Here is a link to the table of contents.

It's been a while since I checked, but it looks as though the site administrators have just posted the table of contents for the last two issues -- 2:1 and 2:2. In any case, both issues look to be chock full of interesting papers!

## Quick Links

- Book
- 200 (or so) Arguments for Atheism
- Index: Assessing Theism
- Why Mainstream Scholars Think Jesus Was A Failed Apocalyptic Prophet
- What's Wrong With Plantinga's Proper Functionalism?
- Draper's Critique of Behe's Design Argument
- The Failure of Plantinga's Free Will Defense
- 100 Arguments for God Answered
- Thomistic Arguments for God Answered
- On a Common Apologetic Strategy
- On Caring About and Pursuing Truth
- A Priori Naturalism, A Priori Inerrantism, and the Bible

### New Survey Article on The Skeptical Theist Response to the Evidential Problem of Evil

...in Philosophy Compass, here. And if a copy of the article should somehow find its way to my email inbox (exhumehume@gmail.com), I wouldn't mind it in the least. Just throwin' that out there....

### Post Index: A Critique of Craig's Philosophical Arguments Against a Beginningless Past

1. Setup: A Brief Exposition of Craig's Kalam Cosmological Argument

2. First Philosophical Argument: The Impossibility of Actual Infinites

2.1 Version 1: Parts as large as the whole

2.2 Version 2: Adding without making larger

2.3 Version 3: Making Room in a Fully-Occupied Region Without Removing Elements; Filling Up a Region Without Adding Elements

2.4 Version 4: Subtracting the Same Amount Yields Contradictory Results

2.5 Version 5: More Objects than Natural Numbers

2.6 Version 6: Craig’s Diagnostic Argument

3. Second Philosophical Argument: The Impossibility of Traversing an Actual Infinite by Successive Addition

2. First Philosophical Argument: The Impossibility of Actual Infinites

2.1 Version 1: Parts as large as the whole

2.2 Version 2: Adding without making larger

2.3 Version 3: Making Room in a Fully-Occupied Region Without Removing Elements; Filling Up a Region Without Adding Elements

2.4 Version 4: Subtracting the Same Amount Yields Contradictory Results

2.5 Version 5: More Objects than Natural Numbers

2.6 Version 6: Craig’s Diagnostic Argument

3. Second Philosophical Argument: The Impossibility of Traversing an Actual Infinite by Successive Addition

### Craig's Case Against the Existence of Actual Infinites, Part Six

**Craig’s Diagnostic Argument: "The Deeper Problem"**

(The rest of the posts in this series can be found here.)

According to Craig, the supposedly counter-intuitive implications we've discussed arise in virtue of the following set of claims:

1. A set is larger than any of its proper subsets. (a version of Euclid’s Maxim)

2. If two sets can be put into a one-to-one correspondence with one another, then neither set is larger than the other -- i.e., each set is just as large as the other. (The Principle of Correspondence)

3. Actually infinite sets are possible.

Craig points out that (1)-(3) jointly entail a contradiction.[1] Which proposition should go? Craig thinks that it should be (3). But why should we follow him in thinking this? Well, it's not clear that we should. For yet again, Craig equivocates on a key term.[2] For there are at least two interpretations of the notion of a set A being "larger than" one of its subsets B. On one interpretation, A is larger than B in the sense that A has a larger cardinal number than B. But if

*this*is how "larger than" is to be understood, then while (2) seems true, it's not clear that (1) is true. For if actual infinites are possible, then they have the same cardinal number as some of their proper subsets.

According to another interpretation, A could be "larger than" its proper subset B in the sense that A has all the elements of B

__and others besides__. But if

*this*is how "larger than" is to be understood, then while (1) seems true, it's not clear that (2) is true. For actually A has all the members of B, and others besides. So, for example, consider the set A of natural numbers: {1, 2, 3, ...}. This set has all the members had by its proper subset B of all the even numbers: {2, 4, 6, ....}. And while the two sets can be put into a one-to-one correspondence with one another, it remains true that A has all the members had by the set of even numbers and more besides, viz., the set of all the odd numbers: {1, 3, 5, ...}. And since this is so, it looks as though (2) is unjustified.

Now Craig appeals to the supposed absurdities we've discussed in previous posts as a basis for rejecting (3) over (1) and (2). The idea is that the absurdities arise because Euclid's Maxim and The Principle of Correspondence -- statements (1) and (2) above, respectively -- come into conflict when we apply them to infinite sets. Therefore, given the intuitive plausibility of (1) and (2), the best explanation is that (1) and (2) are true for all possible concrete sets, and that therefore actual infinites are impossible, i.e., (3) is false. This is Craig's proposed diagnosis of the cause of the supposed absurdities that arise in the cases we've discussed in previous posts. But the problem is that, as we've seen in other posts, all of the apparent absurdities are generated by an illicit equivocation on the notion of two sets having "just as many" members; once the equivocation is noted and the different senses of "just as many" are disambiguated, the apparent absurdities disappear. It looks, then, that Craig's diagnostic argument, like the others, fails to support Craig's claim that actually infinite sets of concrete entities are impossible.

------------------

[1] To see this, suppose an actual infinite set S is possible. Then by the definition of denumerable infinites, S can be put into a one-to-one correspondence with one of its proper subsets. For example, the set of natural numbers {1, 2, 3, ...} can be put into a one-to-one correspondence with its proper subset of the even numbers {2, 4, 6, ...}. Then by (2), S's proper subset of the evens is just as large as S itself. But by (1), S is larger than its proper subset of the evens. So S both is and isn't just as large as one of its proper subsets, which is a contradiction.

[2] The following sort of criticism can be found in the following articles: Morriston, Wes. "Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 153-155; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 299-300; Draper, Paul. "A Critique of the Kalam Cosmological Argument", in Pojman, Louis and Rea, Michael (eds.), Philosophy of Religion, an Anthology, 5th ed. (Wadsworth, 2008). (Notes on Draper's paper can be found here and here.)

### Craig's Case Against the Existence of Actual Infinites, Part Four

**Fourth Version: Subtracting the Same Amount Yields Contradictory Results**

(The rest of the posts in this series can be found here.)

*Hilbert's Hotel Illustration*:

Imagine a hotel with an infinite number of rooms and an infinite number of guests. Such a hotel would have some surprising features. So, for example, suppose all the guests in the even-numbered rooms check out. Then the hotel would still have infinitely many occupied rooms, viz., the odd-numbered rooms. On the other hand, suppose all the guests in rooms 4, 5, 6, ... check out of the hotel. Then the hotel will no longer have infinitely many occupied rooms -- it will have three. But how can this be? Just as many guests checked out in the first scenario as in the second scenario. This is because (i) the infinite set of even-numbered rooms (2, 4, 6, …) can be put into a one-to-one correspondence with the set of rooms numbered 4 and higher (4, 5, 6, …), and (ii) any two sets that can be put into a one-to-one correspondence with one another are the same “size” – each set has just as many members as the other. But this is absurd; subtracting the same amount cannot yield different amounts. Therefore, an infinite hotel is metaphysically impossible. But the absurdity here isn't particular to just infinite hotels. Rather, they generalize to every denumerably infinite set of concrete entities whatsover. Therefore, concrete actual infinites are metaphysically impossible.

We can express the argument illustrated above as follows:

1. If concrete actual infinites are possible, then a hotel with infinitely many rooms and infinitely many guests is possible.

2. If a hotel with infinitely many rooms and infinitely many guests is possible, then it’s possible to remove just as many guests from the hotel and get different amounts of remaining guests.

3. It’s impossible to remove just as many guests from the hotel and get different amounts of remaining guests.

4. Therefore, concrete actual infinites are impossible.

What to make of this argument?[1] By now the problem with Craig's argument is a familiar one: there is an equivocation on the locution "just as many" in (2) and (3). On the one hand, it might mean that "just as many" guests were removed in the two scenarios in the sense that the two sets of rooms

*have the same cardinal number*. Then while (2) is true, it’s not obvious that (3) is true. For while the two sets of rooms have the same cardinal number, one set has all the members of the other set and then some, viz., the odd-numbered rooms. Referring to the coarse-grained similarity of the two sets (viz., sameness of cardinal number) conceals this crucial difference. But once one shifts one's focus from the sameness of cardinality to the finer-grained difference in the particular members of the two sets, the intuitiveness of premise (3) disappears. For then one would expect different results when the sets removed in the two scenarios are different in this crucial way.

On the other hand, one might mean that "just as many" guests were removed in the two scenarios in the sense that neither set of guests has members besides the ones they share in common. Then while (3) is true on that reading, (2) is not. For of course the set of guests removed in the second scenario has all the guests that were removed in the first scenario and others besides.

Yet again, the upshot is that the argument gets its intuitive appeal by equivocating on the notion of “just as many”. Thus, once the different readings of the expression are disambiguated, the argument loses its force.

-------------------------------

[1] For more careful and rigorous versions of the following sort of criticism of the argument at hand, see the following articles: Morriston, Wes. "Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 151-152; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 300-301. See also Dever, Josh. "Worlds Apart: On the Possibility of an Actual Infinity", esp. pp. 8-10.

### Craig's Case Against the Existence of Actual Infinites, Part Three

Third Version: Making Room in a Fully-Occupied Region Without Removing Occupants; Filling Up an Unoccupied Region Without Adding Occupants

(The rest of the posts in this series can be found here.)

Hilbert's Hotel Illustration:

Imagine a hotel with an actually infinite number of rooms. If such a hotel were possible, then it would have some remarkable features. For example, suppose every one of its rooms were full. Then one could create vacancies in some of the rooms even if none of the guests left. So, for example, the hotel manager could have the guests in room 1 move to room 2, the guests in room 2 move to room 3, and so on for all the rest of the rooms. In this way, room 1 would could become vacant for a new guest, even though not a single guest left the hotel.

In fact, one could create infinitely many vacant rooms without removing a single guest. For one could move the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, and in general, for each room n, move the guest in n to 2n. The result would be that an infinite number of odd-numbered rooms would become vacant for more guests to occupy.

Furthermore, one could also remove any vacancies -- i.e., fill all the rooms - without adding a single new guest (again, assuming one has an infinite number of guests to move around). So, for example, one could just reverse the movings of guests stated in the previous examples, and thereby go from an infinite number of vacancies to a fully-occupied hotel.

These features of an infinite hotel illustrate that no such hotel could exist -- not even an omnipotent God could create one. For it is metaphysically impossible for a spatial region to go from full to not-full merely by moving its occupants around; without removing at least some of what occupies a region, that region will remain occupied by something or other. Similarly, it is metaphysically impossible for a spatial region to go from not-full to full merely by moving its occupants around; without adding at least some new occupants to a given region, that region cannot become fully occupied.

We can express the argument in the illustration above as follows:

1. If concrete actual infinites are possible, then a hotel with an infinite number of rooms and guests is possible.

2. If a hotel with an infinite number of rooms and guests is possible, then a spatial region can go from full to not-full without removing any of its occupants, and it can go from not-full to full without adding occupants (i.e., merely moving occupants around is sufficient).

3. No spatial region can go from full to not-full without removing occupants, and no spatial region can go from not-full to full without adding occupants (i.e., merely moving occupants around isn't sufficient).

4. Therefore, concrete actual infinites are impossible.

What to make of this argument? One might have doubts about premise (3): Granted, no fully-occupied finite region of space, containing a finite number of occupants, could go from full to not-full, or from not-full to full, by just moving some of its occupants around. But are these things obviously true for infinitely many elements within infinite regions of space?[1]

In any case, I'd like to set aside concerns about (3) to focus on a concern about the justification for (1). For the supposed absurdities of Hilbert's Hotel don't obviously generalize to all actual infinites.[2] More to the point, the supposedly problematic features of Hilbert's Hotel aren't a part of sets of past events – whether infinite or not. For the supposed problems only arise when one combines infinites with co-present regions, and only when the occupants of such regions can be rearranged. But of course, temporal regions are not co-present (at least not according to Craig’s preferred A-theory of time). Nor can the events that occupy the temporal regions be rearranged or moved around. Therefore, even if the combination of features in Hilbert’s Hotel and Craig’s infinite library give rise to absurdities, such a combination of features is not present in an infinite past. But if not, then Craig's justification for premise (1) is undercut.

---------------------------------

[1] Here are some considerations that make me doubt premise (3). What is the source of the intuition that seems to support it? Perhaps a simpler case than Hilbert's Hotel will help us get clear on this. Thus, imagine three people standing in a line, each holding an iPhone. So everyone's "phone hand" is "fully occupied". Now imagine that they want to free up some space -- free up a "phone hand" -- for a new iPhone by having each person hand their phone to the person to the right. Now let us stipulate that if a person in the line has no one to the right to hand their phone to, then they must hold onto it. Given these conditions, they won't be able to free up a "phone hand" and make space for a new phone. Why not? Because the person on the right end has no one to the right to which they could hand their phone. And since that person can't get rid of their phone and free up their phone hand, no one else can, either.

So far, we have failed to free up a phone hand for a new phone just by moving phones around. And the reason seems to be that at least one person can't move their phone down the line, since they have no one to hand it to. But now suppose we change the scenario a bit. Thus, imagine that the line of people holding phones extends (literally) without end to your right. Finally, suppose their plan to free up space for a new phone, and their rules for permissible passing of phones, are the same as before. The result this time is that the first person in the line is now without a phone! What changed to make this possible? This: Despite the fact that, as with the first scenario, everyone started out holding phones -- all the "space" was occupied -- in the new scenario, every person in the line has someone to hand their phone to (since the line of people has no end). And because of this, a hand was freed up to receive a new phone. It appears, then, that the factors that generated our intuition against freeing up space to add a new occupant just by moving things around aren't present in the second case. But that case is relevantly similar to Hilbert's Hotel. Therefore, it's no longer clear that it's impossible to make room for new occupants in a fully occupied region by moving things around when infinite regions are involved.

Here's another way to come at the point. Perhaps the concern stems from a sense that filling a region of space without adding new occupants sounds like "magic", in the sense that we're getting something from nothing. Relatedly, one might have the sense that making room in a fully-occupied region of space without taking something out sounds like "magic", in the sense that it would require an occupant to pop out of existence. However, remember that in the original example of Hilbert's Hotel, we were dealing with a limitless, inexhaustible supply of rooms and a limitless, inexhaustible supply of guests. If so, then it's not clear why we'd need more guests to fill any unoccupied rooms, and it's not clear why we would need less guests to make a new vacancy.

[2] Wes Morriston has made this sort of criticism in (e.g.), "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 295-298.

(The rest of the posts in this series can be found here.)

Hilbert's Hotel Illustration:

Imagine a hotel with an actually infinite number of rooms. If such a hotel were possible, then it would have some remarkable features. For example, suppose every one of its rooms were full. Then one could create vacancies in some of the rooms even if none of the guests left. So, for example, the hotel manager could have the guests in room 1 move to room 2, the guests in room 2 move to room 3, and so on for all the rest of the rooms. In this way, room 1 would could become vacant for a new guest, even though not a single guest left the hotel.

In fact, one could create infinitely many vacant rooms without removing a single guest. For one could move the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, and in general, for each room n, move the guest in n to 2n. The result would be that an infinite number of odd-numbered rooms would become vacant for more guests to occupy.

Furthermore, one could also remove any vacancies -- i.e., fill all the rooms - without adding a single new guest (again, assuming one has an infinite number of guests to move around). So, for example, one could just reverse the movings of guests stated in the previous examples, and thereby go from an infinite number of vacancies to a fully-occupied hotel.

These features of an infinite hotel illustrate that no such hotel could exist -- not even an omnipotent God could create one. For it is metaphysically impossible for a spatial region to go from full to not-full merely by moving its occupants around; without removing at least some of what occupies a region, that region will remain occupied by something or other. Similarly, it is metaphysically impossible for a spatial region to go from not-full to full merely by moving its occupants around; without adding at least some new occupants to a given region, that region cannot become fully occupied.

We can express the argument in the illustration above as follows:

1. If concrete actual infinites are possible, then a hotel with an infinite number of rooms and guests is possible.

2. If a hotel with an infinite number of rooms and guests is possible, then a spatial region can go from full to not-full without removing any of its occupants, and it can go from not-full to full without adding occupants (i.e., merely moving occupants around is sufficient).

3. No spatial region can go from full to not-full without removing occupants, and no spatial region can go from not-full to full without adding occupants (i.e., merely moving occupants around isn't sufficient).

4. Therefore, concrete actual infinites are impossible.

What to make of this argument? One might have doubts about premise (3): Granted, no fully-occupied finite region of space, containing a finite number of occupants, could go from full to not-full, or from not-full to full, by just moving some of its occupants around. But are these things obviously true for infinitely many elements within infinite regions of space?[1]

In any case, I'd like to set aside concerns about (3) to focus on a concern about the justification for (1). For the supposed absurdities of Hilbert's Hotel don't obviously generalize to all actual infinites.[2] More to the point, the supposedly problematic features of Hilbert's Hotel aren't a part of sets of past events – whether infinite or not. For the supposed problems only arise when one combines infinites with co-present regions, and only when the occupants of such regions can be rearranged. But of course, temporal regions are not co-present (at least not according to Craig’s preferred A-theory of time). Nor can the events that occupy the temporal regions be rearranged or moved around. Therefore, even if the combination of features in Hilbert’s Hotel and Craig’s infinite library give rise to absurdities, such a combination of features is not present in an infinite past. But if not, then Craig's justification for premise (1) is undercut.

---------------------------------

[1] Here are some considerations that make me doubt premise (3). What is the source of the intuition that seems to support it? Perhaps a simpler case than Hilbert's Hotel will help us get clear on this. Thus, imagine three people standing in a line, each holding an iPhone. So everyone's "phone hand" is "fully occupied". Now imagine that they want to free up some space -- free up a "phone hand" -- for a new iPhone by having each person hand their phone to the person to the right. Now let us stipulate that if a person in the line has no one to the right to hand their phone to, then they must hold onto it. Given these conditions, they won't be able to free up a "phone hand" and make space for a new phone. Why not? Because the person on the right end has no one to the right to which they could hand their phone. And since that person can't get rid of their phone and free up their phone hand, no one else can, either.

So far, we have failed to free up a phone hand for a new phone just by moving phones around. And the reason seems to be that at least one person can't move their phone down the line, since they have no one to hand it to. But now suppose we change the scenario a bit. Thus, imagine that the line of people holding phones extends (literally) without end to your right. Finally, suppose their plan to free up space for a new phone, and their rules for permissible passing of phones, are the same as before. The result this time is that the first person in the line is now without a phone! What changed to make this possible? This: Despite the fact that, as with the first scenario, everyone started out holding phones -- all the "space" was occupied -- in the new scenario, every person in the line has someone to hand their phone to (since the line of people has no end). And because of this, a hand was freed up to receive a new phone. It appears, then, that the factors that generated our intuition against freeing up space to add a new occupant just by moving things around aren't present in the second case. But that case is relevantly similar to Hilbert's Hotel. Therefore, it's no longer clear that it's impossible to make room for new occupants in a fully occupied region by moving things around when infinite regions are involved.

Here's another way to come at the point. Perhaps the concern stems from a sense that filling a region of space without adding new occupants sounds like "magic", in the sense that we're getting something from nothing. Relatedly, one might have the sense that making room in a fully-occupied region of space without taking something out sounds like "magic", in the sense that it would require an occupant to pop out of existence. However, remember that in the original example of Hilbert's Hotel, we were dealing with a limitless, inexhaustible supply of rooms and a limitless, inexhaustible supply of guests. If so, then it's not clear why we'd need more guests to fill any unoccupied rooms, and it's not clear why we would need less guests to make a new vacancy.

[2] Wes Morriston has made this sort of criticism in (e.g.), "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 295-298.

### Craig's Case Against the Existence of Actual Infinites, Part Two

Second Version: Adding Without Making Larger

(The rest of the posts in this series can be found here.)

Library Illustration: Imagine a library containing an actually infinite number of books. In such a library one could add one, two, three, or indeed denumerably infinitely many more books to the shelves without making the library any larger in terms of how many volumes it contains. This is because (i) the original infinite set of books can be put into a one-to-one correspondence with any set of books that could result from such additions, and (ii) any two sets that can be put into a one-to-one correspondence with one another are the same “size” – each is just as large as the other. But this is absurd; the original set of books is a proper subset, or proper part, of each set of books that would result from such additions, and no proper part or subset of a set could be just as large as the set of which it is a part. Therefore, a library containing an actually infinite number of books is impossible. But this problem isn’t particular to just infinite libraries; it applies to any actually infinite collection of things whatsoever; therefore, concrete actual infinites are metaphysically impossible.

We can express the argument above as follows:

1. If concrete actual infinites are possible, then a library with an actually infinite number of books is possible.

2. If a library with an actually infinite number of books is possible, then adding any finite or denumerably infinite amount of books to the library would result in it having just as many books as before.

3. Adding any finite or denumerably infinite amount of books to the library couldn't result in it having just as many books as before.

4. Therefore, concrete actual infinites are impossible.

What to make of this argument? As with the argument in the previous post, the notion of there being “just as many” books is ambiguous in (2) and (3).[1] On the one hand, one might mean that there are "just as many" books before the additions as after in the sense that the former has the same cardinal number as the latter. Then while (2) is true, it’s not obvious that (3) is true. Indeed, an actual infinite is standardly defined as a set that has the same cardinal number as some of its proper subsets. But, presumably, Craig's argument is supposed to persuade those who already know what an actual infinite is supposed to be, and yet remain at least agnostic as to whether actually infinite sets of things are metaphysically possible. It seems, then, that one wouldn’t accept (3) unless one already rejected the possibility of actual infinites.

Now as noted in the previous post, Craig thinks he can provide independent support for (3) via other thought experiments besides the one under consideration. Again, we'll consider those in later posts in this series. For now, though, we’ll just note again that on the current interpretation of "just as many", (3) so far lacks the independent support it needs for the current argument to convince those who are "on the fence" about the possibility of actual infinites.

On the other hand, one might mean that there are "just as many" books before the additions as after in the sense that neither set of books has members besides the ones they share in common. Then while (3) is true on that reading, (2) is not. For of course the resultant collection of books after each addition is one that has other books besides the original set, viz., the ones added.

As with the argument discussed in the previous post, the upshot is that the argument gets its intuitive appeal by equivocating on the notion of “just as many”. Thus, once the different readings of the expression are disambiguated, the argument loses its force.

-------------------------------

[1] As noted in the previous post, at least two philosophers have raised this sort of criticism. See (e.g.) Morriston, Wes. "Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 154 (bottom)-155; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 295-298; Draper, Paul. "A Critique of the Kalam Cosmological Argument", in Pojman, Louis and Rea, Michael (eds.), Philosophy of Religion, an Anthology, 5th ed. (Wadsworth, 2008). (Notes on Draper's paper can be found here and here.)

(The rest of the posts in this series can be found here.)

Library Illustration: Imagine a library containing an actually infinite number of books. In such a library one could add one, two, three, or indeed denumerably infinitely many more books to the shelves without making the library any larger in terms of how many volumes it contains. This is because (i) the original infinite set of books can be put into a one-to-one correspondence with any set of books that could result from such additions, and (ii) any two sets that can be put into a one-to-one correspondence with one another are the same “size” – each is just as large as the other. But this is absurd; the original set of books is a proper subset, or proper part, of each set of books that would result from such additions, and no proper part or subset of a set could be just as large as the set of which it is a part. Therefore, a library containing an actually infinite number of books is impossible. But this problem isn’t particular to just infinite libraries; it applies to any actually infinite collection of things whatsoever; therefore, concrete actual infinites are metaphysically impossible.

We can express the argument above as follows:

1. If concrete actual infinites are possible, then a library with an actually infinite number of books is possible.

2. If a library with an actually infinite number of books is possible, then adding any finite or denumerably infinite amount of books to the library would result in it having just as many books as before.

3. Adding any finite or denumerably infinite amount of books to the library couldn't result in it having just as many books as before.

4. Therefore, concrete actual infinites are impossible.

What to make of this argument? As with the argument in the previous post, the notion of there being “just as many” books is ambiguous in (2) and (3).[1] On the one hand, one might mean that there are "just as many" books before the additions as after in the sense that the former has the same cardinal number as the latter. Then while (2) is true, it’s not obvious that (3) is true. Indeed, an actual infinite is standardly defined as a set that has the same cardinal number as some of its proper subsets. But, presumably, Craig's argument is supposed to persuade those who already know what an actual infinite is supposed to be, and yet remain at least agnostic as to whether actually infinite sets of things are metaphysically possible. It seems, then, that one wouldn’t accept (3) unless one already rejected the possibility of actual infinites.

Now as noted in the previous post, Craig thinks he can provide independent support for (3) via other thought experiments besides the one under consideration. Again, we'll consider those in later posts in this series. For now, though, we’ll just note again that on the current interpretation of "just as many", (3) so far lacks the independent support it needs for the current argument to convince those who are "on the fence" about the possibility of actual infinites.

On the other hand, one might mean that there are "just as many" books before the additions as after in the sense that neither set of books has members besides the ones they share in common. Then while (3) is true on that reading, (2) is not. For of course the resultant collection of books after each addition is one that has other books besides the original set, viz., the ones added.

As with the argument discussed in the previous post, the upshot is that the argument gets its intuitive appeal by equivocating on the notion of “just as many”. Thus, once the different readings of the expression are disambiguated, the argument loses its force.

-------------------------------

[1] As noted in the previous post, at least two philosophers have raised this sort of criticism. See (e.g.) Morriston, Wes. "Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 154 (bottom)-155; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 295-298; Draper, Paul. "A Critique of the Kalam Cosmological Argument", in Pojman, Louis and Rea, Michael (eds.), Philosophy of Religion, an Anthology, 5th ed. (Wadsworth, 2008). (Notes on Draper's paper can be found here and here.)

### Craig's Case Against the Existence of Actual Infinites, Part One*

(The rest of the posts in this series can be found here.)

William Lane Craig argues that the universe had a beginning as part of an argument that it had a divine "Beginner" or divine temporal First Cause of the universe. He offers four arguments for a finite past: two a priori and two a posteriori. The first is that an infinite past is impossible because infinites are impossible in general; the second is that even if concrete actual infinites are possible, they can't be traversed by successive addition; the third argues that the evidence for the Big Bang indicates an absolute beginning to the universe; and the fourth argues that the 2nd Law of Thermodynamics indicates that the universe is running down, in which case it must've been "wound up" with an initial input of matter-energy, and the latter was the beginning of the universe. In the current series of posts, I will consider Craig's versions of the first argument.

First Version: Parts as Large as the Whole

Infinite Library illustration: Imagine a library containing an actually infinite number of books, such that exactly one unique natural number is written on the spine of each book. Such a library would have just as many even-numbered books as it would have odd- and even-numbered books combined. This is because (i) the set of even numbers (2, 4, 6, …) can be put into a one-to-one correspondence with the set of natural numbers (1, 2, 3, …), and (ii) any two sets that can be put into a one-to-one correspondence with one another are the same “size” – each set is just as large as the other. But this is absurd; the set of even-numbered books is a proper subset, or proper part, of the set of all the books in the library, and no proper part or subset of a set could be just as large as the set of which it is a part. Therefore, a library containing an actually infinite number of books is impossible. But this problem isn’t particular to just infinite libraries; it applies to any actually infinite collection of things whatsoever; therefore, concrete actual infinites are metaphysically impossible.

We can express the argument above as follows:

1. If concrete actual infinites are possible, then a library with an actually infinite number of books (each labeled with a unique natural number) is possible.

2. If a library with an actually infinite number of books (each labeled with a unique natural number) is possible, then there would be just as many even-numbered books as odd- and even-numbered books combined.

3. There couldn’t be just as many even-numbered books as odd- and even-numbered books combined.

4. Therefore, concrete actual infinites are impossible.

What to make of this argument? The first thing to notice is that there is an ambiguity in the notion of there being “just as many” books in (2) and (3).[1] On the one hand, one might mean that there are just as many even-numbered books as odd- and even-numbered books together in the sense that the former has the same cardinal number as the latter. Then while (2) is true, it’s not obvious that (3) is true. Indeed, an actual infinite is standardly defined as a set that has the same cardinal number as some of its proper subsets. But, presumably, Craig's argument is supposed to persuade those who already know what an actual infinite is supposed to be, and yet remain at least agnostic as to whether actually infinite sets of things are metaphysically possible. It seems, then, that one wouldn’t accept (3) unless one already rejected the possibility of actual infinites. Now Craig thinks he can provide independent support for (3) via other thought experiments besides the one under consideration. We will consider those in later posts in this series. For now, though, we’ll just note that on the current interpretation of "just as many", (3) so far lacks the independent support it needs for the current argument to convince those who are "on the fence" about the possibility of actual infinites.

On the other hand, one might mean that there are just as many even-numbered books as odd- and even-numbered books together in the sense that neither set of books has members besides the ones they share in common. Then while (3) is true on that reading, (2) is not. For of course the set of odd- and even-numbered books has other books besides the set of books they have in common, viz., the even-numbered books.

The upshot is that the argument gets its intuitive appeal by equivocating on the notion of “just as many”. Thus, once the different readings of the expression are disambiguated, the argument loses its force.

------------------

[*] Thanks to Mike Almeida for pointing out a problem in a previous draft of this post.

[1] At least two philosophers have raised this sort of criticism. See (e.g.) Morriston, Wes. "Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 154 (bottom)-155; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 295-298; Draper, Paul. "A Critique of the Kalam Cosmological Argument", in Pojman, Louis and Rea, Michael (eds.), Philosophy of Religion, an Anthology, 5th ed. (Wadsworth, 2008). (Notes on Draper's paper can be found here and here.)

William Lane Craig argues that the universe had a beginning as part of an argument that it had a divine "Beginner" or divine temporal First Cause of the universe. He offers four arguments for a finite past: two a priori and two a posteriori. The first is that an infinite past is impossible because infinites are impossible in general; the second is that even if concrete actual infinites are possible, they can't be traversed by successive addition; the third argues that the evidence for the Big Bang indicates an absolute beginning to the universe; and the fourth argues that the 2nd Law of Thermodynamics indicates that the universe is running down, in which case it must've been "wound up" with an initial input of matter-energy, and the latter was the beginning of the universe. In the current series of posts, I will consider Craig's versions of the first argument.

First Version: Parts as Large as the Whole

Infinite Library illustration: Imagine a library containing an actually infinite number of books, such that exactly one unique natural number is written on the spine of each book. Such a library would have just as many even-numbered books as it would have odd- and even-numbered books combined. This is because (i) the set of even numbers (2, 4, 6, …) can be put into a one-to-one correspondence with the set of natural numbers (1, 2, 3, …), and (ii) any two sets that can be put into a one-to-one correspondence with one another are the same “size” – each set is just as large as the other. But this is absurd; the set of even-numbered books is a proper subset, or proper part, of the set of all the books in the library, and no proper part or subset of a set could be just as large as the set of which it is a part. Therefore, a library containing an actually infinite number of books is impossible. But this problem isn’t particular to just infinite libraries; it applies to any actually infinite collection of things whatsoever; therefore, concrete actual infinites are metaphysically impossible.

We can express the argument above as follows:

1. If concrete actual infinites are possible, then a library with an actually infinite number of books (each labeled with a unique natural number) is possible.

2. If a library with an actually infinite number of books (each labeled with a unique natural number) is possible, then there would be just as many even-numbered books as odd- and even-numbered books combined.

3. There couldn’t be just as many even-numbered books as odd- and even-numbered books combined.

4. Therefore, concrete actual infinites are impossible.

What to make of this argument? The first thing to notice is that there is an ambiguity in the notion of there being “just as many” books in (2) and (3).[1] On the one hand, one might mean that there are just as many even-numbered books as odd- and even-numbered books together in the sense that the former has the same cardinal number as the latter. Then while (2) is true, it’s not obvious that (3) is true. Indeed, an actual infinite is standardly defined as a set that has the same cardinal number as some of its proper subsets. But, presumably, Craig's argument is supposed to persuade those who already know what an actual infinite is supposed to be, and yet remain at least agnostic as to whether actually infinite sets of things are metaphysically possible. It seems, then, that one wouldn’t accept (3) unless one already rejected the possibility of actual infinites. Now Craig thinks he can provide independent support for (3) via other thought experiments besides the one under consideration. We will consider those in later posts in this series. For now, though, we’ll just note that on the current interpretation of "just as many", (3) so far lacks the independent support it needs for the current argument to convince those who are "on the fence" about the possibility of actual infinites.

On the other hand, one might mean that there are just as many even-numbered books as odd- and even-numbered books together in the sense that neither set of books has members besides the ones they share in common. Then while (3) is true on that reading, (2) is not. For of course the set of odd- and even-numbered books has other books besides the set of books they have in common, viz., the even-numbered books.

The upshot is that the argument gets its intuitive appeal by equivocating on the notion of “just as many”. Thus, once the different readings of the expression are disambiguated, the argument loses its force.

------------------

[*] Thanks to Mike Almeida for pointing out a problem in a previous draft of this post.

[1] At least two philosophers have raised this sort of criticism. See (e.g.) Morriston, Wes. "Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 154 (bottom)-155; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 295-298; Draper, Paul. "A Critique of the Kalam Cosmological Argument", in Pojman, Louis and Rea, Michael (eds.), Philosophy of Religion, an Anthology, 5th ed. (Wadsworth, 2008). (Notes on Draper's paper can be found here and here.)

### Jeremy Gwiazda on Swinburne's Case for Theism

Via Prosblogion, I just learned that Jeremy Gwiazda (a recent PhD from CUNY) wrote a critique of Richard Swinburne's case for theism for his dissertation. The title: Probability, Simplicity, and Infinity: A Critique of Richard Swinburne’s Argument for Theism. Thus, for those seeking to understand and evaluate Swinburne's case, you might well want to order a copy of his dissertation (via the link above). Bonus: Here's a free preview to whet your appetite.

Also: a number of his papers can be found via a search on Phil Papers.

UPDATE: John D over at Philosophical Disquisitions has since explicated a number of Gwiazda's criticisms of Swinburne's work. Here is the link.

Also: a number of his papers can be found via a search on Phil Papers.

UPDATE: John D over at Philosophical Disquisitions has since explicated a number of Gwiazda's criticisms of Swinburne's work. Here is the link.

###
Maitzen's Argument Against the Existence of a *Sensus Divinitatis*

I'm currently re-reading Stephen Maitzen's excellent paper, "Divine Hiddenness and the Demographics of Theism" (Religious Studies 42 (2006), pp. 177-191). As the title suggests, Maitzen appeals to the data of the demographics of theism to defend the argument from divine hiddenness. However, in the same paper, he also offers a powerful argument against the existence of a sensus divinitatis (i.e., a natural capacity to form properly basic belief in God in a wide variety of circumstances or "triggering-conditions", such as (e.g.) looking at the starry heavens, reading the Bible, etc.) postulated by reformed epistemologists of the likes of Alvin Plantinga and others. Below is a summary of that portion of Maitzen's article.[1]

The argument summarized: The sensus divinitatis is supposed to be an innate capacity or faculty in all humans. Now innate human faculties tend to be evenly distributed among humans (e.g., hearing, sight, etc.). But the kind of belief that’s supposed to issue from the sensus divinitatis is very unevenly distributed among humans (e.g., Saudia Arabia contains 26 million people, and 95 percent of them are theists. By contrast, Thailand contains 65 million people, and only 5 percent of them are theists). We thus wouldn’t expect this data of “geographic patchiness” with respect to theistic belief if humans were endowed with a sensus divinitatis. However, such data is readily and naturally explained purely in terms of anthropological, sociological, political and economic factors. Therefore, the data of the demographics of theism provides good evidence against the existence of a sensus divinitatis within humans.

Objection: Christian theism entails that the sensus divinitatis was damaged – and perhaps even rendered inoperable – by human sin, whether the sin in question is the original sin from the Fall or the particular sins of humans in their earthly lives. Furthermore, Christian theism entails that it requires the work of the Holy Spirit (e.g., Plantinga’s notion of the internal instigation of the Holy Spirit) to repair and restore the sensus divinitatis in a given individual. Therefore, the fact that some people -- or even very many -- lack belief in the theistic god doesn’t provide good evidence against the existence of a sensus divinitatis within humans.

Reply: This objection misses the point. The problem isn’t the non-universality of theistic belief; the problem is the uneven distribution of theistic belief. As Maitzen puts it, “why has God bestowed this restorative grace so unevenly, contributing to a pattern that, coincidentally, social scientists say they can explain entirely in terms of culture?”[2] Thus, appeal to human sin and divine grace fails to adequately mitigate the criticism.

----------------

[1] To see Maitzen's excellent criticism of a molinist response to problems related to the demographics of theism, see the exchange between Maitzen and Jason Marsh in the following two papers: (i) Marsh, Jason. "Do the Demographics of Theistic Belief Disconfirm Theism? A Reply to Maitzen", Religious Studies 44 (2008), pp. 465-471; (ii) Maitzen, Stephen. "Does Molinism Explain the Demographics of Theism?", Religious Studies 44 (2008), pp. 473-477.

[2] Maitzen, "Divine Hiddenness and the Demographics of Theism", 187.

The argument summarized: The sensus divinitatis is supposed to be an innate capacity or faculty in all humans. Now innate human faculties tend to be evenly distributed among humans (e.g., hearing, sight, etc.). But the kind of belief that’s supposed to issue from the sensus divinitatis is very unevenly distributed among humans (e.g., Saudia Arabia contains 26 million people, and 95 percent of them are theists. By contrast, Thailand contains 65 million people, and only 5 percent of them are theists). We thus wouldn’t expect this data of “geographic patchiness” with respect to theistic belief if humans were endowed with a sensus divinitatis. However, such data is readily and naturally explained purely in terms of anthropological, sociological, political and economic factors. Therefore, the data of the demographics of theism provides good evidence against the existence of a sensus divinitatis within humans.

Objection: Christian theism entails that the sensus divinitatis was damaged – and perhaps even rendered inoperable – by human sin, whether the sin in question is the original sin from the Fall or the particular sins of humans in their earthly lives. Furthermore, Christian theism entails that it requires the work of the Holy Spirit (e.g., Plantinga’s notion of the internal instigation of the Holy Spirit) to repair and restore the sensus divinitatis in a given individual. Therefore, the fact that some people -- or even very many -- lack belief in the theistic god doesn’t provide good evidence against the existence of a sensus divinitatis within humans.

Reply: This objection misses the point. The problem isn’t the non-universality of theistic belief; the problem is the uneven distribution of theistic belief. As Maitzen puts it, “why has God bestowed this restorative grace so unevenly, contributing to a pattern that, coincidentally, social scientists say they can explain entirely in terms of culture?”[2] Thus, appeal to human sin and divine grace fails to adequately mitigate the criticism.

----------------

[1] To see Maitzen's excellent criticism of a molinist response to problems related to the demographics of theism, see the exchange between Maitzen and Jason Marsh in the following two papers: (i) Marsh, Jason. "Do the Demographics of Theistic Belief Disconfirm Theism? A Reply to Maitzen", Religious Studies 44 (2008), pp. 465-471; (ii) Maitzen, Stephen. "Does Molinism Explain the Demographics of Theism?", Religious Studies 44 (2008), pp. 473-477.

[2] Maitzen, "Divine Hiddenness and the Demographics of Theism", 187.

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### What God Would Have Known...

...is the title of J.L. Schellenberg's forthcoming book , which offers a large number of novel arguments against Christian theism. I...