## Monday, January 28, 2008

## Sunday, January 27, 2008

### Follow-up: "Interesting but Inconsistent?"

While looking at the discussion after my last post--for those who missed it, I made a really dumb attribution mistake that was thankfully corrected, and it was forcefully brought to my attention that my original example relied on an understanding of conditionals deeply controversial among precisely the same sort of people likely to find Impressive Scientist X examples compelling, but I argued that the same point could be made without bringing in the material conditional, and that there are simpler explanations of e.g. Bohr's failure to derive random crazy things from his inconsistent atomic theory than that he was somehow unknowingly working with an "underlying paraconsistent logic"--something closely related occurred to me.

There's a phrase that's used a lot in these discussions as if we all knew what it meant. I'm not sure that it's so clear. That's "interesting but inconsistent theory."

(The phrase "non-trivial," often appended to "interesting but inconsistent," adds bupkis. All it means is that not everything will follow from the theory, i.e. that interesting but inconsistent theories should be reasoned about paraconsistently or not at all. OK. If you believe that the moon is both made of green cheese and not made of green cheese, that theory will be "non-trivial" in precisely the same sense, but I doubt anyone would call it "interesting" in the relevant sense.)

Well, what is an 'interesting' theory? I mean, I think I know what it means before modified with "but inconsistent," but after that's there, I'm not so sure any more. Normally, when talking about consistent theories, I would take "interesting theory" (in the sense that seems to be driven at, not "interesting" as in "crazily unexpected" or anything like that) to mean "plausible theory," i.e. one that might very well turn out to be true. Or, applied to out-dated theories, one that it would have been rational to regard as quite possibly true given the evidence available at the time, even if we now understand that it is false.

Now, full-blown dialetheists banding about the phrase "interesting but inconsistent theory" might mean exactly this, since they think it's possible in principle for something to be inconsistent but true. What I'm interested in at the moment is what this "interesting but inconsistent phrase" means to people who bandy it around who are on the 2nd Grade of Paraconsistent Involvement discussed in the last post, the "I'm not a dialetheist, but..." crowd who are still holding back from the 3rd Grade where you admit that some of these theories may be true.

So, if you aren't a dialetheist, you still believe in the Law of Non-Contradiction, hence you believe that it is categorically impossible for an inconsistent theory to turn out to be true and there has never been a situation where it would have been rational on the basis of any sort of empirical evidence to believe that an inconsistent theory was true, what does "interesting but inconsistent theory" mean?

Interestingness also can't just boil down to a degree of predictive accuracy*, right? If so, interesting-but-inconsistent theories would be too easy to generate and lose the aura of respectability they gain from Bohr-type examples. After all, for any hotly disputed scientific area, where one theory predicts a bunch of effects with a certain amount of experimental support, and an obviously logically inconsistent competitor theory predicts a bunch of other effects, and there are a certain amount of experimental support for those two, if someone blandly asserted that the disputed phenomenon both existed and didn't exist, and was thus able to claim the experimental successes of both competitors for this claim (like, "if X exists, we expect to see some Y's and if it doesn't, we'd expected to see some Z's, so I expect to see both Y's and Z's..."), would that make the new conjunctive theory "interesting"?

Anyway, I'm throwing this open to the floor. On the assumption that it's never rational to believe an inconsistent theory to be actually true, what does it mean to call one 'interesting'?

*...although why we'd be interested in predictive accuracy, except as an indicator of truth, is for the most part mysterious to me.

There's a phrase that's used a lot in these discussions as if we all knew what it meant. I'm not sure that it's so clear. That's "interesting but inconsistent theory."

(The phrase "non-trivial," often appended to "interesting but inconsistent," adds bupkis. All it means is that not everything will follow from the theory, i.e. that interesting but inconsistent theories should be reasoned about paraconsistently or not at all. OK. If you believe that the moon is both made of green cheese and not made of green cheese, that theory will be "non-trivial" in precisely the same sense, but I doubt anyone would call it "interesting" in the relevant sense.)

Well, what is an 'interesting' theory? I mean, I think I know what it means before modified with "but inconsistent," but after that's there, I'm not so sure any more. Normally, when talking about consistent theories, I would take "interesting theory" (in the sense that seems to be driven at, not "interesting" as in "crazily unexpected" or anything like that) to mean "plausible theory," i.e. one that might very well turn out to be true. Or, applied to out-dated theories, one that it would have been rational to regard as quite possibly true given the evidence available at the time, even if we now understand that it is false.

Now, full-blown dialetheists banding about the phrase "interesting but inconsistent theory" might mean exactly this, since they think it's possible in principle for something to be inconsistent but true. What I'm interested in at the moment is what this "interesting but inconsistent phrase" means to people who bandy it around who are on the 2nd Grade of Paraconsistent Involvement discussed in the last post, the "I'm not a dialetheist, but..." crowd who are still holding back from the 3rd Grade where you admit that some of these theories may be true.

So, if you aren't a dialetheist, you still believe in the Law of Non-Contradiction, hence you believe that it is categorically impossible for an inconsistent theory to turn out to be true and there has never been a situation where it would have been rational on the basis of any sort of empirical evidence to believe that an inconsistent theory was true, what does "interesting but inconsistent theory" mean?

Interestingness also can't just boil down to a degree of predictive accuracy*, right? If so, interesting-but-inconsistent theories would be too easy to generate and lose the aura of respectability they gain from Bohr-type examples. After all, for any hotly disputed scientific area, where one theory predicts a bunch of effects with a certain amount of experimental support, and an obviously logically inconsistent competitor theory predicts a bunch of other effects, and there are a certain amount of experimental support for those two, if someone blandly asserted that the disputed phenomenon both existed and didn't exist, and was thus able to claim the experimental successes of both competitors for this claim (like, "if X exists, we expect to see some Y's and if it doesn't, we'd expected to see some Z's, so I expect to see both Y's and Z's..."), would that make the new conjunctive theory "interesting"?

Anyway, I'm throwing this open to the floor. On the assumption that it's never rational to believe an inconsistent theory to be actually true, what does it mean to call one 'interesting'?

*...although why we'd be interested in predictive accuracy, except as an indicator of truth, is for the most part mysterious to me.

## Wednesday, January 23, 2008

### A Thought About Underlying Logics

My apologies for the weeks between this and the last post. I'm back in Miami, slugging through my reading list, and I should be posting at a much more frequent clip from now on.

Meanwhile, I have a thought (not really a full thought, but at least the beginning of one), not about dialetheism per se but about what Graham Priest calls the "second grade of paraconsistent involvement."

Just for future reference, his "grades" are:

1st: "Gentle-strength paraconsistency" (you reject the principle that anything follows from a contradiction)

2nd: "Full-strength paraconsistency" (you think there some inconsistent but interesting, non-trivial theories)

3rd: "Industrial-strength paraconsistency" (some of those theories may be true)

4th: "Dialetheism" (some of those theories *are* true)

#

Standard apologetics for the usefulness of paraconsistent logic often include historical examples of inconsistent but non-silly theories. It generally goes something like this. "Impressive Scientist X believed P and he also believed Theory Q, and he knew that Theory Q entails that not-P, but he didn't derive just any claim R, so the underlying logic he was using was clearly not classical."

Now, on the face of it the only sort of explosive logical rule that this situation would be any kind of challenge to would be a claim in the language of epistemic logic that Bxp & Bx~p entailed Bxq for any q. [Of course, (Bxp & Bx~p) is not a contradiction. Only (Bxp & ~Bxp) would be a contradiction.]

Sadly, this has nothing to do with what the historical examples are there for. The historical claim here is not that according to any sort of classical logic, it should follow from Scientist X believing P and ~P that Scientist X will believe that Q for any Q. Rather, the point is that if Scientist X believes that P and he believes that ~P, then classical logic would give him permission to draw the conclusion that Q. The fact that he never exercised this privilege is then seen as evidence that he was (albeit unconsciously, when we're talking about figures who predated the development of paraconsistent logic) operating according to paraconsistent rules of inference in which there are strict limits on what you can derive from a contradiction.

Now, I have severe doubts about the very idea that either scientific practices (or, worse yet, as is sometimes claimed, natural languages) have "underlying logics," but for the moment I'm going to put that to one side.

Instead, let's go for a simple analogy.

Einstein believed in the Special Theory of Relativity.

Einstein never drew the conclusion that "if it is not the case that it is not the case that it is not the case that it is not the case that it is not the case that the Special Theory of Relativity is true, then it is not the case that it is not the case that it is not the case that it is not the case either that Hitler won World War II or that the national flag of Australia is a pirate flag containing a jewel, that, under magnification, is revealed to be a detailed map of the surface of Mars with an X marking the spot where Elvis has high tea every afternoon at four with the Mad Hatter and a ghostly semi-physical representative of the Hegelian world-spirit."

But wait!

According to classical logic, the entire quoted claim is entailed by the truth of the Special Theory of Relativity. (Unless, of course, I slipped up while counting the number of negations.) If the STR is true then it follows that if you put an odd number of negation signs in front of the proposition that it is true, you have a false claim, and any conditional statement with a false antecedent is true, regardless of whether the consequent is true or false.

...so Einstein never drew that conclusion, or any of the infinite number of other similar conclusions classical logic would have given him permission to draw. Does it therefore follow that his "underlying logic" must have been some alternative non-classical logic, where strict rules are in place to reign in the sorts of consequents that can be put on these conditionals?

Meanwhile, I have a thought (not really a full thought, but at least the beginning of one), not about dialetheism per se but about what Graham Priest calls the "second grade of paraconsistent involvement."

Just for future reference, his "grades" are:

1st: "Gentle-strength paraconsistency" (you reject the principle that anything follows from a contradiction)

2nd: "Full-strength paraconsistency" (you think there some inconsistent but interesting, non-trivial theories)

3rd: "Industrial-strength paraconsistency" (some of those theories may be true)

4th: "Dialetheism" (some of those theories *are* true)

#

Standard apologetics for the usefulness of paraconsistent logic often include historical examples of inconsistent but non-silly theories. It generally goes something like this. "Impressive Scientist X believed P and he also believed Theory Q, and he knew that Theory Q entails that not-P, but he didn't derive just any claim R, so the underlying logic he was using was clearly not classical."

Now, on the face of it the only sort of explosive logical rule that this situation would be any kind of challenge to would be a claim in the language of epistemic logic that Bxp & Bx~p entailed Bxq for any q. [Of course, (Bxp & Bx~p) is not a contradiction. Only (Bxp & ~Bxp) would be a contradiction.]

Sadly, this has nothing to do with what the historical examples are there for. The historical claim here is not that according to any sort of classical logic, it should follow from Scientist X believing P and ~P that Scientist X will believe that Q for any Q. Rather, the point is that if Scientist X believes that P and he believes that ~P, then classical logic would give him permission to draw the conclusion that Q. The fact that he never exercised this privilege is then seen as evidence that he was (albeit unconsciously, when we're talking about figures who predated the development of paraconsistent logic) operating according to paraconsistent rules of inference in which there are strict limits on what you can derive from a contradiction.

Now, I have severe doubts about the very idea that either scientific practices (or, worse yet, as is sometimes claimed, natural languages) have "underlying logics," but for the moment I'm going to put that to one side.

Instead, let's go for a simple analogy.

Einstein believed in the Special Theory of Relativity.

Einstein never drew the conclusion that "if it is not the case that it is not the case that it is not the case that it is not the case that it is not the case that the Special Theory of Relativity is true, then it is not the case that it is not the case that it is not the case that it is not the case either that Hitler won World War II or that the national flag of Australia is a pirate flag containing a jewel, that, under magnification, is revealed to be a detailed map of the surface of Mars with an X marking the spot where Elvis has high tea every afternoon at four with the Mad Hatter and a ghostly semi-physical representative of the Hegelian world-spirit."

But wait!

According to classical logic, the entire quoted claim is entailed by the truth of the Special Theory of Relativity. (Unless, of course, I slipped up while counting the number of negations.) If the STR is true then it follows that if you put an odd number of negation signs in front of the proposition that it is true, you have a false claim, and any conditional statement with a false antecedent is true, regardless of whether the consequent is true or false.

...so Einstein never drew that conclusion, or any of the infinite number of other similar conclusions classical logic would have given him permission to draw. Does it therefore follow that his "underlying logic" must have been some alternative non-classical logic, where strict rules are in place to reign in the sorts of consequents that can be put on these conditionals?

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