Sunday, February 8, 2009

Why Gil Harman Should Embrace Paraconsistent Logic (If He Wants To Make Any Sense At All)

Last month, I got a chance to see Gilbert Harman speak at my University. We hold a big ol’ Epistemology Conference every year and Harman was the invited keynote speaker. I was excited to get a chance to interact with him philosophically, since a few months earlier—after I gave a Miami Forum presentation on "What You Don’t Need Paraconsistency or Pluralism For"—my friend Brian* recommended that I read the chapter on logic in Harman’s book, “Change in View.” I read it, and thought that while parts of it were interesting and valuable, his view on the role of logic in belief revision ultimately didn’t work. The discussion here in Miami drove the point home even more for me.

Harman’s position is that logic has far less to do with reasoning that is generally supposed. “Logic,” he said in Miami, “is not a theory of reasoning.” Believing the logical consequences of our views isn’t rationally mandatory, and sometimes is rationally impermissible, and it can be perfectly acceptable to believe inconsistencies. Some of his reasons for that make sense to me, although I would draw a far more modest conclusion from them, which is that there’s a lot more to rationality than logic.

There are two ways in which we tend to think logic constrains good reasoning. The first is that it’s irrational to continue believe two inconsistent things, and the second is that if it’s rational to believe P, and P entails Q, we have a rational obligation to believe Q. Harman rejects both claims. He isn’t a dialetheist, so he doesn’t think inconsistent pairs of claims are ever both true, but he thinks that sometimes we have better things to do than sit around resolving contradictions within our belief set, and in any case everyone’s belief sets are likely to contain all sorts of contradictions that we aren’t aware of.

(The latter claim, at least, seems obviously true to me, but it also seems obvious that it has no consequences pertaining to the relationship with logic and reasoning. If more orthodox assumptions about logic and reasoning are correct, you have a duty to reject inconsistent beliefs once you realize that they are inconsistent, just as you have a rational duty to reject false beliefs in general once you realize that they’re false, but inadvertent inconsistency is another matter entirely.**)

Moreoever, he rejects the claim that we have any sort of rational duty to go around believing everything that follows from your beliefs logically. Put in terms of explicit, conscious belief, this makes perfect sense. First of all, there’s the matter of what Harman calls “clutter avoidance”—even if you only start out believing one claim, an infinite number of uninteresting things follow from that one claim, and you don’t want to clutter up your mind with all sorts of irrelevancies that happen to follow from things you believe. (Think about the infinite number of things that could be inferred from “Snow is white” using only, e.g. Double Negation, or Disjunction Addition.) Plus, you want to do other things with your time—climb mountians, solve math problems, drink whiskey, get laid, paint murals, contemplate the inevitability of the heat death of the universe and so on***—that compete with “making inferences from stuff you believe” for your time and energy. Again, all that seems obviously true to me.

When Harman goes beyond this to the claim that it is sometimes irrational to believe things that logically follow from beliefs we are entitled to hold, however, from my perspective his position tips over into incoherence. Combining the two positions just mentioned—that you don’t have to believe everything that follows from everything you believe, and that it’s sometimes OK to knowingly let inconsistencies stay in your belief set—he says that, although he believes that classical logic gives us the right account of the logical consequence relation (see below) and hence that from a contradiction, anything follows—you have a positive rational duty to refrain from “irresponsibly” exploiting your contradictory beliefs (beliefs that, keep in mind, Harman thinks you are entitled to hold) in order to draw spurious conclusions (conclusions that, remember, Harman thinks really do logically follow from those beliefs.)

Now, to be fair to Harman, intuitively there’s something compelling about the sort of case he has in mind. To take an example from the chapter on logic in “Change in View,” the Liar Paradox—
This sentence is not true.
--is a very hard problem for the philosophy of logic. After all, if it’s true, it isn’t, if it isn’t it is, if it’s false, it isn’t, so it is, and if it’s “neither true nor false,” then it isn’t true, so it is. The intuitive reaction of most non-specialists on hearing it is that, roughly, “there’s clearly something wrong with it, but I have no idea what. Where’s the trick?” Specialists fill shelves with thick books arguing for various solutions, and most of them aren’t terribly convincing. Perhaps, Harman seems to suggest, it is acceptable to tentatively believe that it both is and isn’t true, since that’s where all the options seem to lead. Still, even if we have good reason to believe this particular contradiction, we surely don’t have good reason to believe that, say, Hitler won World War II, or that the sky is green, or that Gil Harman goes out at night to fight crime in a spandex costume. The fact that these conclusions can be inferred from the premise that the Liar sentence both is and isn’t true in Classical Logic surely doesn’t make it rational to believe them, even if it is rational to believe that premise.

Many bright logicians have shared this intuition, and crafted systems of logic (“paraconsistent” or inconsistency-tolerant logics) in which the inference from contradictions to any and every aribtrary conclusion is reigned in. Harman, however, will have none of this. He seems to reject non-classical logics in general, and in particular claimed in his Miami talk that paraconsistent logic arises from “a confusion between logic and reasoning.” Sure, it’s not rational to believe that Hitler won World War II (even though it is rational to believe some contradictions), and sure, the former follows from the latter in Classical Logic, but that doesn’t mean that there’s anything wrong with Classical Logic, it just means that logic and reasoning are two different things.

This doesn’t work.

Even if you grant Harman for the sake of argument, or because you think he’s right (as I tend to on much of this) that there is a distinction between logic and reasoning, and that logic is not a theory of reasoning, and that in particular people don’t have a general obligation to conclude everything that logically follows from their beliefs (clutter avoidance, the fact that they may have better things to do with their time, etc.), none of that gets you out of accepting trivialism (the position that everything is true) as a result of marrying the claim that it is sometimes rationally acceptable to have explicitly inconsistent beliefs and the claim that Classical Logic gives us the uniquely correct account of valid argumentation.

After all, I think, even if he’s entirely right about clutter avoidance, time conflicts and the rest of it, I think a large sense remains in which logic imposes an important constraint on reasoning. If you believe P, you are rationally entitled to do continue to do so, and you (correctly) believe that Q logically follows from it, this is brought to your attention and the question arises of whether or not you should accept Q (i.e. in conditions in which you choose to think about the question, don’t have anything better to do, etc., but have, say, chosen to devote an hour to arguing about logic with a friend from graduate school), I can’t see any remotely plausibly grounds on which anyone could deny that you have a rational obligation to assent to Q. (Or, put more carefully, you are rationally obliged to either stop believing in P or start believing in Q—as always, as Tim McGrew used to remind us in his seminars back at Western Michigan, one man’s Modus Ponens is another’s Modus Tollens.) In this situation, where clutter avoidance doesn’t come up (and indeed, the sorts of things you are concluding aren’t uninteresting consequences of your beliefs like iterations of Disjunction Addition on some uninteresting original claim, but extremely interesting claims like “Hitler Won World War II” or “Gil Harman will wake up tomorrow, finding himself transformed into a giant cockroach”), and you happily choose to spend the time and energy on considering the question, if it is rational for you to believe the question, and the conclusion really does follow, I can see no principled basis whatsoever on which anyone could claim that you are doing anything irrational by embracing the conclusion. The package of views that he holds:
(1) Classical Logic is the uniquely correct account of the logical consequence relation,
(2) It is sometimes reasonable to believe contradictions, &
(3) It would be irrational to infer just anything from these contradictions,
…simply does not add up to a sensible, internally coherent position. If he wants to hold a minimally rationally defensible position, Harman should either abandon key features of his theory of reasoning, or embrace paraconsistency.

*aka “The Riverboat Rambler”
** The former claim is one that I disagree with in a certain sense, although spelling out the alternative is a bit tricky. My claim is basically, that it’s sometimes perfectly rational to believe, for example, that some inconsistent claim is “almost true” or “as close to the truth as we can get right now” or “the sort of thing instrumentally useful to pretend that it’s true for certain purposes,” and that of course we may not have any rational duty to drop everything and devote our lives to discovering the exact (consistent) truth of the matter, but that none of that adds up to it ever being rational to knowingly believe something you know to be inconsistent.
***My examples, not Harman’s.

6 comments:

Deleet said...
This comment has been removed by the author.
Deleet said...

Interesting. I also with many of the thoughts. Although I would draw the distinction in another way. (Between logic/strict reasoning and pragmatic reasoning.)

Keep it up!

PS. You seem to have posted this post twice. They are nearly identical.

-Deleet

Gil said...

Dear Ben,

Thanks! I really enjoyed the discussion in Miami. I think I agree with almost everything you say here. On the other hand, I do not think that a paradox like the liar paradox makes it reasonable to believe an explicit contradiction of the form P&~P. One thing interesting about the liar paradox is that it may seem hard (at least at first) to see what assumptions are involved that imply the contradiction. And that makes it hard to figure out how to avoid the inconsistency. It also remains obscure to me why you take your discussion to provide a reason for a paraconsistent logic!

Gil

Ben said...

Deleet,

Sorry...deleted the redundant post. Thanks for catching it.

Dr. Harman,

In the chapter on "Logic and Reasoning" in "Change of View," you talk about the "Logical Inconsistency Principle"--"Logical inconsistency is to be avoided." (p. 11) Later on in the chapter, in a section entitled "Unavoidable Inconsistency and the Liar Paradox," you say:

"To see that the Logical Inconsistency Principle has its exceptions, observe that sometimes one discovers one's views are inconsistent and does not know how to revise them in order to avoid inconsistency without great cost. In that case the best response may be to keep the inconsistency and try to avoid inferences that exploit it. This happens in everyday life whenever one simply does not have time to figure out what to do about a discovered inconsistency. It can also happen on more reflective occasions. For example, there is the sort of inconsistency that arises when one believes that not all one's beliefs could be true. One might well be justified in continuing to believe that and each of one's other beliefs as well.

"There are also famous logical paradoxes..." (pp. 15-16)

In the discussion afterwards, you discuss the Liar Paradox and say that your favored solution is to say that there are exceptions to the T-Schema. My reading of it--why you brought it up in this context--was that your idea was that someone who hadn't discovered this solution yet might be entitled to tentatively continue to hold inconsistent beliefs about the Liar Paradox--that this might be a case of "unavoidable inconsistency" as indicated by the title of the section. I gather from your comment that I misread your view, and that you don't think anyone is entitled to hold inconsistent beliefs about the Liar...?

In any case, at least as of the time you wrote "Change of View" (and, unless I seriously misunderstood your talk in Miami, this is your current view as well), your position is that there are exceptions to the principle that logical inconsistency is to be avoided. I.e. it is rationally permissible for me to both believe that all of my beliefs are true and that at least some of them must not be true.

In these cases of permissibly inconsistent beliefs, then, your view is that (as quoted above) "the best response may be to keep the inconsistency and try to avoid inferences that exploit it." My problem with this, given your hostility to paraconsistent logics and apparent belief in the unique correctness of Classical Logic, is that I don't see what grounds you could have for denying that I am entitled to make inferences that "exploit" my inconsistent beliefs to derive irrelevant things from them. Given that I'm allowed to believe inconsistencies, and that classical logic really does give us the correct account of logical consequence, if I reason from the inconsistencies I believe in to any conclusion I so choose (since, after all, I can prove anything if I start from inconsistent premises and use the rules of classical logic), am I being unreasonable in some way? Do you have any grounds to criticize me for doing so?

After all, the limits you put on the Logical Implication Principle--clutter avoidance, time conflicts, etc. (and on all of that, I think you make your case well, and I agree with you) don't seem to provide any grounds for anyone to deny that I'm entitled, in any reflective moments that I choose to devote to thinking about this, to draw any conclusions that I like based on my inconsistent beliefs....after all, according to you (a) I'm entitled to hold those beliefs and (b) those arbitrary conclusions really do follow from those beliefs, as they must, given that you think that classical logic gives us the correct account of logical consequence and non-classical, inconsisteny-tolerant logics are a result of confusion and, as I think you said in Miami, a less worthwhile way to spend one's time than playing Sodoku. I have a hard time seeing how, if (a) and (b) are both were both true, I could fail to be entitled to "exploit" my inconsistent beliefs to derive anything that I like. On the other hand, if you altered your views to the extent of endorsing a paraconsistent account of the logical consequence relation, according to which only some things and not others follow from any particular inconsistency, this problem would disappear.

Ben

Gil said...

Dear Ben,

Two comments:

"I gather from your comment that I misread your view, and that you don't think anyone is entitled to hold inconsistent beliefs about the liar."

No. All I said was: "I do not think that a paradox like the liar paradox makes it reasonable to believe an explicit contradiction of the form P&~P." But the liar paradox may show someone that his or her beliefs are inconsistent and the person may continue to be justified in holding those beliefs.

"Given that I'm allowed to believe inconsistencies and that classical logic really does give us the correct account of logical consequence, if I reason from the inconsistences I believe in to any conclusion I so choose (since, afer all, I can prove anything if I start from inconsistent premises and use the rules of classical logic), am I being unreasonable in some way?"

Yes! You are conflating logical principles with principles of reasoning. You shouldn't do that. That's my point! Without the assumption that logical principles are principles of reasoning, the premise that inconsistent beliefs imply any conclusion you choose, it does not follow that from inconsistent beliefs you can legitimately come to believe any conclusion you choose.

Without the assumption that logical principles are principles of reasoning, there is no motivation for paraconsistent logic.

Gil

Gil said...

Oops, my next to last sentence should read as follows:

"Without the assumption that logical principles are principles of reasoning, it does not follow that from inconsistent beliefs you can legitimately come to believe any conclusion you choose."