tag:blogger.com,1999:blog-3891434218564545511.post6799547102410727750..comments2024-03-27T20:37:09.185-05:00Comments on Alexander Pruss's Blog: Propositions that never become true but are probableAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger92125tag:blogger.com,1999:blog-3891434218564545511.post-64999954584251050572016-06-07T18:26:12.943-05:002016-06-07T18:26:12.943-05:00The argument at the heart of this blog post is a c...The argument at the heart of this blog post is a central part of a paper coming out in Faith and Philosophy.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-45548276836896616792014-11-20T17:09:26.063-06:002014-11-20T17:09:26.063-06:00ah, too bad. thought we were right there. ok.ah, too bad. thought we were right there. ok.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-49638560341605883622014-11-20T16:30:05.138-06:002014-11-20T16:30:05.138-06:00I am afraid that we're not making progress. I ...I am afraid that we're not making progress. I suggest a bit of silence on this thread, unless there is something radically new to say.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-16801403760007614872014-11-20T14:03:05.473-06:002014-11-20T14:03:05.473-06:00Alex,
If for every finite number x, there is a w...Alex, <br /><br />If for every finite number x, there is a world in which S <i>counts up to</i> x, and there are infinitely many worlds, then for every finite number y, S counts up to y in some world. But then there cannot be a finite number that S does not <i>count up to</i> in some world. As you get closer and closer to the infinite limit on worlds, you get closer and closer to counting every finite number. At the limit, you've done it.<br /><br />Compare a plant that is one centimeter taller in each successive world in an infinite sequence of worlds. At the limit the plant has grown by an infinite number of increments. I'm pretty sure that's uncontroversial.<br /><br />Compare an infinite sequence of worlds that increase by one turp of evil per world. At the limit we have a world that includes infinitely many turps of evil. <br /><br />These are all the same argument essentially. None is especially controversial.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-86976608426443441952014-11-20T13:48:09.610-06:002014-11-20T13:48:09.610-06:00Ok, Michael, thanks.Ok, Michael, thanks.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-49791591358892095992014-11-20T13:35:01.976-06:002014-11-20T13:35:01.976-06:00Mike:
I'm not saying infinity is a number. I&...Mike:<br /><br />I'm not saying infinity is a number. I'm saying that accomplishing a task has to include the ability to truthfully say "I just did X". If you can't ever truthfully say "I just finished counting to infinity" then you can't ever get to the point where you have counted to infinity. Whether Pruss or anyone else says a sentence including the phrase "count to every finite number", it's still a totally incoherent phrase. There is no highest number, so it doesn't mean anything to say that you've counted every finite number. It's like saying "if you could catch the horizon, you would indeed fall of the edge of the Earth".Michael Gonzalezhttps://www.blogger.com/profile/05279261871735286117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-31146677629731025092014-11-20T10:17:52.865-06:002014-11-20T10:17:52.865-06:00We're reaching the point of diminishing utilit...We're reaching the point of diminishing utility in this discussion. Alex has already agreed, if you count to every finite number, then you have counted to infinity. See the post above (Nov. 19, 4:44). I will repost here.<br />---<br />Blogger Alexander R Pruss said...<br />Agreed: if you count to every finite number, you count to infinity<br />----<br /><br />It's just confused to say that you must say "infinity, there I've reached it". You seem to think that infinity is a number, and it isn't. You seem to think there is an infinite number in the naturals, and there isn't. <br /><br />I think I can and have made some progress in the discussion with Alex; but there's otherwise too much work for too little pay. Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-22667511766744175692014-11-20T09:15:06.492-06:002014-11-20T09:15:06.492-06:00Mike:
I really think you're missing a fundame...Mike:<br /><br />I really think you're missing a fundamental point. For every finite number, there is always a larger one. There is no finite number which does not have a number larger than it. Therefore your statement, " ~☐(∀y)(∃x)(x is greater than all the counted naturals y)", is false. It is necessarily true that, in any case of counting, no matter how far you've gotten, there is still a number you have not yet counted.<br /><br />Now you try to re-word the case in terms of worlds. You say "☐(∀x)(there is some world in which the counter has counted to x)." But, even if that's true, there is always another number that that person did not count to. The point is that, for any person to actually <b>accomplish the task</b> of counting to infinity, they do indeed have to be able to say "... infinity! There! I finished." If they can't truthfully say that, then there is no sense in which they have counted to infinity. In every world, I can always say to the counter, "add 1".<br /><br />Saying you can count to ANY finite number does not mean you can count ALL of them. Heath and Pruss are using intuitive cases to get at that.Michael Gonzalezhttps://www.blogger.com/profile/05279261871735286117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-28348648866733605762014-11-20T08:06:07.197-06:002014-11-20T08:06:07.197-06:00Alex,
I hope I didn't commit an elementary qu...Alex,<br /><br />I hope I didn't commit an elementary quantifier switch fallacy. I think I conceded that there is no world where it is true that every finite number has been counted. That is (2') above. What I claimed was that there is no finite number to which S does not count in some world. That shows, I'm urging, that S can count to infinity though there is no world in which he does. You would think that there is something mistaken about this claim only if you think that everything that happens, happens in a world. Imagine that S is a modal continuant, if that will make the case easier to think about. The modal continuant S counts to infinity. Now ask: is it really necessary to think of S as a modal continuant to reach the conclusion that (i) S counted to infinity and (ii) there is no world in which S counts to infinity? I think the answer to that is no.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-68174046286678313542014-11-20T01:10:36.027-06:002014-11-20T01:10:36.027-06:00Hi Alexander, I do not want to assume anything, so...Hi Alexander, I do not want to assume anything, so I ask:<br /><br />Do you still think that any of this discussion disproves the possibility of an open future?<br /><br />Also, if your answer is yes, then do you think it also disproves the possibility of a closed future?James Goetzhttps://www.blogger.com/profile/02412501436355228925noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-75902242428585332842014-11-19T16:44:51.931-06:002014-11-19T16:44:51.931-06:00Agreed: if you count to every finite number, you c...Agreed: if you count to every finite number, you count to infinity.<br /><br />But you cannot infer from:<br /> For all n: Possibly: you count to n.<br />To:<br /> Possibly: For all n: you count to n.<br /><br />This discussion reminds me of the difference between consistency and omega-consistency.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-66476626241946498972014-11-19T15:42:34.695-06:002014-11-19T15:42:34.695-06:00Alex,
Let me alter your example slightly. Suppos...Alex, <br /><br />Let me alter your example slightly. Suppose there is a world in which there is a countably infinite number of times, t1, t2, t3, t4,...,tn. Now suppose it is true that, for each finite time tn, you will count to tn. There is no tn in the sequence of times such that you do not count to tn at some point. This too may be a wish God has granted to you. <br /><br />CLAIM: If for every finite time tn in the sequence you count to that time (1 for t1, 2 for t2, n for tn), then you will count to infinity. <br /><br />Suppose you did not thereby count to infinity. Then there would be some finite time tn such that you did not count to tn. But by hypothesis that is false. Therefore, you counted to infinity. <br /><br />Yes, counting to every finite time is counting to infinity. The illusion that counting to infinity entails reaching the point where you say "...infinity, there I did it!!" really needs to be resisted. It is a badly misleading representation of what's required. All of the naturals are finite, count all the finite numbers and you have counted to infinity.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-92054194880214401452014-11-19T15:17:48.254-06:002014-11-19T15:17:48.254-06:00I think I can make Alex’s counterexample simpler, ...I think I can make Alex’s counterexample simpler, at least as an analogy. Suppose God offers me an arbitrarily long life, after which I will die. I can specify any natural number and God will let me live that number of years. So there is a possible world W1 in which I live 1 year, a world W2 in which I live 2 years, etc. So for any (and every) natural number, it is possible that I live that long. <br /><br />But from the fact that it is possible that I live any length of time, it does not follow that I am possibly immortal. And from the fact that I can count to any finite number, it does not follow that I can count to infinity.<br />Heath Whitehttps://www.blogger.com/profile/13535886546816778688noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-73168927223618860522014-11-19T15:02:40.230-06:002014-11-19T15:02:40.230-06:00Mike:
Here's a counterexample that seems to w...Mike:<br /><br />Here's a counterexample that seems to work. Suppose that there is no afterlife, but God offers me as long a finite life as I want. I just have to specify the number of years. He'll even give me extra time to specify the length of life, as long as I spend that time doing nothing else than specifying that number (in particular, I'm not allowed to count).<br /><br />So, for any finite number n, I can count to n. How? I first specify to God how long it will take for me to count to n, and then I count to n.<br /><br />But I cannot count through all the natural numbers, because that would require me to have eternal life, and in the story God hasn't offered me that option.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-47912150942014798072014-11-19T12:55:55.162-06:002014-11-19T12:55:55.162-06:00I think the argument can be made even stronger. I ...I think the argument can be made even stronger. I could concede the intuition that (2') is true and depend on (3) alone for my claim.<br /><br />(2') ☐(Vy)(Ex)(x is greater than all the counted naturals y)<br />This says necessarily, for all the counted numbers y, there is some greater uncounted number x.<br /><br />(2') just concede what I did above that in every world the counter counts to a finite number. But (3) guarantees that the counter nonetheless counts to infinity. Again quantifying over the naturals.<br /><br />(3). ☐(Vx)(there is some world in which the counter has counted to x).<br />This says that every finite number is counted to in some world or other.<br /><br />That accommodates the intuition that in no world does the counter say "infinity, there I counted to infinity" (i.e. he counts all and only finite numbers). And it is nonetheless true that there is no finite number that is not counted in some world.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-83927573726538877292014-11-19T12:32:04.101-06:002014-11-19T12:32:04.101-06:00Heath,
You seem to be attributing to my argument ...Heath,<br /><br />You seem to be attributing to my argument the claim in (1) above, which I am not using to support my claim that you can count to infinity. I took the time to point this our above. What I am using is (2).<br />(2) ~☐(Vy)(Ex)(x is greater than all the counted naturals y)<br />This says that it's not necessary that, for all the counted numbers y, there is some greater uncounted number x.<br /><br />That is, I am claiming that it is false that, necessarily, for any number you count to, there is some greater number or other that you did not count to. I understand why someone might think that (2) is false. They might be thinking that in order to count to infinity you have to arrive at some possible world where someone utters the words "...infinity, there I did it, I counted to infinity!". But that is not what it means to count to infinity. There is no such number "infinity". Craig gives the deeply misleading impression that this is what must happen in order to count to infinity, and that gives his argument some misleading force. <br /> In order to count to infinity you have to count all the finite numbers. You of course do not reach the number "infinity", since there is no such number. But then how do you count to infinity? Consider an countably infinite set of possible worlds all numbered w0, w1, . . .,wn. Let these worlds overlap with respect to a "counter" who counts to 1 in w1, counts to 2 in w2, 3 in w3...all the way up. It is easy to map the naturals to worlds and a counter in this way. If the "counter" counts to 1 in w1, 2 in w2, 3 in w3, and so on upward, and the possible worlds are infinite in cardinality, then the counter has counted to infinity. He has counted to infinity though <i>there is no possible world in which he says "infinity! there I did it!!". There is no possible word in which he does not count to a finite number!</i>. Rather, he has counted all of the finite numbers. It follows from this that it is true in every world that he can count to infinity, since in each of infinitely many possible worlds wn he has counted to the finite number n.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-78359310958316006682014-11-19T12:16:01.081-06:002014-11-19T12:16:01.081-06:00Preface to my previous comment, cut off by Blogger...Preface to my previous comment, cut off by Blogger for some reason:<br /><br />"It seems to me one should reject Mike’s argument but for a rather subtle reason."Heath Whitehttps://www.blogger.com/profile/13535886546816778688noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-35265048944953061652014-11-19T11:48:43.732-06:002014-11-19T11:48:43.732-06:00He is arguing that from
(a) I can’t count all the...He is arguing that from<br /><br />(a) I can’t count all the natural numbers<br />(b) All the natural numbers are finite numbers<br /><br />We can infer<br /><br />(c) There is some finite number to which I cannot count<br /><br />Which of course seems wrong, and suggests that we should abandon the first premise.<br /><br />But suppose we talk about ‘naming’ numbers rather than ‘counting to’ them, because counting to a number n is just naming all the numbers 1..n sequentially. Now intuitively,<br /><br />(a’) I can’t name all the natural numbers<br /><br />is still true but<br /><br />(c’) There is some finite number I cannot name<br /><br />is even more obviously false. That suggests the inference from (a’) to (c’) is not what Mike’s original argument makes it out to be, and that Alex’s 50% strategy for defeating it is on the wrong track, because it won’t work for naming.<br /><br />The obvious thing to say is that I can name ANY number, but not ALL of them. This means that Mike’s inference is invalid. I am not sure how to represent this formally (and I am bothered that I am not sure) but the idea is simple. “I can’t grade all these papers in an hour” doesn’t mean that there is some paper that takes longer than an hour to grade but that the collective set of papers can’t be graded in the time allotted. If we argue <br /><br />(a”) I can’t grade in an hour an infinite set of papers<br />(b”) All these papers are three pages long<br />(c”) Therefore, there is some three-page paper I can’t grade in an hour<br /><br />the fallacy is obvious.<br />Heath Whitehttps://www.blogger.com/profile/13535886546816778688noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-79429022809864014172014-11-19T11:20:08.767-06:002014-11-19T11:20:08.767-06:00I should point out that there is some quantifier a...I should point out that there is some quantifier ambiguity in the argument, but it does not affect the conclusion. Here are the options.<br /><br />(1) ~☐(Ex)(Vy)((x is greater than all the counted naturals y).<br />This just says that it's not necessary that some uncounted number x is greater than all the counted numbers y. I take (1) to be obviously true, but less than the argument needs.<br /><br />(2) ~☐(Vy)(Ex)(x is greater than all the counted naturals y)<br />This says that it's not necessary that, for all the counted numbers y, there is some greater uncounted number x.<br /><br />(2) is what the argument needs, and it is less obvious than (1). But (2) is false only if there is some finite number n such that for no world w is n counted to in w. But that's not credible. For every finite number n there is some world w such that n is counted to in w. Indeed, you can map the worlds in which each natural is counted to 1 to w1, 2 to w2, all the way up. So there are worlds in which the naturals are counted. But that is equivalent to saying that it is possible to count to infinity.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-15802729533504923492014-11-19T11:07:15.598-06:002014-11-19T11:07:15.598-06:00I meant that I'm not convinced by the countera...I meant that I'm not convinced by the counterargument. I like your argument, and it neatly dovetails with my attempts to fix my account of modality.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-12241063466091114782014-11-19T11:06:34.480-06:002014-11-19T11:06:34.480-06:00Mike:
I like your argument, but let me try this p...Mike:<br /><br />I like your argument, but let me try this potential counterargument against you. I am not convinced by it myself:<br /><br />As long as you have non-zero probability of success, then you have the ability to do something. But when the probability of success is zero assuming you keep on trying. <br /><br />Suppose that each time you count a number, you have a 50% chance of losing your voice. Suppose you're determined to count as far as possible. Then for any finite number n, the probability that you'll count to n is 1/2^(n-1). But the probability that you'll count all finite numbers is zero. So you can count to each finite number but can't count them all.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-40715492940901907162014-11-19T10:52:12.534-06:002014-11-19T10:52:12.534-06:00Counting is a sequential activity, and no amount o...<i>Counting is a sequential activity, and no amount of sequential additions is going to take you from talking about finite sums to talking about infinite ones.</i><br /><br />I don't think I'm missing the point; Craig made that point long ago and has since made it over and over again. I'm really familiar with it. I think you might be underestimating the argument. The natural numbers include only finite numbers. Every number in the set is finite. The naturals is our smallest infinite set. It follows straightforwardly that you cannot count all of the naturals only if there is some finite number n to which you cannot count. Put positively, if you can count to every finite number n, then you can count all of the naturals. But then you can count to infinity. But there is no argument--no argument I know of--which even suggests that there is some finite number to which, and past which, we cannot count. So there is no argument that shows that we cannot count all of the finite numbers in the naturals. But then there is no argument which shows that we cannot count the naturals. So no argument that we cannot count all of the elements in the smallest infinite set of numbers.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-90287922056724812212014-11-19T10:37:40.002-06:002014-11-19T10:37:40.002-06:00Mike: You're missing my point. Even if someone...Mike: You're missing my point. Even if someone could count to any finite number (though I agree that that would become less and less plausible as the numbers got bigger), they would always have some further finite number to count to. Counting is a sequential activity, and no amount of sequential additions is going to take you from talking about finite sums to talking about infinite ones. You just never cross that threshold, which is specifically why potential infinity is used to define the "limits" in mathematics. No matter how long someone counts, they will always be on some finite number.Michael Gonzalezhttps://www.blogger.com/profile/05279261871735286117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-26164770535417144122014-11-19T06:43:32.271-06:002014-11-19T06:43:32.271-06:00Recognizing that infinity may come in different si...Recognizing that infinity may come in different sizes, I would want to start counting a small infinity. For instance what about time? It began about fourteen billion years ago and is now at the point where it grows sequentially one year every year like a potentially infinite series of whole numbers. An hour is composed of i guess finite units of the present but how many present moments are in an hour?Mark Rogershttps://www.blogger.com/profile/12691324025964108341noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-58416407162308162762014-11-18T15:08:09.177-06:002014-11-18T15:08:09.177-06:00Alex,
It depends on what's being asserted whe...Alex,<br /><br />It depends on what's being asserted when it's claimed that 'it's impossible to count to infinity'. I took that claim as a restricted de re modal claim, viz., that there are no human beings who have the property of possibly counting to infinity. So, my claim is that (being very explicit now) there is some human being x who is such that possibly x counts to infinity. So, the counterpart of x who does the counting would be one who does not tire easily, who lives an infinitely long time, etc. Of course, you may deny that any human being has such a counterpart. I think that's all good, since what is being claimed is something stronger anyway. They are saying that it is impossible that anything counts to infinity. All we need now is some possible being in some world that counts to infinity. For my part, I'd immediately urge that God can do it; but there is no need to go that far up the hierarchy of beings.Mike Almeidahttps://www.blogger.com/profile/12001511002085064198noreply@blogger.com