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.
Sunday, February 8, 2009
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.
Thursday, August 7, 2008
The Standard for Question-Begging
In a context of trying to refute the case for dialetheism, it's common and (I'd argue) entirely reasonable to accuse proponents of various solutions to the semantic paradoxes of begging the question when they assume consistency and work from there. Something I'm less sure of is when the opposite is the case...if, in making the case for dialetheism, the dialetheist assumes at least the conceptual possibility of true contradictions, when, if ever, does this beg the question against the orthodox camp, which, after all, firmly disbelieves in this conceptual possibility? And if it doesn't, why not? Or if so, how can one make sure one isn't doing this when making the case for dialetheism?
Please discuss.
Please discuss.
Monday, August 4, 2008
Can Dialetheists Make Sense of Monaletheias?
A few weeks ago I was in Australia, at the Fourth World Congress of Paraconsistency, and I saw Graham Priest give a talk entitled "Inclosures, Vagueness and Self-Reference." The point of the talk was, as far as I remember, was that he now thinks the paradoxes of vagueness are in the same 'family' of paradoxes as the so-called 'paradoxes of self-reference' (i.e. the semantic and set-theoretic paradoxes), since (he now thinks) they all fit within the Inclosure Schema. I won't be talking about that in this post, but for anyone who's interested in finding out what the IS is or how it fits into Priest's views, I'd recommend that they read his book "Beyond the Limits of Thought."
Before getting to the point of this post, let's define some terms. Since true contradictions are called 'dialetheias,' and the theory that true contradictions exist is called dialetheism, my friend Ryan decided that a good word for the opposite view, that there are no true contradictions, would be 'monaletheism.' Dialetheists believe that a single statement can have as many as two truth-values, whereas a monaletheist is someone who believes that, whether or not there are gappy statements with zero truth-values, the maximum number of truth-values that any one statement can have is one. I prefer to talk about 'monaletheism,' rather than the Law of Non-Contradiction, since if the LNC is just the formula that (∀α)¬(α ∧ ¬α), or even (∀α)¬◊(α ∧ ¬α), then a dialetheist can happily accept that the LNC is true. It just means that every time they assert a contradiction (α ∧ ¬α), they also have to accept the truth of the further contradiction [(α ∧ ¬α)] ∧ ¬(α ∧ ¬α)]. Moreover, in many paraconsistent logics--including Priest's favored logic LP--(∀α)¬(α ∧ ¬α) is a basic rule, derivable from other important basic rules like the law of identity. In fact, in Australia, in a talk entitled "Making Sense of Paraconsistent Logic - Classical Logic, Paraconsistent Logic and the Nature of Logic," Koji Tanaka said that this was a feature of all the paraconsistent logics that he approved of. When I presented my paper later that afternoon, I saw that Koji was in the audience, so when I got to this point I described this as "a feature of all logics that Koji likes," and he nodded enthusiastically.
Anyway, that's monaletheism. So by analogy to true contradictions as "dialetheias," I'll be calling statements with only one truth value 'monaletheias.' A monaletheist doesn't believe in the existence of dialetheias, but a (non-trivialist) dialetheist certainly believes in the existence of monaletheias. (So does the trivialist, technically, but of course the trivialist also disbelieves in monaletheias, which the regular dialetheist does not.) In fact, dialetheists typically believe that the vast majority of statements are monaletheias. For example, 'the Axis powers won World War II' seems to be just false and not true, and 'the Allied powers won World War II' seems to be just true and not false, so those are both monaletheias. I won't be talking much about them in this post, but we could analogously call statements without truth-values (if, unlike Priest, you think that a statement can be gappy without being glutty) nonaletheias.
OK, back to the Priest talk on vagueness and inclosure. One of his examples kind of bugged me, although at the time I couldn't quite articulate why. I'm going to take this out of context, because I'm not interested in the point he was ultimately building toward in that paper, but in whether this this example is actually a serious problem for the dialetheist. Moreover, I'm doing this from memory, so I may be misrepresenting Priest's comments. Still, whether Priest said it or not, it's an interesting point.
One of his power point slides contained a Liar-type sentence that said of itself that it was a false monaletheia. "This sentence is false and not true." If I'm remembering correctly, Priest did a few quick logical derivations to reduce this to a standard Liar, and concluded that it was both true and false. Although of course he wasn't using this terminology, the implicit problem he was dealing with was that this result seems to show that the sentence in question is both a monaletheia and not a monaletheia. (To get the same effect even more starkly, one could try standard Liar paradox reasoning on "this sentence is not true and not a dialetheia." If you think that the usual reasoning from Liars to true contradictions is sound and if, like Priest, you believe that gaps entail gluts, so the only way to not be a dialetheia is to be a monaletheia, "this sentence is true and not a dailetheia" both is and is not a monaletheia.) Priest's response in the Australia talk was to say that he didn't think this was a problem for dialetheism. "After all, the point of dialetheism was never to be consistent, it was just to contain inconsistencies..."
I think this response may be too quick, and miss the real potential objection here, which is roughly this:
If you want to even be able to express the claim that most statements are monaletheias (i.e. that 'the rate of true contradictions is very low, so we are entitled to assign a very low epistemic probability to any particular contradiction,' which is the foundation stone of the 'classical recapture,' the prohibition against multiplying contradictions beyond necessity and whatever plausibility moderate dialetheism has) you need to be able to make sense of the idea of a monaletheia, i.e. that we can assert the truth of some statements in a way that rules out our also accepting their falsity, or vice versa. If, on the other hand, you believe, based on the usual reasoning, that we can derive contradictions from Liar-like sentences, then applying uniform standards means that you have to allow the possibility that something can both be a monaletheia and not be a monaletheia, as in the two cases discussed above. If that's a possibility, then there is no meaningful concept of a monaletheia.
Anyone else see a problem here? If you do, do you see any way that the dialetheist could (non-arbitrarily) get around it without sacrificing a hugely important part of the traditional case for dialetheism? It looks to me like they can't.
Before getting to the point of this post, let's define some terms. Since true contradictions are called 'dialetheias,' and the theory that true contradictions exist is called dialetheism, my friend Ryan decided that a good word for the opposite view, that there are no true contradictions, would be 'monaletheism.' Dialetheists believe that a single statement can have as many as two truth-values, whereas a monaletheist is someone who believes that, whether or not there are gappy statements with zero truth-values, the maximum number of truth-values that any one statement can have is one. I prefer to talk about 'monaletheism,' rather than the Law of Non-Contradiction, since if the LNC is just the formula that (∀α)¬(α ∧ ¬α), or even (∀α)¬◊(α ∧ ¬α), then a dialetheist can happily accept that the LNC is true. It just means that every time they assert a contradiction (α ∧ ¬α), they also have to accept the truth of the further contradiction [(α ∧ ¬α)] ∧ ¬(α ∧ ¬α)]. Moreover, in many paraconsistent logics--including Priest's favored logic LP--(∀α)¬(α ∧ ¬α) is a basic rule, derivable from other important basic rules like the law of identity. In fact, in Australia, in a talk entitled "Making Sense of Paraconsistent Logic - Classical Logic, Paraconsistent Logic and the Nature of Logic," Koji Tanaka said that this was a feature of all the paraconsistent logics that he approved of. When I presented my paper later that afternoon, I saw that Koji was in the audience, so when I got to this point I described this as "a feature of all logics that Koji likes," and he nodded enthusiastically.
Anyway, that's monaletheism. So by analogy to true contradictions as "dialetheias," I'll be calling statements with only one truth value 'monaletheias.' A monaletheist doesn't believe in the existence of dialetheias, but a (non-trivialist) dialetheist certainly believes in the existence of monaletheias. (So does the trivialist, technically, but of course the trivialist also disbelieves in monaletheias, which the regular dialetheist does not.) In fact, dialetheists typically believe that the vast majority of statements are monaletheias. For example, 'the Axis powers won World War II' seems to be just false and not true, and 'the Allied powers won World War II' seems to be just true and not false, so those are both monaletheias. I won't be talking much about them in this post, but we could analogously call statements without truth-values (if, unlike Priest, you think that a statement can be gappy without being glutty) nonaletheias.
OK, back to the Priest talk on vagueness and inclosure. One of his examples kind of bugged me, although at the time I couldn't quite articulate why. I'm going to take this out of context, because I'm not interested in the point he was ultimately building toward in that paper, but in whether this this example is actually a serious problem for the dialetheist. Moreover, I'm doing this from memory, so I may be misrepresenting Priest's comments. Still, whether Priest said it or not, it's an interesting point.
One of his power point slides contained a Liar-type sentence that said of itself that it was a false monaletheia. "This sentence is false and not true." If I'm remembering correctly, Priest did a few quick logical derivations to reduce this to a standard Liar, and concluded that it was both true and false. Although of course he wasn't using this terminology, the implicit problem he was dealing with was that this result seems to show that the sentence in question is both a monaletheia and not a monaletheia. (To get the same effect even more starkly, one could try standard Liar paradox reasoning on "this sentence is not true and not a dialetheia." If you think that the usual reasoning from Liars to true contradictions is sound and if, like Priest, you believe that gaps entail gluts, so the only way to not be a dialetheia is to be a monaletheia, "this sentence is true and not a dailetheia" both is and is not a monaletheia.) Priest's response in the Australia talk was to say that he didn't think this was a problem for dialetheism. "After all, the point of dialetheism was never to be consistent, it was just to contain inconsistencies..."
I think this response may be too quick, and miss the real potential objection here, which is roughly this:
If you want to even be able to express the claim that most statements are monaletheias (i.e. that 'the rate of true contradictions is very low, so we are entitled to assign a very low epistemic probability to any particular contradiction,' which is the foundation stone of the 'classical recapture,' the prohibition against multiplying contradictions beyond necessity and whatever plausibility moderate dialetheism has) you need to be able to make sense of the idea of a monaletheia, i.e. that we can assert the truth of some statements in a way that rules out our also accepting their falsity, or vice versa. If, on the other hand, you believe, based on the usual reasoning, that we can derive contradictions from Liar-like sentences, then applying uniform standards means that you have to allow the possibility that something can both be a monaletheia and not be a monaletheia, as in the two cases discussed above. If that's a possibility, then there is no meaningful concept of a monaletheia.
Anyone else see a problem here? If you do, do you see any way that the dialetheist could (non-arbitrarily) get around it without sacrificing a hugely important part of the traditional case for dialetheism? It looks to me like they can't.
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