A Variation on the Grim Reaper Paradox
In one of my earlier posts, I addressed the Grim Reaper paradox and offered my input on a possible resolution of the thought experiment’s curious implications. However, some of my readers may have been dissatisfied with my answer, thinking that it sidestepped around the issue rather than addressing the conundrum directly. A few people asked me why I thought that obscure philosophy on the nature of Time might have any relevance to the question, in the first place. To that end, I have decided to offer a bit more clarification and to attempt to illustrate why I think the Grim Reaper paradox is inherently flawed.
Consider this slightly modified version of the thought experiment…
Fred is sitting in a room at 8:00 am. There exist four Grim Reapers along with Fred, each of which is currently dormant. When any individual Grim Reaper becomes activated, if Fred is not going to be killed by the next Reaper in the order, then this Reaper will instantaneously kill Fred; otherwise, this Reaper will return to a dormant state and continue to do nothing. Each of the Grim Reapers is timed to activate at a specific time after 8:00 am. The first Reaper will activate at 8:15 am. The second activates at 8:30 am. The third activates at 8:45 am. The fourth activates at 9:00 am.
Now, 8:15 arrives and the first Reaper activates. Does it kill Fred or not? If it does kill Fred, because the second Reaper is not going to kill Fred, then the 3rd Reaper in the line is not going to kill Fred– it can’t, obviously, since Fred is already dead. However, if that’s the case, then the second Reaper is going to kill Fred (since those conditions are met) and the first Reaper’s conditions are no longer valid. So, even though we started assuming that the first Reaper killed Fred, we’ve learned that this cannot be the case. Indeed, the same holds true for the second Reaper– if the second Reaper kills Fred, then the fourth Reaper cannot kill Fred meaning that the third Reaper should kill Fred, violating our initial assumption. So, we see that the second Reaper is not going to kill Fred. But if the second Reaper isn’t going to kill Fred, then the first Reaper should– except that we’ve already seen this cannot happen.
Unlike Pruss’s formulation of the paradox, this problem cannot be resolved by simply claiming that actual infinites cannot exist. We’re not relying on actual infinities, here. We are looking at a finite number of Grim Reapers. Nor does is seem reasonable to come to the sort of conclusion which Pruss does in his proposed solution to the paradox. If a person tried to claim that the number “four” cannot actually be a number which applies to the real world because of this paradox, we would all laugh in their faces.
It’s a little bit easier to see the point I was trying to make in my other post, now. Regardless of whether one is an A-Theorist or a B-Theorist as far as Time is concerned, both camps agree that events which lie in the future do not alter the ontology of events in the present. On the A-Theory view of things, I cannot make a decision based upon a future which has not yet been actualized. Things which are not yet actual cannot affect that which is actual, and as such, it is clear that my version of the Grim Reaper Paradox violates this view of things.
Similarly, on the B-Theory, causality is a description of a relation between two events, but it doesn’t affect the ontology of those events. So an event in the future cannot alter the ontology of something in the present. Both events are actualized and static, and my version of the Grim Reaper Paradox violates this precept. However, this also means that events in the present do not alter the ontology of events in the future. The future is just as actual and static as are the past and present, on the B-Theory. As such, it becomes immediately clear that Pruss’ version of the Grim Reaper Paradox violates this same precept, since it is dependent upon the idea that an event can affect the ontology of future events.
I do not think that Pruss’ version of the Grim Reaper paradox shows that actual infinities are inapplicable to the real world any more than my version of this thought experiment shows that the number “four” is inapplicable to the real world. In fact, it seems to me that the paradox is best resolved by abandoning an antiquated and untenable idea of the nature of Time. Apologists like William Lane Craig have attempted to cite the Grim Reaper paradox in order to support the Kalam Cosmological Argument. Ironically, it may be the case that the Grim Reaper Paradox actually undermines the KCA, since that argument is entirely dependent upon the tensed A-Theory of Time.
Boxing Pythagoras 
Hey BP, long time my friend. This is great! I have no objections to your thoughts but i do wonder how they fit with current quantum mechanics which seems to be pointing to an idea that future events do determine past events. I’ll try to find a link (i actually read the article in print (weird right?). Anyway, the QM test basically shot a particle through a solid wheel with a hole. If the particle appeared on the other side of the wheel then it made it through the hole. But what they found was the place where the particle was started (the event in the past) changed based on where the particle was found (the futuristic end point).
Not sure if any of that makes sense as I’m paraphrasing but I’m hoping the point still comes through. If QM does indicate that future events have sub atomic influence on past events (in this case, grim reaper 4 really won’t attack because grim reaper 2 did) how does that affect your position (or will it?)
Thank you.
Hey, Roger! Unfortunately, life has gotten in the way of my blogging, in recent months, but I finally managed to get some time to post.
The concept to which you are referring is known as “retrocausality.” It’s actually one of the prime implications of my favorite interpretation of QM (Two-State Vector Formalism). Again, it comes down to ontology, though. On the B-Theory, the starting and ending positions of the particle, though unknown, are both actual. The term “causality,” then, is not a reference to actualizing a potentiality, as it is on the A-Theory of Time. Rather, if refers to a shared relationship between the two events.
Imagine a number line, for a moment. The fact that 3 lies before 7 on the number line doesn’t imply that 3 therefore actualizes 7. Both are actual numbers regardless of their ordering. Causality represents a similar concept for the relationship of events over time, on the B-Theory.
First off, I have to say that I agree with the general assessment of the Paradox. The GRP is no more problematic than any other proposed contradictory system. If you define a set of rules that are inconsistent, then of course they won’t all work as intended. This solution, however, works out regardless of which theory of time you choose, which was the point I made earlier in your last post.
“Similarly, on the B-Theory, causality is a description of a relation between two events, but it doesn’t affect the ontology of those events…However, this also means that events in the present do not alter the ontology of events in the future. The future is just as actual and static as are the past and present, on the B-Theory. As such, it becomes immediately clear that Pruss’ version of the Grim Reaper Paradox violates this same precept, since it is dependent upon the idea that an event can affect the ontology of future events.”
I’m a bit confused by this. You mention that the B-theory has causality but argue that the events in the present do not “alter” the future. But isn’t that what causality is about?
As far as I can tell, I assume by “affect the ontology” you are referring to some A-theoretic form of causality. But Pruss’s GRP doesn’t rely upon this idea at all. All it requires is that events in the future are dependent upon events in the past (or alternatively, if you’re a B-theorist, that later events are dependent upon earlier ones), which is something that is true under both the A and B theory of time.
Sorry I missed this until just now! Thanks for the further input on the subject!
The question is “where can the inconsistency in the GRP be found?” On the A-Theory, what is inconsistent about a situation in which a Reaper chooses one action over another? What is inconsistent about the ordering of these Reapers or their numbering? Where does the GRP contradict itself, on the A-Theory?
Not on the B-Theory. On this sort of Tenseless time, causality takes on a very different definition. It becomes something more like a description of an ordering relation rather than the actualization of potentiality discussed by A-Theorists.
Consider the simple linear function f(x)=x. When we graph this simple function, we notice that the point (1,1) is certainly on the line, and as we progress along the x direction we’ll find other points on the line, as well: (2,2) and (98/43,98/43) and (π,π), for example. All of the points on this line are co-extant with one another– they are all just as actual as any which comes at a lower x value. And it’s not as if (1,1) is the reason for (2,2)’s existence or actuality– (1,1) does not “cause” (2,2) in the A-Theoretical sense of the word.
On the B-Theory, this sort of ordering relation is precisely the same, just in (at least) four dimensions rather than the two of my example. Events which we say have “causal relation” to one another carve out contiguous regions of space-time, in exactly the same way that f(x) carves out a contiguous line in a Cartesian plane. We order these causal relations between events with respect to Time, just as we were ordering the points in our function’s graph with respect to the x-axis. On this view, one event does not cause a subsequent event to exist or become actualized– as in the A-Theory– but we can distinguish a contiguous ordinal priority between events with respect to Time.
It’s actually not true under the B-Theory. Future events are every bit as actualized as past or present events, on the B-Theory. They do not “depend” on prior events any more than (2,2) depends upon (1,1) in my Cartesian simile.
“The question is “where can the inconsistency in the GRP be found?” On the A-Theory, what is inconsistent about a situation in which a Reaper chooses one action over another? What is inconsistent about the ordering of these Reapers or their numbering? Where does the GRP contradict itself, on the A-Theory?”
The inconsistency lies in the fact that every Reapers’s actions are constrained by another Reaper. Unlike the other paradoxes of the infinite, the Reapers do not behave independently of the others..
If every Reaper acts according to the rules, then they should not be able to kill the man when they activate because another Reaper before them should’ve already done so. Of course, if a Reaper did kill a man at their time, then that would suggest that one of the Reaper’s before them did not do their job. So much as every Reaper has another Reaper before them, then none of them should act, meaning the man would stay alive by the end of the night (which should be impossible). No matter what scenario you end up with, it seems at least one or more of the Grim Reapers would not be able to fulfill their duties.
A similar problem arises when I try to create a system where two competitors are tasked to win a game against the other. Because one cannot win without the other one losing, it is impossible to expect them both to be able to fulfil their duties, and we should not be surprised when they don’t. Here you don’t even need to invoke Grim Reapers at all, and such a setup doesn’t violate any particular theory of time at all. I can run this scenario under the A-theory and I imagine the B-theory, and the results are the same in each.
”
Consider the simple linear function f(x)=x. When we graph this simple function, we notice that the point (1,1) is certainly on the line, and as we progress along the x direction we’ll find other points on the line, as well: (2,2) and (98/43,98/43) and (π,π), for example. All of the points on this line are co-extant with one another– they are all just as actual as any which comes at a lower x value. And it’s not as if (1,1) is the reason for (2,2)’s existence or actuality– (1,1) does not “cause” (2,2) in the A-Theoretical sense of the word.
On the B-Theory, this sort of ordering relation is precisely the same, just in (at least) four dimensions rather than the two of my example. Events which we say have “causal relation” to one another carve out contiguous regions of space-time, in exactly the same way that f(x) carves out a contiguous line in a Cartesian plane. We order these causal relations between events with respect to Time, just as we were ordering the points in our function’s graph with respect to the x-axis. On this view, one event does not cause a subsequent event to exist or become actualized– as in the A-Theory– but we can distinguish a contiguous ordinal priority between events with respect to Time.”
I would look at it this way. Assume the block universe was laid out altogether and that there is a beginning of time. The events and conditions at the first time can be defined independently of any other time, because there is no time that is had earlier. Every other time that comes later, however, is dependent upon all of the times that come before it. We cannot, for instance, have certain later events if the events earlier are a particular way. So it seems there is a sense in which later events in this series are dependent upon the earlier ones. Even if they are all laid out, we can still say that one time has priority over the others and from there we get a sense of causality and order among them.
This does not seem to be the case, on the A-Theory. There is no inconsistency in constraining one Reaper’s actions by those of another, as can be quite simply illustrated by utilizing a finite number of Reapers instead of an infinite number. If that constraint was the inconsistency, then it should produce a paradox without regard to the finitude of the number of Reapers.
This is a perfect example of what I mean. In this scenario, it does not matter whether you are talking about two competitors, or seventeen competitors, or some Hyperinteger X of competitors. Since the paradox is found in a place other than the quantity of the competitors, it is quite clear that the paradox has nothing at all to do with quantity. This is not the case in the GRP, on the A-Theory.
Again, this really isn’t true. Ordinal priority does not imply any sort of dependence. Moments of time subsequent to the first are no more dependent upon that initial moment than the first moment of time is dependent upon all those subsequent, on the B-Theory. They are all equally extant regions of a complete space-time geometry.
Again, I’ll look to a 2-dimensional analogy. Consider the equation r=θ when graphed on a polar plane.
http://www.wolframalpha.com/input/?i=polar+plot+r%3Dtheta
This produces the most simple spiral curve in mathematics. This graph also has an ordinally first point, with respect to r at (0,0). There are no points in this plane– let alone in the graph of the equation– which are ordinally prior to (0,0) with respect to r. This does not imply that the points (1,1) and (4,4) and (2π, 2π) are at all dependent upon (0,0), despite the fact that they are ordinally subsequent points on the same graph.
Again, space-time is no different. Even if there is an ordinally first moment of time, that does not– in any way– imply that subsequent moments are dependent upon the first.
“This does not seem to be the case, on the A-Theory. There is no inconsistency in constraining one Reaper’s actions by those of another, as can be quite simply illustrated by utilizing a finite number of Reapers instead of an infinite number. If that constraint was the inconsistency, then it should produce a paradox without regard to the finitude of the number of Reapers.”
Of course there is no inconsistency in the constraining condition by itself. There also isn’t anything wrong in making a square or making a circle by itself. However, that does not mean we can create square circles together. Also although it is possible for an individual person to win a game on its own, it is not possible for two people to win a game simultaneously against one another.
It is only under certain conditions that the constraining becomes problematic. When you set up an interdependent system in a manner like the GRP, then it simply won’t work as it should, as I just illustrated earlier. I’m sorry, but I still don’t see how this is any different from any other inconsistent system, like the competitor example I gave. So much as the latter poses no problems for any theory of time, then I do not think the GRP favours any theory of time either.
“This is a perfect example of what I mean. In this scenario, it does not matter whether you are talking about two competitors, or seventeen competitors, or some Hyperinteger X of competitors. Since the paradox is found in a place other than the quantity of the competitors, it is quite clear that the paradox has nothing at all to do with quantity. This is not the case in the GRP, on the A-Theory.”
Not really. It should be noted that while your finite variation of the paradox demonstrates an inconsistent set of rules, it is not a finite version of Pruss’s GRP, since it involves a different set of rules for each Reaper. A proper finite version would be if a finite number of Reapers were set up to kill the man in his sleep according to the rules set out by the paradox. Under that scenario, it is clear that the first Reaper would kill him while the others would not and this is done without contradiction. And this is where infinity comes into play for the paradox. In the GRP, the Grim Reapers are set up such that every Grim Reaper has another one before it, so there is no such thing as a “first” Reaper. So much as the Reaper’s actions are made to be dependent upon their prior counterparts, then the paradox arises.
“Again, this really isn’t true. Ordinal priority does not imply any sort of dependence. Moments of time subsequent to the first are no more dependent upon that initial moment than the first moment of time is dependent upon all those subsequent, on the B-Theory. They are all equally extant regions of a complete space-time geometry.”
And again I can’t say I agree here that your analogy holds, but it feels like we would have to agree to disagree on which interpretation is more appropriate.
Let me try to tackle this another way. Under the B-theory, you cannot have certain times together with other ones. For instance, having a ball on a slope at t=0 cannot be grouped together with a ball going up the slope at t=1, because that would be inconsistent with gravity. In that sense, they are interdependent, and this I think is sufficient for the GRP. We cannot have a time (any time will do) that is like the setup of the paradox at 8:00 A.M., because then we cannot group it up with any other possible later times that would be consistent with the fact that every Grim Reaper fulfills its duties. The times that are in the B-theory, though they are all actualized, do not exist completely independent of one another, which I feel is part of the reason why you believe the B-theory avoids the paradox.
Before I reply, let me note that I am incredibly appreciative for your input, here, and that I am elated that you find this topic as interesting as I do.
It’s fairly easy to show why these two examples are inconsistent. By a rather simple exploration of the definitions of “square” and “circle,” one can show that it is not possible to produce a single plane figure in Euclidean space which satisfies both definitions. As regards the game, if we are discussing a game in which only one person can win, then it is an entirely trivial task to show that having more than one winner contradicts our premise.
By contrast, what explanation do we have for the contradiction generated by the GRP? Everyone is in agreement that the contradiction exists, given the premises of the thought experiment. So which premises are invalid? Alexander Pruss argues that the premise that there can be an actually infinite number of things is invalid. I argue that the premise that events can actually affect the ontology of a future event is invalid. If you are saying that neither of these is invalid, which premise would you say is wrong?
I never claimed that this was a finite version of Pruss’ GRP. Indeed, I explicitly noted that this is my own variation on the GRP. That said, however, I’m curious as to what you mean by that last part. There’s not a different set of rules for each Reaper. Each Reaper in my scenario operates by the exact same rule: if Fred is not going to be killed by the next Reaper in the order, then this Reaper kills Fred.
In exactly the same way, if we have a ball on a slope at t=1, we wouldn’t have the ball further down the slope at t=0, so once again, the ordinally subsequent time is no more dependent on the prior than the prior is on the subsequent. Again, both events are entirely coextant. Nothing about the ontology of an event at t=1 is altered by an event at t=0 because that would imply a dynamism which simply does not exist on the B-Theory.
Thanks. I’m glad to know that you appreciate my comments on the matter.
“By contrast, what explanation do we have for the contradiction generated by the GRP? Everyone is in agreement that the contradiction exists, given the premises of the thought experiment. So which premises are invalid? Alexander Pruss argues that the premise that there can be an actually infinite number of things is invalid. I argue that the premise that events can actually affect the ontology of a future event is invalid. If you are saying that neither of these is invalid, which premise would you say is wrong?”
I have already explained how the GRP is inconsistent. No matter what happens, the system of Grim Reapers won’t be able to all work together according to the rules set out for them. If any Reaper kills the man when they activate, then that means that a previous Reaper didn’t do their job, and if they all don’t kill the man, then the man would stay alive. I have been hinting at what I believe was the main issue with the paradox, but if you want me to be clear, then the problem is that we expect the system of Grim Reapers to all be able to fulfill their duties perfectly.
Like I said earlier, the difference between this particular paradox of the infinite and something like supertasks is that the GRP involves an infinite system of interdependent parts. When we refer to the tasks of something like Thomson’s lamp, every action is done completely independently of the other, there are just an infinite number of them. The same, however, cannot be said of the Reapers. Their actions can affect what the other one does, and I believe this is what is key here.
If we assume that every actor in a system would be able to fulfill their duties, then we can see how we can derive a similar paradox to my other examples. Two people cannot both succeed in winning their competition when pitted against the other, but if we insist that they both should be successful, then of course there will be a contradiction. How do we find ourselves out of this problem? Simply understand that one of more of these actors will fail their duties. And that is how I believe the GRP will play out if we were to ever enact it. There is no problem with the setup of the paradox itself. Whether it be A or B time, if we watch the scenario play out on its own the system of Reapers will fail either way. At least one or more Reapers will simply not be able to fulfill their duties, but this should be no more surprising than in the case of the competitors.
“I never claimed that this was a finite version of Pruss’ GRP. Indeed, I explicitly noted that this is my own variation on the GRP. That said, however, I’m curious as to what you mean by that last part. There’s not a different set of rules for each Reaper. Each Reaper in my scenario operates by the exact same rule: if Fred is not going to be killed by the next Reaper in the order, then this Reaper kills Fred.”
I never claimed you did. I am just saying that the rules that you apply for your variation is different from those of Pruss’s. Compare:
– If Fred is not going to be killed by the next Reaper in the order, then this Reaper will instantaneously kill Fred.
Versus,
– If Fred is still alive, then that Reaper will instantaneously kill Fred.
Both rules are not the same. And this is the difference between your paradox and the GRP. If you consider a finite version of the GRP following the latter rules, you will see that there will be no paradox at all. Not so for the infinite case, which is the point of the paradox and the reason why I disagreed with you when you said that the quantity was irrelevant to the GRP. Your variation may work without the need for infinity, but Pruss’s GRP certainly needs it.
“In exactly the same way, if we have a ball on a slope at t=1, we wouldn’t have the ball further down the slope at t=0, so once again, the ordinally subsequent time is no more dependent on the prior than the prior is on the subsequent. Again, both events are entirely coextant. Nothing about the ontology of an event at t=1 is altered by an event at t=0 because that would imply a dynamism which simply does not exist on the B-Theory.”
Of course there is no ontology alteration, but the point is that we cannot have certain events together, and this is all that is needed for the paradox. Some events can’t exist alongside others, and it seems that none can exist with the setup of the paradox at 8:00 A.M (again, assuming that every Grim Reaper follows their rules).
I think you are conflating the observation of a contradiction for the explanation of that contradiction. Again, everyone agrees that Pruss’ GRP results in a contradiction. The question is why does it result in a contradiction. Why is it that the contradiction occurs? Which premise is being violated in order to result in contradictory behavior.
Looking at the two examples you’ve given, I’ll illustrate what I mean:
Creating a square circle in Euclidean space
1. A circle is a plane figure consisting of all those points which are a single, specified distance from a given point, called the center.
2. A square is a regular tetragon– that is, a four sided plane figure whose sides are all of equal length and whose internal angles are all equal.
3. Let ABCD describe a square.
4. Plane figure ABCD is a circle.
5. In square ABCD, segments AC and BD are the diagonals of the square.
6. The intersection of AC and BD is the center of the square, which we will call point O.
7. Let the midpoint of AB be point M.
8. Segment OA is exactly √2 times the length of segment OM.
9. According to (1), OA and OM ought to be equal in length.
10. Therefore ABCD is not a circle.
Therefore, we arrive at our contradiction. It is clear that Premise (1) is violated by any square in Euclidean space, and therefore either Premise (3) or Premise (4) must be invalid. The explanation of why (10) creates a contradiction is not the simple fact that (10) is a contradiction. Rather, it is that (4) clearly violates (3).
We can do the same thing with your game, but much more easily:
Two winners in a game
1. Game X is a contest which can have only one winner.
2. Two players, One and Two, both win a single instance of X.
3. Therefore, X is a contest which can have more than one winner.
It’s quite clear that (2) violates (1), in this scenario, and one of the two premises must therefore be invalid. Again, the contradiction at (3) is not explained by the fact that (3) is a contradiction. Rather, it is explained by the fact that (2) violates (1).
In exactly the same way, the fact that the GRP leads to a contradiction does not, in any way, explain that contradiction. Which premise of the GRP violates a prior premise or is otherwise invalid? That revelation would explain, rather than simply illustrate, the contradiction.
I never said that the quantity was irrelevant to the GRP. I said that the quantity is irrelevant to the validity of constraining one Reaper’s actions by those of another, and I noted that this is demonstrated precisely because a finite quantity of Reapers produces no contradiction. If it were simply the case that one cannot constrain a Reaper’s actions by those of a prior Reaper, then it should not matter whether the number of total Reapers is infinite or finite– either case should equally produce a contradiction.
The fact that the GRP produces no contradiction when a finite number of Reapers is utilized instead of an infinite number tells us that the constraint on each Reaper is not the inconsistent premise of the thought experiment.
Again, though, why is it the case that we cannot have these infinite Reaper events together? In the case of your ball-on-a-slope example, those two events could not exist together because it would violate one of our premises– that of Gravity. So, again, which premise of the GRP is being violated to result in the contradiction?
” I think you are conflating the observation of a contradiction for the explanation of that contradiction. Again, everyone agrees that Pruss’ GRP results in a contradiction. The question is why does it result in a contradiction. Why is it that the contradiction occurs? Which premise is being violated in order to result in contradictory behavior.
…
In exactly the same way, the fact that the GRP leads to a contradiction does not, in any way, explain that contradiction. Which premise of the GRP violates a prior premise or is otherwise invalid? That revelation would explain, rather than simply illustrate, the contradiction. ”
Forgive me, but I am not sure what about my explanation of the paradox was insufficient to you, or really how your examples of explanations are any different from mine. I have not just said that the paradox was problematic because it is contradictory. I have showed where and how the premises are all inconsistent regardless of how it plays out. In addition, I have explained how eliminating the one premise that the Reapers fulfill all of their duties removes such contradiction (similar to how Pruss chooses to eliminate Actual Infinity as a premise). What more do you want?
Look, I will try to spell out my understanding of the paradox one more time in terms of premises. First let’s assume some of the basic premises of Pruss’s GRP:
(1) There are an actual infinite number of Grim Reapers all set to activate at 8:30, 8:15, 8:07.5 …
(2) The Grim Reapers are such that, without exception, if a grim reaper is activated and the man is alive at that instant, then the reaper will kill the man at that instant, otherwise it will do nothing.
___________________________________________
Now, let’s consider the implications of these premises:
(4) The man will die by 9:00 by a Grim Reaper’s hands(Proof: If the man was alive by 8:30, then the last reaper should kill him, from 1,2)
(5) If a Reaper kills the man, then there will always exist another Reaper who activated prior to that one who did not kill him while he was still alive (from 1)
(6) Some Reapers will not kill the man while he is alive when they activate (from 4,5)
(7) Contradiction (6 violates 2)
So there is a contradiction between the premises regardless of how we choose to look at it. The contradiction boils down to the adoption of the basic premises 1 and 2. As we can see from the reasoning above, they would both lead to a contradiction if we try to accept them both. In other words, (1) violates (2) and vice-versa. Thus, it appears that one of our basic premises is incorrect and that we should reject one or both of them.
Pruss’s move (I believe) was to reject the possibility of 1. There are no actual infinites so we cannot have such a setup of Grim Reapers according to 1. However, what I believe was the faulty premise, and the one that I have tried to argue should be rejected, is 2. Not all Grim Reapers will behave as the the second premise dictates so in turn the contradiction is avoided in doing so.
As you can see, the above is not just an observation, but a demonstrations in a manner similar to your own of why there is a contradiction, outlining the relevant premises and how they are both inconsistent. If you feel this is lacking in some way, then please do tell me what is missing. In addition, if you still don’t think this analysis is sufficient, then I would appreciate it if you were to tell me how you would go about analyzing the paradox yourself, telling me your premises and how they would lead to a contradiction.
” I never said that the quantity was irrelevant to the GRP. I said that the quantity is irrelevant to the validity of constraining one Reaper’s actions by those of another, and I noted that this is demonstrated precisely because a finite quantity of Reapers produces no contradiction. If it were simply the case that one cannot constrain a Reaper’s actions by those of a prior Reaper, then it should not matter whether the number of total Reapers is infinite or finite– either case should equally produce a contradiction.
The fact that the GRP produces no contradiction when a finite number of Reapers is utilized instead of an infinite number tells us that the constraint on each Reaper is not the inconsistent premise of the thought experiment. ”
And again, I must say that constraints in themselves are not contradictory, any more than the concept of infinity is by itself, or the concept of being a square or a circle is in themselves. In that sense, there is no such thing as an “inconsistent premise” because most premises by themselves are merely statements. It is only when combined with other conditions that we can derive a contradiction, for that is usually how contradictions and inconsistencies come about. If we try to say that a square is also circular for instance, than we have a contradiction. In the case of the GRP, it is clear that infinity does play a role in the production of a paradox when combined with the constraining condition. I have shown how this is in the case above (the infinity introduced in premise 1 is inconsistent with the constraining condition of premise 2) and in previous posts as well, so I will not repeat myself here.
” Again, though, why is it the case that we cannot have these infinite Reaper events together? In the case of your ball-on-a-slope example, those two events could not exist together because it would violate one of our premises– that of Gravity. So, again, which premise of the GRP is being violated to result in the contradiction? ”
Like I keep saying, it would not be consistent with the fact that every Grim Reaper fulfills its duties successfully. Really, you can try it out yourself. Assume the a time t=0 on the block universe is equivalent to the setup of GRP at 8:00, with an infinite number of Reapers set to kill the man in his sleep. Now try to fill in the times following that one. What will be contained on the time corresponding to 9:00?
There are a few problems in the logic of your layout of Pruss’ GRP. As you have it laid out, (5) is incorrect– which means that (6) and (7) are also invalid, as they build upon (5).
Allow me to suggest the following revisions:
1. There are an actual infinite number of Grim Reapers in a room with a man which are all set to activate at 8:30, 8:15, 8:07.5 …
2. The Grim Reapers are such that, without exception, if a grim reaper is activated and the man is alive at that instant, then the reaper will kill the man at that instant, otherwise it will do nothing.
3. The man must be killed by a Reaper before 9:00, since the final Reaper activates at 8:30. (1,2)
4. For any Reaper that activates, there exists an infinite number of Reapers which activated previously. (1)
5. For any Reaper that activates, the man cannot be alive, as if he had survived until the instant a previous Reaper activated, he ought to have been killed by that previous Reaper. (2, 5)
6. Any Reaper that activates will do nothing. (2, 6)
So, now we have reached our contradiction. It is clear that (3) and (6) stand in opposition to one another, since no Reaper can have killed the man and yet the man must have been killed by a Reaper. However, it is not the case that any of (3), (4), (5), or (6) violates either of our major premises (1) and (2).
If every individual Reaper activates and does nothing because the man is not alive, as (6) tells us, then it is absolutely true that every Reaper, without exception, fulfilled its duties as defined in (2). So there is no inconsistency there, despite your protestations to the contrary.
“There are a few problems in the logic of your layout of Pruss’ GRP. As you have it laid out, (5) is incorrect– which means that (6) and (7) are also invalid, as they build upon (5).”
You haven’t pointed out why (5) is false though. Here is a reason why it is true. Every Grim Reaper has a Reaper prior to them. If that Reaper activates and kills the man, then it means that the man must be alive in order to kill him. That must mean that the man was not killed by any other individual and thus the Reapers that activated prior did nothing.
Regardless, even if you choose to interpret it differently (that no Grim Reapers did anything instead), there would still be a contradiction. I didn’t want to bring up the idea that the Reapers all do nothing (an alternative version of premise (6) can be built based upon (5) and (2) if we wanted), because that would unnecessarily complicate things (it would just be another contradiction), but the results are pretty much the same. There is a contradiction. Now what are we going to do with it?
” So, now we have reached our contradiction. It is clear that (3) and (6) stand in opposition to one another, since no Reaper can have killed the man and yet the man must have been killed by a Reaper. However, it is not the case that any of (3), (4), (5), or (6) violates either of our major premises (1) and (2).”
I think you’re missing the bigger picture here. The illustration of the contradiction demonstrates the basic premises violate one other. We cannot have (1) and (2) true together. Why? Well, we just gave a structured proof showing why! So much as you admit that there is indeed a contradiction here, it seems like the next move would be to reject one or both of our basic premises if we want to avoid any contradiction.
Your text of (5) is false because it is not the case that “if a Reaper kills the man, then there will always exist another Reaper who activated prior to that one who did not kill him while he was still alive.” That does not follow logically from any of the previous premises, and is– in fact– explicitly ignoring (2). There’s a difference between observing a contradiction implicit in a set of premises and forcing one.
A Reaper that activates does nothing only if the man is not alive. So, for any Reaper that activates, for the man to have survived until that point, he must have been alive for the previous Reaper’s activation. If that had been the case, he would have immediately been killed by that previous Reaper, so he cannot be alive for the activation of the current Reaper.
No Reaper activates and does nothing despite the fact that the man is still alive. Rather, every Reaper activates and does nothing because the man is not alive.
Yet again, no one has denied this. Everyone agrees that the GRP presents a contradiction. What is not resolved is the reason that this contradiction is present, and which of our prior premises must therefore be invalid.
You are again saying that a contradiction arises from the conjunction of (1) and (2) because we observe that a contradiction arises from the conjunction of (1) and (2). That does not answer the “why?” question, it simply restates the problem.
A square cannot be a circle because it is necessarily true from the definition of a square that some points which compose the plane figure are closer to its center than others.
A game which can only be won by a single person cannot be won by more than one person because that violates the rules of the game.
Nothing in (1) or (2) is being violated, in our further exploration of the GRP. We have not contradicted either of our major premises, and as such, the “why?” question remains. The observation of a contradiction is good enough to tell us that it cannot be done, but it is not good enough to tell us why it cannot be done.
You know what, I think I can show how your (6) contradicts a basic premise. It is not one of the main two premises but it is rather a background assumption made in the context of the GRP:
(3*) The man is alive at 8:00.
This premise I believe is simple and reasonable in the context of the paradox. There are other background assumptions that can be made as well. One in which I have thought about is that the man can only die from a Grim Reaper, which will prevent the man’s death from other irrelevant causes. The reason why most of these assumptions go unnoticed is that most of these premises are usually either too obvious to mention or irrelevant. But here, I feel like it should be noted.
Of course, what reason is there for (3) to be true? In order for the paradox to even get off the ground, we have to assume that the man was alive by the beginning of the paradox. Else he would be dead and all the Grim Reapers would fulfill their duties by doing nothing, which would be missing the point of the paradox entirely. You know where I’m going with this.
So let’s go back to (6):
(6) Every Grim Reaper that activates will not kill the man because he is already dead.
This was the premise that you insisted upon, but if we try to follow up on this premise we get:
(7) There is no time that can be given after 8:00 that will be before every Reaper’s activation (Proof: Given (1), we cannot provide any time after 8:00 that will be before every Reaper’s activation. For any time you give me, no matter how miniscule, I will be able to provide a Reaper that will activate moments before).
(8) The man is dead at 8:00 (Proof: We know from (6) that every Reaper that activates will find the man dead. We also know from (7) that there is no time after 8:00 that is before every Reaper’s activation. Thus in order for either of this to work, the man has to already be dead by or before 8:00)
(8) Contradicts (3*), therefore a major premise is contradicted. The man will have to be dead, but we have already said that he should be alive.
Now, of course, in the face of this contradiction, we can choose to now reject the fact that the man is alive at 8:00. However, that would be a sneaky way out of things. So we are forced to deal with either (1) or (2).
I agree that this is a major premise of the argument which we neglected to enumerate earlier. Personally, I think it would be easier to label it as Premise (0), since my list already included a (3) and since this major premise is significant to both of our formulations of the GRP.
So, for clarity’s sake, I’ll restate the major premises of the argument as being:
0. There is a man who is alive in a room at 8:00.
1. There exists an actually infinite number of Grim Reapers in the room with the man which are all set to activate at 8:30, 8:15, 8:07.5 …
2. The Grim Reapers are such that, without exception, if a grim reaper is activated and the man is alive at that instant, then the reaper will kill the man at that instant; otherwise, it will do nothing.
Strictly speaking, this is not actually true. The number of Reapers in the room is infinite, to be sure, but it is countably infinite. Presuming that time is continuous and not discrete (another inherent assumption of Pruss’ formulation of the GRP, mind you) the number of moments of time between 8:00 and 9:00 is uncountably infinite. I’m not sure if you are at all familiar with the mathematics of the infinite, but it’s actually not difficult to determine a moment of time which would be after 8:00 but before any of the Reapers activate.
The Reapers in the room activate in a very predictable sequence of inverted half intervals after 8:00. The last one activates after a half hour. The second to last after a quarter hour. The third to last after an eighth hour, et cetera, et cetera. In general, the nth from last Reaper will activate at 1/(2^n) hours after 8:00. If we let X be a Hyperinteger equal to the number of Reapers in the room, then no Reaper will activate prior to 1/(2^X) hours after 8:00. There still remains an uncountably infinite number of moments which are prior to that particular moment– for example, 1/(3^X) hours after 8:00.
Even though there exist an infinite number of Reapers in the room, there still remain an infinite number of moments after 8:00 but before any Reaper activates.
All that said, even if it was actually the case that we could not list any time after 8:00 but before the activation of a Reaper, your 8th premise does not logically follow after this.
Even if you had been right about there not being any time after 8:00 which is before every Reaper’s activation, the only logical step from that point would be to say that there is no time after 8:00 at which the man is alive. It is an unjustifiable leap of logic to claim that since there is no time after 8:00 at which the man is alive, therefore the man was not alive at 8:00. There are no logical rules of inference which could lead to such a conclusion.
I’ll conclude by restating my formulation of the GRP to include the two new major premises which we’ve discussed in these last few posts.
MAJOR PREMISES
(-1) Time is continuous, and not discrete.
(0) There exists a man who is alive in a room at 8:00.
(1) There exists an actually infinite number of Grim Reapers in the room with the man which are all set to activate at 8:30, 8:15, 8:07.5 …
(2) The Grim Reapers are such that, without exception, if a Reaper is activated and the man is alive at that instant, then the Reaper will kill the man at that instant; otherwise, it will do nothing.
IMPLICATIONS
(3) The man must be killed by a Reaper before 9:00, since the final Reaper activates at 8:30. (1, 2)
(4) For any Reaper that activates, there exists an infinite number of Reapers which activated previously. (1)
(5) For any Reaper that activates, the man cannot be alive, as if he had survived until the instant a previous Reaper activated, he ought to have been killed by that previous Reaper. (2, 4)
(6) Any Reaper that activates will do nothing. (2, 5)
“Nothing in (1) or (2) is being violated, in our further exploration of the GRP. We have not contradicted either of our major premises, and as such, the “why?” question remains. The observation of a contradiction is good enough to tell us that it cannot be done, but it is not good enough to tell us why it cannot be done.”
To be quite honest, your insistence on a particular explanation of a contradiction still confuses me and this is part of the reason why I feel frustrated with your responses. The fact that we do not explain a contradiction in the form of a direct violation with our basic premises does not seem to me to be any more fruitful then explaining an inconsistency of premises in general, since in the end, they are inconsistencies that should be rejected. Perhaps what I mean to ask you is what the point of trying to understand the paradox in a particular manner is and how this will help us in addressing the GRP.
From my interpretation of the paradox, we have our basic premises and an understanding that they all lead to a contradiction if had together. If we wish to merely solve the paradox itself, then the next move would simply be to reject one or more of those premises as being false. Perhaps you believe that understanding the contradiction as undermining one basic premise in the argument will determine which one is the “wrong” one. But to this, I will have to differ. The fact is, rejecting any premise will be sufficient in solving the paradox, and in the end, I do not see how is this insufficient in order to explain why the contradiction cannot happen. For instance, if we reject actual infinities then we will not be able to establish any of the contradictory conclusions. Whether or not we understand the paradox in your preferred way or mine doesn’t matter at all and doesn’t undermine it as a solution to the paradox.
The entire point of thought experiments like the GRP is to attempt to isolate particular assumptions which we have about reality which may not be true. When Zeno of Elea introduced his paradoxes to ancient Greek philosophers, he did so to challenge their assumptions about motion. When Albert Einstein presented his thought experiments underlying Relativity, he did so to challenge the assumptions his colleagues held about space and time. When Alexander Pruss formulated his particular version of the GRP, he did so to challenge our notion of the applicability of infinite numbers to reality.
To observe the occurrence of a contradiction without attempting to understand what caused that contradiction to occur doesn’t really afford us a better comprehension of reality– which is pretty much the entire goal of philosophy.
”
To observe the occurrence of a contradiction without attempting to understand what caused that contradiction to occur doesn’t really afford us a better comprehension of reality– which is pretty much the entire goal of philosophy.”
I think you are mixing up the “cause” of the contradiction with an understanding of “which (basic) premise is being violated in order to result in contradictory behavior”. Understanding the latter isn’t going to give us any way to determine the former, for it certainly can be the case that the latter and the former can differ.
For me, the answer to what caused the contradiction, if we are to speak of a “cause” at all, is actually addressed in the rejection of one of the basic premises. With any contradiction, we agree that there are a certain set of premises that cannot occur together. At least one or more of these premises would be false because we have seen that they cannot work together. Which one of course is open to debate, but no doubt whatever one isolates as the wrong premise will be seen to be the “cause” of the given contradiction. In the case of Zeno, the “cause” was the premise that motion was possible, but someone else might think that the offending “cause” was the idea that space is continuous. And as we have seen in the GRP, Pruss’s idea of the “cause” was the premise that there can be an actual infinity to reality. In the end, we certainly acquire a better understanding of reality, or more specifically what reality is not, and so much as the goals of philosophy are to determine what is and is not true, then there should be no objections there.
However, again I must note that none of this requires any understanding of the contradiction by knowing “which (basic) premise is being violated in order to result in contradictory behavior”. For instance, if Pruss were to conclude based upon my reasoning that the basic premise of an actual infinity is false, why should his conclusion be undermined by whether we understand the contradiction in your terms? So far as you prefer to understand the contradiction in that manner, I don’t see it as anything more substantial than that.
These thought experiments do not stand in complete isolation. They are intended to supplement other ideas, positions, discoveries, and arguments which are aimed at understanding the manner in which reality works.
For example, William Lane Craig likes to support his claim that the physical cosmos was divinely created by arguing that the physical cosmos cannot have a past-infinite history. One of the means by which he argues this is to claim that actual infinites cannot exist, and he supports this claim with a number of thought experiments– including the GRP. Now, if it were the case that the contradiction inherent in the GRP could only be due to its inclusion of actual infinites, Dr. Craig might have a stronger position. However, by exploring the thought experiment, we find that there are other possibilities which could explain the occurrence of the contradiction. Time might be discrete, in which case it would be impossible to order an infinite number of Reapers in the manner described in (1). Time might be tenseless, in which case (2) must be understood as a descriptive rule rather than a prescriptive one, leading to an easier understanding of why it must be false. It may simply be that this particular ordering of an infinite set of objects is invalid, though we do not yet understand why, preserving the possibility that other things could still be infinite in number.
Pruss and Craig are A-Theorists who reject actual infinites, so naturally they prefer to see the existence of an actually infinite number of Reapers as being the problematic element. I am a B-Theorist who actively supports the notion of actual infinites, so naturally I prefer to resolve the problem by eliminating the idea that a Reaper’s action is only a potentiality under it activates. Neither Pruss nor Craig nor I can rely upon the GRP to cement our positions, since there are multiple paths to resolving the problem. However, we each must be able to show that the problem can be resolved in a manner which is consistent with the rest of the positions we espouse. To acknowledge that we hold to paradoxical irrationalities would rather undermine our claim to be driven by logic to our beliefs.
Okay, I can see your point now. Simply saying that the infinite system of Grim Reapers as does not work just because the logic says so is not a satisfying solution when applied generally. I originally assumed that the system of Grim Reapers was simply a special case, but it seems like that isn’t true. So I am willing to back off on that solution for now at least.
_______________________________________________________
In any case, going back to the topic of time, I still don’t see how the B-theory fares any better in solving the paradox, despite your claims to the contrary. In your current blog post, you’ve stated that the problem with the GRP is that it implicitly assumes that events in the present must “alter events in the future” in an A-theoretic sense. I disagreed with your claim of course and have shown earlier that we can have a sense of interdependence between events that is compatible with the B-theory of time, one which was sufficient to establish the GRP. I would like to draw attention to it again because I do not think your response to it was satisfactory.
Here it is in case you don’t remember:
“Under the B-theory, you cannot have certain times together with other ones. For instance, having a ball on a slope at t=0 cannot be grouped together with a ball going up the slope at t=1, because that would be inconsistent with gravity. In that sense, they are interdependent, and this I think is sufficient for the GRP. We cannot have a time (any time will do) that is like the setup of the paradox at 8:00 A.M., because then we cannot group it up with any other possible later times that would be consistent with the fact that every Grim Reaper fulfills its duties. The times that are in the B-theory, though they are all actualized, do not exist completely independent of one another, which I feel is part of the reason why you believe the B-theory avoids the paradox.”
On the B-Theory of time, these events would be related, to be certain, by virtue of forming a contiguous region of space-time. However, as I stated earlier, this does not imply that they are interdependent insofar as their ontology is concerned, as is the case on the A-Theory.
On the A-Theory, (2) is a prescriptive rule, meaning that any individual Reaper’s actions are only a potential until that Reaper activates. At that point, one of the two potential results of (2) becomes actualized based upon the actualized situation at that point. As such, the ontology of that Reaper’s action is dependent upon the previous Reapers. If we cleave off all the Reapers prior to any given Reaper from the experiment, then that given Reaper’s action will be different.
On the B-Theory, (2) is a descriptive rule, meaning that every individual Reaper’s actions are actualized and we are just trying to find a shorthand way to describe them in total. Whether any individual Reaper kills the man is not altered by the actions of any previous Reaper. Those actions are actualized. If we cleave off all the Reapers prior to any given Reaper from the experiment, then that given Reaper’s action cannot be different– it is already actualized and cannot change.
“On the B-Theory of time, these events would be related, to be certain, by virtue of forming a contiguous region of space-time. However, as I stated earlier, this does not imply that they are interdependent insofar as their ontology is concerned, as is the case on the A-Theory.”
Whether or not the events are already actualized or not doesn’t seem to matter. I should note that when I say “interdependence”, I do not mean it like dependence in in an “ontological” sense. I simply mean that certain sets of events are not compatible with one another and can only be compatible with others. And like I keep saying this doesn’t require the A-theory of time. We can describe a spatial analogy where certain regions of space are “interdependent” on one another in much the same way. For instance if we find a planet on one region of space, then we can deduce that there is a star nearby (assuming that there are strict rules of object placement akin to temporal causality of course). All the objects in space can be said to be “actualized” but that does not mean that they are not “interdependent” in the sense above. So it seems like can say that, if anything, there is a strict relationship between the events in the B-theory of time.
Now, again, try assuming that there is a time in the block universe that matches up with 8:00 in the paradox. Can we think of any time that is compatible with it? If you’re like me, then you would agree that that is not possible (if not, then you can tell me what times). Any event we place leads to a contradiction. So there is something inherently wrong here. But what? At this point I am posing the same question you asked me: what is the inconsistency here? It surely cannot be the fact that we are assuming an A-theory of time since we have just stated the same problem under the B-theory.
Also I wanted to raise another point that I neglected to mention before. Earlier when I asked you to model your interpretation of the paradox in terms of premises, I expected you to provide one that involves your claimed implicit premise of the A-theory. So far as you have given me your major premises:
”
MAJOR PREMISES
(-1) Time is continuous, and not discrete.
(0) There exists a man who is alive in a room at 8:00.
(1) There exists an actually infinite number of Grim Reapers in the room with the man which are all set to activate at 8:30, 8:15, 8:07.5 …
(2) The Grim Reapers are such that, without exception, if a Reaper is activated and the man is alive at that instant, then the Reaper will kill the man at that instant; otherwise, it will do nothing.”
It seems like none of these deal explicitly with a notion of an A-theory of time. Of course, you may have just left out that premise here, but it does not seem like the premises above single out any particular theory of time. So much as you agree that this leads to a contradiction, then it seems like the A-theory is irrelevant. If you want to try reformulating it again, then I suggest you:
– Include the implicit premise of the A-theory as one of the major premises
– Show how the paradox contradicts one of the major premises directly
Let me illustrate using another situation which is precisely analogous to the GRP on the B-Theory.
(1) Let X be a set which contains an infinite number of elements.
(2) There is an ordering relation on X such that for any n∈X there exists either some m∈X<n or else some m∈X>n.
(3) For any n∈X, n=1 if and only if there exists no m∈X<n; otherwise, n=0.
Given these premises, it is exceedingly clear that (3) violates (2). The only set which can satisfy (3) is one in which every element is equal either to 0 or else to 1. However, if every element is equal to every other, then there is no element of the set which is less than or greater than any given element– which is a necessary condition of (2).
These premises map precisely onto the situations which arise from the GRP. Each element of the set describes the state of the room at a time in which a Reaper activates. A value of 0 represents the man being alive. A value of 1 represents the man being dead. These are the only two possible values for any element of the set, so at any time in which a Reaper activates the man must either be alive or else dead.
We can see by (2) that if the set contains a time in which the man is alive, then it must contain a time in which the man is dead; and the converse is also true– if the set contains a time in which the man is dead, then it must contain a time in which he is alive. This corresponds precisely to the manner in which the Reapers are directed to act. A Reaper can only kill the man if he is alive to kill. And if the man is alive to kill, a Reaper must kill him.
However, we can see by (3) that if the set contains a time in which the man is dead, it cannot contain a time in which the man is alive; and if the set contains a time in which the man is alive, it cannot contain a time in which the man is dead. This is the exact behavior which we witness in the GRP. If the man is dead upon the activation of any particular Reaper, then he must have been dead upon the activation of the previous Reaper, as well.
This approach only works if we view all of the times in which a Reaper activates as a single, completed set– as is the case on the B-Theory. On the A-Theory, this would not be the case until after the GRP has already run its course, either putting us in the absurd position of having to affirm that the GRP is not a paradox until after it has already occurred, or else leaving us to find another explanation for the contradiction.
” (1) Let X be a set which contains an infinite number of elements.
(2) There is an ordering relation on X such that for any n∈X there exists either some m∈Xn.
(3) For any n∈X, n=1 if and only if there exists no m∈X<n; otherwise, n=0."
Okay, before I dive in, I have to say that I am not intimately familiar with the symbolic language, so it would be best to use plain English so that I can follow what you say. The above seems easy enough to read though if by ∈ you mean "element of", so I can at least follow you there.
"These premises map precisely onto the situations which arise from the GRP. Each element of the set describes the state of the room at a time in which a Reaper activates. A value of 0 represents the man being alive. A value of 1 represents the man being dead. These are the only two possible values for any element of the set, so at any time in which a Reaper activates the man must either be alive or else dead.
We can see by (2) that if the set contains a time in which the man is alive, then it must contain a time in which the man is dead; and the converse is also true– if the set contains a time in which the man is dead, then it must contain a time in which he is alive. This corresponds precisely to the manner in which the Reapers are directed to act. A Reaper can only kill the man if he is alive to kill. And if the man is alive to kill, a Reaper must kill him."
May not be that important, but (2) doesn't exactly mean there are only times in which the man is dead or alive (or times having the values of 0 or 1). For instance, it could be that the only values are 1 and 2, or -94 and -5673 which is also compatible with (2) as stated. You can say that this would ignore (3) which restricts the values to 0 or 1, but it seems (2) ignores (3) already by directly contradicting it.
That aside, as I see it, this is how your idea of this situation would work for the GRP:
(1) There are an infinite number of times between 8:00 and 9:00
(2) There is an ordering relation on the times between 8:00 and 9:00 such that for any time between 8:00 and 9:00 in which the man is dead, then there will be a time in which he is alive, and for any time in which he is alive, there is a time in which he is dead(this seems to be what you want to say).
(3) For any time between 8:00 and 9:00, the man is alive if any only if there exists no other time between 8:00 and 9:00 in which the man is dead, otherwise the man is dead.
The problem with these premises, apart from the first one, is that they do not match at all the basic premises that we have established in the GRP. So much as they are premises, they seem to be derived from the major ones. In that sense, your approach involves assuming the contradictory derived premises as basic premises in themselves and then deriving a contradiction from them. But essentially this is no different from me saying that (6) contradicts (3) in our other discussion. According to you, that type of contradiction is not satisfactory since it does involve a direct contradiction with a major premise. So much as you demonstrate that a contradiction occurs in the GRP, you do not explain why it occurs.
Also, you mention that the A-theory of time is incapable of accommodating this approach because of the set of times between 8:00 and 9:00 being incomplete and unactualized. I presume this is because, as you have stated earlier, you believe the future is a potential one under the A-theory and that there are no actualized facts about the future. However, I do not believe this is the case. The future being potential is compatible with the A-theory, but only if one assumes indeterminism in their beliefs. Indeterminism is not necessarily tied to the A-theory though. We can assume something like fatalism, for instance, and under that theory it seems as though the A-theory can indeed accomodate future facts (only that they will be tensed facts about what will be but they are facts nonetheless). From there, it seems like we can create a completed set of future facts, from which we can create an analogous situation to yours:
(1) There are an infinite number of times that will happen between 8:00 and 9:00
(2) There is an ordering relation on the times that will happen between 8:00 and 9:00 such that for any time that will happen between 8:00 and 9:00 in which the man will be dead, then there will be a time in which he will be alive, and for any time in which he will be alive, there is a time in which he will be dead
(3) For any time that will happen between 8:00 and 9:00, the man will be alive if any only if there exist no other times between 8:00 and 9:00 in which the man will be dead, otherwise the man will be dead.
The basic content is the same, I just replaced the "is" with "will be" in order to indicate the passage of time. So it seems like the approach can be had on the A-theory as well as the B-theory. So much as your approach even works, it seems like the B-theory is not the solution.
In any case, it seems like your approach to the paradox falls victim to the same problems that you raised against mine. You simply demonstrated that a contradiction exists for the GRP, but you have not explained why. At best, we can say that there is a problem with one of our basic premises which give rise to the contradiction, but you have not identified which one it is any more than I have.
Indeed, the “∈” symbol means “is an element of.”
I did not say that this was an implication of (2). The fact that the elements of X can only be valued 0 or 1 is an implication of (3).
When exploring a logical construction, it is well within reason to conclude that two premises are necessarily contradictory. However, it is not within reason to introduce ideas which do not follow from our premises. Indeed, it is strictly opposed to reason to do so. So, while there is an infinite range of values besides 0 and 1 which could satisfy (2) in the absence of (3), there are no premises in our discussion which would permit us to define the elements of X with such values.
As I explained in my earlier post, they map precisely to the states which arise from the GRP.
Let’s have another look at them to see that this is true. I’ll use a GRP prefix to denote premises from the original GRP argument, and an SM prefix to denote premises from our set mapping of the argument.
(GRP -1) Time is continuous, and not discrete.
(GRP 0) There exists a man who is alive in a room at 8:00.
(GRP 1) There exists an actually infinite number of Grim Reapers in the room with the man which are all set to activate at 8:30, 8:15, 8:07.5 …
(GRP 2) The Grim Reapers are such that, without exception, if a Reaper is activated and the man is alive at that instant, then the Reaper will kill the man at that instant; otherwise, it will do nothing.
The implication of (GRP -1) and (GRP 1) together is that there exists an actually infinite number of moments in which a Reaper will activate after 8:00. This is precisely akin to (SM 1) which defines an infinite set.
When (GRP 0) and (GRP 2) are examined, it directly follows that in order for the man to be dead at some moment in which a Reaper activates, he must have been alive at some moment in which a Reaper activates; and if the man is alive at some moment in which a Reaper activates, he must be dead at some moment in which a Reaper activates. This is the exact ordering relation which was laid out in (SM 2).
If we take (GRP 1) and (GRP 2) in combination, it is implied that if the man is dead at any moment in which a Reaper activates, he must have been dead in the previous moment in which a Reaper activates, and therefore he must have been dead in every previous moment. This conforms to (SM 3) from our set mapping.
The goal of the set mapping was not to create premises which were exact restatements of those of the GRP, but with different language. The goal was to describe exactly the same set of possible states in another way.
Indeterminism doesn’t mean the future is any more real on the A-Theory. It’s still just an unrealized potentiality, despite the fact that (given perfect knowledge) we would be able to predict that future perfectly.
That’s not true at all. I demonstrated precisely why (SM 1) and (SM 2) are contradictory. The properties which are required of the set by (SM 1) necessarily contradict the properties which are required of the set by (SM 2), in exactly the same way that the properties of a square necessarily require that plane figure does not conform to the properties of a circle.
In our original discussion of the GRP, the contradiction was between two different implications of our major premises. This was good enough to tell us that our major premises cannot stand in conjunction with one another, but it failed to tell us which necessary properties of one of our major premises violate the necessary properties of another. This is not the case with our set mapping. There, the contradiction arises directly from the major premises themselves, and we can see exactly which properties of each premise stand opposed to another.
“I did not say that this was an implication of (2). The fact that the elements of X can only be valued 0 or 1 is an implication of (3).”
I was responding to this part of your post:
“We can see by (2) that if the set contains a time in which the man is alive, then it must contain a time in which the man is dead; and the converse is also true– if the set contains a time in which the man is dead, then it must contain a time in which he is alive. ”
So much as you refer to dead and alive as corresponding to 0 and 1, then that is just false.
“When exploring a logical construction, it is well within reason to conclude that two premises are necessarily contradictory. However, it is not within reason to introduce ideas which do not follow from our premises. Indeed, it is strictly opposed to reason to do so. So, while there is an infinite range of values besides 0 and 1 which could satisfy (2) in the absence of (3), there are no premises in our discussion which would permit us to define the elements of X with such values.”
But the point of the contradiction is that we cannot introduce any ideas that follow from all of our premises, at least not without being irrational. There is nothing unreasonable to looking at the logical implications of a premise in itself, especially if we are going to show how they disagree with the implications of another premise. So I would not say that that is “strictly opposed to reason”.
“Let’s have another look at them to see that this is true. I’ll use a GRP prefix to denote premises from the original GRP argument, and an SM prefix to denote premises from our set mapping of the argument.”
This all just goes to prove my point. The two contradictory premises you establish are derived from more basic premises, just like how we derived (3) and (6) from our basic premises. But that in itself isn’t sufficient. So much as we say that these premise are contradictory, we still have not explained what basic premises are being directly violated.
“The goal of the set mapping was not to create premises which were exact restatements of those of the GRP, but with different language. The goal was to describe exactly the same set of possible states in another way.”
And again, I must say that your set mapping approach is no different from the manner in which we derive our secondary premises. We create an imaginary scenario in which the GRP is executed and, given the premises about what will happen, we then we explore its implications and find that they contradict one another. Really, your (SM2) and (SM3) are really no different from the (3) and (6) in which we have deduced from the same method. And of course, none of these exercises in imagination requires any loyalty to a theory of time, despite your claims to the contrary.
“Indeterminism doesn’t mean the future is any more real on the A-Theory. It’s still just an unrealized potentiality, despite the fact that (given perfect knowledge) we would be able to predict that future perfectly.”
I never said indeterminism meant that the future was any more real. Of course, the future isn’t actual. It still has yet to pass. But the fact is, rejecting indeterminism means more than just being able to predict the future. We have solid facts about what will happen just as much as we do facts about the past and facts about the present (the only difference is that they are tensed facts). Under the GRP, it has been established as a premise that the Grim Reapers will activate and will behave a certain way without exception, which is how we come to the contradiction in the first place. Just like we cannot have a contradiction between a set of present and past facts, we cannot have a contradiction between a set of future ones and the GRP goes to demonstrate that.
”
That’s not true at all. I demonstrated precisely why (SM 1) and (SM 2) are contradictory. The properties which are required of the set by (SM 1) necessarily contradict the properties which are required of the set by (SM 2), in exactly the same way that the properties of a square necessarily require that plane figure does not conform to the properties of a circle.”
You have not demonstrated anything at all! You have demonstrated why (SM 1) and (SM 2) are contradictory just as much as I have demonstrated precisely why (3) and (6) are contradictory (in fact, they both represent the same thing). The properties required by (3) necessarily contradict those required by (6) in much the same way that a properties of a square contradicts those of a circle. But that just isn’t good enough. SM2 and SM3, as you have admitted, are the byproducts of more basic premises. You have yet to show how these premises are directly violated, and more importantly which of these premises are faulty. And until you do, then I simply cannot distinguish your approach from mine.
“In our original discussion of the GRP, the contradiction was between two different implications of our major premises. This was good enough to tell us that our major premises cannot stand in conjunction with one another, but it failed to tell us which necessary properties of one of our major premises violate the necessary properties of another. This is not the case with our set mapping. There, the contradiction arises directly from the major premises themselves, and we can see exactly which properties of each premise stand opposed to another.”
And what necessary properties are that? What properties are being directly violated here and how?
Again, the implication that elements of X can only be valued 0 or 1 arises from (SM 3). Given this implication, then (SM 2) implies that if the set contains a time in which the man is alive, then it must contain a time in which the man is dead; and if the set contains a time in which the man is dead, then it must contain a time in which he is alive.
Examining a single premise in the absence of some or all the others is perfectly well and good. Drawing conclusions about the nature of the whole argument based upon a single premise in the absence of some or all the others is wholly irrational.
They are not derived from more basic premises. It is, in fact, quite clear that the SM premises are more basic than the GRP premises, as they are more generalized and employ fewer entities. The SM premises stand in parallel to the GRP premises. They are entirely independent of the GRP’s premises, but result in equal state spaces.
Premises (SM 1), (SM 2), and (SM 3) can stand upon their own in the complete absence of any of the GRP’s premises. We cannot say the same for (GRP 3) and (GRP 6), nor does the contradiction in (GRP 3) and (GRP 6) make it clear precisely which properties of which premises are being violated by that contradiction.
I can grant this. Shifting focus from the actual events to the facts about the events would yield a similar completed set, given Determinism. That completed set would be just as bound to the implications of the set mapping as would the actual events, themselves.
Premise (SM 2) requires that the set contains at least two elements which are not equal to one another. Premise (SM 3) requires that the set contains only elements which are equal to one another. These are major premises of the set mapping, and not implications based upon the combination of earlier premises. There are a whole host of ways to resolve the GRP– denial of actual infinites, denial of time as a continuum, denial of the particular arrangement of Reapers, et cetera. There are only two ways to resolve the Set Mapping: denial of (SM 2) or denial of (SM 3). In fact, our Set Mapping isn’t even reliant upon actual infinities. We could very easily rephrase (SM 1) as, “Let X be a non-empty set,” and the exact same contradiction would arise whether we have a finite number of elements or an infinite one.
Imagine that there is a man who is attempting to construct a frame from four straight wooden boards of equal length which he joins together at right angles. After constructing the frame, he adds two metal support rods equal in length to one another so that the span exactly across both of the frame’s diagonals and intersect with one another. When the project is complete, the distance from the intersection of the support rods to any point on the outer edge of the frame is exactly equal to the distance from that intersection to every other point on the outer edge of the frame.
Now, which is preferable: to say that this situation is impossible because this particular situation ultimately leads to a paradox, or to say that this situation is impossible because, in general, the properties of a square stand in direct contradiction to those of a circle?
The GRP is no different. Sure, anyone can see the production of a paradox and simply cease inquiry, at that point, satisfied in the knowledge that this particular configuration of things cannot occur. However, isn’t it far more preferable to seek out something which is more generally true about the universe, as a whole, to explain the paradox? Of what use is the GRP otherwise?
This is what I was attempting to do with the Set Mapping. Just as the geometric proof which I gave earlier in the thread regarding the impossibility of a square circle stands as a more general parallel of the very specific wood-and-metal-frame which I presented here, so too does the Set Mapping stand as a more general parallel of the very specific scenario put forth in the GRP.
“They are not derived from more basic premises. It is, in fact, quite clear that the SM premises are more basic than the GRP premises, as they are more generalized and employ fewer entities. The SM premises stand in parallel to the GRP premises. They are entirely independent of the GRP’s premises, but result in equal state spaces.
Premises (SM 1), (SM 2), and (SM 3) can stand upon their own in the complete absence of any of the GRP’s premises. We cannot say the same for (GRP 3) and (GRP 6), nor does the contradiction in (GRP 3) and (GRP 6) make it clear precisely which properties of which premises are being violated by that contradiction.”
Yes, but in real life no one would assume that (SM2) and (SM3) are true in themselves, especially if they blatently contradict. If that were the case, then the response would be: “So what?”. But the purpose of your illustration was to compare them to the situation in the GRP. If you look closely (SM2) and (SM3) represent (3) and (6) respectively. The problem with the GRP is that both premises are derived from a set of more seemingly innocuous assumptions about the world. There is nothing wrong with assuming there are actual infinites or that there are a set of Grim Reapers acting a particular way. But they all lead to a contradiction, so we must give something up.
“I can grant this. Shifting focus from the actual events to the facts about the events would yield a similar completed set, given Determinism. That completed set would be just as bound to the implications of the set mapping as would the actual events, themselves.”
So the A-theory can accommodate set-mapping. Good.
“Premise (SM 2) requires that the set contains at least two elements which are not equal to one another. Premise (SM 3) requires that the set contains only elements which are equal to one another. These are major premises of the set mapping, and not implications based upon the combination of earlier premises. There are a whole host of ways to resolve the GRP– denial of actual infinites, denial of time as a continuum, denial of the particular arrangement of Reapers, et cetera. There are only two ways to resolve the Set Mapping: denial of (SM 2) or denial of (SM 3). In fact, our Set Mapping isn’t even reliant upon actual infinities. We could very easily rephrase (SM 1) as, “Let X be a non-empty set,” and the exact same contradiction would arise whether we have a finite number of elements or an infinite one.”
And how would such a solution work in the case of the GRP? The problem with the GRP is that the premises (SM2) and (SM3) arise from more basic premises. Premise (3) requires that the man be alive for at least one moment and dead for at least one moment. Premise (6) requires that the man be dead at all times. They contradict each other in the exact same way as (SM2) and (SM3). But here we cannot just simply one of the contradictory on its own because in order to get rid of the contradiction because they follow from more fundamental assumptions about the world. We must get rid of their foundations or else there is nothing stopping the paradox from rising up again.
“The GRP is no different. Sure, anyone can see the production of a paradox and simply cease inquiry, at that point, satisfied in the knowledge that this particular configuration of things cannot occur. However, isn’t it far more preferable to seek out something which is more generally true about the universe, as a whole, to explain the paradox? Of what use is the GRP otherwise?
This is what I was attempting to do with the Set Mapping. Just as the geometric proof which I gave earlier in the thread regarding the impossibility of a square circle stands as a more general parallel of the very specific wood-and-metal-frame which I presented here, so too does the Set Mapping stand as a more general parallel of the very specific scenario put forth in the GRP.”
I still do not believe you have done any better here. You create a separate simple paradox that assumes the contradictory derived premises of the GRP as basic ones, then proceed to solve that particular contradiction. That doesn’t get to the core of the what worries us about the GRP at all. It only dances around the surface of the issue.
There is a non-paradoxical outcome here: the first reaper to awake kills Fred, which ensures that no other reaper will, as Fred will be dead whenever the others awake.
Thank you for taking the time to read and reply, but I think you might not have understood the Grim Reaper Paradox quite well.
There is no “first reaper to awake.” Any given reaper in the sequence is preceded by an infinite number of other reapers. None of them can be the first reaper, in exactly the same way as there is no first rational number which is greater than 0.
Ah, I see I should have made it clear that I was referring to your alternative thought experiment, in which there is indubitably a first reaper: as you yourself wrote, the first reaper will activate at 8:15 am.