The last post was about counterpossible conditionals as applied to questions about moral rules and "ought implies can." It occurs to me that there's another obvious application of such conditionals, in arguments for atheism.
As is (for better or for worse) standard for discussions about God in analytic philosophy of religion, I'll assume that by "God" we mean an entity that combines the three properties of omnipotence, omniscience and omnibenevolence. There seem to me to be two excellent arguments against the existence of such an entity...one from the "Stone Paradox" and one from the problem of evil.
Start with the "Stone Paradox," taken from the classical formulation about stones too heavy for God to lift. (My favorite formulation of it is, of course, the Simpsons version: "Could Jesus microwave a burrito so hot that he himself could not eat it?") If you say that there are no limits to what God can do, you certainly can't turn around and say that He can't make a stone of a certain level of heaviness. If you say "but there is no such level of heaviness!", you've just abandoned the original premise that God is omnipotent....surely, an omnipotent being can make stones as heavy, burritos as hot, as He likes.
The Stone Paradox for omnipotence seems to exact structurally parallel Russell's Paradox for unrestricted comprehension in naive set theory. If the latter shows either that naive set theory is a hopelessly inconsistent theory that must be abandoned (as the orthodox story goes, although I'm sure that some readers of the blog would beg to differ with the "must be abandoned" part), then the Stone Paradox shows the same about the theory that something is omnipotent.* In both cases, ad hoc restrictions to save consistency--"well, maybe for every *logically possible* property, there's a set of things that have that property....", etc.--fall pretty flat.
The second excellent argument against the existence of God** is the argument from evil. A being that was omniscient would know about the Holocaust, one that was omnipotent would have it in His power to send bolts of lightning down onto the train tracks leading to Auschwitz, and one that was omnibenevolent would certainly want to do so. The structure of the argument looks pretty clear: If God existed, gratuitous evil wouldn't exist, but it does exist, so God doesn't.
The various traditional defenses all, it seems to me, fail pretty spectacularly. Many of them are morally repugnant on their face, some of them seem obviously inapplicable to natural evil, or rest on un-argued and non-obvious assumptions about the nature of good and evil, or on strange accounts of free will that seem to go well beyond the most wild-eyed libertarian accounts out there, or in some other way seem implausible on their face. Even the ones that seem establish that X wouldn't exist without evil and X is a good thing never come even close to clearing the bar of showing that both (a) X is more good that Auschwitz, etc., are bad, so that the trade is worth it, and (b) in order to have X, we need anything even close to *as much* suffering as in fact exists. To anyone who's ever taken an Intro to Philosophy class, the details of this argument are fairly familiar, so, if you agree with me about this, you're the one I'm talking to, and we'll move on from here.***
So, both of the major, classical, historically influential arguments from atheism mentioned here seem to be good ones.
But wait.
The formulation of the argument from evil necessarily involves a counterfactual premise..."If God existed, then such-and-such would be the case."
But earlier, didn't we establish that it's not logically possible that God could exist?****
So, if you understand counterfactuals in the Lewissian way, in terms of truth-preservation over the closest *possible* worlds in which the antecedent is true, then that conditional is completely trivial, since there are no possible worlds in which God exists. "If God existed, then everything would be exactly the way that it is now" and "if God existed, there would be a planet full of unicorns" would have precisely the same status as "if God existed, then gratuitous evil would not exist." That doesn't seem to be the case. That last conditional seems to be clearly, importantly and non-trivially true. Some one who is talked out of theism by the argument with that conditional as its first premise, far from being taken to have been irrationally suckered in by a silly argument from something trivial, should be praised for having taken a long stride in the direction of rationality.*****
If that's true, you'd better be willing to extend your logic to include a non-trivial counterpossible conditional connective, and, if you have a possible-worlds analysis of counterfactuals, you'd better be willing to entertain impossible worlds as well.
*Small prediction, BTW...someone correct me if I'm wrong here...but to the best of my knowledge, there aren't any Christian Dialetheists yet, but sooner or later, there will be. I'm sure that somewhere down the line young theist grad student interested in formal logic at Notre Dame, too honest to deny the contradictions entailed by his theology, will have a light bulb go off over his head during a phil of logic seminar, and decide to marry the teachings of Thomas Aquinas and Graham Priest.
**Remember, we're talking about the traditional omni-omni-omni God here. The average northern college town Episcopal Priest or Reform Rabbi would describe the object of their worship in a much different way. Whether any such liberal theologies are themselves plausible or not is an entirely different subject.
***If you don't agree with me, I'd be happy to follow up in the comments section, but right now, for the sake of making the logical point being built up to, I'm just gesturing at the atheistic arguments, and trying to make my case (about conditionals) to those who already accept them.
****Hey there, future Christian Dialetheists going back to read this blog post from 2009 predicting your existence! Say hi to everyone in the future for me!
*****Of course, you might be tempted to say something like the following. 'Look, given the paradoxes of omnipotence, that conditional is trivial, but it's a useful second-level-of-attack, since *if* those paradoxes could be satisfactorily solved, then the argument from evil would be an excellent objection to theism.' To which I'd have to reply, 'OK, but dude, given your assumptions, that itself was a counterpossible conditional! There's no escaping them.'
Monday, May 25, 2009
Sunday, May 17, 2009
Counterpossible Conditionals and Unbreakable Moral Rules
As we were all taught as baby philosophers, ought implies can. As we grew up, a few of us had little adolescent rebellions in which we denied this principle, but most of us kept the faith. There really does seem to be a deep absurdity involved in telling people that they should have done things that they couldn't have possibly done.
If you find this line of thought compelling, you're likely to be sympathetic to the corollary principle that we might want to call "ought implies can fail to." Norms that it's impossible to violate might seem just as irrational as norms that it's impossible to carry out. For one thing, neither can guide our actions, which might seem central to the idea of a norm. For another, if it seems unfair to blame people for things outside of their control, it might plausibly seem equally unreasonable to praise people for things outside of their control. The objects of blame in the former case have done nothing to deserve the blame, and the objects of praise in the latter case have done nothing to deserve the praise.*
In any case, in most discussions I've heard about the relationship between free will and moral responsibility, the two principles seem to be systematically run together into something like an "ought implies both can and can fail to" principle, the standard line being that it's only rational to praise or blame people for things that are matters of choice.
So there seems to be at least some reason to suppose that we should reject in principle the notion that there could be moral rules that we can't possibly break. Neil Sinhababu, however, has argued against this**, giving the example "don't kill God." Now, I'm not sure about that example. (Personally, I'm kinda sympathetic to Bakunin's claim that if God existed, it would be necessary to overthrow him.) However, nothing much hinges on this. We can substitute in the rule "don't beat people to death with round square blunt objects."
Now, OA entails ~P~A, so if we say that there is no rule against beating people to death with round square blunt objects (since there can't be unbreakable rules), it follows by Modus Tollens and Double Negation that it's permissible to beat people to death with round square blunt objects. That might seem terribly counter-intuitive. Surely (to the extent that we can imagine it***), when we imagine someone being beaten to death with a round square blunt object, we feel instinctive moral revulsion about it. Intuitively, it seems terribly wrong to say that such an action would be morally acceptable.
I'd argue that there's a good, principled way to have our cake and eat it too here. We can keep "ought implies can fail to" *and* accommodate the relevant moral intuition. A nice, simple way of doing both would be to say that, as a matter of fact, we have no such obligation, but if it *were* possible to bludgeon people to death with round square objects, then we *would* have an obligation to refrain from doing so.
The problem, of course, is that a certain type of orthodox logician would tell you that this conditional is trivial, so it's equally true that "if it were possible to bludgeon people to death with round square objects, we would have an obligation to kill as many people as possible that way." If we understand counterfactual conditionals (as Lewis did) in terms of truth-preservation across possible worlds, and if there are no possible worlds at which the antecedent is true, then the conditional is true regardless of what the consequent is.
In his article "Impossible Worlds: A Modest Approach," Daniel Nolan has a nice argument against both this Lewissian account and against more radical approaches according to which we have to abandon classical logic in order to reason about counterpossibles. On the one hand, he argues, if there are no actually existing counter-examples to, for example, Disjunctive Syllogism, there's no reason to revise our logic in order to deprive it of this bit of inferential power instead of extending it with a counterpossible conditional connective. On the other hand, we clearly *do* non-trivially reason about counterpossibles all the time. In fact, a few moments of thought should show that it would be damned hard to do philosophy at all if we couldn't do so. In interesting debates about truth, reference, metaphysics, epistemology and a lot of other things, it's surely routinely the case that if our side of the argument is right, the claims of the other side are not merely false but *necessarily* false. That said, an integral part of what it means to argue against opposing positions is that we reason about the commitments of the other theories and draw out unacceptable consequences. ("Well, if Meingongianism were true, then...") We act as if some such conditions were true, and others were false.**** Moreover, we can even, if we're so inclined, adapt Lewis' account of counterfactuals and tweak it slightly, so that counterfactual conditionals are true so long as, in the closest worlds (whether possible or impossible) at which their antecedents are true, their consequents are as well. Presumably, "possible worlds are always closer than impossible ones, all else being equal" is a plausible principle here, where "closer" means something like "similar in the contextually relevant respects." Of course, there may be tricky issues about how to cash all this out, but it doesn't seem *trickier* in the impossible cases than the possible ones.
In any case, I'd argue that the plausibility of the "ought implies can fail to" principle, combined with the intuitive force of the claim that it would be morally wrong to do certain impossible things, gives us yet another excellent reason to endorse non-trivial counterpossible conditionals.
*Of course, it could be objected that we could get around this by de-linking the notion of deserving praise from the notion of fulfilling obligations. After all, we don't normally go around go around praising each other for passively fulfilling obligations which we have no reason to expect each other to break. The following conversation, for example, would be fairly strange. Mark: "Hey, did you kill anyone on your way to the Philosophy Department today?" Ryan: "Um.....No." Mark: "Wow, man, you're so awesome. God, you're awesome!") The obvious response is that "praise" in the relevant sense means not "making a big deal of the fact that someone has done something good" but something more like "thinking that someone has done something good." The former is something we only do when people have done something good that we don't expect of them as a matter of course....we don't make a big fuss about praising people for things we would be horrified if they *didn't* do, just as we don't make a big fuss about blaming people for trivial human failures to fulfill very difficult obligations. That doesn't suffice to say that at least mild praise and blame *in the relevant sense* don't apply in both cases.
**In a talk at the University of Miami entitled "How Double-Humeans Can Make Room For Error"
***Of course, if one takes "imagining" to mean "having a detailed mental picture of how it would look," we can't imagine beating someone to death with a thousand-sided object either, but that seems "imaginable" in the relevant sense.
****It could be objected that what's going on is that for the sake of argument, we're assuming that the other side's position *was* possibly true, so as not to beg the question by starting from the position that it was impossible. Fine. This would not, however, get around Nolan's point. After all, "if Meingongianism were possibly true, then..." *is itself* a counterpossible conditional if one takes non-existent objects to be impossible.
If you find this line of thought compelling, you're likely to be sympathetic to the corollary principle that we might want to call "ought implies can fail to." Norms that it's impossible to violate might seem just as irrational as norms that it's impossible to carry out. For one thing, neither can guide our actions, which might seem central to the idea of a norm. For another, if it seems unfair to blame people for things outside of their control, it might plausibly seem equally unreasonable to praise people for things outside of their control. The objects of blame in the former case have done nothing to deserve the blame, and the objects of praise in the latter case have done nothing to deserve the praise.*
In any case, in most discussions I've heard about the relationship between free will and moral responsibility, the two principles seem to be systematically run together into something like an "ought implies both can and can fail to" principle, the standard line being that it's only rational to praise or blame people for things that are matters of choice.
So there seems to be at least some reason to suppose that we should reject in principle the notion that there could be moral rules that we can't possibly break. Neil Sinhababu, however, has argued against this**, giving the example "don't kill God." Now, I'm not sure about that example. (Personally, I'm kinda sympathetic to Bakunin's claim that if God existed, it would be necessary to overthrow him.) However, nothing much hinges on this. We can substitute in the rule "don't beat people to death with round square blunt objects."
Now, OA entails ~P~A, so if we say that there is no rule against beating people to death with round square blunt objects (since there can't be unbreakable rules), it follows by Modus Tollens and Double Negation that it's permissible to beat people to death with round square blunt objects. That might seem terribly counter-intuitive. Surely (to the extent that we can imagine it***), when we imagine someone being beaten to death with a round square blunt object, we feel instinctive moral revulsion about it. Intuitively, it seems terribly wrong to say that such an action would be morally acceptable.
I'd argue that there's a good, principled way to have our cake and eat it too here. We can keep "ought implies can fail to" *and* accommodate the relevant moral intuition. A nice, simple way of doing both would be to say that, as a matter of fact, we have no such obligation, but if it *were* possible to bludgeon people to death with round square objects, then we *would* have an obligation to refrain from doing so.
The problem, of course, is that a certain type of orthodox logician would tell you that this conditional is trivial, so it's equally true that "if it were possible to bludgeon people to death with round square objects, we would have an obligation to kill as many people as possible that way." If we understand counterfactual conditionals (as Lewis did) in terms of truth-preservation across possible worlds, and if there are no possible worlds at which the antecedent is true, then the conditional is true regardless of what the consequent is.
In his article "Impossible Worlds: A Modest Approach," Daniel Nolan has a nice argument against both this Lewissian account and against more radical approaches according to which we have to abandon classical logic in order to reason about counterpossibles. On the one hand, he argues, if there are no actually existing counter-examples to, for example, Disjunctive Syllogism, there's no reason to revise our logic in order to deprive it of this bit of inferential power instead of extending it with a counterpossible conditional connective. On the other hand, we clearly *do* non-trivially reason about counterpossibles all the time. In fact, a few moments of thought should show that it would be damned hard to do philosophy at all if we couldn't do so. In interesting debates about truth, reference, metaphysics, epistemology and a lot of other things, it's surely routinely the case that if our side of the argument is right, the claims of the other side are not merely false but *necessarily* false. That said, an integral part of what it means to argue against opposing positions is that we reason about the commitments of the other theories and draw out unacceptable consequences. ("Well, if Meingongianism were true, then...") We act as if some such conditions were true, and others were false.**** Moreover, we can even, if we're so inclined, adapt Lewis' account of counterfactuals and tweak it slightly, so that counterfactual conditionals are true so long as, in the closest worlds (whether possible or impossible) at which their antecedents are true, their consequents are as well. Presumably, "possible worlds are always closer than impossible ones, all else being equal" is a plausible principle here, where "closer" means something like "similar in the contextually relevant respects." Of course, there may be tricky issues about how to cash all this out, but it doesn't seem *trickier* in the impossible cases than the possible ones.
In any case, I'd argue that the plausibility of the "ought implies can fail to" principle, combined with the intuitive force of the claim that it would be morally wrong to do certain impossible things, gives us yet another excellent reason to endorse non-trivial counterpossible conditionals.
*Of course, it could be objected that we could get around this by de-linking the notion of deserving praise from the notion of fulfilling obligations. After all, we don't normally go around go around praising each other for passively fulfilling obligations which we have no reason to expect each other to break. The following conversation, for example, would be fairly strange. Mark: "Hey, did you kill anyone on your way to the Philosophy Department today?" Ryan: "Um.....No." Mark: "Wow, man, you're so awesome. God, you're awesome!") The obvious response is that "praise" in the relevant sense means not "making a big deal of the fact that someone has done something good" but something more like "thinking that someone has done something good." The former is something we only do when people have done something good that we don't expect of them as a matter of course....we don't make a big fuss about praising people for things we would be horrified if they *didn't* do, just as we don't make a big fuss about blaming people for trivial human failures to fulfill very difficult obligations. That doesn't suffice to say that at least mild praise and blame *in the relevant sense* don't apply in both cases.
**In a talk at the University of Miami entitled "How Double-Humeans Can Make Room For Error"
***Of course, if one takes "imagining" to mean "having a detailed mental picture of how it would look," we can't imagine beating someone to death with a thousand-sided object either, but that seems "imaginable" in the relevant sense.
****It could be objected that what's going on is that for the sake of argument, we're assuming that the other side's position *was* possibly true, so as not to beg the question by starting from the position that it was impossible. Fine. This would not, however, get around Nolan's point. After all, "if Meingongianism were possibly true, then..." *is itself* a counterpossible conditional if one takes non-existent objects to be impossible.
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.
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.
Wednesday, October 8, 2008
Classical Logic and Inconsistent but Non-Trivial Fiction
This is a short excerpt from my paper on "What You Don't Need Paraconsistency or Pluralism For." The basic claim of the paper, as mentioned in the last post, was that while it is certainly true that if dialetheism is right, classical logic is wrong, since rules like Disjunctive Syllogism won't be universally truth-preserving, none of the traditional (non-dialetheist) motivations for paraconsistency give the classical monist (i.e. the logical monist who thinks that classical logic is the correct logic) any reason to abandon that position in favor of either logical pluralism or paraconsistent monism, since they can happily accept the relevant claims and accommodate them in their classical monist framework. Here's the bit about fiction:
"Start with inconsistent fiction. That it is possible for the writers of fiction to create inconsistencies, whether on purpose (as in Graham Priest’s whimsical short story ‘Sylvan’s Box,’ in which Priest himself, appearing as a character, finds that the late Richard Routley owned a box that both contained and did not contain a statuette, at the same time and in the same sense) or, as is vastly more common, by accident, is clear enough. To take a classic example, in Arthur Conan Doyle’s Sherlock Holmes stories, Watson’s war wound is always referred to in the singular, strongly suggesting that he only has one of them, but sometimes it is on his left shoulder (and thus, presumably not on his right shoulder) and sometimes it is on his right shoulder (and thus not on his left shoulder). Moreover, in both the Conan Doyle case and the Priest case, not everything is true in the worlds of the stories. It is not true in the world of ‘Sylvan’s Box’ that Graham Priest agrees with everything in this paper, and it is not true in the world of Conan Doyle’s Sherlock Holmes stories that Professor Moriarty’s ultimate aim was to raise up Cthulu and Azathoth to destroy humanity.
"How can the classical monist make sense of this? Easily, and in pretty much the same way that a conscientious paraconsistentist would have to make sense of it. Regardless of what your view is on fictional characters--whether they have no ontological status at all, or are abstract objects, or are Meinongian objects that fail to have the property of existing but have all sorts of other properties, like ‘being referred to in philosophy papers’--statements like, for example, “Sherlock Holmes lives on 221B Baker Street” are, strictly speaking, false. Go to 221B Baker Street in London, and you will not find Holmes or Watson there, but a bank, one with a special employee in charge of responding to correspondence written to Holmes. To the extent that we regard such statements as true--and this is true regardless of our stance on the ontological status of fictional characters--it is because we read them as containing implicit fiction operators. Sherlock Holmes, if he exists as an abstract object, has, for example, the property of being written about by Arthur Conan Doyle, but he most certainly does not have the property of actually living on 221B Baker Street. He only has this property in fiction.
"Once we realize this, however, the apparent conflict between classical monism and our obvious ability to non-trivially reason about fictional inconsistencies disappears. If Dr. Watson had the property of both being written about by Arthur Conan Doyle and not being written about by Conan Doyle, in precisely the same sense, that would be a real contradiction symbolized as (α ∧ ¬α), that would indeed entail any and every β in classical logic. If, however, Dr. Watson has the property of having in his war wound on his left shoulder and having it on his right (i.e. not on his left) shoulder in fiction, then acknowledging that that’s true commits us not to (α ∧ ¬α), but F(α ∧ ¬α), which is a very different thing.
"When we extend classical logic, to include such operators I see no reason why we should mimic the rules about what goes on outside of them when we make the rules for what goes on within their scope. A classical monist can happily admit that there are cases where F(α ∨ β) and F(¬α) are both true, but F(β) is not. More radically still, we can allow inferences within the scope of the fiction operators to be not only ‘paraconsistent’ but non-adjunctive. That is to say, F(α) and F(¬α) need not always entail F(α ∧ ¬α). This preserves the intuitive difference between the ‘Sylvan’s Box’ case, in which it is true in the world of the story that the box is both empty and non-empty, and the Sherlock Holmes case, in which both ‘Watson’s war wound is on his left shoulder rather than his right shoulder’ and ‘Watson’s war wound is on his right shoulder rather than his left shoulder’ are both true, but ‘Watson’s war wound is on both of his shoulders’ does not seem to be true. Notice that even here, we have not revised classical logic one whit, as evidenced by the fact that even if F(α) and F(¬α) do not jointly entail F(α ∧ ¬α), they continue to jointly entail (F(α) ∧ F(¬α))."
"Start with inconsistent fiction. That it is possible for the writers of fiction to create inconsistencies, whether on purpose (as in Graham Priest’s whimsical short story ‘Sylvan’s Box,’ in which Priest himself, appearing as a character, finds that the late Richard Routley owned a box that both contained and did not contain a statuette, at the same time and in the same sense) or, as is vastly more common, by accident, is clear enough. To take a classic example, in Arthur Conan Doyle’s Sherlock Holmes stories, Watson’s war wound is always referred to in the singular, strongly suggesting that he only has one of them, but sometimes it is on his left shoulder (and thus, presumably not on his right shoulder) and sometimes it is on his right shoulder (and thus not on his left shoulder). Moreover, in both the Conan Doyle case and the Priest case, not everything is true in the worlds of the stories. It is not true in the world of ‘Sylvan’s Box’ that Graham Priest agrees with everything in this paper, and it is not true in the world of Conan Doyle’s Sherlock Holmes stories that Professor Moriarty’s ultimate aim was to raise up Cthulu and Azathoth to destroy humanity.
"How can the classical monist make sense of this? Easily, and in pretty much the same way that a conscientious paraconsistentist would have to make sense of it. Regardless of what your view is on fictional characters--whether they have no ontological status at all, or are abstract objects, or are Meinongian objects that fail to have the property of existing but have all sorts of other properties, like ‘being referred to in philosophy papers’--statements like, for example, “Sherlock Holmes lives on 221B Baker Street” are, strictly speaking, false. Go to 221B Baker Street in London, and you will not find Holmes or Watson there, but a bank, one with a special employee in charge of responding to correspondence written to Holmes. To the extent that we regard such statements as true--and this is true regardless of our stance on the ontological status of fictional characters--it is because we read them as containing implicit fiction operators. Sherlock Holmes, if he exists as an abstract object, has, for example, the property of being written about by Arthur Conan Doyle, but he most certainly does not have the property of actually living on 221B Baker Street. He only has this property in fiction.
"Once we realize this, however, the apparent conflict between classical monism and our obvious ability to non-trivially reason about fictional inconsistencies disappears. If Dr. Watson had the property of both being written about by Arthur Conan Doyle and not being written about by Conan Doyle, in precisely the same sense, that would be a real contradiction symbolized as (α ∧ ¬α), that would indeed entail any and every β in classical logic. If, however, Dr. Watson has the property of having in his war wound on his left shoulder and having it on his right (i.e. not on his left) shoulder in fiction, then acknowledging that that’s true commits us not to (α ∧ ¬α), but F(α ∧ ¬α), which is a very different thing.
"When we extend classical logic, to include such operators I see no reason why we should mimic the rules about what goes on outside of them when we make the rules for what goes on within their scope. A classical monist can happily admit that there are cases where F(α ∨ β) and F(¬α) are both true, but F(β) is not. More radically still, we can allow inferences within the scope of the fiction operators to be not only ‘paraconsistent’ but non-adjunctive. That is to say, F(α) and F(¬α) need not always entail F(α ∧ ¬α). This preserves the intuitive difference between the ‘Sylvan’s Box’ case, in which it is true in the world of the story that the box is both empty and non-empty, and the Sherlock Holmes case, in which both ‘Watson’s war wound is on his left shoulder rather than his right shoulder’ and ‘Watson’s war wound is on his right shoulder rather than his left shoulder’ are both true, but ‘Watson’s war wound is on both of his shoulders’ does not seem to be true. Notice that even here, we have not revised classical logic one whit, as evidenced by the fact that even if F(α) and F(¬α) do not jointly entail F(α ∧ ¬α), they continue to jointly entail (F(α) ∧ F(¬α))."
Saturday, October 4, 2008
What You Don't Need Paraconsistency or Pluralism For
Yesterday, I gave a Miami Forum talk entitled "What You Don't Need Paraconsistency Or Pluralism For." Here's the abstract:
"If dialetheism is right, classical logic is wrong, at least for the context of reasoning about the domain of the inconsistent. As such, classical monism--the claim that there is One True Logic, and that logic is classical--must be rejected in favor of either paraconsistent monism or some sort of logical pluralism. Many paraconsistent logicians, however, reject the claim that there are real contradictions ‘out there in the world,’ but think that there are good reasons short of that to reject classical monism in favor of some sort of paraconsistent approach. I argue that standard defenses of this claim fail to hit their target, examining and rejecting claims that classical logic somewhow gets negation wrong, and showing how standard motivations for paraconsistency from inconsistent fiction, counterpossible conditionals and so on offered by Greg Restall, J.C. Beall, Newton Da Costa and others can be accommodated in a classical monist framework."
"If dialetheism is right, classical logic is wrong, at least for the context of reasoning about the domain of the inconsistent. As such, classical monism--the claim that there is One True Logic, and that logic is classical--must be rejected in favor of either paraconsistent monism or some sort of logical pluralism. Many paraconsistent logicians, however, reject the claim that there are real contradictions ‘out there in the world,’ but think that there are good reasons short of that to reject classical monism in favor of some sort of paraconsistent approach. I argue that standard defenses of this claim fail to hit their target, examining and rejecting claims that classical logic somewhow gets negation wrong, and showing how standard motivations for paraconsistency from inconsistent fiction, counterpossible conditionals and so on offered by Greg Restall, J.C. Beall, Newton Da Costa and others can be accommodated in a classical monist framework."
Thursday, August 21, 2008
Along The Same Lines
Chris Mortensen, in his book "Inconsistent Mathematics," says this about motivations for dialetheism (or 'strong paraconsistency,' in Mortensen's terminology):
"...the idea finds roots in an older view, that change especially motion is contradictory, which can be traced through Engels and Hegel to Zeno and Heraclitus, and has recently been revived, e.g. by Priest..."
The "e.g." strikes me as a bit odd. Who else besides Priest advocates this view, exactly?
"...the idea finds roots in an older view, that change especially motion is contradictory, which can be traced through Engels and Hegel to Zeno and Heraclitus, and has recently been revived, e.g. by Priest..."
The "e.g." strikes me as a bit odd. Who else besides Priest advocates this view, exactly?
Monday, August 11, 2008
Wikiality?
The wikipedia page on dialetheism contains* the following, from the section on "Formal Consequences," immediately after a brief run-through of the standard proof that contradictions explode in classical logic:
"Any system in which any formula is provable is trivial and uninformative; this is the motivation for solving the semantic paradoxes. Dialethesists solve this problem by rejecting the principle of explosion, and, along with it, at least one of the more basic principles that lead to it, e.g. disjunctive syllogism or transitivity of entailment, or disjunction introduction."
Now, on a nit-picky level, I object to say that "this is the motivation for solving the semantic paradoxes," as if there weren't any other motivations, but I suppose that's debatable. (It's certainly not *my* motivation for wanting to solve them, but one could maybe argue that I'm just eccentric that way and that the use of the definite article there is still basically accurate.) More importantly, though, I'm pretty sure that the last bit is flat-out wrong. Certainly, there are paraconsistent logics (e.g. the ones most used for the computer database applications) where disjunction introduction is eliminated, but (a) there seems to be no obvious reason why dialetheism being true would mean that disjunction introduction wasn't universally truth-preserving, and (b) to the best of my knowledge, there aren't any dialetheists who reject disjunction introduction (much less the transitivity of entailment), whereas all the ones I know about reject disjunctive syllogism, for obvious reasons...if a statement P can be both true and false, then P could be true, P v Q would also be true (since one of its disjuncts would be true), ~P could be true and Q could just be false. On the face of it, it seems hard to see how disjunctive syllogism *could* be valid given the assumption that there are true contradictions, or, given this, what motivation there would be for a dialetheist to reject disjunction introduction. In fact, even if some pragmatically useful formal systems disregard it, I don't know of the existence of s*any* logical monist, dialetheist or otherwise, who don't think that disjunction introduction isn't present in whatever they think the One True Logic is, or any logical pluralists who think that there aren't any logics adequate for at least some contexts that contain disjunction introduction.
Anyone have any information otherwise? Is this just a gap in my knowledge of the field? Any dialetheists out there who reject disjunction introduction? Anyone know about any that reject it?
*'Contains,' in this context, of course means 'contains on Monday, August 11th, 2008.' We are talking about Wikipedia here, so it could be edited to say something entirely different at any mnute.
"Any system in which any formula is provable is trivial and uninformative; this is the motivation for solving the semantic paradoxes. Dialethesists solve this problem by rejecting the principle of explosion, and, along with it, at least one of the more basic principles that lead to it, e.g. disjunctive syllogism or transitivity of entailment, or disjunction introduction."
Now, on a nit-picky level, I object to say that "this is the motivation for solving the semantic paradoxes," as if there weren't any other motivations, but I suppose that's debatable. (It's certainly not *my* motivation for wanting to solve them, but one could maybe argue that I'm just eccentric that way and that the use of the definite article there is still basically accurate.) More importantly, though, I'm pretty sure that the last bit is flat-out wrong. Certainly, there are paraconsistent logics (e.g. the ones most used for the computer database applications) where disjunction introduction is eliminated, but (a) there seems to be no obvious reason why dialetheism being true would mean that disjunction introduction wasn't universally truth-preserving, and (b) to the best of my knowledge, there aren't any dialetheists who reject disjunction introduction (much less the transitivity of entailment), whereas all the ones I know about reject disjunctive syllogism, for obvious reasons...if a statement P can be both true and false, then P could be true, P v Q would also be true (since one of its disjuncts would be true), ~P could be true and Q could just be false. On the face of it, it seems hard to see how disjunctive syllogism *could* be valid given the assumption that there are true contradictions, or, given this, what motivation there would be for a dialetheist to reject disjunction introduction. In fact, even if some pragmatically useful formal systems disregard it, I don't know of the existence of s*any* logical monist, dialetheist or otherwise, who don't think that disjunction introduction isn't present in whatever they think the One True Logic is, or any logical pluralists who think that there aren't any logics adequate for at least some contexts that contain disjunction introduction.
Anyone have any information otherwise? Is this just a gap in my knowledge of the field? Any dialetheists out there who reject disjunction introduction? Anyone know about any that reject it?
*'Contains,' in this context, of course means 'contains on Monday, August 11th, 2008.' We are talking about Wikipedia here, so it could be edited to say something entirely different at any mnute.
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