Philosophers are divided on whether or not an actual infinite can exist and, secondly, on whether an actual infinite (if possible in and of itself) can be formed by successive addition. Karl Popper, for example, objects to Craig's argument that an actual infinite cannot be formed by successive addition. Popper states that the problem only arises if we assume that there is a point in the infinite past. However, there is no point in the infinite past, since infinity is not a number per se, but a set of numbers.
Even assuming that this objection is satisfactory, it does nothing to undermine the Aristotelian argument against an infinite regress of essentially ordered causes. Let's assume that the past is infinite. Keep in mind that the infinity of past events is still composed of finite periods of time. Now, for each finite period of time, a new regress of sustaining causes begins to be instantiated:
..., -5, -4, -3, -2, -1, 0
So, at -5 we have a new regress of sustaining causes; and the same is true for -4, -3, and so on.
Cn Cn Cn
C3 C3 C3
C2 C2 C2
C1 C1 C1
-5, -4, -3, . . .
Before -4 can begin its instantiation of sustaining causes, all of the sustaining causes of -5 must be instantiated. However, beginning with C1 of -5, the regress of sustaining causes cannot arrive at infinity, since we are in fact beginning with a point (more specifically, a cause). This means the Aristotelian argument is immune to Popper's criticism. After all, there is an obvious disparity between an infinite past and an infinite regress of sustaining causes. While the former does not have a first point, the latter most certainly does.
Wednesday, November 23, 2011
Thursday, October 27, 2011
Contingency and Design Arguments
An acceptance of the PSR leads one to believe that the whole of contingent reality C has an explanation of its existence. Now, an explanation can be of one of two types: a thing's explanation is found either in the necessity of its own nature or in an external cause. Since C does not exist by a necessity of its own nature, it follows that C's explanation is found in an external cause. This external cause must be a necessary and eternal entity N, as well as very powerful in order to cause something as vast as C.
Let's suppose the atheist not only denies that N is God, but that there is any such entity as N. The atheist could do this, for example, by denying the PSR. The problem is that this presents the atheist with a new challenge. For, design arguments often present the uniformity of nature or the fine-tuning of the universe's initial conditions within the context of a trilemma:
1. The laws of nature are either due to necessity, chance, or design. (Premise)
One would be hard-pressed to think of a fourth alternative to premise (1). Keeping in mind the atheist's denial of the argument from contingency, especially the existence of N, we are led to premise (2):
2. The laws of nature are not due to necessity or chance. (Premise)
From these two premises, it follows that:
3. Therefore, the laws of nature are due to design. (From 1 and 2)
But surely the atheist does not want to affirm (3)! Yet, he already denies one of his alternatives, which leaves him with only one option: the affirmation that the laws of nature (and/or the universe's fine-tuning) are due exclusively to chance. Remember, there is no N at all according to this radical position, so the laws of nature cannot be the products of a combination of necessity and chance.
Obviously, if the chance hypothesis doesn't pan out (and there are good reasons to think it doesn't), one must accept that N exists and/or that the laws of nature are designed. Whichever route is taken, a large chunk of either the contingency or design arguments must be affirmed as a matter of consistency.
Let's suppose the atheist not only denies that N is God, but that there is any such entity as N. The atheist could do this, for example, by denying the PSR. The problem is that this presents the atheist with a new challenge. For, design arguments often present the uniformity of nature or the fine-tuning of the universe's initial conditions within the context of a trilemma:
1. The laws of nature are either due to necessity, chance, or design. (Premise)
One would be hard-pressed to think of a fourth alternative to premise (1). Keeping in mind the atheist's denial of the argument from contingency, especially the existence of N, we are led to premise (2):
2. The laws of nature are not due to necessity or chance. (Premise)
From these two premises, it follows that:
3. Therefore, the laws of nature are due to design. (From 1 and 2)
But surely the atheist does not want to affirm (3)! Yet, he already denies one of his alternatives, which leaves him with only one option: the affirmation that the laws of nature (and/or the universe's fine-tuning) are due exclusively to chance. Remember, there is no N at all according to this radical position, so the laws of nature cannot be the products of a combination of necessity and chance.
Obviously, if the chance hypothesis doesn't pan out (and there are good reasons to think it doesn't), one must accept that N exists and/or that the laws of nature are designed. Whichever route is taken, a large chunk of either the contingency or design arguments must be affirmed as a matter of consistency.
Friday, October 21, 2011
De Ente et Essentia and the First Way
Thomas argues in De Ente et Essentia that God is Pure Act, or existence itself subsisting. His metaphysical argument is the ground of the first four of his five ways in proof of God's existence. The argument from motion, the first of these, specifically recalls the transition of a thing's potentiality to actuality. For example, an acorn exists as an acorn in actuality and as an oak tree in potentiality.
Since no potentiality can actualize itself (an acorn needs water, sunlight, etc), it follows that some actuality is needed to actualize a potentiality. Since Pure Act just is existence itself, it's not hard to see why Thomas associates Pure Act with God. All potentialities are possibly actualized, and nothing is actualized apart from Pure Act (existence), so Pure Act ultimately has power over all potentialites and is therefore omnipotent.
Since no potentiality can actualize itself (an acorn needs water, sunlight, etc), it follows that some actuality is needed to actualize a potentiality. Since Pure Act just is existence itself, it's not hard to see why Thomas associates Pure Act with God. All potentialities are possibly actualized, and nothing is actualized apart from Pure Act (existence), so Pure Act ultimately has power over all potentialites and is therefore omnipotent.
Wednesday, October 12, 2011
The Euthyphro Dilemma
I'm of the opinion that Euthyphro should have been the one asking Socrates the questions. Roughly, the modern defender of the alleged dilemma asks, "is something good because God wills it, or does God will something because it is good?" Let's tackle this question by separating each part of the disjunction:
A. X is good because God wills it.
B. God wills X because X is good.
Now, in order for this to be a true dilemma, B must be the negation (or the equivalent to the negation) of A. Otherwise, the disjunction that Socrates presents is a false dilemma. Euthyphro should have asked Socrates why A and B are contradictory. Why can it not be the case that both A and B?
In debating the question over the years, it has become clear to me that defenders of the dilemma are making a very crucial assumption. What the Euthyphro Dilemma requires in order to work properly is the implication that B entails independence of God. A and B should really be rephrased like this:
A'. X, which is good, is dependent on God.
B'. X, which is good, is independent of God.
Obviously, A' and B' are mutually incompatible, but this raises an even more obvious question: why not simply state the dilemma like this? The answer is likely that Euthyphro would have simply affirmed A'. Hence, there is no dilemma for him to consider. What Socrates and his modern counterpart have to defend is that B entails B'. Are there any forthcoming arguments to support this? I doubt it. In any case, the theist should not accept the burden of proof in trying to explain away the (false) dilemma. Rather, the dilemma's defender ought to accept responsibility for arguing that B and B' are ultimately identical.
A. X is good because God wills it.
B. God wills X because X is good.
Now, in order for this to be a true dilemma, B must be the negation (or the equivalent to the negation) of A. Otherwise, the disjunction that Socrates presents is a false dilemma. Euthyphro should have asked Socrates why A and B are contradictory. Why can it not be the case that both A and B?
In debating the question over the years, it has become clear to me that defenders of the dilemma are making a very crucial assumption. What the Euthyphro Dilemma requires in order to work properly is the implication that B entails independence of God. A and B should really be rephrased like this:
A'. X, which is good, is dependent on God.
B'. X, which is good, is independent of God.
Obviously, A' and B' are mutually incompatible, but this raises an even more obvious question: why not simply state the dilemma like this? The answer is likely that Euthyphro would have simply affirmed A'. Hence, there is no dilemma for him to consider. What Socrates and his modern counterpart have to defend is that B entails B'. Are there any forthcoming arguments to support this? I doubt it. In any case, the theist should not accept the burden of proof in trying to explain away the (false) dilemma. Rather, the dilemma's defender ought to accept responsibility for arguing that B and B' are ultimately identical.
Monday, September 26, 2011
A Leibnizian Kalam Argument?
Staying with my recent approach of tossing up new arguments and seeing what sticks, I thought I'd give this a try. I've defended several modal cosmological arguments before, but this time I want to take some Leibnizian and kalam considerations into account.
1. Possibly, everything that exists has an explanation of its existence, either in the necessity of its own nature or in an external cause. (Premise, W-PSR)
2. Necessarily, whatever begins to exist is contingent. (Premise)
3. Possibly, the universe began to exist. (Premise)
4. Possibly, the universe is contingent. (From 2 and 3)
5. Hence, the universe is contingent. (From 4 and S5)
6. Possibly, whatever is contingent has an external cause. (Premise, implied by W-PSR)
7. Possibly, the universe has an external cause. (From 5 and 6)
8. Necessarily, if the universe has an external cause, that cause is a timeless, changeless, immaterial, and very powerful first cause. (Premise)
9. Possibly, a timeless, changeless, immaterial, and very powerful first cause exists. (From 7 and 8)
10. Necessarily, a first cause is either necessary or contingent. (Definition)
11. Necessarily, a first cause cannot be explained by anything else. (Premise)
12. Necessarily, a first cause is explained by a necessity of its own nature. (From 1 and 11)
13. Therefore, a timeless, changeless, immaterial, and very powerful first cause exists. (Implied by 12)
1. Possibly, everything that exists has an explanation of its existence, either in the necessity of its own nature or in an external cause. (Premise, W-PSR)
2. Necessarily, whatever begins to exist is contingent. (Premise)
3. Possibly, the universe began to exist. (Premise)
4. Possibly, the universe is contingent. (From 2 and 3)
5. Hence, the universe is contingent. (From 4 and S5)
6. Possibly, whatever is contingent has an external cause. (Premise, implied by W-PSR)
7. Possibly, the universe has an external cause. (From 5 and 6)
8. Necessarily, if the universe has an external cause, that cause is a timeless, changeless, immaterial, and very powerful first cause. (Premise)
9. Possibly, a timeless, changeless, immaterial, and very powerful first cause exists. (From 7 and 8)
10. Necessarily, a first cause is either necessary or contingent. (Definition)
11. Necessarily, a first cause cannot be explained by anything else. (Premise)
12. Necessarily, a first cause is explained by a necessity of its own nature. (From 1 and 11)
13. Therefore, a timeless, changeless, immaterial, and very powerful first cause exists. (Implied by 12)
Monday, September 19, 2011
Modalizing Descartes
1. Whatever is conceivable C is possibly caused by an existing C's impression. (Premise)
2. A perfect being is conceivable. (Premise)
3. Hence, the conception of a perfect being is possibly caused by a perfect being's impression. (From 1 and 2)
4. Hence, a perfect being possibly exists. (Implied by 3)
5. Therefore, a perfect being exists. (From 4 and S5)
Besides the usual unpackaging of (4), this argument is defensible. Descartes held that in order to conceive of something, there had to be an impression of that conception caused by the entity being conceived. That's a bit of a mouthful, but the idea can be illustrated by connotation, as opposed to strict denotation. The reason I can conceive of a horse is because I have seen an existing horse. Even imaginary entities, such as a unicorn, are just an amalgamation of existing things, e.g. a horse (which exists) and a horn (which exists on other animals).
Now, instead of contending that Descartes' strong principle is correct, we can actually weaken it and still arrive at the same conclusion. If it is even possible for a conception to be caused in the manner described above, it follows that a perfect being (omnipotent, omniscient, and morally perfect in every possible world) possibly exists. All that's left is a defense of the rather uncontroversial S5 axiom, and we have a successful ontological argument for God's existence.
Of course, I don't expect anyone to be persuaded by this argument if they're not persuaded by similar ones, e.g. the modal third way, that have arguably even more modest premises.
2. A perfect being is conceivable. (Premise)
3. Hence, the conception of a perfect being is possibly caused by a perfect being's impression. (From 1 and 2)
4. Hence, a perfect being possibly exists. (Implied by 3)
5. Therefore, a perfect being exists. (From 4 and S5)
Besides the usual unpackaging of (4), this argument is defensible. Descartes held that in order to conceive of something, there had to be an impression of that conception caused by the entity being conceived. That's a bit of a mouthful, but the idea can be illustrated by connotation, as opposed to strict denotation. The reason I can conceive of a horse is because I have seen an existing horse. Even imaginary entities, such as a unicorn, are just an amalgamation of existing things, e.g. a horse (which exists) and a horn (which exists on other animals).
Now, instead of contending that Descartes' strong principle is correct, we can actually weaken it and still arrive at the same conclusion. If it is even possible for a conception to be caused in the manner described above, it follows that a perfect being (omnipotent, omniscient, and morally perfect in every possible world) possibly exists. All that's left is a defense of the rather uncontroversial S5 axiom, and we have a successful ontological argument for God's existence.
Of course, I don't expect anyone to be persuaded by this argument if they're not persuaded by similar ones, e.g. the modal third way, that have arguably even more modest premises.
Friday, September 16, 2011
An Epistemological Argument Against a Beginningless Universe
Suppose that a beginningless universe is metaphysical possible. With this assumption in mind, it may still be shown that it is irrational to believe in such a universe.
1. If the universe is beginningless, then its past is infinite. (Premise)
2. An infinite past yields infinitely-many actualities. (Premise)
3. If there are infinitely-many actualities, then it is epistemically impossible to weigh any probability. (Premise)
4. Probability can be epistemically weighed. (Premise)
5. Therefore, it is irrational to believe in an infinite universe. (From 1 - 4)
The logic of the argument as stated is pretty loose, but one can follow the reasoning of the premises to the conclusion fairly naturally, I think. (1) is obviously true, and (2) is almost entirely uncontroversial. (4) must be true in order for probability and induction to survive, and these principles are indispensable toward rationality. This leaves us with (3).
Imagine the lines of evidence both for and against a beginningless universe. Further, imagine that the evidence on both sides is inexhaustible. We can represent the evidence for a beginningless universe with the set of all odd numbers {1, 2, 3, . . . n}, and the evidence against a beginningless universe with the set of all even numbers {2, 4, 6, . . . 2n}. This scenario should be expected, given the existence of infinitely-many actualities.
But if this is the case, how can the probability of either hypothesis be reasonably assessed? For every line of evidence for a beginningless universe, there is an equally strong line of evidence against it. Yet, we all know that the probability of a hypothesis can be reasonably assessed. If this is correct, then it follows that it is inconsistent with probability for there to be infinitely-many actualities to take into account. Because of the epistemic warrant for probability, then, it seems that the rejection of infinitely-many actualities, and therefore a beginningless universe, is preferable.
1. If the universe is beginningless, then its past is infinite. (Premise)
2. An infinite past yields infinitely-many actualities. (Premise)
3. If there are infinitely-many actualities, then it is epistemically impossible to weigh any probability. (Premise)
4. Probability can be epistemically weighed. (Premise)
5. Therefore, it is irrational to believe in an infinite universe. (From 1 - 4)
The logic of the argument as stated is pretty loose, but one can follow the reasoning of the premises to the conclusion fairly naturally, I think. (1) is obviously true, and (2) is almost entirely uncontroversial. (4) must be true in order for probability and induction to survive, and these principles are indispensable toward rationality. This leaves us with (3).
Imagine the lines of evidence both for and against a beginningless universe. Further, imagine that the evidence on both sides is inexhaustible. We can represent the evidence for a beginningless universe with the set of all odd numbers {1, 2, 3, . . . n}, and the evidence against a beginningless universe with the set of all even numbers {2, 4, 6, . . . 2n}. This scenario should be expected, given the existence of infinitely-many actualities.
But if this is the case, how can the probability of either hypothesis be reasonably assessed? For every line of evidence for a beginningless universe, there is an equally strong line of evidence against it. Yet, we all know that the probability of a hypothesis can be reasonably assessed. If this is correct, then it follows that it is inconsistent with probability for there to be infinitely-many actualities to take into account. Because of the epistemic warrant for probability, then, it seems that the rejection of infinitely-many actualities, and therefore a beginningless universe, is preferable.
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