tag:blogger.com,1999:blog-20067416.post347968456132560673..comments2021-06-05T04:17:55.413-04:00Comments on Recursivity: William Lane Craig Does MathematicsUnknownnoreply@blogger.comBlogger108125tag:blogger.com,1999:blog-20067416.post-41906985870559683432013-01-14T14:11:35.692-05:002013-01-14T14:11:35.692-05:00From what I can see the GR paradox is dependent up...From what I can see the GR paradox is dependent upon the assumption that the infinite amount of universes are causually connected. If the assumption isn't met then the paradox doesn't occur. Sebbenoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-56221657600642537282011-05-22T03:38:12.255-04:002011-05-22T03:38:12.255-04:00I'm not a very smart guy, I'm not even a p...I'm not a very smart guy, I'm not even a professional philosopher, but I wonder if this MIGHT solve the Grim Reaper paradox, sort of;-): <br /><br />The Grim Reaper paradox shows that there must be an indivisible unit of time. Okay, so, lets say that God wants to find out what the smallest possible unit of time is. He visits a World, W1, and finds that the Planck time, P1, is equal to 10^-43 seconds. But it is logically possible that P is one order smaller and generate no logical paradox at al. So he goes to W2 and finds P2 is 10^-44 seconds. But of course, God can keep jumping to possible worlds, W, and find P, one order smaller than the previous where no logical paradox is generated. In fact there is no W where P could not be one order smaller and not generate a logical paradox. Therefore, time is infinitely divisible. But that can't be right, right? <br /><br />I think, with great trepidation, that any way the skeptic of infinite temporal divisibility answers, the skeptic of the Grim Reaper paradox can answer the same, which would mean the the argument against temporal infinity from the Grim Reaper paradox is false. Here are some examples:<br /><br />Objection 1: It's just a brute fact that there is a W where P is indivisible.<br /><br />Well, it could also be a brute fact that one of the GR kills Fred. <br /><br />Objection 2: God cannot create a W where the indivisible is divided as he cannot do the logically impossible.<br /><br />Okay, then one of the GR must kill Fred as it is logically impossible that there as a W where none of the GR kills Fred. <br /><br />Objection 3: It is logically necessary that there is a W where P is indivisible, even though it appears as though there could be an infinity of W where P is one order smaller. <br /><br />Same again. Its logically necessary that a GR kills Fred even though it looks like no GR kills Fred. <br /><br />Objection 4: The paradox is, in some way, incorrectly presented and isn't analogous. <br /><br />Again, same could be said of the GR paradox: in some way it isn't analogous to a backwards temporally infinite sequence. How? For example, a backwards temporally infinite sequence has no beginning, yet in the GR paradox, Fred is alive at 11:00pm, which means, somehow, the GR at that instance, didn't kill Fred instantaneously, which isn't analogous to a universe with a beginningless past. <br /><br />Anyway, I'm not confident enough to think I am certainly right. But this does seem a reasonable conclusion, at least to me.mpghttps://www.blogger.com/profile/00445199879510273357noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-44426743289312855462011-05-16T11:29:07.060-04:002011-05-16T11:29:07.060-04:00So the suggestion is that we can take Grim Reaper ...So the suggestion is that we can take Grim Reaper types of arguments to show that there is a minimum unit of time. That's a really controversial conclusion.<br /><br />Moreover, it is a conclusion that helps the Kalaam arguer. For when we combine the claim that there is a minimum unit of time with the arguments that the past has only finite length (whether based on the Big Bang or on <a href="http://arxiv.org/abs/grqc/0110012" rel="nofollow">inflationary considerations</a>), we get the claim that there were only finitely many past moments, and hence there cannot be a pastwards infinite regress. That certainly would help Craig.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-50354373726539030392011-05-04T20:42:40.318-04:002011-05-04T20:42:40.318-04:00While I’m sorry to hear that you’re not interested...While I’m sorry to hear that you’re not interested in continuing, I thank you for your input and would like to introduce a stronger version of your argument (a supertask that relies on arbitrarily small time intervals and <i>does</i> yield a self-contradiction). But first…<br /><br /><i>I've already pointed out that even Pruss admits his arguments needs the infinite divisibility of time to work, and I already presented an argument which I consider rigorous to show, using your own methods, that this is not sensible.</i><br /><br />Well, not <i>quite</i> my own methods. Compared to the Grim Reaper argument I put forward, you were substantially less analytical and less rigorous than I ended up being (which I find a bit ironic since you’re the one who earned a Ph.D. in mathematics). Recall that I constructed a deductively valid argument with clearly listed premises and I later proved the argument’s validity via symbolic logic. Not only did you not use symbolic logic, you didn’t even clearly delineate your argument’s premises even after I asked you to. That said, I’ll try to introduce a strengthened form of your argument anyway.<br /><br />It is true that Pruss’s argument requires infinite divisibility of time in the sense that it requires that arbitrarily small time intervals between events be metaphysically possible (more specifically, premise 2 requires this). One could claim that “arbitrarily small time intervals between events cannot exist in any possible world,” but I find such a metaphysical constraint very strange and implausible, akin to saying that an infinite number of marbles can exist but subtracting from an infinite marble set is metaphysically impossible (with no apparent motivation other than to avoid the paradoxes of infinity subtraction while still believing in the actual infinite).<br /><br />So why on earth think that arbitrarily small time intervals between events are metaphysically impossible? Your approach, if I understand it correctly, goes something like this:<br /><br />(1) If arbitrarily small time intervals between events are metaphysically possible, then supertask X is metaphysically possible.<br />(2) Supertask X is not metaphysically possible.<br />(3) Therefore, arbitrarily small time intervals between events are not metaphysically possible.<br /><br />The key problem with the premise (2) is that the supertask you’ve chosen (Thomson’s lamp) <a href="http://plato.stanford.edu/entries/spacetime-supertasks/#ThoImpArg" rel="nofollow">doesn’t yield the self-contradiction that you’ve claimed</a>. Not to worry though, because there are supertasks that rely on arbitrarily small time intervals between events that <i>do</i> yield self-contradictions. A quick example is the Grim Reaper paradox. If it makes you feel any more comfortable, we can modify the paradox to be a “Duo” version in which Grim Reaper 1 checks on Fred at 11 a.m. + 1/(2n) minutes and Grim Reaper 2 checks on Fred at 11 a.m. + 1/(2n+1) minutes. Still, let’s assume arguendo that the Grim Reaper Duo supertask and your own lamp supertask both yield self-contradictions. The problem is that these self-contradictions arise <i>only when there is an infinite sequence of past events</i>. No paradox is generated when only a finite quantity of such events occur (e.g. the Grim Reaper Duo checks on Fred only finitely many times). So claiming that the argument against an infinite number of past events fails because arbitrarily small time intervals between events are metaphysically impossible—and then justifying that claim by pointing out self-contradictions resulting from cases where there are an infinite number of past events—isn’t going to sound terribly convincing.Maverick Christianhttps://www.blogger.com/profile/08874266433779343710noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-84301448633185783472011-05-01T20:03:16.884-04:002011-05-01T20:03:16.884-04:00Wade:
I'm sorry, but I'm not interested a...Wade:<br /><br />I'm sorry, but I'm not interested at all in continuing. I've already pointed out that even Pruss admits his arguments needs the infinite divisibility of time to work, and I already presented an argument which I consider rigorous to show, using your own methods, that this is not sensible.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-10818441828652832682011-05-01T19:38:16.614-04:002011-05-01T19:38:16.614-04:00And you also provided no reason to think my argume...<i>And you also provided no reason to think my argument with the light is incorrect.</i><br /><br />I linked to the SEP, but if you want me to come up with something tailor-made to your argument, please delineate your argument with clearly defined premises and conclusion (as I did for the Grim Reaper argument).<br /><br /><i>The Stanford article claims "what can be deduced logically from this way of acting will apply only to instants in the t-series". This is clearly false, if we also specify our model of time explicitly: namely, if the lamp is in state s at time t, and no action is taken between time t and time t' > t, then the lamp is still in state s at time t'. With this proviso, it is clear that "what can be deduced" includes other times than those in the t-series alone.</i><br /><br />It’s hard to see how that assertion applies to Thomson’s lamp or saves your argument, because there is no last “let’s flip the switch” time t where there is a time t’ > t for the lamp to be still in state s at time t’ and for there not to be another time t’’ > t’ that is a member of the <i>t</i>-series and thus in which another flip-the-switch action is done at t’’, and if that’s the case “we only act on it at instants in the <i>t</i>-series, and so what can be deduced logically from this way of acting will apply only to instants in the <i>t</i>-series” as the SEP article says. Now it may be possible that I’m not correctly understanding your argument’s train of thought here, but if so I again request that you put your argument in the form of clearly listed premises and conclusion.<br /><br />A popular style of philosophy is known as analytic philosophy, which emphasizes clarity, rigor and precision in its practice. For example, an analytic philosopher might clearly list out the premises of an argument (as I have) and use symbolic logic to prove the argument’s validity (as I have) or attack a specific premise when his or her opponent clearly lists out the premises of an argument (as I have on other occasions). I recommend we adopt this style not only because of your mathematical background, but because it makes communication and dialogue a bit more fruitful.Maverick Christianhttps://www.blogger.com/profile/08874266433779343710noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-72401716616570009962011-05-01T19:35:34.420-04:002011-05-01T19:35:34.420-04:00Jeffrey Shallit, for the Grim Reaper argument I’ll...Jeffrey Shallit, for the <a href="http://alexanderpruss.blogspot.com/2009/10/from-grim-reaper-paradox-to-kalaam.html" rel="nofollow">Grim Reaper argument</a> I’ll finally use some symbolic logic that might help understand the situation better. But first…<br /><br /><i>Honestly, I don't see any value in simply repeating my arguments. I've already said over and over again what I think is wrong with the argument.</i><br /><br />Again, have you given any rebuttal that I haven’t already refuted? I apologize if I missed a rebuttal, but I don’t believe I have. As I said, your rebuttals tended to rely on misunderstandings of the argument, and it wasn’t until relatively recently that you correctly understood the argument (assuming you correctly understand it now). If you have a rebuttal I have missed, by all means please point it out to me.<br /><br /><i>I even provided a counterexample universe for which the argument fails, but you simply dismissed it for no good reason I can see.</i><br /><br />My reason for dismissing it was that your counterexample is logically impossible. Don’t you consider that to be a good reason? Perhaps it’s best if I finally use some symbolic logic, though I’ll have to approximate the symbols with text.<br /><br />Let []-> represent the counterfactual/subjunctive conditional, e.g. A []->B translates into “If A were true, then B would be true.” Let <> represent the modal operator for possibility, e.g. <>A is “A is possibly true.” The tilde will represent negation, e.g. ~A translates into “not-A” or “A is false.” Let <i>G</i> represent “the Grim Reaper paradox occurs,” <i>I</i> represent “an infinite past sequence of events exists” and <i>H</i> stand for “Hilbert’s Hotel exists.” Thus, since we’re using metaphysical possibility, <>H stands for “Hilbert’s Hotel is metaphysically possible” (or to use possible world semantics, “Hilbert’s Hotel exists in at least one possible world”). The proof can go as follows:<br /><br />(1) <>I []-> <>H<br />(2) <>H []-> <>G<br />(3) ~<>G<br />(4) ~<>H, 2, 3, <i>modus tollens</i><br />(5) ~<>I, 1, 4, <i>modus tollens</i><br /><br />So as you can hopefully see now, your counterexample of a possible world in which <i>I</i> is true and yet is consistent with everything in Pruss's argument (e.g. the first three premises of the argument) is logically impossible. The conclusion that no such possible world exists follows logically and inescapably from the first three premises.<br /><br />Because the argument is provably valid, the only way for the argument to be unsound is for a premise to be false. There’s simply no way to get around this. Incidentally, do you concede that the premises are more plausible than their denials?<br /><br />I’ll comment on your lamp-argument next.Maverick Christianhttps://www.blogger.com/profile/08874266433779343710noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-64656859560173204102011-05-01T18:28:15.690-04:002011-05-01T18:28:15.690-04:00"a Grim Reaper swings his scythe and it’s all...<i> "a Grim Reaper swings his scythe and it’s all over for Fred": well, if *that* is what looks to you like a description of *how* a Reaper does its job then (1) your infinite-converging-swarm-of-Reapers scenario is "metaphysically impossible" for reasons that have nothing to do with a general impossibility of infinities -- namely, "scythe" denotes a particular sort of object that can only exist in a universe very like this one, and in this sort of universe Reapers are physically impossible -- and (2) an equally good answer to "how do the Reapers collectively kill Fred?" is "They all stand their with their scythes ready to kill him, and as a result he dies".</i><br /><br />For (1), it isn’t clear why scythes are incompossible with infinitely many entities that can kill Fred at any positive nonzero time interval. In any case, if need be we can use “scythe” in a somewhat more metaphorical way to refer to a weapon that is shaped like a scythe but when it touches Fred it emits a lethal energy field capable of killing Fred in any positive nonzero time interval. Point (2) doesn’t seem to work; if they none of them actually <i>use</i> their scythes to kill Fred, how do they collectively kill Fred? And let’s not forget my “hit point” variant of the paradox to illustrate the problem more clearly. In that modified scenario, all cumulative Grim Reaper events are the sum of their individual hit points, e.g. three Grim Reapers each doing a scythe attack collectively take off 300 hit points, and three Grim Reapers doing nothing collectively take away 0 hit points. If none of the Grim Reapers do anything, they collectively take away 0 hit points away from Fred, and therefore they do not collectively kill Fred.Maverick Christianhttps://www.blogger.com/profile/08874266433779343710noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-63703397107838513782011-05-01T18:22:20.366-04:002011-05-01T18:22:20.366-04:00For "If backward-infinite series of events po...<i> For "If backward-infinite series of events possible, then Hilbert Hotel possible": "If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed." *This is just a restatement of the premise*</i><br /><br />No it’s not, the premise is “If there could be a backwards infinite sequence of events, Hilbert's Hotel would be possible” and the argument goes into further detail as to <i>why</i> that would be true (it describes how the hotel could be created).<br /><br /><i> For "if Hilbert hotel possible, then paradoxical grim reapers possible": "If Hilbert's Hotel were possible, each room in it could be a factory in which a GR is produced. Moreover, it is surely possible that the staff in room n should set the GR to go off at 11 am + 1/n minutes." Again, the alleged argument is basically just a restatement of the premise.</i><br /><br />Again, the argument allows us to go into greater detail; it explains <i>why</i> Hilbert’s Hotel being possible would entail the Grim Reaper paradox being possible, and I found such an explanation illuminative. Of course, both arguments for the premises are very short, and we could quibble over how “basically” it is like the premises. At the end of the day though, if we want to say the arguments for the premises are unsuccessful one ought to have a good counter-argument (it seems the arguments work unless some counter-argument is available).<br /><br />I haven’t seen any good counterargument against the premises or the (albeit brief) justification for the premises. The closest you offered was this:<br /><br /><i> Worse: consider a universe with the same rules as ours that's spatially and temporally infinite. (Our actual universe may or may not be such a universe.) In such a universe, (a) the Hilbert hotel scenario is possible but (b) the GR "paradox" is not. So Pruss's intuition for what the Hilbert Hotel staff could "surely" do is demonstrably broken.</i><br /><br />Suppose it’s true that in such a universe, (a) is possible but (b) is not. This doesn’t relevantly attack the premise or the justification given for it. The argument’s second premise doesn’t say that “For any world W, If Hilbert’s Hotel were nomically possible in W, then the Grim Reaper paradox would be nomically possible in world W.” Rather, the claim is more modest, something like “If Hilbert’s Hotel were metaphysically possible (if were to exist in at least one possible world), then the Grim Reaper paradox would be metaphysically possible (it would exist in at least one possible world).” So the fact that in some possible worlds the antecedent is nomically possible and the consequent isn’t doesn’t relevantly affect the veracity of this premise.<br /><br />The second premise also appears more plausible than its denial; if Hilbert’s Hotel is metaphysically permissible, it doesn’t appear that any of our known metaphysical constraints would prohibit e.g. each room having a Grim Reaper factory. If you think otherwise and have a good argument for thinking that this is would be metaphysically impossible if Hilbert’s Hotel were metaphysically possible, I’d very much like to hear it.Maverick Christianhttps://www.blogger.com/profile/08874266433779343710noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-88200975631517016222011-05-01T12:21:19.696-04:002011-05-01T12:21:19.696-04:00Sorry, maybe one final comment is worth saying. T...Sorry, maybe one final comment is worth saying. The Stanford article claims "what can be deduced logically from this way of acting will apply only to instants in the t-series". This is clearly false, <i>if</i> we also specify our model of time explicitly: namely, if the lamp is in state s at time t, and no action is taken between time t and time t' > t, then the lamp is still in state s at time t'. With this proviso, it is clear that "what can be deduced" includes other times than those in the t-series alone. So if that's the counter-argument, it is not convincing.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-17437207774243032792011-05-01T12:12:50.018-04:002011-05-01T12:12:50.018-04:00Wade:
Honestly, I don't see any value in simp...Wade:<br /><br />Honestly, I don't see any value in simply repeating my arguments. I've already said over and over again what I think is wrong with the argument. I even provided a counterexample universe for which the argument fails, but you simply dismissed it for no good reason I can see. <br /><br />And you also provided no reason to think my argument with the light is incorrect. <br /><br />So at this point I don't have much more to say.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-56865665177138846422011-05-01T11:49:34.644-04:002011-05-01T11:49:34.644-04:00Wade says that Pruss presented arguments for the m...Wade says that Pruss presented arguments for the most dubious premises in his argument, but he really didn't. Here are Pruss's "arguments".<br /><br />For "If backward-infinite series of events possible, then Hilbert Hotel possible": "If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed." *This is just a restatement of the premise*. "If p is possible, then p-and-q is possible". It's ridiculous.<br /><br />For "if Hilbert hotel possible, then paradoxical grim reapers possible": "If Hilbert's Hotel were possible, each room in it could be a factory in which a GR is produced. Moreover, it is surely possible that the staff in room n should set the GR to go off at 11 am + 1/n minutes." Again, the alleged argument is basically just a restatement of the premise. Worse: consider a universe with the same rules as ours that's spatially and temporally infinite. (Our actual universe may or may not be such a universe.) In such a universe, (a) the Hilbert hotel scenario is possible but (b) the GR "paradox" is not. So Pruss's intuition for what the Hilbert Hotel staff could "surely" do is demonstrably broken.<br /><br />I have no idea what the point is of asking me to say what's wrong with Pruss's arguments: Pruss doesn't offer any arguments. He just states his premises, states them again, and apparently thinks he's given good support for them.<br /><br />"a Grim Reaper swings his scythe and it’s all over for Fred": well, if *that* is what looks to you like a description of *how* a Reaper does its job then (1) your infinite-converging-swarm-of-Reapers scenario is "metaphysically impossible" for reasons that have nothing to do with a general impossibility of infinities -- namely, "scythe" denotes a particular sort of object that can only exist in a universe very like this one, and in this sort of universe Reapers are physically impossible -- and (2) an equally good answer to "how do the Reapers collectively kill Fred?" is "They all stand their with their scythes ready to kill him, and as a result he dies". Which is, yes, a pathetic excuse for an explanation: just like yours, which I cannot believe you intended seriously.<br /><br />The point of the thing about continuous time, differential equations, etc., is: The GR "paradox" depends on intuitions like "if none of them kills Fred then all-of-them-collectively do not kill Fred", and no matter how you bluster the fact is that this is just a statement about your intuitions for what various configurations of things can do. Those intuitions might be of some use when it comes to the actual world in which they were formed (though, as I've pointed out a couple of times and been ignored, the actual world is host to actual phenomena that run counter to exactly those intuitions) but should obviously not be trusted when it comes to universes where (e.g.) time and causality are very different from in ours. And it sure seems like the nature of time and causality would have to be very different in a universe in which the possibility of completed infinities sufficed to enable the Reaper "paradox": and *that* is the point of what I was saying. It wasn't meant to attack any specific premise of any deductive argument because no one's given any non-laughable deductive argument for the Reapers: the only such arguments you've so much as waved in the general direction of are based on very dubious intuitions. (If you are content to call an argument a valid deductive argument simply because you've packaged the invalid steps into premises, fair enough; there's some value to doing that. But the value in doing it isn't that the argument isn't stronger that way.)Gareth McCaughanhttps://www.blogger.com/profile/05377158305586280009noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-68740999658575343222011-05-01T11:40:35.937-04:002011-05-01T11:40:35.937-04:00As for the Grim Reaper, I've said over and ove...<i>As for the Grim Reaper, I've said over and over what I think is wrong with it, so I think it is fruitless to continue with that.</i><br /><br />Is there any objection you’ve made that I haven’t refuted? Remember, the Grim Reaper argument can fail to be sound only in one of two ways: it is deductively invalid or a premise is false. Since the argument is deductively valid, the only way it can fail to be sound is if a premise is false. Yet your objections to a premise of the argument tended to rely on a misunderstanding. Assuming you correctly understand the argument now, notice that it’s apparently not until very recently that you really understood the argument (you seemed to think that Pruss was referring to a fixed universe through all lines of the argument, but this was mistaken). With such misunderstanding it’s perhaps no surprise then that you have yet to provide any good, relevant objection to any premise of the argument. If you think I’m mistaken here, please cite one specific example.<br /><br />Remember, the Grim Reaper argument can fail to be sound only in one of two ways: it is deductively invalid or a premise is false. If you believe the argument to be unsound, do you believe it to be invalid? Do you concede that the argument’s premises are more plausible than their denials?<br /><br /><i>“Pruss could argue that your intuition of such a world being metaphysically possible is faulty, and the argument for it being faulty would be the Grim Reaper argument itself.”<br /><br />The world I have constructed is completely consistent with everything in Pruss's argument - and it invalidates the Grim Reaper paradox by denying the possibility of the Grim Reapers' existence.</i><br /><br />I find your claim that your construction of the world is consistent with everything in Pruss’s argument very surprising, because it is logically impossible for “the world I have constructed is completely consistent with everything in Pruss's argument” to be true. Why? Because it is logically impossible for the premises of Pruss’s argument to be true and the conclusion (that a world in which an actual infinite sequence of past events exists is <i>not</i> metaphysically possible) to be false. If you don’t believe me, I’ll happily prove so via symbolic logic.<br /><br /><br /><i>With reference to the lamp, what you linked to was about its *logical* impossibility, but you're the one claiming that logical impossibility is not the same as metaphysical impossibility.</i><br /><br />That is true, but your argument seemed to suggest that the proposition in question is metaphysically impossible as a result of yielding a self-contradictory situation (which of course is a logical impossible state of affairs).<br /><br /><i>So what flaw do </i>you<i> see in my argument?</i><br /><br />The same one that the Stanford Encyclopedia of Philosophy entry sees: the end result isn’t a genuine self-contradiction. If you want me to be more analytical about it, please frame your argument with clearly delineated premises and conclusion, and I’ll be happy to tell you if I believe the argument is deductively invalid or if the argument has a false premise.Maverick Christianhttps://www.blogger.com/profile/08874266433779343710noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-78440071156231941752011-05-01T11:03:15.563-04:002011-05-01T11:03:15.563-04:00With reference to the lamp, what you linked to was...With reference to the lamp, what you linked to was about its *logical* impossibility, but you're the one claiming that logical impossibility is not the same as metaphysical impossibility. Anyway, I found the argument there to be without any merit. <br /><br />So what flaw do <i>you</i> see in my argument? You have not provided any. <br /><br />Of course, I see my argument as having the same epistemological worth as the Grim Reaper argument: none. You can hypothesize all you like about "possible worlds", but the conclusions you ultimately draw have to be tied to what is possible in <i>this</i> world. <br /><br />As for the Grim Reaper, I've said over and over what I think is wrong with it, so I think it is fruitless to continue with that.<br /><br /><i>Pruss could argue that your intuition of such a world being metaphysically possible is faulty, and the argument for it being faulty would be the Grim Reaper argument itself.</i><br /><br />The world I have constructed is completely consistent with everything in Pruss's argument - and it invalidates the Grim Reaper paradox by denying the possibility of the Grim Reapers' existence. I find that completely convincing.<br /><br />Even Pruss seems to recognize this, since he says "The argument has an additional premise, namely that time is not necessarily discrete."Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-4350927610056433872011-05-01T10:41:03.836-04:002011-05-01T10:41:03.836-04:00I am still unsure what your claim 4 is intended to...<i>I am still unsure what your claim 4 is intended to mean.</i><br /><br />Recall that I used possible worlds semantics for modal logic (if you’re unfamiliar with modal logic or symbolic logic in general, I recommend reading up on it; a mathematician such as yourself might enjoy it). Since we’re using possible world semantics, line (4) means “It is not possible that an infinite sequence of past events exists,” or, since the type of possibility we’re using is metaphysical possibility, line (4) means “It is not metaphysically possible that an infinite sequence of past events exists.”<br /><br /><i>Is it:<br /><br />4. It is not the case that P holds, where P is the proposition "there exists a possible world W having an infinite sequence of past events".<br /><br />Because if it is, I can create a possible world where this happens: namely a world W like our world, existing infinitely far in the past, where at every time -t in the past a hotel of size 1/2 comes into existence at position t.</i><br /><br />Pruss could argue that your intuition of such a world being metaphysically possible is faulty, and the argument for it being faulty would be the Grim Reaper argument itself. Your intuitions may tell you that the conclusion of the Grim Reaper argument (line 4) is wrong, but don’t you agree that the argument is both deductively valid and that the premises are justifiably true? If you do, then you can reject the conclusion only on pain of irrationality. If you do not believe the premises are true, which one do you believe is false and why? If you have any good objections to the premises of the argument I would be genuinely interested in hearing them, because so far I haven’t heard any.<br /><br />To recap the argument (again assuming Pruss was using metaphysical possibility):<br /><br />(1) If in infinite sequence of past events were metaphysically possible, then Hilbert’s Hotel would be metaphysically possible.<br />(2) If Hilbert’s Hotel were metaphysically possible, then the Grim Reaper paradox would be metaphysically possible.<br />(3) The Grim Reaper paradox is not metaphysically possible.<br />(4) Therefore, an infinite sequence of past events is not metaphysically possible.<br /><br /><a href="http://alexanderpruss.blogspot.com/2009/10/from-grim-reaper-paradox-to-kalaam.html" rel="nofollow">Pruss’s justification</a> for lines (1) and (2) are as follows:<br /><br />Argument for (1): If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed. An infinite number of hotel rooms would then be the result. (Recall that the “could be” presumably refers to <i>metaphysical</i> possibility; not physical possibility)<br /><br />Argument for (2): If Hilbert's Hotel were possible, each room in it could be (again, think metaphysical possibility) a factory in which a Grim Reaper is produced. Moreover, it is surely possible that the staff in room <i>n</i> should set the Grim Reaper to go off at 11 am + 1/<i>n</i> minutes. And that would result in the Grim Reaper paradox.<br /><br /><br /><i>Now at time t = 1/2, 3/4, 7/8, ... I flip the switch. Question: is the light on or off at time t = 1? It has to be one or the other because of my imposed requirements.</i><br /><br />This is actually a famous thought experiment called Thomson’s lamp, but <a href="http://plato.stanford.edu/entries/spacetime-supertasks/#ThoImpArg" rel="nofollow">I don’t think it yields a self-contradiction</a>.<br /><br />Meanwhile, back to the Grim Reaper argument, do you concede that the premises are more plausible than their denials? If not, which one do you think isn’t and why?Maverick Christianhttps://www.blogger.com/profile/08874266433779343710noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-29490708518758525562011-05-01T10:29:56.618-04:002011-05-01T10:29:56.618-04:00“Yes I have, but the interpretation that the cat i...<i>“Yes I have, but the interpretation that the cat is both alive and not alive at the same time and in the same context is just bad philosophy of science (for that matter, interpreting superpositions in general as violations of the law of noncontradiction is bad philosophy of science).”<br /><br />I'm not so sure about that.</i><br /><br />Violations of the law of noncontradiction are (by definition) logically impossible. Unsurprisingly then, I have yet to see a good argument for this bizarre metaphysical interpretation of quantum mechanics. Of course, one could say that asking whether an electron is spin-up or spin-down when the electron is in a certain superposition is a category error, like asking whether the number 2 tastes like chicken or roast beef. But this conception of a superposition, odd as it might be compared to the macro-world, would be very different from the lunacy of accepting the logically impossible as possible. You can’t get much more irrational than believing in logically impossible propositions.<br /><br />Incidentally, the adherent of the Copenhagen interpretation need not accept that the cat is in a superposition of alive/dead; it seems that the most sensible Copenhagen adherent would conclude that the wave function collapses before then (say, by the time the radioactive particle detector determines whether to release the poison gas).Maverick Christianhttps://www.blogger.com/profile/08874266433779343710noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-91259930219913495772011-05-01T10:17:13.576-04:002011-05-01T10:17:13.576-04:00David:
That is a non-sequitur.
If you’re saying ...David:<br /><br /><i>That is a non-sequitur.</i><br /><br />If you’re saying the Grim Reaper argument is a non sequitur, say the word and I will happily prove otherwise via symbolic logic.Maverick Christianhttps://www.blogger.com/profile/08874266433779343710noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-23247760810909898202011-05-01T09:48:12.355-04:002011-05-01T09:48:12.355-04:00I like these metaphysical "proofs" becau...I like these metaphysical "proofs" because the rules for deciding what you are allowed to do, and what constitutes a contradiction, are so vague, it seems you can prove almost anything.<br /><br />Let's "prove" that one of the requirements of the grim reaper paradox - that you have a killing machine that can both test if you are dead and kill you if you aren't, within an <i>arbitrarily short time period</i> - is metaphysically impossible.<br /><br />Suppose in some possible world all those grim reapers exist. They function in arbitrarily small time intervals, and act simultaneously, so in some possible world we can also turn lights on and off within an arbitrarily small time period. In my possible world lights are either on or off and they change state instantaneously.<br /><br />I will just use a single light switch and a light. When the light switch is flipped, the light goes on (if it was off) or off (if it was on). Initially it is off.<br /><br />Now at time t = 1/2, 3/4, 7/8, ... I flip the switch. Question: is the light on or off at time t = 1? It has to be one or the other because of my imposed requirements.<br /><br />Well, if it is on at t = 1, this can only be - since it was initially off - from the fact that it was turned on at some point. Which point? If you say because it was turned on at 1 - 2^{-n}, I counter by saying, no, it was turned off at the later time 1 - 2^{-(n+1)}. <br /><br />Now it is turned on at t = 1/2, so by the same argument, if it is off at time t = 1, this can only be because it was turned off at some point after t = 1/2. Which point? Same argument applies.<br /><br />Therefore it cannot be either on or off, a contradiction. So arbitrarily small time periods are metaphysically impossible.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-24763784155007712542011-05-01T06:40:11.704-04:002011-05-01T06:40:11.704-04:00Yes I have, but the interpretation that the cat is...<i>Yes I have, but the interpretation that the cat is both alive and not alive at the same time and in the same context is just bad philosophy of science (for that matter, interpreting superpositions in general as violations of the law of noncontradiction is bad philosophy of science).</i><br /><br />I'm not so sure about that. Maybe quantum mechanics says something deeper about the structure of our universe. Maybe some propositions about the physical world have no absolute truth value, but only a probabilistic distribution of truth. <br /><br />But I guess this is just a side distraction from our current discussion.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-16404052518490107842011-05-01T06:24:46.847-04:002011-05-01T06:24:46.847-04:00David:
There is some confusion here. If the conc...David:<br /><br />There is some confusion here. If the conclusion is "it is not the case that P holds", then this conclusion can be defeated by producing a single counterexample. This is what I have done. Talking about worlds other than the one I have suggested is not relevant.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-52964923431952127422011-05-01T05:31:33.240-04:002011-05-01T05:31:33.240-04:00That is a non-sequitur. P does not imply any such...That is a non-sequitur. P does not imply any such capability of your creating arbitrary possible worlds like the one you propose, and indeed I can define<br /><br />P and not quantized (and/or not relativized)<br /><br />This world satisfies P but precludes your construction.<br /><br />I can further construct such a possible world P that is not assumed to have any minimum quantum of time by merely relaxing the quantum assumption in our known world, e.g. by defining a new unit of time the silisecond such that time in siliseconds forward or back from the time of this comment are defined by an integral of t^-2 seconds<br />such that the first silisecond forward or backward is of duration 1/1 second, the second silisecond is of duration 1/4 second, the third is of duration 1/8 second.<br /><br />Our whole infinite universe forward and backward now fits into ±2 seconds, where the infinite number of events of interest in either direction are defined to occur as each integral number of seconds passes.<br /><br />Another infinite universe takes places in achileseconds, each successive achillesecond being the time it takes for Achilles to cover the lead of the tortoise after the previous achillesecond.<br /><br />At the time Achilles overtakes the tortoise there is a past of an infinite number of such achillesecond events in Xeno's paradox.Davidhttps://www.blogger.com/profile/12896150438306760193noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-55539493761057473422011-05-01T04:42:23.700-04:002011-05-01T04:42:23.700-04:00I am still unsure what your claim 4 is intended to...I am still unsure what your claim 4 is intended to mean. Is it:<br /><br />4. It is not the case that P holds, where P is the proposition "there exists a possible world W having an infinite sequence of past events".<br /><br />Because if it is, I can create a possible world where this happens: namely a world W like our world, existing infinitely far in the past, where at every time -t in the past a hotel of size 1/2 comes into existence at position t. Further, in such a world, time and space are quantized and relativity applies. In such a world the grim reaper "paradox" is not possible, because the hypotheses you demand cannot be put into place, as we've discussed. So W seems a counterexample to 4.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-47845496820372991772011-05-01T01:17:28.455-04:002011-05-01T01:17:28.455-04:00I repeat my question: which of (1) - (3) do you ta...<i>I repeat my question: which of (1) - (3) do you take issue with?</i><br /><br />If you’ll recall I took issue with “Pruss's argument is either existential or universal” which was (1). The reason is that some parts of the argument are existential and some parts of it are universal (assuming you’re referring to the possible world semantics of modal logic). (1) as it was didn’t seem to make sense (hence my repeated questions of clarification); it was as if you weren’t understanding the argument correctly. On that note, let’s visit this:<br /><br /><i> I'm going to restate Pruss as I understand him. If you disagree with something let me know:<br /><br />1. Fix a possible universe U. If U allows an infinite sequence of events, Hilbert's Hotel would be possible, in U.<br /><br />2. If U permits the construction of Hilbert's hotel, then U permits the possibility of constructing all those grim reapers.<br /><br />3. But the grim reapers allow someone to be killed by all without being killed by any individual grim reaper. Pruss and you find this absurd.<br /><br />4. Therefore, there cannot be a backwards infinite sequence of events in U.</i><br /><br />This misunderstands Pruss’s argument. There is no fixed possible universe U running through all four lines of the argument, and I think this misunderstanding was what was impeding communication between us. I tried to clear up what Pruss’s argument was in a previous comment but I’ll try again, bringing out the possible world semantics more explicitly this time. Assuming Pruss was using metaphysical possibility in his argument, we get the following:<br /><br />(1) If an infinite sequence of past events were to exist in at least one possible world, then Hilbert’s Hotel would exist in at least one possible world.<br /><br />(2) If Hilbert’s Hotel were to exist in at least one possible world, then the Grim Reaper paradox would exist in at least one possible world.<br /><br />(3) It is not the case that the Grim Reaper paradox exists in at least one possible world (the Grim Reaper paradox is metaphysically impossible).<br /><br />(4) Therefore, it is not the case that an infinite sequence of past events exists in at least one possible world.<br /><br />I preferred dispensing with the more explicit use of possible world semantics since I feared they might foster misunderstanding (e.g. perhaps the reasoning behind <a href="http://alexanderpruss.blogspot.com/2009/10/from-grim-reaper-paradox-to-kalaam.html" rel="nofollow">Pruss’s justification for the premises</a> would be less graspable), but in this case perhaps it’ll help you correctly understand the argument. Again, there is no fixed universe running through all four lines of the argument, or even in the first two lines.Maverick Christianhttps://www.blogger.com/profile/08874266433779343710noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-21495020197910857892011-05-01T00:53:39.834-04:002011-05-01T00:53:39.834-04:00Gareth McCaughan, regarding the Grim Reaper argume...Gareth McCaughan, regarding the <a href="http://alexanderpruss.blogspot.com/2009/10/from-grim-reaper-paradox-to-kalaam.html" rel="nofollow">Grim Reaper argument</a>,<br /><br /><i> Wade, it isn't possible to answer the question "how do they kill Fred?" since we've got no information about how even a single Reaper would kill him.</i><br /><br />Sure we do; a Grim Reaper swings his scythe and it’s all over for Fred. But if none of the Grim Reapers do anything, how do they collectively kill Fred? It doesn’t seem very plausible that they would.<br /><br />Perhaps modifying the scenario in the following way will make the problem more visible: suppose Fred has 100 hit points and is dead only when his hit point level is less than or equal to 0 hit points. Each Grim Reaper has a “scythe attack” that can take off 100 hit points from Fred, thereby killing him if he isn’t already dead. Cumulative Grim Reaper attacks are additive, e.g. three Grim Reapers each doing a scythe attack collectively take off 300 hit points. At 11 a.m. + 1/<i>n</i> minutes, Grim Reaper <i>n</i> checks on Fred to see if he is alive. If Fred is alive, the Grim Reaper does his scythe attack, taking 100 hit points away from Fred and thereby killing him. If Fred is already dead, the Grim Reaper does nothing and thereby takes away exactly 0 hit points away from Fred. Yet it seems for any Grim Reaper <i>n</i>, that Grim Reaper doesn’t do anything (i.e. takes away 0 hit points) because there’s another Grim Reaper <i>n + 1</i> before him. And if none of the Grim Reapers do anything (they each swipe away 0 hit points), they collectively take away 0 hit points from Fred, thereby not killing him. The idea that the Grim Reapers collectively kill Fred just doesn’t work.<br /><br /><br /><i>I have not said that "continuous time makes it metaphysically impossible [etc.]"; please do not misrepresent me.</i><br /><br />I’m sorry if I misunderstood you, but then it’s hard to see how your objection could relevantly attack any premise of the argument.<br /><br /><br /><i> Steps 1 and 2 in the argument you present are (or at least) require arguments themselves. Neither you nor Pruss have presented a deductively valid argument for either of them. I think it's a bit off to say "This is a deductively valid argument" when what you're really doing is taking an invalid argument and collapsing the invalid steps into premises.</i><br /><br />The argument as Pruss presented it is deductively valid, and the fact that the premises themselves require arguments (and I agree they do) doesn’t change this. Nonetheless, Pruss presented arguments for the premises, and I have yet to see a good rebuttal for them. If you know of any please let me know.Maverick Christianhttps://www.blogger.com/profile/08874266433779343710noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-69980048783591445932011-04-30T20:25:13.934-04:002011-04-30T20:25:13.934-04:00I'm going to restate Pruss as I understand him...I'm going to restate Pruss as I understand him. If you disagree with something let me know:<br /><br />1. Fix a possible universe U. If U allows an infinite sequence of events, Hilbert's Hotel would be possible, in U.<br /><br />2. If U permits the construction of Hilbert's hotel, then U permits the possibility of constructing all those grim reapers.<br /><br />3. But the grim reapers allow someone to be killed by all without being killed by any individual grim reaper. Pruss and you find this absurd.<br /><br />4. Therefore, there cannot be a backwards infinite sequence of events in U.<br /><br />Once we make it clear what universe we are talking about, suddenly the supposed "contradiction" disappears, and claim 2 is just wrong. It could well be that U permits the construction of a Hilbert hotel, yet does not permit the construction of the grim reapers with the properties they are supposed to have. Indeed, U = our universe is a possible counterexample. <br /><br />Claim 3 is also suspect, as previously pointed out.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.com