the formation of sentences. Past several levels of embedding, there just is no way to make distinctions. You probably can't express "(A or (A and not (B or (B and (C or D) and E) or F) and not A))" in English. Yet English does not have a deviant logic. It's just that English's logic is likely incomplete.

There are two ways of looking at this. One way is to say that what I have offered is a family of genuinely deviant logics that don't change the subject, and hence that Quine's argument against deviant logics fails. The other way—and it is what I prefer—is to say that what I have given is in an important sense a family of non-deviant, and even classical, logics, but one that differs from First Order Logic.

I think it could be a good thing to define the connectives in terms of valid inference (perhaps understood in terms of entailment). For instance, one might say that:

3. A partially-defined functor C that takes a pair of sentences p and q into a new sentence C(p,q) is a conjunction if and only if you can validly infer p as well as q from C(p,q) and C(p,q) from the pair of premises p and q whenever C(p,q) is defined.

(We also need an extension to wffs.) If we do this, then excluded middle is true by definition in the following sense:

4. Whenever p is a disjunction of q with a negation of q, then p is true.

But no claim is made that every sentence has a negation or that every pair of sentences has a disjunction. That would be a substantive claim. But whenever a sentence has a negation and can be disjoined with that negation, the result of the latter disjunction is true. That is a claim that is true by definition of "negation" and "disjunction".

This also lets one stipulate into place new connectives like tonk. Tonk is a connective such that one can infer q from "p tonk q" and "p tonk q" from p. The problem with tonk is that once one has the connective, it seems one derive anything (e.g., 1+1=2, so 1+1=2 tonk q, so q, for any q). But not quite. One can only derive everything with tonk if one adds the additional thesis that sufficiently many pairs of sentences have tonks. For instance, if we grammatically restrict tonking so that one is only allowed to tonk a sentence with itself, we can continue to have a sound logic.

Why care about such logics? Well, they might be helpful with the Liar Paradox. They might provide a way of doing the sort of thing that Field does to resolve the Liar by invoking a deviant logic but within a logic that has all the classical rules of inference.

I think Sorensen's "The Metaphysics of Words" [PDF] is very relevant to the above.

Logical fatalism -- the options

This post is just an attempt by me to work something out for myself. Maybe it doesn't interest anybody else.

People who accept an Aristotelian open future because of concerns about logical fatalism do so on the strength of the intuition that if I freely do something, it was possible for me not to do it, and that:
(*) If it is now the case that I will do A, then it is now necessary that I will do it.

So, the question is: What are the options for getting out of the argument from logical fatalism. One option is just to deny (*). This, I think, is by far the best option. Call someone who accepts (*) an "Open Futurist". What logical options does an Open Futurist have?

Well, to see that, let's sketch a logical fatalism argument based on (*):

1. If it is now necessary that I will do A, then I will not be freely doing A. (Premise, justified via Principle of Alternate Possibilities)
2. If it is now necessary that I will not do A, then I will not be freely refraining from doing A. (Premise, same justification)
3. If it is now necessary that I will do A, then I will not be freely refraining from doing A. (Premise)
4. If it is now necessary that I will not do A, then I will not be freely doing A. (Premise)
5. If it is now necessary that I will do A, then I will not be freely doing A and I will not be freely refraining from doing A. (By (1) and (3) and the principle that if p→q and p→r, then p→q&r.)
6. If it is now necessary that I will not do A, then I will not be freely doing A and I will not be freely refraining from doing A. (By (2) and (4) and the principle that if p->q and p→r, then p→q&r.)
7. If I will do A, then I will not be freely doing A and I will not be freely refraining from doing A. (By (5) and (*).)
8. If I will not do A, then I will not be freely doing A and I will not be freely refraining from doing A. (By (6) and (*).)
9. Either it is the case that I will do A or it is not the case that I will do A. (By Law of Excluded Middle)
10. If it is not the case that I will do A, then I will not do A. (Premise)
11. Therefore, either it is the case that I will do A or it is the case that I will not do A. (By (9) and (10) and the principle that if p or q, and q→r, then p or r.)
12. Therefore, I will not be freely doing A and I will not be freely refraining from doing A. (By (7), (8) and (11), together with the principle that if p→r and q→r and (p or q), then r.)

So what are the at all plausible options if we accept (*) (which we shouldn't)?
(I) Compatibilism: Deny (1) (and (2)--they are surely in the same boat).
(II) Intuitionistic Logic: Deny the Law of Excluded Middle, thereby allowing the denial of (9).
(III) Not-will / will-not distinction: Deny (10), holding on to Law of Excluded Middle.
(IV) Deny the principle that if p→r and q→r and (p or q), then r.
(V) Deny one of the other rules of inference used.

I think (V) is not attractive--all the other rules of inference seem really hard to deny. Option (IV) is pretty radical. It means that we will no longer accept arguments like: "If Bob is telling the truth, George is guilty. If Fred is telling the truth, George is guilty. Either Bob or Fred is telling the truth. So, George is guilty." The principle denied in (IV) follows from the axioms of intuitionistic logic, and I think is also going to hold in supervaluationist settings.

If this is right, then an exhaustive list of our at all plausible options with regard to the logical argument for fatalism is:

1. Denial of free will
2. Denial of (*)
3. Compatibilism
4. Denial of excluded middle
5. Denial of the claim that not-will implies will-not

I rank the attractiveness of these as follows, in order of most to least attractive: 2, 3, 5, 4, 1. Why list 1 last? Because free will is central to the things that matter most in life. How to justify the rest of the ordering? Well, we should be least willing to give up general rules of all reasoning, like excluded middle. We should be more willing to give up rules about particular kinds of reasoning, such as the tensed logic rule that not-will implies will not. We should be more willing yet to give up intuitions about modal or concrete concepts, since there things get difficult by everybody's lights, and so 2 and 3 are even more attractive as options than 5, 4 and 1. Why take 2 as more attractive than 3? Well, that's a judgment call on my part--I find compatibilism deeply implausible, and I suspect that most people who find (*) plausible find the denial of compatibilism even more plausible.

Excluded Middle and an Open Future

Some people deny the Law of Excluded Middle (LEM--for all p, p or not-p) because they are convinced it leads to fatalism. But they really shouldn't deny LEM.

Suppose Helga is convinced that utilitarianism is true. You offer Helga a reductio argument against utilitarianism on the assumption that the hedonistic theory of happiness holds and another reductio argument against utilitarianism on the assumption that the hedonistic theory of happiness does not hold. Helga accepts both reductios and comes to deny hedonistic utilitarianism and non-hedonistic utilitarianism, but continues to accept utilitarianism. Pressed on how Helga's new position squares with logic, Helga asserts that based on her belief that utilitarianism holds, and her new beliefs that if hedonism holds, utilitarianism is not true, and if hedonism doesn't hold, utilitarianism is not true, she has concluded that LEM does not hold. There seems to be something irrational about this. Surely, she should either find fault with at least one of the reductios or abandon her belief in utilitarianism. It is hard to imagine premsies whose plausibility should trump LEM.

Arguments the depend on LEM are not, I think, uncommon in philosophy. If Molinism is true, evil and the existence of God are compatible (by Plantinga's free will defense). If Molinism is not true, evil and the existence of God are compatible (by Adams' free will defense). Hence, evil and the existence of God are compatible. It is pretty likely that Helga uses LEM-based arguments in other contexts, and it is pretty likely that the defender of the Open Future who denies LEM also uses LEM in other contexts.

Could they both say that LEM applies in some contexts (e.g., non-normative ones in Helga's case, or in ones that do not involve the future in the freedom case) but not others? Yes. But once we denied the plausible view that LEM follows from the meaning of the words "or" and "not", and denied the general intuition that between p and not-p tertium non datur, it seems that we have undercut the grounds we could have for thinking LEM holds even in those contexts in which it is supposed to hold in. Besides, the defender of the Open Future who denies LEM presumably does so on the basis of something like a temporalized modal logic according to which if p already holds, then not-p is no longer possible. But surely the principles of classical non-temporal non-modal logic are more plausible and more deeply embedded in our thinking than those of temporalized modal logic.

Anyway, it seems much better to hold on to LEM, and just deny the principle that if not(will(p)), then will(not-p), where "will(p)" means p will hold. The principle that if not(will(p)), then will(not-p) is a dubious one if we see "will" as a modal-type operator, maybe akin to "would" except for being a one-place operator, and that is precisely how we will see "will" if we have presentist or growing-block intuitions. Moreover, it is a principle that is less central to our thinking than LEM, particularly because it applies only to our thinking about the future, while LEM applies to all our thinking. It seems clear to me that this is what the person impressed by the argument for logical fatalism should say, boldly holding that there is a fact of the matter whether Jones will mow the lawn tomorrow: it is false that Jones will mow the lawn tomorrow, just as it is false that he will fail to mow the lawn tomorrow. And God's omniscience will be unrestricted: he knows that it is false that Jones will mow the lawn and that it is false that he will not mow the lawn.

Of course, it's best to hold on to both LEM and if not(will(p)), then will(not-p).