In Mark Colyvan's article "Who's Afraid Of Inconsistent Mathematics?", he starts things off with a snarky "five line proof" of Fermat's Last Theorem.

1. The Russell Set is both a member of itself and not a member of itself.

2. (From 1, and Conjunction-Elimination): The Russell Set is a member of itself.

3. (From 2, and Disjunction-Addition): Either the Russell Set is a member of itself or Fermat's Last Theorem is true.

4. (From 1, and Conjunction-Elimination): The Russell Set is not a member of itself.

5. (From 3, 4 and Disjunctive Syllogism): Fermat's Last Theorem is true.

Now, he argues, the first premise is easily demonstratable given naive set theory. Why is it, then, that no mathematician ever tried to prove FLT in this way, and that FTL was considered un-proved until Andrew Wiley came along with a proof running over a hundred pages and employing all kinds of sophsiticated, recently developed mathematical machinery?

Part of the explanation is that the first premise relies on naive set theory, and mathematical orthodoxy has, in no small part *because* of Russell's Paradox, abandoned the naive conception of sets in favor of the hierchical conception of ZFC set theory. Fair enough, but what about the three decades between the discovery of Russell's Paradox and the consistent re-formulation of set theory in terms of ZFC's cumulative hierarchy? Why didn't anyone try to prove FTL this way during the lag period?

The lesson Colyvan draws is more or less standard for historical examples used in apologias for paraconsistency:

Mathematicians during the lag period implicitly treated the contradiction exposed by Russell's Paradox as entailing some conclusions but not others. In other words, the standards of reasoning in play in the mathematical community of the time are best captured by paraconsistent logic rather than classical logic. Moreover, so goes the story, they were not being irrational in implicitly employing these standards of reasoning. Therefore, paraconsistent logic represents the appropriate logic for at least some domains of inquiry, at least sometimes.

I'm picking on Colyvan's paper because it's an extraordinarily clear, clean, chemically pure example of its type, but this sort of thing is a very common manuever in the literature on paraconsistency. We can call these Retroactive Implicit Paraconsistency (RIP) claims. To make an RIP claim is to say that, in the past, some investigator or community of investigators about some topic reasoned in a way best codified by paraconsistent logic. (Not, of course, that they actually reasoned using formally explicit deductive arguments of any kind, or that they were aware that their practices were, in some, sense in conflict with the norms of classical reasoning--generally speaking, RIP claims are made about people who predate the explicit formulation of paraconsistent logics--but that the implicit standards of reasoning their practices seemed to conform to were ones that fit better with a paraconsistent logical consequence relationship than with one in which anything can be derived from any contradiction.) In most cases, a normative element is at least implicit--do we really want to say that these people *should* have concluded everything from the contradicitons inherent in their theories?--but for the sake of simplicity, let's put that aside and just deal with RIP claimss on a purely descriptive level.

I should have some more to say about this soon, but for now, I just want to note that the standards we use to evaluate RIP claims should be a bit more rigorous than the standards that tend to be appealed to in standard presentations of them.

Theorists often take it to be sufficient for an RIP claim about Person X that:

(a) Person X accepted some overall package of theories involving inconsistencies between its elements, or from which some sort of contradiction could be derived, or even is just used some sort of theoretical tools that they know *could* be used to generate contradictions, but:

(b) They didn't use the relevant contradiction to prove random arbitrary conclusions.

I think this is grossly insufficient. As a first stab as something a bit more substantive, I'd argue that we also need:

(c) That they were aware of the entailment of inconsistency (note that this is a standard that the Colyvan example meets, but which other historical examples used in this literature may not) and, crucially, that

(d) That they explicitly believed the contradiction, and also crucially, that

(e) They used both halves of the contradiction (conjunctively, or even one half at a time) as a premise in at least some of their reasoning about the subject.

Also, on an even more basic level, and as an absoltuely minimal standard, I'd suggest:

(f) That, in general or at *least* when dealing with the relevant subject matter (e.g. in other set-theoretic reasoning during the lag time Colyvan talks about), they did *not* implicitly reason according to classical rules of derivation that are paraconsistently invalid, like Disjunctive Syllogism and Reudctio Ad Absurdum.

Thoughts?

## Wednesday, October 28, 2009

## Sunday, October 25, 2009

### Absolutely Unrestricted Quantification, the Classical Principle, and Non-Existence Solutions to the Liar Paradox

Graham Priest has argued that the structure of paradoxes like the Liar (and similar paradoxes, like Russell’s Paradox for naïve set theory) is best understood in terms of his ‘Inclosure Schema.’ The three elements of the Schema are:

(1) Existence

(2) Closure

(3) Transcendence

(1) is the condition that the relevant object (like the Liar Sentence, or the Russell Set)

A type-(2) solution to the Liar Paradox would be standard truth-value gap proposals, that say that the key place where the argument from the Liar to inconsistency goes wrong is the assumption that it is either true or false. We can keep the assumption that it’s true iff it’s false, but deny that it’s either one of those things. Presto. Done!

Of course, famously, while this works for the Simple Liar:

"This sentence is false."

….it runs into problems for the Strengthened Liar:

"This sentence is not true."

A type-(3) solution to the Liar Paradox would be a denial of the relevance instance of the Biconditional Truth Schema. (For example, Gil Harman takes this line in

Type-(1) solutions to the Liar Paradox are a bit trickier to formulate. To see why, consider the contrast to Russell’s Paradox. The standard, orthodox mathematical and philosophical response to it is simply to abandon the assumption that sets like the Russell Set exist. And, after all, that’s sort of plausible. If the Russell Set did exist, how would we ever know? If the best reason to think that

Whatever one thinks of this argument, my point here is not to defend it but to underline the contrast to the Liar Paradox. There’s no denying that sentence tokens stringing together the words “this," "sentence," "is," and "false” in the right order exist, or even that, in some cases, the intended referent of the “this” is that sentence itself. (By contrast, of course, you can point at a sentence about some empirical subject written on a chalkboard and say “this sentence is false” without any sort of paradox being in the offing.) Non-existence claims, therefore, aren’t quite literally claims that the relevant sentences don’t exist. They’re claims that the relevant truth-bearers don’t exist. If you think that the bearers of truth are propositions, the non-existence claim is the claim that “this sentence is false” fails to express a proposition. If you think that sentences are the bearers of truth (or sentences-in-contexts, or precissified sentences, or any of the many variants of sentence theory out there), then the non-existence claim is that, for whatever reason, sentences like the Liar are actually meaningless. Exactly how to motivate this claim, respond to the obvious objections (don’t such sentences

I do, however, want to note that any remotely plausible non-existence story will apply just as well to the Truth-Teller:

"This sentence is true."

…as it does to the Liar. If a sentence can still be meaningful, or express a proposition, or whatever, when it isn’t trying to assert anything above and beyond its own truth, why not the same for falsity? Any non-existence solution that doesn’t treat the two cases similarly starts to look entirely too much like an ad hoc maneuver to save consistency. So a non-existence solution that even begins to be plausible must apply to

OK, but wait a second. Presumably, part of the motivation for any non-existence solution is likely a desire to defend the following principle about how truth and falsity relate to each other, which we can call the Classical Principle (CS), and, for the sake of simplicity, express in terms of “statements.” When reading “statement,” substitute in whatever your favorite candidate for the role of truth-bearers may be.

CS: "Every statement is either true or false (but not both)."

Stop. Wait. Is the statement of CS a “statement” (i.e. does the sentence express a proposition, or is it meaningful, or whatever)? It seems like it had better be, if we want to express our anti-gap, anti-glut commitments by asserting it! But if so, is it mixing use and mention in the context of talking about truth, and thus violating the requirement that truth-talk ultimately ground out in a subject other than truth? If so, it looks like, on the assumptions that got us to non-existence, it’s not eligible for truth-bearing status.

To get a grip on what’s going on here, it’s useful to start thinking about the general semantics of claims that involve quantification (some, all, etc.) For example, think about the following statements:

(1) All of the milk is gone!

(2) Everyone but John and Jack was eliminated in the first round of the game.

(3) All sentences are composed of words.

(4) Everything the Pope says is true.

In (1) and (2), interpretive charity should warn us away from seeing the domain over which the “all” and “everyone” are intended to quantify as absolutely unrestricted, including everything whatsoever, and context gives us good, solid clues about what the intended domains are. In the case of (1), if we take the domain being quantified over to be universal and unrestricted, then this is an obviously, bizarrely false claim, given the existence of full bottles and cups of milk in grocery stores and houses all around the world. Obviously, however, we take the intended domain to be something more like the objects that are inside the house of the people who are having the conversation. For parallel reasons, we shouldn’t take the “everyone” in (2) to quantify over all persons everywhere, but instead take it to quantify over the restricted domain of people who were playing the game with John and Jack on the particular occasion in question.

(3), on the other hand, seems to be a different sort of case. There’s no reason

How about (4)? Well, in the logical translation “for every x, if x is something the pope said, then x is true”, if we take the domain that “every x” is quantifying over to include questions the Pope asks, commands he gives, meaningless jumbles of word thats he murmurs in his sleep, etc, etc., then (4) is meaningless. Moreover, if he endorses Papal infallibility himself (as Popes are prone to do), then he might do so (oddly and pretentiously enough) in the third person, and thus himself utter (4). Whatever considerations led us to a non-existence solution to the Liar should rule out (4)-as-uttered-by-the-Pope-when-the-domain-over-which-he-is-quantifying-includes-his-assertion-of-(4). But it seems implausible that we should take any of these considerations to render standard statements of Papal infallibility not just false but

None of this limits the Pope’s expressive power. He can express any bit of information that strikes his holy fancy. He can’t, however, do it all at once—even if he was infallible, he still wouldn't be capable of the miracle of expressing all of his beliefs with one short sentence.

The road from this to a plausible claim about the CS and non-existence solutions should be starting to get pretty clear. Before we get there, though, one reminder of the stakes.

When the antecedent and the consequent of a conditional are both true, the conditional itself is true. (Material conditionals work that way, anyway. The question of whether the material conditional is the right analysis of the conditional of ordinary language is actually going to be irrelevant to the problem I'm getting at…all I’ll need is the assumption that the conditional we end up with is at least

MP: "For every statement, if that statement is true, then McCain won the election."

If you take the domain over which “every statement” quantifies to include the statement of the MP itself, then it asserts (among other things) that if

Similarly for the CS. If, for example, the gap theorist expresses their view by saying that “all statements are either true, false or neither” and we (as non-existence theorists) take this as a meaningful, truth-bearing statement (since we think it’s false), we won’t take the domain over which the gap theorist’s statement is quantifying to include itself. Thus, our counter-statement that “every sentence is either true or false” will be presumed to quantify over exactly the same domain, and we’ve succeeded in disagreeing with each other. Moreover, when someone asks about the truth-status of the sentence we constructed to express our disagreement with the gap theorist, we can attribute truth to it, and when someone asks about the truth-status of the sentence attributing truth to

Now, note that I’m not actually asserting the non-existence solution here, or defending it against the multitude of

There’s no special objection to the non-existence solution lurking in a non-existence advocate uttering the sentence “every sentence is true or false (but not both),” or in the fact that they are using that sentence to express their disagreement with gap and glut theorists despite the fact that it doesn't quantify over itself. Given that they wouldn’t be asserting it if they didn’t take it to be true, we have really excellent third person justification for believing that their attitude to the truth-status of

*....and, moreover, that

(1) Existence

(2) Closure

(3) Transcendence

(1) is the condition that the relevant object (like the Liar Sentence, or the Russell Set)

*exists*. (2) is the condition that it be a member of some category (being true, being a member of itself), and (3) is the condition that it also ‘transcend’ (i.e. fail to be part of) that category (being untrue, being a non-member of itself). Whatever one thinks of the Inclosure Schema as an analysis of what’s going on with Liar-type paradoxes, and whatever one thinks of Priest’s ‘Principle of Unified Solution’ whereby all paradoxes modeled by the Inclosure Schema should be solved in the same way, it’s at least a useful way of classifying various solutions.A type-(2) solution to the Liar Paradox would be standard truth-value gap proposals, that say that the key place where the argument from the Liar to inconsistency goes wrong is the assumption that it is either true or false. We can keep the assumption that it’s true iff it’s false, but deny that it’s either one of those things. Presto. Done!

Of course, famously, while this works for the Simple Liar:

"This sentence is false."

….it runs into problems for the Strengthened Liar:

"This sentence is not true."

A type-(3) solution to the Liar Paradox would be a denial of the relevance instance of the Biconditional Truth Schema. (For example, Gil Harman takes this line in

*Change Of View*.) That way, we get to keep the classical assumption that it’s either true or false, but ditch the assumption that it’s true iff it’s false, and thus avoid the contradiction. A problem with this is that it seems terribly ad hoc, and, when we approach the inference from Liar to inconsistency as an argument for dialetheism, it begs the question. If all you can say in response to an argument for a position that you don’t like is that some extremely basic, obviously true-sounding conceptual principle must have exceptions, because if it applied in this case, we’d get that conclusion, you’re not going to win any arguments.Type-(1) solutions to the Liar Paradox are a bit trickier to formulate. To see why, consider the contrast to Russell’s Paradox. The standard, orthodox mathematical and philosophical response to it is simply to abandon the assumption that sets like the Russell Set exist. And, after all, that’s sort of plausible. If the Russell Set did exist, how would we ever know? If the best reason to think that

*any*sets exist is (a la Quine) because they’re indispensible to our best scientific theories, then the restricted universe of ZFC hierarchical set theory should be quite sufficient. Given that, all else being equal, it seems un-parsimonious to*also*postulate a multitude of other sets, outside of the ZFC hierarchy.Whatever one thinks of this argument, my point here is not to defend it but to underline the contrast to the Liar Paradox. There’s no denying that sentence tokens stringing together the words “this," "sentence," "is," and "false” in the right order exist, or even that, in some cases, the intended referent of the “this” is that sentence itself. (By contrast, of course, you can point at a sentence about some empirical subject written on a chalkboard and say “this sentence is false” without any sort of paradox being in the offing.) Non-existence claims, therefore, aren’t quite literally claims that the relevant sentences don’t exist. They’re claims that the relevant truth-bearers don’t exist. If you think that the bearers of truth are propositions, the non-existence claim is the claim that “this sentence is false” fails to express a proposition. If you think that sentences are the bearers of truth (or sentences-in-contexts, or precissified sentences, or any of the many variants of sentence theory out there), then the non-existence claim is that, for whatever reason, sentences like the Liar are actually meaningless. Exactly how to motivate this claim, respond to the obvious objections (don’t such sentences

*seem*meaningful?, etc.), get off the hook of various revenge paradoxes and so on is a terribly difficult and complicated matter, and I don’t want to get into ay of that here.I do, however, want to note that any remotely plausible non-existence story will apply just as well to the Truth-Teller:

"This sentence is true."

…as it does to the Liar. If a sentence can still be meaningful, or express a proposition, or whatever, when it isn’t trying to assert anything above and beyond its own truth, why not the same for falsity? Any non-existence solution that doesn’t treat the two cases similarly starts to look entirely too much like an ad hoc maneuver to save consistency. So a non-existence solution that even begins to be plausible must apply to

*all*sentences that do nothing but talk about their own truth-status, or the truth-status of some other sentence (as in, say, the Yablo series), without ever "grounding out" in some other subject.OK, but wait a second. Presumably, part of the motivation for any non-existence solution is likely a desire to defend the following principle about how truth and falsity relate to each other, which we can call the Classical Principle (CS), and, for the sake of simplicity, express in terms of “statements.” When reading “statement,” substitute in whatever your favorite candidate for the role of truth-bearers may be.

CS: "Every statement is either true or false (but not both)."

Stop. Wait. Is the statement of CS a “statement” (i.e. does the sentence express a proposition, or is it meaningful, or whatever)? It seems like it had better be, if we want to express our anti-gap, anti-glut commitments by asserting it! But if so, is it mixing use and mention in the context of talking about truth, and thus violating the requirement that truth-talk ultimately ground out in a subject other than truth? If so, it looks like, on the assumptions that got us to non-existence, it’s not eligible for truth-bearing status.

To get a grip on what’s going on here, it’s useful to start thinking about the general semantics of claims that involve quantification (some, all, etc.) For example, think about the following statements:

(1) All of the milk is gone!

(2) Everyone but John and Jack was eliminated in the first round of the game.

(3) All sentences are composed of words.

(4) Everything the Pope says is true.

In (1) and (2), interpretive charity should warn us away from seeing the domain over which the “all” and “everyone” are intended to quantify as absolutely unrestricted, including everything whatsoever, and context gives us good, solid clues about what the intended domains are. In the case of (1), if we take the domain being quantified over to be universal and unrestricted, then this is an obviously, bizarrely false claim, given the existence of full bottles and cups of milk in grocery stores and houses all around the world. Obviously, however, we take the intended domain to be something more like the objects that are inside the house of the people who are having the conversation. For parallel reasons, we shouldn’t take the “everyone” in (2) to quantify over all persons everywhere, but instead take it to quantify over the restricted domain of people who were playing the game with John and Jack on the particular occasion in question.

(3), on the other hand, seems to be a different sort of case. There’s no reason

*not*to take the domain which the “all” is quantifying over—“for all x, if x is a sentence, x is composed of words”—to include itself. Use and mention seem to be innocently combined here.How about (4)? Well, in the logical translation “for every x, if x is something the pope said, then x is true”, if we take the domain that “every x” is quantifying over to include questions the Pope asks, commands he gives, meaningless jumbles of word thats he murmurs in his sleep, etc, etc., then (4) is meaningless. Moreover, if he endorses Papal infallibility himself (as Popes are prone to do), then he might do so (oddly and pretentiously enough) in the third person, and thus himself utter (4). Whatever considerations led us to a non-existence solution to the Liar should rule out (4)-as-uttered-by-the-Pope-when-the-domain-over-which-he-is-quantifying-includes-his-assertion-of-(4). But it seems implausible that we should take any of these considerations to render standard statements of Papal infallibility not just false but

*meaningless*. As such, it seems reasonable to charitably interpret the domain over which a non-Pope’s assertion of (4) quantifies as failing to include the Pope’s questions, commands, meaningless jumbles of words uttered during his holy sleep, and so on, and the domain over which a Pope’s assertion of it quantifies as also failing to include his own sentence asserting it. Which is not to say, of course, that the Pope doesn’t think his assertion of (4) is also true, but just that to express*that*bit of information, he’s going to need to construct a new sentence. And if he wants to express his belief in the truth of*that*sentence, he’s going to need to construct a new sentence to do it with, and so on.None of this limits the Pope’s expressive power. He can express any bit of information that strikes his holy fancy. He can’t, however, do it all at once—even if he was infallible, he still wouldn't be capable of the miracle of expressing all of his beliefs with one short sentence.

The road from this to a plausible claim about the CS and non-existence solutions should be starting to get pretty clear. Before we get there, though, one reminder of the stakes.

When the antecedent and the consequent of a conditional are both true, the conditional itself is true. (Material conditionals work that way, anyway. The question of whether the material conditional is the right analysis of the conditional of ordinary language is actually going to be irrelevant to the problem I'm getting at…all I’ll need is the assumption that the conditional we end up with is at least

*false*.) Now, say that someone has the false belief that McCain won the 2008 election. (Perhaps they just woke up from a coma they’ve been in since before the election, and the last polling data they saw showed McCain in the lead.) Given that they have that belief, and standard classical beliefs about the semantics of conditionals, and the plausible belief that “P” and “”P’ is true” are universally intersubstitutable, they (validly) conclude the following McCain Principle (MP):MP: "For every statement, if that statement is true, then McCain won the election."

If you take the domain over which “every statement” quantifies to include the statement of the MP itself, then it asserts (among other things) that if

*it*is true, then McCain won the election. As anyone who’s familiar with the Curry Paradox knows,*that*claim is one that has some very strange properties (if it's meaningful). Given the Biconditional Truth Schema and any logic strong enough to include Modus Ponens and Conditional Proof, the truth of its consequent can be derived from the simple statement of its truth conditions. If we want the MP to be*false*(and thus meaningful, truth-evaluable, etc.), and we don’t want to believe that it's also true*, then it might be a good idea to take the domain over which the MP quantifies as one that does not include itself. (Or, rather, to say that it's only meaningful if the domain over which it is supposed to be quantifying fails to include itself, and that when we say that it is*false*, we're reading it in this way.) Of course, this isn’t to say that the person who mistakenly believes McCain to have won the election doesn’t believe about the statement of the MP itself that if*that*statement is true, then McCain won the election, or that they can’t or shouldn't express this belief of theirs. They will, however, have to construct a new statement to express it with. It’s not always possible to say everything you want to say all at once.Similarly for the CS. If, for example, the gap theorist expresses their view by saying that “all statements are either true, false or neither” and we (as non-existence theorists) take this as a meaningful, truth-bearing statement (since we think it’s false), we won’t take the domain over which the gap theorist’s statement is quantifying to include itself. Thus, our counter-statement that “every sentence is either true or false” will be presumed to quantify over exactly the same domain, and we’ve succeeded in disagreeing with each other. Moreover, when someone asks about the truth-status of the sentence we constructed to express our disagreement with the gap theorist, we can attribute truth to it, and when someone asks about the truth-status of the sentence attributing truth to

*that*sentence, we can attribute truth to*it*, and so on. Again, expressive power is unlimited, but we can’t necessarily express all of our relevant beliefs all at once. Of course, as we've seen, we have excellent independent reasons to think that we'll often be unable to do so.Now, note that I’m not actually asserting the non-existence solution here, or defending it against the multitude of

*other*obvious objections that one could make to it. My point is a much narrower one, which is this:There’s no special objection to the non-existence solution lurking in a non-existence advocate uttering the sentence “every sentence is true or false (but not both),” or in the fact that they are using that sentence to express their disagreement with gap and glut theorists despite the fact that it doesn't quantify over itself. Given that they wouldn’t be asserting it if they didn’t take it to be true, we have really excellent third person justification for believing that their attitude to the truth-status of

*that*sentence isn’t an*exception*to the general pattern of their attitudes about the truth-status of every*other*sentence. However, this extra belief that we should so obviously attribute to the non-existence theorist who claims that "every sentence is either true or false (but not both)" isn't expressed by*that very sentence*. That’s OK. It doesn’t mean they can’t express that belief as well. It just means that they can’t say everything that they have to say all at once.*....and, moreover, that

*every other*false statement is also true, which could be shown with the production of exactly similar sentences for them.## Tuesday, October 20, 2009

### Logical Pluralism, Monism and Logically Impossible Worlds

Logics are either:

(a) Overall theories of reality, relative to the high level of generality and abstractness at which formal languages operate, or

(b) Codifications of what we take to be good reasoning, or

(c) Something else

For the moment, let’s put aside (c) and focus on the debate between (a) and (b). On initial consideration, (b) may sound more natural and obvious. We do, after all, often use “logical” and “reasonable” as if they were synonymous, and the use of terms like “inductive logic” and “informal logic” certainly lends itself most easily to (b). On the other hand, normal talk of “logically possible worlds,” “logical impossibility” and so on most naturally lends itself to (a), and (a) has a long and glorious philosophical lineage, it or something along very much the same lines having been advocated (in one form or another) by figures as diverse in their other views as, for example, Frege, Quine, Penelope Maddy and Graham Priest.*

That list includes two people who believe that logical truths are “analytic” in some sense and two who reject the very notion of analyticity, two who accept the web-of-belief model of logical theory change and two who do not, two people who accept classical logic and two who reject it, one who believes in truth-value gaps and three who do not and one who believes in true contradictions and three who do not. Moreover, those categories and battle lines crisscross in dizzingly complex enough ways just among the overall views of those four thinkers to pretty much demand philosophical baseball cards to keep it all straight....the point is just that, far from being an eccentric consequence of any one perspective, it can fit with a variety of different overall packages of beliefs.

Further, note that (a) is even compatible with a wide variety of views about how logic and reasoning

With the conceptual distinction between (a) and (b) firmly in mind, let’s move on to a related distinction. Logical pluralists believe that various logics are suitable for various purposes, but that there is no such thing as the One True Logic. By contrast, logical monists believe that there is some sense in which one logic is uniquely correct.

(a) seems to pretty much demand logical monism, unless of course one takes reality to be spectacularly inconsistent in a way that it’s hard to make sense of even dialetheically. (Note, for example, that Priest is a monist.) On the other hand, (b) fits very naturally with pluralism. Think about the notion of underlying logics, where “underlying logic” means, very roughly, something like the following.

*Logic L is the underlying logic of some context C iff L formally captures the standards of reasoning that are relevant to C.

Of course, what it means to be “relevant to C” here is importantly non-obvious. Are we talking about the standards implicit in what most people treat as good reasoning about C, or if not most people, then some special class of people (e.g. subject matter experts)? Or are we, maybe, talking about something a bit more objective, like “the standards of reasoning that are the most likely to lead to the truth about C” or “the standards of reasoning that are most likely to lead to progress towards some goal (other than truth) that’s important in C” or something like that?

These are important distinctions, and it’s usually hard to tell, when reading bits of underlying-logic-talk, exactly which position is implicitly being taken on these issues. Fortunately, at the moment, we won't have to clarify them for the purposes

(d) Different contexts of reasoning involve different underlying logics, or

(e) There’s just one logic which is the unique underlying logic of every context of reasoning, or perhaps just of all the ones that are worth engaging with.

After all, any given context of actual human reasoning is going to be finite. (Note that this is true even in the total context of "all reasoning that has ever, or will ever occur in any particular context whatsoever.”) Only so many inferences will be made, and thus only so many inferences will be classified as reasonable or unreasonable relative to that context. Formal logics, on the other hand, are total systems whose rules classify every inference as valid or invalid. As such, a variety of logics—in fact, an

If you start out with a strong committment to (b), the obvious move is to take this as a reductio against logical monism. On the other hand, if you don’t start by making that crucial background assumption, it might make more sense to take it as a reductio against the usefulness of underlying-logic-talk.

With all that kept in mind, let’s backtrack from the special difficulties faced by (b)-plus-(e)-monism (which, obviously, don't touch (a)-monism) to a more general issue.

One might plausibly think that counterpossible reasoning is a problem for

The process of debate between deeply entrenched philosophical views seems to necessarily involve reasoning about cases in which each side takes it not just that the other side’s views are wrong, but that they *couldn’t* be right. (Although this sort of thing goes on all the time in other areas of philosophy, it’s particularly unavoidable in phil of logic, for obvious reasons.) That said, if we’re going to actually reason with each other about which of the disputing views is correct, rather than simply retreating behind our respective barricades to glare at each other, than we’ve got to make the case for our view (in part) by drawing out unappealing consequences of the views we reject. “X is absurd. After all, if X were true, then Y would be true.”

Now, on the most orthodox theory of counterfactual conditionals, “if X were true, Y would be true” just means that the closest possible worlds at which X is true are also worlds at which Y is true. This has the unfortunate consequence that all counterpossible conditionals come out as (vacuously) true. This is both wildly counterintuitive and (for the reasons just discussed) would, if taken seriously, make philosophical debate about basic issues impossible. As such, an attractive move to make here is to tweak the Lewissian rules for evaluating counterfactuals, such that a statement of the form “if X were true, Y would be true” is true iff the closest

So far, so good.

But wait.

One could make the following objection here. “On the old Lewissian picture where only logically possible worlds are allowed, the logical monist can evaluate whether the closest worlds at which X is true are worlds at which Y is true by using whatever they regard as the actually correct logic to reason about whether Y follows from X. Now, the (b)-plus-(d)-pluralist, who changes his logic due to variations of contexts of reasoning as easily as the rest of us change our shirts to due to variations in weather conditions, can handle this in an easy, straightforward way. When reasoning about whether C holds in the closest quantum-logical worlds, he can just reason about these worlds using quantum logic. But the logical monist faces a dilemma. Either she uses her favored (non-quantum) logic to reason about what’s true at the logically impossible world where distribution fails, and, given this inconsistency, she comes to many strange and irrelevant conclusions, or else she uses quantum logic when reasoning about the world where quantum logic is correct, and thus in effect becomes a logical pluralist.”**

Now, there are two things wrong with all of this. The first is that it’s not entirely true, even on the old only-logically-possible-worlds-need-apply model of counterfactual reasoning, that anyone reasoned about whether the closest worlds at which X was true were worlds at which Y was true by “using” the logic they used to determine the conditions for counting as a “logically possible world,” or, for that matter, by using any logic at all. In fact, it looks to me like interesting and informative counterfactual conditionals are very rarely ones in which the consequent is logically derivable from the antecedent. (After all, if Y is logically derivable from X, then Y is true at

Now, *if* we’re OK with underlying logic talk, we can re-phrase the point to get around this issue by saying that at least, there’s no reason to think that the underlying logic of reasoning about the actual world needs to be different from the underlying logic of reasoning about other logically possible worlds, whereas reasoning about what's true at some logically impossible world seems to pretty much demand a different underlying logic. Given this new phrasing, the first problem with the objection disappears. The problem posed for underlying logic talk by the failure of strictly logically valid inferences to exhaust the whole category of intuitively rational inferences doesn’t look like

To get a handle on the second and more important thing that’s wrong with the objection, we need to first answer the following question: Which sort of logical monism is this directed against? If the answer is that it’s a problem for

Taken as an objection to (b)-plus-(e) monists who are interested in counterpossible reasoning, it looks to me like it goes through. On the other hand, it’s a bit redundant. After all, as we saw above, (b)-plus-(e) monism fails on much more basic grounds than this, and the more basic objection works just as well even against (b)-plus-(e) monists who don’t think that it’s possible or desirably to (non-trivially) reason about counterpossible scenarios.

OK, how about (a)-monists? Well, an (a)-monist who accepts the notion of “underlying logics” can cheerfully admit that, when they reason about what’s the case in logically impossible worlds, the underlying logic of their reasoning is something other than the logic they take to be the uniquely correct overall theory of

One way or another, anyone who accepts (a) has no reason to accept the notion that all reasoning involves “using a logic,”

*This is clearest in the cases of Maddy, who explicitly claims in her book

**This is certainly an objection that could be levelled against Graham Priest's overall package of views, given his explicit endorsement of logical monism in

(a) Overall theories of reality, relative to the high level of generality and abstractness at which formal languages operate, or

(b) Codifications of what we take to be good reasoning, or

(c) Something else

For the moment, let’s put aside (c) and focus on the debate between (a) and (b). On initial consideration, (b) may sound more natural and obvious. We do, after all, often use “logical” and “reasonable” as if they were synonymous, and the use of terms like “inductive logic” and “informal logic” certainly lends itself most easily to (b). On the other hand, normal talk of “logically possible worlds,” “logical impossibility” and so on most naturally lends itself to (a), and (a) has a long and glorious philosophical lineage, it or something along very much the same lines having been advocated (in one form or another) by figures as diverse in their other views as, for example, Frege, Quine, Penelope Maddy and Graham Priest.*

That list includes two people who believe that logical truths are “analytic” in some sense and two who reject the very notion of analyticity, two who accept the web-of-belief model of logical theory change and two who do not, two people who accept classical logic and two who reject it, one who believes in truth-value gaps and three who do not and one who believes in true contradictions and three who do not. Moreover, those categories and battle lines crisscross in dizzingly complex enough ways just among the overall views of those four thinkers to pretty much demand philosophical baseball cards to keep it all straight....the point is just that, far from being an eccentric consequence of any one perspective, it can fit with a variety of different overall packages of beliefs.

Further, note that (a) is even compatible with a wide variety of views about how logic and reasoning

*do*relate to each other. That said, we might be able to say some general things about that relationship without bringing in very many extra assumptions. The theory of what it means to say that one thing follows from another that most naturally fits with (a) it the traditional view that validity is truth-preservation. (Roughly: Q follows from P iff P has never been and will never be true without Q being true as well. This gives us an obvious way to connect the claim that logic is all about giving us rules of entailment with the claim that logic is an overall theory about how the world is. For example, a philosopher who takes classical logic to give us the right overall theory of reality will claim that any time the world is configured in such a way as to make a disjunction true and one of its disjuncts false, it will be configured in a way that makes the other disjunct true. The dialetheist, who takes it that sometimes the world makes one disjunct both true and false and the other disjuct false only, a possibility the classical theorist rules out, will disagree with that, and the two sides proceed to engage in an argument about which way the world actually is.) This, in turn, generates at least one obvious point of connection between logic and good reasoning: logically valid arguments will never lead you from true premises to false conclusions, so if it’s reasonable to believe a bunch of premises, it’s also reasonable to (on the basis of the logical connection) believe the conclusions that can be validly inferred from those premises, hence Frege’s observation that although logic is not about how men think, it is about how they must think “if they are not to miss the truth.” So even if logically valid inferences don’t exhaust the category of reasonable inferences (as, on this view, it’s pretty clear that they don’t), they still constitute an important subcategory of it.With the conceptual distinction between (a) and (b) firmly in mind, let’s move on to a related distinction. Logical pluralists believe that various logics are suitable for various purposes, but that there is no such thing as the One True Logic. By contrast, logical monists believe that there is some sense in which one logic is uniquely correct.

(a) seems to pretty much demand logical monism, unless of course one takes reality to be spectacularly inconsistent in a way that it’s hard to make sense of even dialetheically. (Note, for example, that Priest is a monist.) On the other hand, (b) fits very naturally with pluralism. Think about the notion of underlying logics, where “underlying logic” means, very roughly, something like the following.

*Logic L is the underlying logic of some context C iff L formally captures the standards of reasoning that are relevant to C.

Of course, what it means to be “relevant to C” here is importantly non-obvious. Are we talking about the standards implicit in what most people treat as good reasoning about C, or if not most people, then some special class of people (e.g. subject matter experts)? Or are we, maybe, talking about something a bit more objective, like “the standards of reasoning that are the most likely to lead to the truth about C” or “the standards of reasoning that are most likely to lead to progress towards some goal (other than truth) that’s important in C” or something like that?

These are important distinctions, and it’s usually hard to tell, when reading bits of underlying-logic-talk, exactly which position is implicitly being taken on these issues. Fortunately, at the moment, we won't have to clarify them for the purposes

*this*discussion, so let’s put all of those issues to one side, and consider one last distinction. Given that we decide to get into the business of using underlying-logic-talk, either we take it to be the case that:(d) Different contexts of reasoning involve different underlying logics, or

(e) There’s just one logic which is the unique underlying logic of every context of reasoning, or perhaps just of all the ones that are worth engaging with.

*If*we make the background assumption that (b) is the correct view about what logics are, then (d) just*is*logical-pluralism-expressed-in-underlying-logic-talk and (e) is logical-monism-expressed-in-underlying-logic-talk. Now, again relative to the assumption of (b), it’s pretty easy to argue for (d) and against (e).After all, any given context of actual human reasoning is going to be finite. (Note that this is true even in the total context of "all reasoning that has ever, or will ever occur in any particular context whatsoever.”) Only so many inferences will be made, and thus only so many inferences will be classified as reasonable or unreasonable relative to that context. Formal logics, on the other hand, are total systems whose rules classify every inference as valid or invalid. As such, a variety of logics—in fact, an

*infiite*number of possible systems of formal logic—will license all of the inferences regarded as good reasoning in that context. Thus, far from (e) making sense, it doesn’t look like we can specify a singular underlying logic even within a single context.If you start out with a strong committment to (b), the obvious move is to take this as a reductio against logical monism. On the other hand, if you don’t start by making that crucial background assumption, it might make more sense to take it as a reductio against the usefulness of underlying-logic-talk.

With all that kept in mind, let’s backtrack from the special difficulties faced by (b)-plus-(e)-monism (which, obviously, don't touch (a)-monism) to a more general issue.

One might plausibly think that counterpossible reasoning is a problem for

*any*sort of logical monism. Here's why:The process of debate between deeply entrenched philosophical views seems to necessarily involve reasoning about cases in which each side takes it not just that the other side’s views are wrong, but that they *couldn’t* be right. (Although this sort of thing goes on all the time in other areas of philosophy, it’s particularly unavoidable in phil of logic, for obvious reasons.) That said, if we’re going to actually reason with each other about which of the disputing views is correct, rather than simply retreating behind our respective barricades to glare at each other, than we’ve got to make the case for our view (in part) by drawing out unappealing consequences of the views we reject. “X is absurd. After all, if X were true, then Y would be true.”

Now, on the most orthodox theory of counterfactual conditionals, “if X were true, Y would be true” just means that the closest possible worlds at which X is true are also worlds at which Y is true. This has the unfortunate consequence that all counterpossible conditionals come out as (vacuously) true. This is both wildly counterintuitive and (for the reasons just discussed) would, if taken seriously, make philosophical debate about basic issues impossible. As such, an attractive move to make here is to tweak the Lewissian rules for evaluating counterfactuals, such that a statement of the form “if X were true, Y would be true” is true iff the closest

*worlds*, possible or impossible, at which X is true are also worlds at which Y is true. (A natural assumption to make is that possible worlds are just definitionally always closer to the actual world than even the closest impossible worlds.) So, while arguing against quantum logician’s claim that the distributive law (the conjunction of “A” and “B or C” entails the disjunction of “A and B” or “A and C”) fails, a partisan of distribution can say that “if the distributive law failed, then counter-intuitive consequence C would hold,” and we take it that that conditional is true if and only if the closest (impossible) worlds where the conjunctions and disjunctions of the properties of subatomic particles really do fail to distribute in the way the quantum logician claims that they fail to in the actual world are also worlds at which C holds.So far, so good.

But wait.

One could make the following objection here. “On the old Lewissian picture where only logically possible worlds are allowed, the logical monist can evaluate whether the closest worlds at which X is true are worlds at which Y is true by using whatever they regard as the actually correct logic to reason about whether Y follows from X. Now, the (b)-plus-(d)-pluralist, who changes his logic due to variations of contexts of reasoning as easily as the rest of us change our shirts to due to variations in weather conditions, can handle this in an easy, straightforward way. When reasoning about whether C holds in the closest quantum-logical worlds, he can just reason about these worlds using quantum logic. But the logical monist faces a dilemma. Either she uses her favored (non-quantum) logic to reason about what’s true at the logically impossible world where distribution fails, and, given this inconsistency, she comes to many strange and irrelevant conclusions, or else she uses quantum logic when reasoning about the world where quantum logic is correct, and thus in effect becomes a logical pluralist.”**

Now, there are two things wrong with all of this. The first is that it’s not entirely true, even on the old only-logically-possible-worlds-need-apply model of counterfactual reasoning, that anyone reasoned about whether the closest worlds at which X was true were worlds at which Y was true by “using” the logic they used to determine the conditions for counting as a “logically possible world,” or, for that matter, by using any logic at all. In fact, it looks to me like interesting and informative counterfactual conditionals are very rarely ones in which the consequent is logically derivable from the antecedent. (After all, if Y is logically derivable from X, then Y is true at

*all*of the logically possible worlds at which X is true, not juts the closest ones.) For example “if Hitler had won World War II, Germany would be the dominant power in post-War Europe” certainly seems to be true, but not because there’s a logically valid inference connecting the antecedent to the consequent.Now, *if* we’re OK with underlying logic talk, we can re-phrase the point to get around this issue by saying that at least, there’s no reason to think that the underlying logic of reasoning about the actual world needs to be different from the underlying logic of reasoning about other logically possible worlds, whereas reasoning about what's true at some logically impossible world seems to pretty much demand a different underlying logic. Given this new phrasing, the first problem with the objection disappears. The problem posed for underlying logic talk by the failure of strictly logically valid inferences to exhaust the whole category of intuitively rational inferences doesn’t look like

*more*of a problem for ordinary counterfactual reasoning than it does for factual reasoning, and assuming that there’s some sort of good solution to that problem when it comes to factual reasoning, that solution should carry over just as well to counterfactual reasoning.To get a handle on the second and more important thing that’s wrong with the objection, we need to first answer the following question: Which sort of logical monism is this directed against? If the answer is that it’s a problem for

*any*logical monist who wants to engage in counterpossible reasoning, we should still disambiguate monisms and try it out separately as an objection against each kind.Taken as an objection to (b)-plus-(e) monists who are interested in counterpossible reasoning, it looks to me like it goes through. On the other hand, it’s a bit redundant. After all, as we saw above, (b)-plus-(e) monism fails on much more basic grounds than this, and the more basic objection works just as well even against (b)-plus-(e) monists who don’t think that it’s possible or desirably to (non-trivially) reason about counterpossible scenarios.

OK, how about (a)-monists? Well, an (a)-monist who accepts the notion of “underlying logics” can cheerfully admit that, when they reason about what’s the case in logically impossible worlds, the underlying logic of their reasoning is something other than the logic they take to be the uniquely correct overall theory of

*our*reality. This semi-pluralism doesn’t, however, threaten, or indeed have much of anything to do with, the sense of logical monism that they endorse. (I say 'semi-pluralism' because to get to logical pluralism from these considerations, you need to combine your endorsement of (d) with an endorsement of (b).) On the other hand, I’ve already suggested that, if we don’t start out by assuming (b), the problem about uniquely specifying underlying logics sketched out above might be more naturally taken as a reductio against the very idea of underlying logics. An (a)-monist who took that route would not only be able to resist the push towards pluralism on the basis of counterpossible reasoning, but even the push towards the semi-pluralism involved in combining (a) with (d).One way or another, anyone who accepts (a) has no reason to accept the notion that all reasoning involves “using a logic,”

*but*, crucially, for the evaluation of what would and would not be the case in certain counterpossible scenarios, an (a)-monist might find it useful to consult the rules of a logic other than the one they take to the One True Logic. For example, when trying to figure out what’s the case in the failure-of-distribution-worlds that are the closest to the actual world (despite their logically impossibility), an important subcategory of good inferences will be those that are valid in quantum logic. Assuming that classical logic gives us the correct theory of the actual case, quantum logic will play the same role in relationship to our reasoning about what’s true at failure-of-distribution worlds that classical logic plays in relationship to our reasoning about what’s actually true. What of it? Once we reject (b), nothing in that picture gives us any more reason to abandon classical monism about logic than the uses of the details of Lemarckian biological theory for determining what’s true at the closest worlds at which acquired traits are genetically inherited gives us the slightest reason to abandon Darwinian monism about biology.*This is clearest in the cases of Maddy, who explicitly claims in her book

*Second Philosophy*that the laws of logic are like the laws of physics at a different level of generality, and Quine, who famously argues in*Two Dogmas Of Empiricism*that even logic is (rationally) revisable in the light of empirical evidence, and explicitly points to the possibility that classical logic will be falsified by quantum anomolies. Priest and Frege are more difficult cases, given that both men subscribe to some notion of "analytic truth," but at least in Priest's case, given his endorsement of the web-of-belief model in*Two Dogmas Of Quineanism*, and the explicit arguments he has made in various contexts for rejecting classical logic on*empirical*grounds, seems to more or less comfortably fit in the Quine/Maddy camp on this issue. As far as Frege goes, the evidence is weaker, but a very natural reading of claims like "logic is the science of the general laws of truth" and that logic gives us not the laws of "how men think, but of how they must think if they are not to miss the truth" is to see him as a type-(a) theorist, albeit a very un-Quine-like type-(a) theorist who takes logical laws to describe reality at a "third realm," non-empirical level. At the very least, without getting too deeply into thorny questions of Frege-interpretation, we can point out that the quoted statements are ones that a theorist who*was*solidly in the Quine/Maddy/Priest camp would enthusiastically endorse.**This is certainly an objection that could be levelled against Graham Priest's overall package of views, given his explicit endorsement of logical monism in

*Doubt Truth To Be A Liar*and of logically impossible worlds (at which, for example, there are no true contradictions) in*Towards Non-Being*.## Sunday, October 18, 2009

### Possible Girls

Neil Sinhababu's paper Possible Girls may be one of the most interesting things ever published by a respectable philosophical journal. Sinhababu told me about it in conversation at the reception after he spoke at the University of Miami last semester, but I didn't get a chance to read it until last week.

Here's his claim:

Given Lewissian "genuine modal realism," it's possible for him to enter into a long-distance relationship with a girl who lives in a different possible world.

You read that right.

Moreover, using the awe-inspiring power of Pure Reason, Sinhababu demonstrates:

(a) that it's actually possible to exchange love letters across different possible worlds, despite the impossibility of causal interactions between them (read the paper to find out how this can be done!),

(b) that if you expand the Lewissian picture to include concrete impossibilia, we have good reason to think that the girls who live in the impossible worlds have tentacles, and

(c) that his likely otherworldly girlfriend would be "immortal, with eternally youthful beauty, spending each day singing out every fact about my world that differs from hers."

But...wait...isn't this image kind of sad? Aren't you worried that the otherworldy goddess wasting her eternal life by singing out her love for Sinhababu (and, trust me, he has specific, well-worked out reasons for thinking that she's doing so) is going to be horribly crushed if he finds a girlfriend in the actual world? Wouldn't it, then, be cruel for him to dump her for said actual girlfriend? And shouldn't her friends try to talk her out of wasting her life in this way in any case?

Not to worry. All of these concerns are put to rest in the paper.

Still not convinced that this is a paper you need to read? Well, I submit for your approval the single most excellent endnote in the entire history of academic papers on any subject whatsoever:

"6 I thank Dan Korman for raising this issue, and Ted Sider for informing me about the sexual possibilities offered by paraconsistent logic. "

## Wednesday, October 14, 2009

### Graham Priest's Theory Of Change: The Long-Delayed Follow-Up Post

So Graham Priest believes that change is impossible without contradiction. I bogged about that a while back, glossing Priest's argument and presenting three objections to it:

(1) Priest's theory, formulated as it is in terms of paraconsistent tense logic, assumes the A-Theory of time, which conflicts with our best current science, given that Einstein's Special Theory of Relativity seems to entail the B-Theory.

(2) The contradiction theory of change seems to undermine Priest's "classical re-capture," since too many statements would be dialetheias for comfort, given that the re-capture relies on the claim that this is only true of a small minority of statements, &

(3) His argument centrally relies on the intuition that change exists and that Russellian "cinematic change" wouldn't count as real change, but we have no reason from any plausible theory of intuitions to suppose that if unobservable, instantaneous contradictory states of change existed, our intuitions would track them.

All of these points are explained in the original post. One important point to note here is that, unlike other critics of Priest's views about change (see, for example, the Tahko article discussed in this post, or Fracis Jeffry Pelletier's comments a while back in the Bulletin of Symbolic Logic), I take seriously Priest's claim that he's *not* talking about vagueness, either in language or in the world, or problems relating to the extensions of vague predicates.

The discussion in the comment thread focused on (2). Rafal Urbaniak suggested that dialetheism about change might be rendered compatible with the classical re-capture by some mechanism like:

(a) Differentiating between contexts involving change and those that do not involve it, and regarding only the latter as the target for the classical re-capture, and/or

(b) Interpreting ordinary statements that might be dialetheic in the light of Priest's theory of change in a somewhat static way, so that we can reason about them as part of the 'contexts not involving change' category.

Deleet seconded these concerns, while raising a quite separate concern about the whole notion of the classical re-capture, which is that it seems hard to make sense of talking about percentage of infinite categories. What does it mean to say that the percentage of statements which are both true and false is very low?

At the time, I planned to do a follow-up post on all that, and for a variety of reasons (moving back and forth from California to Florida, road-tripping, dissertating, yadayadayada), it never happened, although while I was dithering, the post was more widely noticed and various people said nice things about it.

So...much later, and with more than a little embarrassment, here's a reply to all of that, working backwards from Deleet's point to Rafal's. Here's what I have to say about former in a footnote in my dissertation:

"One might be concerned that, if we are wondering about the proportion of claims that are true and false, or of contradictions that are true, we will face problems about performing statistical calculations on categories with transfinite numbers of members. It’s not clear to me, however, that this is what’s going on. First of all, if one does not accept Platonism about propositions—the claim that claims that no one has ever made and will never made still, in some sense, exist—then the problem goes away, and even if it doesn’t, there may be a fix in terms of looking a the hypothetical limiting frequency of arbitrarily selected members of that transfinite set of claims, or of contradictory claims, or whatever one takes the relevant category to be."

As far as Rafal's points go, I'd say about (a) that the problem is that change is a constant feature of the properties of the sorts of objects that ordinary reasoning is usually about, and so it seems to me that much of the point of the classical re-capture goes is lost if we restrict ourselves to contexts not involving change. What would those be? Perhaps a certain sort of mathematical Platonist would claim that the properties of mathematical objects are eternal and unchanging, but of course Priest postulates all sorts of contradictions involving *those*, from Russell's Paradox in naïve set theory to his incompleteness-theorem-based argument for the inconsistency of arithmetic. Moreover, it is the ordinary, garden-variety reasoning cases to which the classical re-capture is supposed to apply.

For (a) to really work, then, we need (b) to work. It seems to me that there's a deep tension between Zeno's Principle's formulation in terms of *tense* logic--a formulation, moreover, that is not a happenstance but seems to be conceptually basic to the idea--and the claim that statements have their truth-value statically. The whole point of tense logic is that statements that are currently false were true, that statements that are true will be false, etc. In other words, the *very same statement* changes truth value over time. As such, I see no plausible way to avoid the conclusion (on Priest's premises) that an ordinary statement like "Graham is in the room" uttered as he is changing from being in the room to not being in the room, is dialetheic, and the problem persists.

Of course, I'm not confident that this gets us to the point where there are enough true contradictions to definitely invalidate the classical re-capture, but it certainly seems to be too many for comfort for anyone who's hopes for rendering dialetheism compatible with the intuitive role of rules like Disjunctive Syllogism in what we normally regard as good reasoning about garden-variety cases are bound up with the classical re-capture.

(1) Priest's theory, formulated as it is in terms of paraconsistent tense logic, assumes the A-Theory of time, which conflicts with our best current science, given that Einstein's Special Theory of Relativity seems to entail the B-Theory.

(2) The contradiction theory of change seems to undermine Priest's "classical re-capture," since too many statements would be dialetheias for comfort, given that the re-capture relies on the claim that this is only true of a small minority of statements, &

(3) His argument centrally relies on the intuition that change exists and that Russellian "cinematic change" wouldn't count as real change, but we have no reason from any plausible theory of intuitions to suppose that if unobservable, instantaneous contradictory states of change existed, our intuitions would track them.

All of these points are explained in the original post. One important point to note here is that, unlike other critics of Priest's views about change (see, for example, the Tahko article discussed in this post, or Fracis Jeffry Pelletier's comments a while back in the Bulletin of Symbolic Logic), I take seriously Priest's claim that he's *not* talking about vagueness, either in language or in the world, or problems relating to the extensions of vague predicates.

The discussion in the comment thread focused on (2). Rafal Urbaniak suggested that dialetheism about change might be rendered compatible with the classical re-capture by some mechanism like:

(a) Differentiating between contexts involving change and those that do not involve it, and regarding only the latter as the target for the classical re-capture, and/or

(b) Interpreting ordinary statements that might be dialetheic in the light of Priest's theory of change in a somewhat static way, so that we can reason about them as part of the 'contexts not involving change' category.

Deleet seconded these concerns, while raising a quite separate concern about the whole notion of the classical re-capture, which is that it seems hard to make sense of talking about percentage of infinite categories. What does it mean to say that the percentage of statements which are both true and false is very low?

At the time, I planned to do a follow-up post on all that, and for a variety of reasons (moving back and forth from California to Florida, road-tripping, dissertating, yadayadayada), it never happened, although while I was dithering, the post was more widely noticed and various people said nice things about it.

So...much later, and with more than a little embarrassment, here's a reply to all of that, working backwards from Deleet's point to Rafal's. Here's what I have to say about former in a footnote in my dissertation:

"One might be concerned that, if we are wondering about the proportion of claims that are true and false, or of contradictions that are true, we will face problems about performing statistical calculations on categories with transfinite numbers of members. It’s not clear to me, however, that this is what’s going on. First of all, if one does not accept Platonism about propositions—the claim that claims that no one has ever made and will never made still, in some sense, exist—then the problem goes away, and even if it doesn’t, there may be a fix in terms of looking a the hypothetical limiting frequency of arbitrarily selected members of that transfinite set of claims, or of contradictory claims, or whatever one takes the relevant category to be."

As far as Rafal's points go, I'd say about (a) that the problem is that change is a constant feature of the properties of the sorts of objects that ordinary reasoning is usually about, and so it seems to me that much of the point of the classical re-capture goes is lost if we restrict ourselves to contexts not involving change. What would those be? Perhaps a certain sort of mathematical Platonist would claim that the properties of mathematical objects are eternal and unchanging, but of course Priest postulates all sorts of contradictions involving *those*, from Russell's Paradox in naïve set theory to his incompleteness-theorem-based argument for the inconsistency of arithmetic. Moreover, it is the ordinary, garden-variety reasoning cases to which the classical re-capture is supposed to apply.

For (a) to really work, then, we need (b) to work. It seems to me that there's a deep tension between Zeno's Principle's formulation in terms of *tense* logic--a formulation, moreover, that is not a happenstance but seems to be conceptually basic to the idea--and the claim that statements have their truth-value statically. The whole point of tense logic is that statements that are currently false were true, that statements that are true will be false, etc. In other words, the *very same statement* changes truth value over time. As such, I see no plausible way to avoid the conclusion (on Priest's premises) that an ordinary statement like "Graham is in the room" uttered as he is changing from being in the room to not being in the room, is dialetheic, and the problem persists.

Of course, I'm not confident that this gets us to the point where there are enough true contradictions to definitely invalidate the classical re-capture, but it certainly seems to be too many for comfort for anyone who's hopes for rendering dialetheism compatible with the intuitive role of rules like Disjunctive Syllogism in what we normally regard as good reasoning about garden-variety cases are bound up with the classical re-capture.

## Monday, October 12, 2009

### Dissertation and Short Story

A post with a bit more substance than this one should be coming up on Wednesday. Meanwhile, two quick notes:

(1) My alternate history short story Dark Coffee, Bright Light and the Paradoxes of Omnipotence came out in Atomjack Magazine yesterday. It's worth mentioning here only because it contains a classroom scene with an extended discussion of modal logic and possible worlds semantics, and an argument about omnipotence paradoxes. The latter is loosely adapted from a real life incident that happened to my friend David, who's a very hard-working lawyer and family man. When I wrote the first draft of the story, I e-mailed it to David and asked if he minded having part of his life story re-contextualized as part of the life story of a character who as (a) gay, and (b) a terrorist, David wrote back saying no, don't worry about it, it's fine, and hey, how can I be sure that he's neither of those things?

(2) I finally finished my dissertation last night. Or, rather, a complete rough draft of it, which I realize makes me about halfway done with the process in real terms. Still, it's good to be at that point. Weighing in at 221 pages, it's tentatively entitled "Truth Is A One-Player Game: A Defense Of Monaletheism And Classical Logic."

Here's the table of contents:

Introduction: Dialetheism and Monaletheism

Chapter One: But How Can You Use Logic To Argue About Logic

-(I) What The Explosion Proof Isn't

-(II) Dialetheism And Negation

-(III) Negation And Complementation

Chapter Two: Motivations For Dialetheism

-(I) The Paradoxes Of Motion And Change

-(II) Inconsistent Obligations

-(III) Naïve Set Theory

Chapter Three: Liars And Gluts

Chapter Four: Liars And Gaps

Chapter Five: Liars And Meaninglessness

Chapter Six: Meaninglessness and Revenge

Chapter Seven: The Epistemic Consequences Of Dialetheism

-(I) Failures Of Truth?

-(II) Curry

-(III) Epistemic Consequences

Chapter Eight: Monaletheism And Negation

-(I) Folk Dialetheism?

-(II) Semantic Dialetheism

-(III) Keeping Score

-(IV) A Final Thought About Truth

(1) My alternate history short story Dark Coffee, Bright Light and the Paradoxes of Omnipotence came out in Atomjack Magazine yesterday. It's worth mentioning here only because it contains a classroom scene with an extended discussion of modal logic and possible worlds semantics, and an argument about omnipotence paradoxes. The latter is loosely adapted from a real life incident that happened to my friend David, who's a very hard-working lawyer and family man. When I wrote the first draft of the story, I e-mailed it to David and asked if he minded having part of his life story re-contextualized as part of the life story of a character who as (a) gay, and (b) a terrorist, David wrote back saying no, don't worry about it, it's fine, and hey, how can I be sure that he's neither of those things?

(2) I finally finished my dissertation last night. Or, rather, a complete rough draft of it, which I realize makes me about halfway done with the process in real terms. Still, it's good to be at that point. Weighing in at 221 pages, it's tentatively entitled "Truth Is A One-Player Game: A Defense Of Monaletheism And Classical Logic."

Here's the table of contents:

Introduction: Dialetheism and Monaletheism

Chapter One: But How Can You Use Logic To Argue About Logic

-(I) What The Explosion Proof Isn't

-(II) Dialetheism And Negation

-(III) Negation And Complementation

Chapter Two: Motivations For Dialetheism

-(I) The Paradoxes Of Motion And Change

-(II) Inconsistent Obligations

-(III) Naïve Set Theory

Chapter Three: Liars And Gluts

Chapter Four: Liars And Gaps

Chapter Five: Liars And Meaninglessness

Chapter Six: Meaninglessness and Revenge

Chapter Seven: The Epistemic Consequences Of Dialetheism

-(I) Failures Of Truth?

-(II) Curry

-(III) Epistemic Consequences

Chapter Eight: Monaletheism And Negation

-(I) Folk Dialetheism?

-(II) Semantic Dialetheism

-(III) Keeping Score

-(IV) A Final Thought About Truth

## Tuesday, October 6, 2009

### A Thought About Probability Theory And Logical Fallibilism

In classical probability theory, the probability of all excluded middle instances (A v ~A) is 1 and the probability of all contradictions (A & ~A) is 0. This makes sense, given (some other, even more basic assumptions and) the fact that one of the basic organizing principles of classical probability theory is that for every A, P(A) and P(~A) have to add up to 1.

Of course, given some non-classical assumptions about logic, all that falls apart...as Hartry Field points out somewhere (in one of his essays in "Deflationism and Paradox," I think?), the dialetheist should accept that for some A's, the sum of P(A) and P(~A) can be high as 2, and the truth-value gap theorist should accept that it can be as low as 0.* But what I'm concerned with, at the moment, is *classical* logic and *classical* probability theory. This is my worry:

Even if you make thoroughly classical assumptions about logic--no truth-value gaps and no true contradictions--and take those assumptions to be extremely well-justified given current evidence, and the current arguments for and against them, it still seems to make sense to be a *fallibilist* about these matters. To admit that we can't be entirely sure, that considerations sometimes brought to bear against these assumptions (from the paradoxes of self-reference to Aristotle's sea battle problem to issues related to vagueness and "partially-defined predicates" or "over-defined predicates" or whatever) have at least some weight, even if we ultimately find them unconvincing, that we should be open to new arguments and new evidence that might be brought to bear by future dialetheists or by future gap theorists. To, in other words, grant at least *some* rational credence to belief in truth-value gaps and/or true contradictions, even if we firmly reject these beliefs, much the same way we grant at least *some* rational credence, however slight, to Lorentzian absolutism about space and time, or to extreme skeptical possibilities in epistemology, or to belief in ghosts.

Now, if you are a frequency theorist about probability, there's no problem about fallibilistically believing orthodox things about logic on the one hand, and accepting classical probability theory on the other hand, since, if there are no truth-value gaps and true contradictions, the number of each that will show up over the course of any sort of hypothetical limiting frequency will be 0, and thus it will always be correct to say that P(A v ~A)=1 and P(A & ~A)=0. On the other hand, I see no way for the Bayesian epistemologist, who takes probability to be intimately linked to *justification*, not a calculus of frequencies but of rational credences or some such, to have their cake and eat it too on this issue. Unless I'm missing something, it looks to me like the Bayesian has to give up on classical logic, classical probability theory or logical fallibilism, that they can't hold onto all three, given with their view of probability.

Thoughts?

*Note that the sort of "truth-value gap" under discussion here--and this applies to the rest of the post--are exceptions not just to Bivalence but to the Law of the Excluded Middle.

Of course, given some non-classical assumptions about logic, all that falls apart...as Hartry Field points out somewhere (in one of his essays in "Deflationism and Paradox," I think?), the dialetheist should accept that for some A's, the sum of P(A) and P(~A) can be high as 2, and the truth-value gap theorist should accept that it can be as low as 0.* But what I'm concerned with, at the moment, is *classical* logic and *classical* probability theory. This is my worry:

Even if you make thoroughly classical assumptions about logic--no truth-value gaps and no true contradictions--and take those assumptions to be extremely well-justified given current evidence, and the current arguments for and against them, it still seems to make sense to be a *fallibilist* about these matters. To admit that we can't be entirely sure, that considerations sometimes brought to bear against these assumptions (from the paradoxes of self-reference to Aristotle's sea battle problem to issues related to vagueness and "partially-defined predicates" or "over-defined predicates" or whatever) have at least some weight, even if we ultimately find them unconvincing, that we should be open to new arguments and new evidence that might be brought to bear by future dialetheists or by future gap theorists. To, in other words, grant at least *some* rational credence to belief in truth-value gaps and/or true contradictions, even if we firmly reject these beliefs, much the same way we grant at least *some* rational credence, however slight, to Lorentzian absolutism about space and time, or to extreme skeptical possibilities in epistemology, or to belief in ghosts.

Now, if you are a frequency theorist about probability, there's no problem about fallibilistically believing orthodox things about logic on the one hand, and accepting classical probability theory on the other hand, since, if there are no truth-value gaps and true contradictions, the number of each that will show up over the course of any sort of hypothetical limiting frequency will be 0, and thus it will always be correct to say that P(A v ~A)=1 and P(A & ~A)=0. On the other hand, I see no way for the Bayesian epistemologist, who takes probability to be intimately linked to *justification*, not a calculus of frequencies but of rational credences or some such, to have their cake and eat it too on this issue. Unless I'm missing something, it looks to me like the Bayesian has to give up on classical logic, classical probability theory or logical fallibilism, that they can't hold onto all three, given with their view of probability.

Thoughts?

*Note that the sort of "truth-value gap" under discussion here--and this applies to the rest of the post--are exceptions not just to Bivalence but to the Law of the Excluded Middle.

## Monday, October 5, 2009

### Diet Soap Interview

I talk to Doug Lain of the Diet Soap podcast about paraconsistent logic, counterpossible reasoning, analyticity and confirmational holism here. You can also get the episode by just going to iTunes and searching for Diet Soap. It's Episode # 25, entitled "This Podcast Is A Lie."

There's also a short story at the end of the episode. It had to be pretty severely abridged for the podcast, and the print magazine it was originally published in is now defunct, so I put a Creative Commons-licensed version of the whole thing here.

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