## Monday, August 31, 2009

I'm tentatively experimenting with a Facebook page. If the experiment turns out to be worthwhile, I'll keep the page and start linking my blog posts over there as well. Please feel free to search for my page there and "friends" me, if you'd like.

## Sunday, August 30, 2009

###
Review of Sinnott-Armstrong's *Morality Without God*

Mark C. Murphy (Georgetown University) reviews the book for NDPR, here.

## Saturday, August 29, 2009

### Rounding Up Posts: Moreland, the Kalam Argument, and a Beginningless Past

Here is a list of posts critiquing J.P. Moreland's four main arguments against the traversabilty of a beginningless past in Scaling the Secular City:

Part One

Part Two

Part Three

Part Four

Part One

Part Two

Part Three

Part Four

## Wednesday, August 26, 2009

### Moreland, the Kalam Argument, and a Beginningless Past, Part 4

Moreland offers one more argument against beginningless traversals -- one he says was suggested to him by Dallas Willard (his mentor at USC). After a brief discussion of the nature of causal sequences, and how any given event depends on the actualization of every event in the causal sequence that led up to it, he expresses the argument as follows:

"...the present moment has as its ultimate chain of causal antecedents the entire history of the cosmos. If any past event has not already been actualized, then the present moment could not have occurred. This means that the past is actual and contains a specifiable, determinate number of events. This chain of events must have had a first member. Without a first member, there could be no second, third, or nth member in the chain where the nth member is the present event. But an infinite succession of past events would not have a determinate number of members nor would it have a first member. So if the past is actually infinite, the present moment could not have been caused; that is, it could not have come to be."[1]

In short, Moreland argues that the actualization of the present moment entails the finitude of the past. For the actuality of the present moment entails that its entire causal history has been actualized, since that is the causal chain upon which it depends. But the actuality of such a chain requires that it have a determinate, specifiable number of events, and that it have a first event. But if the past were infinite, it would meet neither condition. Therefore, the past is finite, and had a beginning or first event.

It will help to evaluate this argument if we express it a bit more formally:

1. The present moment M is actual.

2. If M is actual, then all the members of the set S of events that constitute M's causal chain have been actualized.

3. Therefore, all the members of S have been actualized. (from 1 and 2)

4. If all the members of S have been actualized, then S is constituted by a specifiable, determinate number of events, and S has a first member.

5. Therefore, S is constituted by a specifiable, determinate number of events, and S has a first member. (from 3 and 4)

6. If the past were beginningless, then S would not be comprised by a determinate number events and S would not have a first member.

7. Therefore, the past is not beginningless (from 5 and 6)

What to make of this argument?

Well, it's valid; so if the premises are true, then the conclusion follows of necessity. Furthermore: (3) follows from (1) and (2), (5) follows from (3) and (4), (7) follows from (5) and (6), and (1) and (2) look to be impeccable. That leaves us with (4) and (6). Why should we accept them?

For my purposes, I'll focus on (4). For I will argue that (4) is without sufficient justification, which is enough to undermine the force of the argument.

On to an evaluation of premise (4), then. Now (4) is a conditional statement, and its consequent has two conjuncts. We can thus split the conditional into two, viz.,

4a. If all the members of S have been actualized, then S is constituted by a specifiable, determinate number of events.

and

4b. If all the members of S have been actualized, then S has a first member.

Start with (4a). To evaluate the conditional, we'll need to know what Moreland means by a "specifiable, determinate number of events". A natural interpretation of the language is that it is meant to denote a number of events that can be specified by some natural number n. This reading is further supported by his use of such language in his discussion of actual infinites a few pages back, on p. 20. For there, he says that "a finite set has a definite number of elements which can be specified by counting the number of members in the set and assigning the appropriate number to that set. Thus, our set A had n=2 elements, and B had n=5." (italics mine) This interpretation is further confirmed by his use of the language in an argument we considered in a previous post -- the one on p. 29. In his argument there against the possibility of counting to infinity, he writes that at any point in such a count, one "can always specify the number he is currently counting. Furthermore, he can always add one more to what he has counted and thereby increase the series by one. Such a series can increase forever without limit, but will always be finite." (italics mine)

Thus, prima facie, it appears that by "specifiable, determinate number of events", Moreland at least means a finite number of events. But if so, then conditional (4a) asserts that the actualization of the causal sequence responsible for the present moment requires that the sequence be finite. But since that's the very point in dispute, Moreland can't just assert without argument that (4a) is true without begging the question against those antecedently convinced of the conclusion.

However, while Moreland doesn't explicitly offer an argument for (4a), he does offer a reason in support of (4b), and that rationale can be used to support (4a) as well. What about (4b), then? Well, that conditional asserts that the actuality of the present moment requires that its causal chain have a first member. But why think that? Recall Moreland's reason from the passage above: it's because "Without a first member, there could be no second, third, or nth member in the chain where the nth member is the present event." We can thus express Moreland's rationale as follows:

1. If the causal series S that led to the present moment lacks a first member, then S lacks a second, third, etc. member.

2. Therefore, S has a first member.

Unfortunately, (2) doesn't follow from (1). To get (2), we must add another premise:

1. If the causal series S that led to the present moment lacks a first member, then S lacks a second, third, etc. member.

1.1 S has a second, third, etc. member.

2. Therefore, S has a first member.

(2) then follows from (1) and (1.1) by Modus Tollens. Unfortunately, one cannot assert without argument that (1.1) is true without without begging the question against those who are antecedently unconvinced of the finitude of the past. For if the past is beginningless, then it has no second, third, etc. member, any more than the series ...-3, -2, 1 has a second, third, etc. member.

I'll belabor the point a bit more. Consider two epistemically possible causal sequences, represented by the following two series of numbers:

A: 1, 2, 3, ....

B: ...-3, -2, -1

Now it's of course true that any causal sequence of the sort that contains second, third, etc. events -- i.e., any sequence like A -- requires a first event. But of course, a beginningless causal sequence -- i.e., a sequence like B -- is precisely the sort that lacks second, third, etc. events. For a beginningless causal sequence is, by definition, a sequence that lacks a beginning, or first, event. Now of course, for all I've said, it could turn out that such an epistemically possible sequence like B is in fact metaphysically impossible. But the problem is that the metaphysical possibility of a causal sequence that lacks a first event -- i.e., the metaphysical possibility of a B-type causal sequence -- is the very issue in dispute here, and so Moreland can't just assume without argument that it's impossible without begging the question against the antecedently unconvinced. Unfortunately, though, that's precisely what Moreland seems to have done here.

===========================

[1] Moreland, Scaling the Secular City, pp. 28-29.

[2] in Moreland and Kai Neilsen's Does God Exist? The Great Debate (Thomas Nelson Publishers, 1990), pp. 197-217. That's where it is in my copy, anyway. The book has since been re-published with Prometheus Press.

[3] Ibid., pp. 203-204.

"...the present moment has as its ultimate chain of causal antecedents the entire history of the cosmos. If any past event has not already been actualized, then the present moment could not have occurred. This means that the past is actual and contains a specifiable, determinate number of events. This chain of events must have had a first member. Without a first member, there could be no second, third, or nth member in the chain where the nth member is the present event. But an infinite succession of past events would not have a determinate number of members nor would it have a first member. So if the past is actually infinite, the present moment could not have been caused; that is, it could not have come to be."[1]

In short, Moreland argues that the actualization of the present moment entails the finitude of the past. For the actuality of the present moment entails that its entire causal history has been actualized, since that is the causal chain upon which it depends. But the actuality of such a chain requires that it have a determinate, specifiable number of events, and that it have a first event. But if the past were infinite, it would meet neither condition. Therefore, the past is finite, and had a beginning or first event.

It will help to evaluate this argument if we express it a bit more formally:

1. The present moment M is actual.

2. If M is actual, then all the members of the set S of events that constitute M's causal chain have been actualized.

3. Therefore, all the members of S have been actualized. (from 1 and 2)

4. If all the members of S have been actualized, then S is constituted by a specifiable, determinate number of events, and S has a first member.

5. Therefore, S is constituted by a specifiable, determinate number of events, and S has a first member. (from 3 and 4)

6. If the past were beginningless, then S would not be comprised by a determinate number events and S would not have a first member.

7. Therefore, the past is not beginningless (from 5 and 6)

What to make of this argument?

Well, it's valid; so if the premises are true, then the conclusion follows of necessity. Furthermore: (3) follows from (1) and (2), (5) follows from (3) and (4), (7) follows from (5) and (6), and (1) and (2) look to be impeccable. That leaves us with (4) and (6). Why should we accept them?

For my purposes, I'll focus on (4). For I will argue that (4) is without sufficient justification, which is enough to undermine the force of the argument.

On to an evaluation of premise (4), then. Now (4) is a conditional statement, and its consequent has two conjuncts. We can thus split the conditional into two, viz.,

4a. If all the members of S have been actualized, then S is constituted by a specifiable, determinate number of events.

and

4b. If all the members of S have been actualized, then S has a first member.

Start with (4a). To evaluate the conditional, we'll need to know what Moreland means by a "specifiable, determinate number of events". A natural interpretation of the language is that it is meant to denote a number of events that can be specified by some natural number n. This reading is further supported by his use of such language in his discussion of actual infinites a few pages back, on p. 20. For there, he says that "a finite set has a definite number of elements which can be specified by counting the number of members in the set and assigning the appropriate number to that set. Thus, our set A had n=2 elements, and B had n=5." (italics mine) This interpretation is further confirmed by his use of the language in an argument we considered in a previous post -- the one on p. 29. In his argument there against the possibility of counting to infinity, he writes that at any point in such a count, one "can always specify the number he is currently counting. Furthermore, he can always add one more to what he has counted and thereby increase the series by one. Such a series can increase forever without limit, but will always be finite." (italics mine)

Thus, prima facie, it appears that by "specifiable, determinate number of events", Moreland at least means a finite number of events. But if so, then conditional (4a) asserts that the actualization of the causal sequence responsible for the present moment requires that the sequence be finite. But since that's the very point in dispute, Moreland can't just assert without argument that (4a) is true without begging the question against those antecedently convinced of the conclusion.

However, while Moreland doesn't explicitly offer an argument for (4a), he does offer a reason in support of (4b), and that rationale can be used to support (4a) as well. What about (4b), then? Well, that conditional asserts that the actuality of the present moment requires that its causal chain have a first member. But why think that? Recall Moreland's reason from the passage above: it's because "Without a first member, there could be no second, third, or nth member in the chain where the nth member is the present event." We can thus express Moreland's rationale as follows:

1. If the causal series S that led to the present moment lacks a first member, then S lacks a second, third, etc. member.

2. Therefore, S has a first member.

Unfortunately, (2) doesn't follow from (1). To get (2), we must add another premise:

1. If the causal series S that led to the present moment lacks a first member, then S lacks a second, third, etc. member.

1.1 S has a second, third, etc. member.

2. Therefore, S has a first member.

(2) then follows from (1) and (1.1) by Modus Tollens. Unfortunately, one cannot assert without argument that (1.1) is true without without begging the question against those who are antecedently unconvinced of the finitude of the past. For if the past is beginningless, then it has no second, third, etc. member, any more than the series ...-3, -2, 1 has a second, third, etc. member.

I'll belabor the point a bit more. Consider two epistemically possible causal sequences, represented by the following two series of numbers:

A: 1, 2, 3, ....

B: ...-3, -2, -1

Now it's of course true that any causal sequence of the sort that contains second, third, etc. events -- i.e., any sequence like A -- requires a first event. But of course, a beginningless causal sequence -- i.e., a sequence like B -- is precisely the sort that lacks second, third, etc. events. For a beginningless causal sequence is, by definition, a sequence that lacks a beginning, or first, event. Now of course, for all I've said, it could turn out that such an epistemically possible sequence like B is in fact metaphysically impossible. But the problem is that the metaphysical possibility of a causal sequence that lacks a first event -- i.e., the metaphysical possibility of a B-type causal sequence -- is the very issue in dispute here, and so Moreland can't just assume without argument that it's impossible without begging the question against the antecedently unconvinced. Unfortunately, though, that's precisely what Moreland seems to have done here.

===========================

[1] Moreland, Scaling the Secular City, pp. 28-29.

[2] in Moreland and Kai Neilsen's Does God Exist? The Great Debate (Thomas Nelson Publishers, 1990), pp. 197-217. That's where it is in my copy, anyway. The book has since been re-published with Prometheus Press.

[3] Ibid., pp. 203-204.

## Tuesday, August 25, 2009

### Moreland, the Kalam Argument, and a Beginningless Past, Part 3

Moreland offers an Aristotelian solution to one of Zeno's paradoxes as the basis of an argument against a beginningless past. Moreland sets up Zeno's Dichotomy paradox as follows:

"...Consider a runner who begins at some point A and who wishes to reach the midpoint between A and B. But before he can reach this midpoint, he must reach the midpoint of the midpoint. In order to move from any point to any other point, a runner must traverse an infinite number of points and this is impossible. Thus, [concludes Zeno] motion is an illusion."[1]

Moreland then argues that a structurally identical paradox applies to the hypothesis of a beginningless universe: he argues that if the past were beginningless, then the prospects of traversing all the events of the past to reach the present moment would be like those of Zeno's runner on the assumption that his task involved the traversal of an actual infinite: one couldn't even begin such a task, much less finish it.[2]

So Moreland thinks Zeno's Dichotomy paradox and his paradox for a beginningless past are structurally similar. His next step is to argue for a solution to the former, and then reason that, by analogy, the solution to the latter is thus similar. Thus, he argues that the most plausible solution to Zeno's Dichotomy paradox is to distinguish between an actual and a potential infinite, and to assert that the racer's task only involves the traversal of a potential infinite. And since all spatial distances that are merely potentially infinite are traversable in principle, the racer can traverse the whole track.[3] Similarly, the set of temporal distances in the universe's past is potentially infinite only, and thus finite. But that solution entails that the universe had a beginning. Therefore, thinks Moreland, the finitude of the past is justified.

What to make of this argument? One sort of worry about it is that it's not clear that the two paradoxes are sufficiently relevantly similar to justify Moreland's conclusion that their solutions are similar. For the runner's traversing task has a beginning or starting point; not so for a beginningless past. And the worry is that the feature that generates the problem in Zeno's Dichotomy parodox -- i.e., that the runner must start a task that has no starting point -- doesn't necessarily apply to a beginningless past. In other words, if the requirement of a start is merely a feature of Zeno's thought experiment, and not an essential property of traversals in general, then the grounds for thinking Zeno's runner's task is impossible do not provide grounds for thinking that traversing a beginningless past is impossible.

Now of course one might reply that it is an essential property of all traversals that they have a starting point. But the problem is that that's the very issue in dispute. For it's part of the very concept of a beginningless past that it involves traversing an infinite without a starting point. Therefore, whether or not such traversals are impossible, one cannot just assert the impossibility of a traversal that lacks a starting point without begging the question against the antecedently unconvinced.

By this time I'm no doubt belaboring the point, but here's a slightly more developed variation on the same worry: Roughly, the problem with the runner's task in Zeno's Dichotomy paradox is generated by the following inconsistent set of propositions.

1. The racetrack consists in a particular infinite open interval of spatial distances (IOISD).

2. The runner can traverse IOISD from the direction of the open "end".

3. The runner must start his traversal of IOISD if he is to engage in it at all.

4. All Infinite open intervals lack a first member from the direction of the open "end".

5. All intervals that can be traversed by starting have a first member.

For given propositions (1) and (3)-(5), Zeno's runner is required to start the traversal if he is to engage in it at all. But given that there is no starting point or beginning to the open "end" of the interval, it follows that the runner can never finish his traversal, on the grounds that he can never begin. He will thus forever remain "outside" of, or "external" to, the segment represented by the negative integers. And if that's right, then (2) is false, i.e., the runner cannot traverse IOISD from the open "end".

Now, at first glance, Moreland's paradox about a beginningless past looks sufficiently similar in structure to warrant a similar solution:

1'. The past consists in a particular infinite open interval of temporal distances (IOITD).

2'. IOITD can be traversed from the direction of the open "end"

3'. The traversal of IOITD must have a start if the traversal is to occur at all.

4. All infinite open intervals lack a first member from the direction of the open "end".

5. All intervals that can be traversed by starting have a first member.

However, at second glance, there are reasons for doubting that the runner's predicament in Zeno's Dichotomy paradox is sufficiently analogous to the case of traversing a beginningless past. For the reason why we're supposed to accept (3) in the former paradox -- i.e., that the runner's task must have a starting point -- is that Zeno stipulated that it have one. Now Moreland may be right that all traversals require a starting point. However, when it comes to the basis for accepting the parallel (3') in the latter paradox, something on the order of a mere stipulative ban on traversals without starting points is not going to cut it for those antecedently unconvinced of a finite past; they need a reason to think that it belongs to the very nature of a traversal that it has a starting point. But as we've seen, Moreland has so far failed to offer such a reason. Pending such a reason, then, traversals without beginnings remain epistemic possibilities for the unconvinced.

I conclude, then, that Moreland's use of Zeno's Dichotomy paradox fails to justify the claim that the past is necessarily finite.

===================

[1] Scaling the Secular City, p. 30.

[2] Ibid.

[3] Ibid.

"...Consider a runner who begins at some point A and who wishes to reach the midpoint between A and B. But before he can reach this midpoint, he must reach the midpoint of the midpoint. In order to move from any point to any other point, a runner must traverse an infinite number of points and this is impossible. Thus, [concludes Zeno] motion is an illusion."[1]

Moreland then argues that a structurally identical paradox applies to the hypothesis of a beginningless universe: he argues that if the past were beginningless, then the prospects of traversing all the events of the past to reach the present moment would be like those of Zeno's runner on the assumption that his task involved the traversal of an actual infinite: one couldn't even begin such a task, much less finish it.[2]

So Moreland thinks Zeno's Dichotomy paradox and his paradox for a beginningless past are structurally similar. His next step is to argue for a solution to the former, and then reason that, by analogy, the solution to the latter is thus similar. Thus, he argues that the most plausible solution to Zeno's Dichotomy paradox is to distinguish between an actual and a potential infinite, and to assert that the racer's task only involves the traversal of a potential infinite. And since all spatial distances that are merely potentially infinite are traversable in principle, the racer can traverse the whole track.[3] Similarly, the set of temporal distances in the universe's past is potentially infinite only, and thus finite. But that solution entails that the universe had a beginning. Therefore, thinks Moreland, the finitude of the past is justified.

What to make of this argument? One sort of worry about it is that it's not clear that the two paradoxes are sufficiently relevantly similar to justify Moreland's conclusion that their solutions are similar. For the runner's traversing task has a beginning or starting point; not so for a beginningless past. And the worry is that the feature that generates the problem in Zeno's Dichotomy parodox -- i.e., that the runner must start a task that has no starting point -- doesn't necessarily apply to a beginningless past. In other words, if the requirement of a start is merely a feature of Zeno's thought experiment, and not an essential property of traversals in general, then the grounds for thinking Zeno's runner's task is impossible do not provide grounds for thinking that traversing a beginningless past is impossible.

Now of course one might reply that it is an essential property of all traversals that they have a starting point. But the problem is that that's the very issue in dispute. For it's part of the very concept of a beginningless past that it involves traversing an infinite without a starting point. Therefore, whether or not such traversals are impossible, one cannot just assert the impossibility of a traversal that lacks a starting point without begging the question against the antecedently unconvinced.

By this time I'm no doubt belaboring the point, but here's a slightly more developed variation on the same worry: Roughly, the problem with the runner's task in Zeno's Dichotomy paradox is generated by the following inconsistent set of propositions.

1. The racetrack consists in a particular infinite open interval of spatial distances (IOISD).

2. The runner can traverse IOISD from the direction of the open "end".

3. The runner must start his traversal of IOISD if he is to engage in it at all.

4. All Infinite open intervals lack a first member from the direction of the open "end".

5. All intervals that can be traversed by starting have a first member.

For given propositions (1) and (3)-(5), Zeno's runner is required to start the traversal if he is to engage in it at all. But given that there is no starting point or beginning to the open "end" of the interval, it follows that the runner can never finish his traversal, on the grounds that he can never begin. He will thus forever remain "outside" of, or "external" to, the segment represented by the negative integers. And if that's right, then (2) is false, i.e., the runner cannot traverse IOISD from the open "end".

Now, at first glance, Moreland's paradox about a beginningless past looks sufficiently similar in structure to warrant a similar solution:

1'. The past consists in a particular infinite open interval of temporal distances (IOITD).

2'. IOITD can be traversed from the direction of the open "end"

3'. The traversal of IOITD must have a start if the traversal is to occur at all.

4. All infinite open intervals lack a first member from the direction of the open "end".

5. All intervals that can be traversed by starting have a first member.

However, at second glance, there are reasons for doubting that the runner's predicament in Zeno's Dichotomy paradox is sufficiently analogous to the case of traversing a beginningless past. For the reason why we're supposed to accept (3) in the former paradox -- i.e., that the runner's task must have a starting point -- is that Zeno stipulated that it have one. Now Moreland may be right that all traversals require a starting point. However, when it comes to the basis for accepting the parallel (3') in the latter paradox, something on the order of a mere stipulative ban on traversals without starting points is not going to cut it for those antecedently unconvinced of a finite past; they need a reason to think that it belongs to the very nature of a traversal that it has a starting point. But as we've seen, Moreland has so far failed to offer such a reason. Pending such a reason, then, traversals without beginnings remain epistemic possibilities for the unconvinced.

I conclude, then, that Moreland's use of Zeno's Dichotomy paradox fails to justify the claim that the past is necessarily finite.

===================

[1] Scaling the Secular City, p. 30.

[2] Ibid.

[3] Ibid.

## Monday, August 24, 2009

## Thursday, August 20, 2009

### Morriston's New Critique of Divine Command Ethics

Wes Morriston's latest paper, "What if God Commanded Something Terrible? A Worry for Divine-Command Meta-Ethics (Religious Studies 45 (2009), pp. 249-67), is now available at his department webpage. Here is the link.

It's worth mentioning that, along the way, Morriston critiques Robert Adams' version of divine command theory in his Finite and Infinite Goods, which is arguably the most sophisticated version of the theory.

It's worth mentioning that, along the way, Morriston critiques Robert Adams' version of divine command theory in his Finite and Infinite Goods, which is arguably the most sophisticated version of the theory.

## Tuesday, August 18, 2009

### Moreland, the Kalam Argument, and a Beginningless Past, Part 2

Here's another argument Moreland offers for the finitude of the past in Scaling the Secular City:

"...suppose a person were to think backward through the series of events in the past...Now he will either come to a beginning or he will not. If he comes to a beginning, then the universe obviously had a beginning. But if he never could, even in principle, reach a first moment, then this means that it would be impossible to start with the present and run backward through all the events in the history of the cosmos...But since events really move in the other direction, this is equivalent to admitting that if there was no beginning, the past could have never been exhaustively traversed to reach the present. Counting to infinity through the series 1, 2, 3, ... involves the same number of steps as does counting down from infinity to zero through the series -5, -4, -3, -2, -1, 0. In fact this second series may be even more difficult to traverse than the first. Apart from the fact that both series have the same number of members to be traversed, the second series cannot even get started. This is because it has no first member!" (p. 29)

Stripped down to its essentials, we can express the core of the argument as follows:

1. If the past is beginningless, then it’s impossible in principle to traverse from the present all the way through the past.

2. If it’s impossible in principle to traverse something in one direction, then it’s impossible in principle to traverse it in the other direction.

3. Therefore, if the past is beginningless, then it’s impossible in principle to traverse the past all the way to the present. (from 1 and 2)

4. But it’s not impossible in principle to traverse the past all the way to the present (after all, here we are!).

5. Therefore, that past is not beginningless. (from 3 and 4)

What to make of this argument? Well, it's formally valid; so if the premises are true, then the conclusion follows of necessity. Furthermore, I grant (1), at least for the sake of argument. Also, (3) follows from (1) and (2), and (4) is undeniable. That leaves us with (2). Why are we supposed to accept it?

One might think that (2) has a lot going for it, since all finite sequences are such that if one direction can be traversed in principle (at least mentally -- leave aside worries about actual traversals into the past), then so can the other direction (again, at least mentally).

However, one might worry that although this is so for finite temporal sequences, it's not obviously so for infinite temporal sequences. After all, there clearly are a number of differences between finite and infinite traversals. and pending further investigation, how can we be sure that no such difference makes a difference? Perhaps, then, we need a little more help before we're able to confidently accept (2).

Thankfully, Moreland doesn't leave us guessing as to his own reasons for accepting (2). For recall that he offered two such reasons. First, he argued that admitting that it's impossible in principle to start with the present and mentally traverse the whole past "is equivalent to admitting that if there was no beginning, the past could have never been exhaustively traversed to reach the present", on the grounds that "Counting to infinity through the series 1, 2, 3, ... involves the same number of steps as does counting down from infinity to zero through the series -5, -4, -3, -2, -1, 0."

What to make of his first argument for (2)? Now Moreland is clearly correct to say that there are the same number of steps in each direction of a beginingless history. However, it isn't clear that sameness in number of steps entails sameness in difficulty of traversal. In fact (and quite unlike finite traversals), there are several asymmetries in direction of traversal that seem relevant to difficulty or ease of traversal in a beginningless past:

(i) Going forward, there is an endpoint to reach; not so going backward.

One might believe that no infinite spatial or temporal distance is crossable on the grounds that it's self-evident that one cannot reach the end of that which has no end. Suppose we grant this. Still, this point only applies to the case of starting at the present and mentally traversing through a beginningless past; it doesn't apply to the crucial case here of a traversal from a beginningless past to the present. For the latter has an endpoint, viz., the current moment. Therefore, whether or not the intuition about the impossibility of reaching the end of that which has no end is correct, it doesn't support Moreland's premise (2).

(ii) Going forward, you don’t have to begin at some point; not so going backward. One might believe that no infinite is crossable on the basis of Moreland's argument discussed in the previous post, viz., that if one begins an infinite count from 0 or 1 to infinity, one will at every point be counting a finite number n, in which case the set that corresponds to n at that point cannot be put into a 1-1 correspondence with any of its proper subsets. Suppose we grant this. Still, this point only applies to the case of starting at the present and mentally traversing through a beginningless past; it doesn't apply to the case of a traversal from a beginningless past to the present moment. For the latter has no starting point. Therefore, unless and until Moreland (or anyone else) can come up with an argument that it's necessary for all traversals to have a beginning (a claim that Moreland repeatedly asserts without argument), Moreland's line of reasoning here is unsupported.

(iii) Going forward, some infinite traversal or other is completed at each point; not so going backward. This point is related to the last. One might believe that no infinite is crossable on the basis of Moreland's argument, discussed in the previous post, that if one tries to count to infinity by beginning at some point -- say, with the number 1 or 0 -- then one will never get over the hurdle of going from having counted a finite set to having counted an infinite set. Grant that this is true. But while this would then apply to the task of starting with the present moment and mentally traversing all the events of a beginningless past, it doesn't apply to the task of never starting -- but always counting -- from a beginningless past and then stopping with the present moment. For there is no such hurdle with such a task. For before every point in a beginningless past, some infinite set of events or other has already been traversed -- one is always on the other side of the hurdle, so to speak. Now as we have seen, Moreland thinks that traversals lacking a starting point are impossible; that is, he thinks that all traversals require a starting point. However, that is the very point at issue in the debate about the possibility of a beginningless past. For part of what it means to say that a beginningless past is possible is to say that a traversal without a beginning is possible. Therefore, Moreland can't just assume without argument that all traversals require a starting point without begging the question. I conclude, then, that Moreland's point here cannot serve as the basis for premise (2) of his argument.

Prima facie, then, there seems to be good reason to think that Moreland is wrong about this: direction of traversal does seem to make a difference with respect to difficulty of traversal. Or, more weakly: Given these asymmetries, at the very least we must say that Moreland owes us an explanation as to why they have no bearing on ease or difficulty of traversing a beginningless past. Pending such an explanation, it appears that Moreland's first argument for (2) is undercut.

What about his second reason? Here's the relevant part of the passage above: "In fact this second series [i.e., counting down the negative integers and ending at 0] may be even more difficult to traverse than the first [i.e., starting with 0 or 1 and then counting through all the natural numbers]. Apart from the fact that both series have the same number of members to be traversed, the second series cannot even get started. This is because it has no first member!"

What to make of this second argument for (2)? Well, we've already seen what is wrong with it in the previous post, so here I'll just summarize that discussion: of course a beginningless traversal could never "get started", but no one is claiming that it could. Rather, by the very nature of the case, a beginningless series has no beginning point from which it “got started”; if such a past is possible -- which is the very issue under dispute -- then it’s always been going, in the sense that prior to every event, there is another event that caused it. Furthermore, while it’s true that in traversing such a series you never get to a point in which a "first" infinite is traversed, this is only because some infinite temporal segment or other is already crossed at every point in a beginningless past. We also saw in a previous post that one is guilty of an illicit quantifier shift if from this one reasons that such a past would absurdly contain an infinite segment that was not formed by successive addition.

It looks, then, that Moreland's case for the crucial premise (2) is undercut. Thus, whether or not the past is finite, the arguments we've discussed from Moreland fail to give us a good reason for thinking so.

"...suppose a person were to think backward through the series of events in the past...Now he will either come to a beginning or he will not. If he comes to a beginning, then the universe obviously had a beginning. But if he never could, even in principle, reach a first moment, then this means that it would be impossible to start with the present and run backward through all the events in the history of the cosmos...But since events really move in the other direction, this is equivalent to admitting that if there was no beginning, the past could have never been exhaustively traversed to reach the present. Counting to infinity through the series 1, 2, 3, ... involves the same number of steps as does counting down from infinity to zero through the series -5, -4, -3, -2, -1, 0. In fact this second series may be even more difficult to traverse than the first. Apart from the fact that both series have the same number of members to be traversed, the second series cannot even get started. This is because it has no first member!" (p. 29)

Stripped down to its essentials, we can express the core of the argument as follows:

1. If the past is beginningless, then it’s impossible in principle to traverse from the present all the way through the past.

2. If it’s impossible in principle to traverse something in one direction, then it’s impossible in principle to traverse it in the other direction.

3. Therefore, if the past is beginningless, then it’s impossible in principle to traverse the past all the way to the present. (from 1 and 2)

4. But it’s not impossible in principle to traverse the past all the way to the present (after all, here we are!).

5. Therefore, that past is not beginningless. (from 3 and 4)

What to make of this argument? Well, it's formally valid; so if the premises are true, then the conclusion follows of necessity. Furthermore, I grant (1), at least for the sake of argument. Also, (3) follows from (1) and (2), and (4) is undeniable. That leaves us with (2). Why are we supposed to accept it?

One might think that (2) has a lot going for it, since all finite sequences are such that if one direction can be traversed in principle (at least mentally -- leave aside worries about actual traversals into the past), then so can the other direction (again, at least mentally).

However, one might worry that although this is so for finite temporal sequences, it's not obviously so for infinite temporal sequences. After all, there clearly are a number of differences between finite and infinite traversals. and pending further investigation, how can we be sure that no such difference makes a difference? Perhaps, then, we need a little more help before we're able to confidently accept (2).

Thankfully, Moreland doesn't leave us guessing as to his own reasons for accepting (2). For recall that he offered two such reasons. First, he argued that admitting that it's impossible in principle to start with the present and mentally traverse the whole past "is equivalent to admitting that if there was no beginning, the past could have never been exhaustively traversed to reach the present", on the grounds that "Counting to infinity through the series 1, 2, 3, ... involves the same number of steps as does counting down from infinity to zero through the series -5, -4, -3, -2, -1, 0."

What to make of his first argument for (2)? Now Moreland is clearly correct to say that there are the same number of steps in each direction of a beginingless history. However, it isn't clear that sameness in number of steps entails sameness in difficulty of traversal. In fact (and quite unlike finite traversals), there are several asymmetries in direction of traversal that seem relevant to difficulty or ease of traversal in a beginningless past:

(i) Going forward, there is an endpoint to reach; not so going backward.

One might believe that no infinite spatial or temporal distance is crossable on the grounds that it's self-evident that one cannot reach the end of that which has no end. Suppose we grant this. Still, this point only applies to the case of starting at the present and mentally traversing through a beginningless past; it doesn't apply to the crucial case here of a traversal from a beginningless past to the present. For the latter has an endpoint, viz., the current moment. Therefore, whether or not the intuition about the impossibility of reaching the end of that which has no end is correct, it doesn't support Moreland's premise (2).

(ii) Going forward, you don’t have to begin at some point; not so going backward. One might believe that no infinite is crossable on the basis of Moreland's argument discussed in the previous post, viz., that if one begins an infinite count from 0 or 1 to infinity, one will at every point be counting a finite number n, in which case the set that corresponds to n at that point cannot be put into a 1-1 correspondence with any of its proper subsets. Suppose we grant this. Still, this point only applies to the case of starting at the present and mentally traversing through a beginningless past; it doesn't apply to the case of a traversal from a beginningless past to the present moment. For the latter has no starting point. Therefore, unless and until Moreland (or anyone else) can come up with an argument that it's necessary for all traversals to have a beginning (a claim that Moreland repeatedly asserts without argument), Moreland's line of reasoning here is unsupported.

(iii) Going forward, some infinite traversal or other is completed at each point; not so going backward. This point is related to the last. One might believe that no infinite is crossable on the basis of Moreland's argument, discussed in the previous post, that if one tries to count to infinity by beginning at some point -- say, with the number 1 or 0 -- then one will never get over the hurdle of going from having counted a finite set to having counted an infinite set. Grant that this is true. But while this would then apply to the task of starting with the present moment and mentally traversing all the events of a beginningless past, it doesn't apply to the task of never starting -- but always counting -- from a beginningless past and then stopping with the present moment. For there is no such hurdle with such a task. For before every point in a beginningless past, some infinite set of events or other has already been traversed -- one is always on the other side of the hurdle, so to speak. Now as we have seen, Moreland thinks that traversals lacking a starting point are impossible; that is, he thinks that all traversals require a starting point. However, that is the very point at issue in the debate about the possibility of a beginningless past. For part of what it means to say that a beginningless past is possible is to say that a traversal without a beginning is possible. Therefore, Moreland can't just assume without argument that all traversals require a starting point without begging the question. I conclude, then, that Moreland's point here cannot serve as the basis for premise (2) of his argument.

Prima facie, then, there seems to be good reason to think that Moreland is wrong about this: direction of traversal does seem to make a difference with respect to difficulty of traversal. Or, more weakly: Given these asymmetries, at the very least we must say that Moreland owes us an explanation as to why they have no bearing on ease or difficulty of traversing a beginningless past. Pending such an explanation, it appears that Moreland's first argument for (2) is undercut.

What about his second reason? Here's the relevant part of the passage above: "In fact this second series [i.e., counting down the negative integers and ending at 0] may be even more difficult to traverse than the first [i.e., starting with 0 or 1 and then counting through all the natural numbers]. Apart from the fact that both series have the same number of members to be traversed, the second series cannot even get started. This is because it has no first member!"

What to make of this second argument for (2)? Well, we've already seen what is wrong with it in the previous post, so here I'll just summarize that discussion: of course a beginningless traversal could never "get started", but no one is claiming that it could. Rather, by the very nature of the case, a beginningless series has no beginning point from which it “got started”; if such a past is possible -- which is the very issue under dispute -- then it’s always been going, in the sense that prior to every event, there is another event that caused it. Furthermore, while it’s true that in traversing such a series you never get to a point in which a "first" infinite is traversed, this is only because some infinite temporal segment or other is already crossed at every point in a beginningless past. We also saw in a previous post that one is guilty of an illicit quantifier shift if from this one reasons that such a past would absurdly contain an infinite segment that was not formed by successive addition.

It looks, then, that Moreland's case for the crucial premise (2) is undercut. Thus, whether or not the past is finite, the arguments we've discussed from Moreland fail to give us a good reason for thinking so.

## Sunday, August 16, 2009

### Moreland, the Kalam Argument, and a Beginningless Past, Part I

J.P. Moreland offers the following argument against a beginningless past in Scaling the Secular City:

"It is impossible to count to infinity. For if one counts forever and ever, he will still be, at every moment, in a place where he can always specify the number he is currently counting. Furthermore, he can always add one more member to what he has counted and thereby increase the series by one. A series formed by successive addition is a potential infinite. Such a series can increase forever without limit, but it will always be finite. This means that the past must have been finite. For the present moment is the last member of the series of past events formed by successive addition. And since one cannot reach infinity one at a time, then if the past was actually infinite, the present moment could not have been reached. For to come to the present moment, an actual infinite would have to have been crossed." (p. 29, my copy. Italics added)

We can express the argument a bit more formally as follows:

1. At every point in the growth of any potential infinite, one can specify its cardinal number via a natural number and increase that number by 1. (premise)

2. If at every point in the growth of any potential infinite, one can specify its cardinal number via a natural number and increase that number by 1, then no potential infinite can be transformed into an actual infinite. (premise)

3. Therefore, no potential infinite can be transformed into an actual infinite. (from 1 and 2)

4. Any series formed by successive addition is a potential infinite. (premise)

5. The past is a series formed by successive addition. (premise)

6. Therefore, the past is a potential infinite. (from 4 and 5)

7. Therefore, the past cannot be actually infinite. (from 3 and 6)

The argument is thus formally valid; so if the premises should turn out to be true, then the conclusion follows of necessity. Why, then, should we accept the premises?

Well, as is indicated above, (3) follows (1) and (2), (6) follows from (4) and (5), and (7) follows from (3) and (6), Furthermore, (5) is beyond dispute, and I'm inclined to accept (1) and (2). That leaves (4), i.e., the claim that any series formed by successive addition is a potential infinite. Why should we accept it?

Now in the passage above, Moreland offers a good reason for thinking that any series formed by successive addition that has a beginning is a potential infinite -- i.e., any series naturally represented by the following series of natural numbers:

1 2 3 ....

Furthermore, he reasoned persuasively that no such series can be transformed from a potential infinite into an actual infinite.

However, that line of reasoning has no immediate bearing against the prospects of a beginningless series formed by successive addition, i.e., a series naturally represented by the following series of negative integers:

...-3 -2 -1

Rather, all that follows from that line of reasoning is the weaker claim that if the latter series is possible, it doesn't involve the transformation of a potential infinite into an actual infinite. But of course, those not antecedently convinced of the claimed necessary finitude of the past agree with that. For if the past should turn out to be beginningless, then some infinite set of events or other exists at each point in the past. But if so, then there is no event in the past that involved going from a state of not having traversed at least one infinite set of events to having traversed at least one such set. And if that's right, then if a beginningless past is possible, then it is a series of events formed by successive addition that does not involve turning a potential infinite into an actual infinite.

Thus, those who are antecedently unconvinced of the necessity of a beginningless past believe that it's at least epistemically possible that (i) the past is actually infinite, (ii) it was formed by successive addition, and (iii) the formation of the past did not involve turning a potential infinite into an actual infinite. But if a past of this sort should turn out to be possible, premise (4) is false. Thus, to adequately support premise (4), Moreland must come up with a line of reasoning that rules out the epistemic possibility expressed by (i)-(iii). But as we've seen, Moreland's reasoning in the above-quoted argument fails to do that; rather, it only rules out the possibility of an actually infinite series formed by successive addition that has a beginning. I conclude, then, that Moreland's grounds for premise (4) are inadequate.

Perhaps, though, Moreland construes a beginningless past in the way he does in an attempt to be charitable? For one might worry that if, in a beginningless past, some infinite set or other is traversed before every event, then such a past has at least one infinite proper subset of events that wasn’t formed by successive addition, which is absurd.

I don't know if this is why Moreland construes a beginningless past in the way he does, but such a worry would be ill-founded, based as it is on an inference involving an instance of the quantifier shift fallacy, reasoning from

1) Every point in a beginningless past is such that there exists an actual infinite set of events that existed prior to it.

to

2) There is an actual infinite set of events, such that it exists prior to every point in a beginningless past.

which is the same illicit pattern of inference involved in reasoning that because every child has a mother who directly gave birth to them, there is a mother who directly gave birth to every child.

No, if the past is beginningless, then while an infinite subset of events exists prior to each event, it's a new infinite every time. To illustrate: pick any event --say, the present day -- and represent it by the integer -1. Then the set of past days traversed for each of the previous days, and including today, can be represented as follows:

...

....

.....

2 days ago: {..., -5, -4, -3}

1 day ago: {..., -5, -4, -3, -2}

Present day: {..., -5, -4, -3, -2, -1}

Thus, if a past of this sort is possible, then as is represented above, the set of days traversed at each day of the past is actually infinite. However, at each day, the set of days traversed is different. So, for example, the set of days traversed today contains, in addition to the set of days traversed yesterday, the new member represented by -1, viz., today. Thus, as I said a moment ago, if the past is beginningless, then while the set of events traversed at each point in the past is actually infinite, it's a new set every time, as each passing event adds a new member to the previous set.

"It is impossible to count to infinity. For if one counts forever and ever, he will still be, at every moment, in a place where he can always specify the number he is currently counting. Furthermore, he can always add one more member to what he has counted and thereby increase the series by one. A series formed by successive addition is a potential infinite. Such a series can increase forever without limit, but it will always be finite. This means that the past must have been finite. For the present moment is the last member of the series of past events formed by successive addition. And since one cannot reach infinity one at a time, then if the past was actually infinite, the present moment could not have been reached. For to come to the present moment, an actual infinite would have to have been crossed." (p. 29, my copy. Italics added)

We can express the argument a bit more formally as follows:

1. At every point in the growth of any potential infinite, one can specify its cardinal number via a natural number and increase that number by 1. (premise)

2. If at every point in the growth of any potential infinite, one can specify its cardinal number via a natural number and increase that number by 1, then no potential infinite can be transformed into an actual infinite. (premise)

3. Therefore, no potential infinite can be transformed into an actual infinite. (from 1 and 2)

4. Any series formed by successive addition is a potential infinite. (premise)

5. The past is a series formed by successive addition. (premise)

6. Therefore, the past is a potential infinite. (from 4 and 5)

7. Therefore, the past cannot be actually infinite. (from 3 and 6)

The argument is thus formally valid; so if the premises should turn out to be true, then the conclusion follows of necessity. Why, then, should we accept the premises?

Well, as is indicated above, (3) follows (1) and (2), (6) follows from (4) and (5), and (7) follows from (3) and (6), Furthermore, (5) is beyond dispute, and I'm inclined to accept (1) and (2). That leaves (4), i.e., the claim that any series formed by successive addition is a potential infinite. Why should we accept it?

Now in the passage above, Moreland offers a good reason for thinking that any series formed by successive addition that has a beginning is a potential infinite -- i.e., any series naturally represented by the following series of natural numbers:

1 2 3 ....

Furthermore, he reasoned persuasively that no such series can be transformed from a potential infinite into an actual infinite.

However, that line of reasoning has no immediate bearing against the prospects of a beginningless series formed by successive addition, i.e., a series naturally represented by the following series of negative integers:

...-3 -2 -1

Rather, all that follows from that line of reasoning is the weaker claim that if the latter series is possible, it doesn't involve the transformation of a potential infinite into an actual infinite. But of course, those not antecedently convinced of the claimed necessary finitude of the past agree with that. For if the past should turn out to be beginningless, then some infinite set of events or other exists at each point in the past. But if so, then there is no event in the past that involved going from a state of not having traversed at least one infinite set of events to having traversed at least one such set. And if that's right, then if a beginningless past is possible, then it is a series of events formed by successive addition that does not involve turning a potential infinite into an actual infinite.

Thus, those who are antecedently unconvinced of the necessity of a beginningless past believe that it's at least epistemically possible that (i) the past is actually infinite, (ii) it was formed by successive addition, and (iii) the formation of the past did not involve turning a potential infinite into an actual infinite. But if a past of this sort should turn out to be possible, premise (4) is false. Thus, to adequately support premise (4), Moreland must come up with a line of reasoning that rules out the epistemic possibility expressed by (i)-(iii). But as we've seen, Moreland's reasoning in the above-quoted argument fails to do that; rather, it only rules out the possibility of an actually infinite series formed by successive addition that has a beginning. I conclude, then, that Moreland's grounds for premise (4) are inadequate.

Perhaps, though, Moreland construes a beginningless past in the way he does in an attempt to be charitable? For one might worry that if, in a beginningless past, some infinite set or other is traversed before every event, then such a past has at least one infinite proper subset of events that wasn’t formed by successive addition, which is absurd.

I don't know if this is why Moreland construes a beginningless past in the way he does, but such a worry would be ill-founded, based as it is on an inference involving an instance of the quantifier shift fallacy, reasoning from

1) Every point in a beginningless past is such that there exists an actual infinite set of events that existed prior to it.

to

2) There is an actual infinite set of events, such that it exists prior to every point in a beginningless past.

which is the same illicit pattern of inference involved in reasoning that because every child has a mother who directly gave birth to them, there is a mother who directly gave birth to every child.

No, if the past is beginningless, then while an infinite subset of events exists prior to each event, it's a new infinite every time. To illustrate: pick any event --say, the present day -- and represent it by the integer -1. Then the set of past days traversed for each of the previous days, and including today, can be represented as follows:

...

....

.....

2 days ago: {..., -5, -4, -3}

1 day ago: {..., -5, -4, -3, -2}

Present day: {..., -5, -4, -3, -2, -1}

Thus, if a past of this sort is possible, then as is represented above, the set of days traversed at each day of the past is actually infinite. However, at each day, the set of days traversed is different. So, for example, the set of days traversed today contains, in addition to the set of days traversed yesterday, the new member represented by -1, viz., today. Thus, as I said a moment ago, if the past is beginningless, then while the set of events traversed at each point in the past is actually infinite, it's a new set every time, as each passing event adds a new member to the previous set.

## Friday, August 14, 2009

### Matt Yglesias on High-Quality Rail and Economic Stimulus

Matt Yglesias offers what I take to be an excellent proposal about stimulating the economy and creating jobs: railways. I've mentioned this before.

Plus, it'd be nice to catch up with the rest of the modern world wrt high-speed, high-quality public transportation.

Plus, it'd be nice to catch up with the rest of the modern world wrt high-speed, high-quality public transportation.

## Wednesday, August 12, 2009

###
A Review of Beilby's *Epistemology as Theology*

Randal Rauser's review of James Beilby's

*Epistemology as Theology: An Evaluation of Alvin Plantinga's Religious Epistemology*appears in a semi-recent (2007) issue of Ars Disputandi. Here's the link.## Thursday, August 06, 2009

### Moreland on the Kalam Cosmological Argument

I recently re-read J.P. Moreland's defense of the kalam cosmological argument in his classic apologetics text, Scaling the Secular City. Good Lord.

Virtually all his arguments against the traversability of actual infinites presuppose that any traversal must have a beginning. In other words, he just blatantly begs the question against the possibility of a beginningless past. At least Craig offers arguments against the possibility of beginningless traversals, viz., his "immortal counter" reductio and his variation on the Tristram Shandy Paradox (which, unfortunately, have been decisively critiqued by (e.g.) Wes Morriston -- here, for example). I hope to discuss Moreland's arguments in detail some time soon.

UPDATE: Here is a link to a detailed discussion of Moreland's arguments against the traversability of a beginningless past.

Virtually all his arguments against the traversability of actual infinites presuppose that any traversal must have a beginning. In other words, he just blatantly begs the question against the possibility of a beginningless past. At least Craig offers arguments against the possibility of beginningless traversals, viz., his "immortal counter" reductio and his variation on the Tristram Shandy Paradox (which, unfortunately, have been decisively critiqued by (e.g.) Wes Morriston -- here, for example). I hope to discuss Moreland's arguments in detail some time soon.

UPDATE: Here is a link to a detailed discussion of Moreland's arguments against the traversability of a beginningless past.

## Wednesday, August 05, 2009

## Tuesday, August 04, 2009

###
A Review of Stewart Goetz's *Freedom, Teleology, and Evil*

Evan Fales (U of Iowa) reviews the book for NDPR, here.

## Saturday, August 01, 2009

### The Last Book in J.L. Schellenberg's Trilogy...

...The Will to Imagine: A Justification of Skeptical Religion, is scheduled to come out this month. As the link above indicates, you can pre-order it now at Amazon.com. Links to reviews of the previous two books in the trilogy can be found here.

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