flink@mimsy.UUCP (Paul V Torek) (12/30/87)
To support my claim that an infinite regress is not sufficient justification for believing something, I present Pollock's proof that ANY proposition may be "justified" using an infinite regress. Let "<--" stand for "is justified by", and let "->" stand for the material conditional. Consider any proposition p, and any propositions q[i]. p <-- q1 , q1 -> p <-- q2 , q2 -> q1 , q2 -> (q1 -> p) <-- q3 , q3 -> q2 , q3 -> (q2 -> q1) , q3 -> (q2 -> (q1 -> p)) <-- etc. Sorry, I couldn't find any more detailed reference than "Pollock", which I think means John L. Pollock; a reference might be available in _Knowledge and Justification_ by that author (which I haven't read). -- "It only hurts when I laugh" --Marx Paul Torek torek@umix.cc.umich.edu
weemba@GARNET.BERKELEY.EDU (Obnoxious Math Grad Student) (12/31/87)
In article <9981@mimsy.UUCP>, flink@mimsy (Paul V Torek) writes: >To support my claim that an infinite regress is not sufficient justification >for believing something, I present Pollock's proof that ANY proposition may >be "justified" using an infinite regress. This could only show one particular infinite regression (or family of such) was invalid. The example you give below does not bear upon the one that I was discussing long ago: can self-reference lead to justified belief? Recall that I made this in analogy with Loeb's theorem, which asserts that the number theoretic sentence that asserts "I am provable within PA" is in fact true and provable within PA, despite the surface appearances that it could go either way. You then asserted that self-reference leads to infinite regress, ergo, such a possibility is apriori untenable. The jump from "some IRs are bad" to "all IRs are bad" is an unforgivable quantifier switch. Your example is noteworthy, but I'm still convinced that most times I hear objections to "infinite regress", I'm actually hearing ancient bugaboos. Note that I don't actually yet have an opinion about IRs and bootstrapped justifications, just that I find the possibility intriguing and not at all obvious one way or another. >Let "<--" stand for "is justified by", and let "->" stand for the material >conditional. Consider any proposition p, and any propositions q[i]. > p <-- > q1 , q1 -> p <-- > q2 , q2 -> q1 , q2 -> (q1 -> p) <-- > q3 , q3 -> q2 , q3 -> (q2 -> q1) , q3 -> (q2 -> (q1 -> p)) <-- > etc. [Pollock example] And this reminds me of the infinite regression inside Lewis Carroll's Achilles and the tortoise story (-> binds tighter than &): Start (q) <-- | [ p , p->q ] <-- | { p , p->q , (p&p->q)->q } <-- V < p , p->q , (p&p->q)->q , p&p->q&(p&p->q)->q > <-- ... ( ... ... ) ... And so, by your reasoning, we should reject modus ponens. How now brown cow? (And is there anyone else who's always been bugged by Carroll's example?) Most curiously, about a month ago I came across an article about 2-3 years old in _The British Journal for the Philosophy of Science_ which argued that Descartes' "cogito, ergo sum" can be justified as a Cantor diagonalization! I only skimmed the article--with neither belief nor disbelief nor realization that I had argued similarly, just astonish- ment--and filed it away mentally for later reference; perhaps someone else would like to look at it. [I unfortunately am busy.] ucbvax!garnet!weemba Matthew P Wiener/Brahms Gang/Berkeley CA 94720 ...those infected with the superstring syndrome have to believe in miracles: not the minor ones like the parting of the Red Sea, but major ones in which six of the ten space-time dimensions become compactified to the Planck scale.
flink@mimsy.UUCP (Paul V Torek) (01/05/88)
Matthew P. Wiener writes:
w>This could only show one particular infinite regression (or family of
w>such) was invalid.
True, but I think it establishes a burden of argument on someone who claims
that some other type of infinite regression is justifying. After all, the
Pollock example has some nice attributes like deductive validity and
psychological nontransparency. (By the latter, I mean that a person might
easily believe the premises without noticing that the conclusion follows.
This would not be true of an argument of the form, "A&B, therefore A", for
example.)
w>The example you give below does not bear upon the
w>one that I was discussing long ago: can self-reference lead to justified
w>belief? Recall that I made this in analogy with Loeb's theorem, which
w>asserts that the number theoretic sentence that asserts "I am provable
w>within PA" is in fact true and provable within PA, despite the surface
w>appearances that it could go either way.
w>You then asserted that self-reference leads to infinite regress, ergo,
w>such a possibility is apriori untenable.
I don't recall saying that -- you might be confusing me with some of the
Objectivists. Anyway, the self-reference of your sentence above doesn't
require infinite regress to verify, so it seems importantly different
from, say, the English sentence "this sentence is true". The latter
sentence *does* require infinite regress to verify, and it seems
objectionable (to me at least) for that reason.
w>[The Pollock example] reminds me of the infinite regression inside Lewis
w>Carroll's Achilles and the tortoise story (-> binds tighter than &):
w>
w> Start (q) <--
w> | [ p , p->q ] <--
w> | { p , p->q , (p&p->q)->q } <--
w> V < p , p->q , (p&p->q)->q , p&p->q&(p&p->q)->q > <--
w> ... ( ... ... ) ...
w>
w>And so, by your reasoning, we should reject modus ponens.
But wait, there's less! Since the first set of premises is *transparently*
implied by the Nth, anyone who accepts the Nth ought to accept the
first, without need to deduce that set of premises from anything else.
Hence, there's no need for the regress. By contrast, in the Pollock example,
at any finite stage of the argument, there could be a point to appealing to
the next set of premises. E.g., someone might actually be convinced that q1
and that q1->p, if it is suggested to him that q2 and q2->q1 and q2->(q1->p).
One can consistently maintain that modus ponens is legitimate (i.e., that
q is justified by p & p->q, provided that p is already justified and so
is p->q) while maintaining that the above infinite regress fails to
justify. NOTE: "justify" is different from "validly deduce" (or at least,
the claim that they are the same needs to be argued for.)
--
"It only hurts when I laugh" --Marx
Paul Torek torek@umix.cc.umich.eduweemba@GARNET.BERKELEY.EDU (Obnoxious Math Grad Student) (01/05/88)
[Some of you might have seen a forgery come by recently. Onwards.] In article <10034@mimsy.UUCP>, flink@mimsy (Paul V Torek) writes: >Matthew P. Wiener writes: >w>This could only show one particular infinite regression (or family of >w>such) was invalid. > >True, but I think it establishes a burden of argument on someone who claims >that some other type of infinite regression is justifying. I'll agree that it establishes the burden on someone who claims that it's "intuitive". I've seen enough strange proofs in my time that I don't think "counterintuitive" has much meaning left in me, unless I kick myself, or get sucked in by somebody's mathematical propaganda. Now why should I think there's some plausibility behind any infinite re- gression arguments? Well, thinly disguised versions of these are ban- died about quite freely in modern set theory--or at least at one time they used to be--to "justify" large cardinal axioms (the higher infin- ities). This activity makes no sense to formalists, but to a Platonist like myself it seems a necessary phase one goes through. And not just an infinite regression, but transfinite regressions. I've seen people say things to the effect that, "0# is harmless, and why not, 0## will be too (which justifies 0# by the by), and hey, so will 0###, (which justifies 0##) and so on by transfinite induction of thin-air assumptions. But this Pollock-like reasoning, while plausible--to some people at least--is generally considered too suspicious or em- barrassing even, so one rethinks the issue and discovers a single higher principle that covers all the 0##...##..##...s. The details of this example don't matter, my point is that it HAS been a form of reasoning that has been taken seriously by a good number of mathematicians. Whether the future will come to a consensus is unknown, but I am reminded of the groping infinitary arguments that went on to "justify" the axiom of choice at the turn of the century, contrasted with an almost universal acceptance of its "intuitiveness" today. (And the intuitive introductions one sees today, of course, are still as in- finitary as ever. The explicators are just not abashed anymore, or even worried that anyone might object.) > Anyway, the self-reference of your sentence above doesn't >require infinite regress to verify, so it seems importantly different >from, say, the English sentence "this sentence is true". The latter >sentence *does* require infinite regress to verify, and it seems >objectionable (to me at least) for that reason. Ah, but the mathematical sentence "this sentence is provable" does not require an infinite regress to verify! That's the surprising point of Loeb's theorem. >w>[The Pollock example] reminds me of the infinite regression inside Lewis >w>Carroll's Achilles and the tortoise story >w>.... >w>And so, by your reasoning, we should reject modus ponens. > >But wait, there's less! Since the first set of premises is *transparently* >implied by the Nth, anyone who accepts the Nth ought to accept the >first, without need to deduce that set of premises from anything else. But the tortoise snickers, "Uh uh uh, Mr T, you've just made another one of those darned assumptions! Better put it down in your little notebook!" >Hence, there's no need for the regress. In formal logic, this is correct, since one is studying certain mathem- atical sequences, and the very first step of the regression is blocked: one's reasoning here is not one of those sequences. It is translatable into one of those sequences; that is how the Goedelian and Loebian proofs work. But as for ordinary language, ordinary beliefs? I don't think anyone can claim to know. > By contrast, in the Pollock example, >at any finite stage of the argument, there could be a point to appealing to >the next set of premises. E.g., someone might actually be convinced that q1 >and that q1->p, if it is suggested to him that q2 and q2->q1 and q2->(q1->p). Right. And in the Carroll example, there's also a point--albeit a twisted point--in appealing to the higher premise! Someone might not yet be con- vinced that q, yet be convinced of p and p->q, because he has intuitionis- tic style doubts about modus ponens. And one can then up argument this ad infinitum. >One can consistently maintain that modus ponens is legitimate (i.e., that >q is justified by p & p->q, provided that p is already justified and so >is p->q) while maintaining that the above infinite regress fails to >justify. If I read your references correctly, you are saying one can be convinced that the Carroll example is an unnecessary attempt to appeal to the re- gression while the Pollock example isn't. I agree: a pure formalist is just such a person. ucbvax!brahms!weemba Matthew P Wiener/Brahms Gang/Berkeley CA 94720 A mathematician's wife overhears her husband muttering the name 'Nancy'. She wonders whether Nancy, the thing to which her husband referred, is a woman or a Lie group. --Saul Kripke
franka@mmintl.UUCP (Frank Adams) (01/09/88)
In article <8712310840.AA03892@garnet.berkeley.edu> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: >And this reminds me of the infinite regression inside Lewis Carroll's >Achilles and the tortoise story (-> binds tighter than &): > > Start (q) <-- > | [ p , p->q ] <-- > | { p , p->q , (p&p->q)->q } <-- > V < p , p->q , (p&p->q)->q , p&p->q&(p&p->q)->q > <-- ^^^^^^^^^^^^^^^^^^ (That should be: ((p&p->q)&((p&p->q)->q))->q.) > ... ( ... ... ) ... >And so, by your reasoning, we should reject modus ponens. No, we just note that this isn't a proof of it. It seems to me that the fallacy in Carroll's story lies in confusing the statement of modus ponens from modus ponens itself. "Whenever p and p->q, then we can conclude q" is a statement of modus ponens. But when one applies modus ponens, one does not refer to that statement at all. One merely notes that one has p and p->q, and concludes q. "(p&p->q)->q" isn't modus ponens at all, although it has more than a passing resemblence to it. -- Frank Adams ihnp4!philabs!pwa-b!mmintl!franka Ashton-Tate 52 Oakland Ave North E. Hartford, CT 06108
franka@mmintl.UUCP (Frank Adams) (01/09/88)
In article <8712310840.AA03892@garnet.berkeley.edu> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes: >can self-reference lead to justified belief? (I will address primarily the question, can infinite regress lead to justified belief.) Consider, for the moment, an infinite regress: a series of statements a[1], a[2], ..., such that a[i+1]->a[i]. In order for the regress to provide better reason to believe a[1] than some subsequence a[n],...,a[1], each a[i+1] must be a priori more plausible than a[i]. (Actually, there need only be an infinite subset of the a[i] with this property, but we can consider only that subset without loss of generality.[1]) Now, consider some plausible complexity measure on sentences, such as "the number of letters in the sentence". (Yes, I have jumped from statements to sentences. I believe that jump is justified.) The basic property we require from this measure is that the number of sentences as complex or less complex than a given sentence is finite. I think this is a pretty minimal requirement for a complexity measure. There must be a subset of the a[i] which are of non-decreasing complexity; again, without loss of generality we may take this to be the entire sequence. If the complexity remains finite, then we have in fact circular reasoning. Thus, we must have an infinite sequence of statements, with strictly increasing complexity, and strictly increasing a priori plausibility. I find this completely implausible. This does not deal directly on the case of self reference, since in this case we have the complexity (and hence the number of distinct statements) remaining finite. It does seem to rule out infinite regress in any case *except* for circular reasoning, however, so we are left with the question: can circular reasoning lead to justified belief? I would say no. [1] Consider the sequence b[], which consists of those elements of a[] which are in the subset, in the order they occur in a[]. Thus b[i]=a[j[i]], where j[] is sequence of integers satisfying j[i]<j[i+1]. Then b[i+1] = a[j[i+1]] -> a[j[i+1]-1] -> ... -> a[j[i]] = b[i], so b[] is an infinite regress with the required property. -- Frank Adams ihnp4!philabs!pwa-b!mmintl!franka Ashton-Tate 52 Oakland Ave North E. Hartford, CT 06108
flink@mimsy.UUCP (Paul V Torek) (01/30/88)
Matthew P. Wiener writes:
w> And not just an infinite regression, but transfinite
w> regressions. I've seen people say things to the effect
w> that, "0# is harmless, and why not, 0## will be too (which
w> justifies 0# by the by), and hey, so will 0###, (which
w> justifies 0##) and so on by transfinite induction of thin-air
w> assumptions. But this Pollock-like reasoning, while
w> plausible--to some people at least--is generally considered
w> too suspicious or embarrassing even, so one rethinks the
w> issue and discovers a single higher principle that covers
w> all the 0##...##..##...s.
Sounds like plain old scientific-style generalization to me.
These people have all these intuitions about the O#'s, and ask
themselves "what do all these have in common that makes them so
appealing?" Whereupon they discover the single higher principle
you refer to. Kinda like, "why do all these fruits (banana, apple,
cherry ...) taste sweet?" "Ah, sugar." Perhaps the implicatory
relations among the O#'s are incidental to their appeal?
t> Anyway, the self-reference of your sentence above doesn't
t> require infinite regress to verify, so it seems importantly
t> different from, say, the English sentence "this sentence is
t> true". The LATTER sentence *does* require ...
w> Ah, but the mathematical sentence "this sentence is provable"
w> does not require an infinite regress to verify!
I just said that myself (capitalized emphasis added).
w> [much deleted...]
w> But as for ordinary language, ordinary beliefs? I don't think
w> anyone can claim to know.
Well sure we can! Psychologists have studied this. Dan Osherson,
for one, tested what inferences people would draw from each of
many sets of premises. 100% of his subjects used modus ponens.
(Most of them drew inferences by affirming the consequent, too;
and in some experiments, they fail to use modus tollens.)
--
"The philosopher's dictionary defines `outSmarting
the opposition' as accepting the conclusions of their
reductio ad absurdum arguments" --Jerry Fodor
Paul Torek torek@umix.cc.umich.edu