William Lane Craig

I have collected here a small source of some responses to Criag's material. If one would be interested in reading this collection, follow the links.

Firstly, here is a Counter-Apologetics wiki. It has many responses to common arguments from apologists.

Here are a few things pertaining to Craig:

• After a debate with Craig, Physicist Lawrence Krauss wrote about what he thought of their encounter.
• To be continued....

Kalam Cosmological

This is a quick resource of material on the Kalam Cosmological Argument, popularized by William Lane Craig.

1."The Kalam Cosmological Argument" by Dante Alighieri, 2008

Authors Hidden text
The last thirty years has seen a resurgence in the cosmological argument, with various new versions, both causal and explanatory, being offered across the philosophical landscape. Of particular interest is William Lane Craig's particular kalam cosmological argument, which derives from modifications to the ancient argument by medieval Muslim mutakallim engaged in kalam (speech), in this context, philosophy. The argument goes simply as (1) Whatever begins to exist had a cause (2) The universe began to exist (3) The universe had a cause. Craig argues extensively for (2), somewhat neglects a rigorous defense of (1), and argues from (3) that the cause is God.

The author conducts a comprehensive critique of Craig's argument, ranging from his views on (1), to his support for (2) and the alleged relevance of (2), up to Craig's arguments that the cause is God. The author also examines Craig's views on cosmology and his liberal use of intuition

The Kalam Cosmological Argument

I would like to open a discussion with respect to the argument. Now, in my sincere opinion, it does not prove that God exists. What follows is a summary of my views on the argument.1. Causation
The first premise of the Kalam Cosmological Argument (henceforth KCA) discusses the notion of a causal principle. I intend to examine it, as well as the intersection of causation and physics. Strangely enough, this particular premise is rarely objected. The only three prominent philosophers I have known in the literature to write about this subject are Wesley Morriston, Adolf Grunbaum, and Quentin Smith. In my opinion, they miss a number of crucial points, points I wish to address below.

1.1. Time

The first crucial notion that we must discuss the notion of causation that W.L. Craig employs. Let us consider the causal principle (henceforth CAP) that Craig sets out:

Whatever begins to exist has a cause of its existence.

Craig offers three general lines of argument to support this thesis, namely that (1) it is constantly confirmed empirically (2) it is supported by the axiom ex nihilo nihil fit and (3) Johnathan Edward's argument shows that it is inexplicable that everything does not come to exist uncaused if uncaused events can occur.

Also, crucial to note is the definition of "begins to exist." Craig typically takes it to mean "x begins to exist iff (i) x exists at an interval t (ii) there are no intervals of time t' such that x exists at t' and (iii) t'> t." Of course, this leads to a few questions.

Under Craig's relativistic cosmology, time had a beginning (or at least a first interval) at the Big Bang Singularity. Now, if God existed at the first moment of time, since there are no times prior to which God exists (because there are no prior times), it follows that God begins to exist. This is problematic for Craig for this entails that God has a cause of His existence.

So, Craig amends the definition to "x begins to exist iff (i) x exists at an interval t (ii) there are no intervals of time t' such that x exists at t' (iii) t'> t and (iv) there are no timeless states of affairs involving x's existence." Since Craig argues for a timeless God, this frees God from beginning to exist.

Moreover, since Craig regards time itself to have a beginning, and hence a cause of its existence, He supports timeless causation. More importantly, he supports simultaneous causation, in which cause and effect occur at the same time.

With these definitions in hand, we can now proceed to examine the problems with this CAP.

The problem with the CAP is that it is simply incoherent. Note that by the definition of "begins to exist" a topology of time involving a first interval of time would "begin to exist" and require a cause i.e. it would require that timeless causation is coherent. Moreover, Craig supports the idea of simultaneous causation.

Now, the question is whether or not either of these is really coherent. Let us examine the latter first. Now, what is time? Typically speaking, time is defined as, or at least entails, an ordered sequence of states of affairs by the dyadic relata before, simultaneous with, and after, along with the monadic irreducible properties past, present, and future. Craig agrees with this understanding of time as he is a relationist i.e. he believes that the first moment of time was the first event of time. So, we have in effect an event sequence.

Now, a number of points arise. First, Craig cannot coherently support the idea of simultaneous causation with a timeless cause (God), because simultaneity is an explicitly temporal relation. Now, simultaneous causation is impossible due to the nature of causation. First, causation explicitly involves the notion of priority and asymmetry, although Craig denies that the priority and asymmetry is necessarily temporal. Let us operate under that assumption; and it is certainly true that the priority and asymmetry of causation is at least ontic.

Now, causes are ontically prior to their effects. That is to say that causes bring about their effects; the effects come into existence. In other words, the effect does not exist and then exists due to the cause, because the cause brings it about or actualizes the effect. So, cause and effect are related as such: At the causal position, the effect does not exist, but the cause exists. At the effectual position, the effect exists. In other words: {C&~E, E}. So, we have an ontic ordering as such. So, causation explicitly involves the notion of change, of moving (if we wish to speak Thomistically) from potentiality to actuality.

Now, this cannot occur in the space of a single moment of time, for that would mean that the causal position (CP) and effectual position (EP) are simultaneous. But, the causal position and effectual positions are incompatible: in the former, the effect does not exist, and in the latter, the effect does. So, we are left with the contradiction (E & ~E). In other words, change cannot occur in a single moment.

So, simultaneous causation is impossible. This is true by definition since time involves, as we said earlier, sequenced states of affairs. So, by definition, the ontic priority of the cause is temporally prior, since it too involves a sequence of states of affairs. So, simultaneous causation cannot occur.

Craig's examples of simultaneous causation betray this confusion. For instance, balancing a pencil on one's hand does not constitute simultaneous causation due to that each state is continuously changing to the next state; and that the impression is brought about in the open interval succeeding the balancing of the pen. There is an interaction of particles here, one that cannot be simultaneous, because that would require a change in a single moment of time. If time is continuous and the cause happened at t and the effect concluded at t', it may be said that the effect occurs from the interval (t, t'].

Moreover, actually timeless causation is impossible too. If God created the universe, and time itself, then the act of creation was timeless. For God's creating the universe is causally prior to the universe and time itself; therefore, the act of creation was timeless. First, by definition, time involves the event-sequence as we pointed out and since causes involve such event-sequences, causation is by definition temporal. Moreover, since the causal and effectual positions are incompatible, neither could ever be timeless. If one was timeless and the other temporal, or even both timeless, nonetheless, both would be actual and there would be a contradiction. Suppose that the EP was temporal. Corresponding to the existence of either the EP is the proposition expressing its actuality, p. Now, if the EP is temporal, then it exists in a specific interval or space of time. The proposition expressed by this state of affairs is, "For t, p." Now, suppose that the CP is timeless. Timeless states of affairs always exist, so propositions expressing their existence are always true (but they don't have truthmakers in time). So, since the CP is incompatible with the EP, corresponding to the CP is "For all t, ~p." Now, since something that is true for all times is also true for a specific moment or interval of time, it follows that "For all t, ~p" entails "For t, ~p."

But, given the propositions for the CP and EP, we get, "For t, (p & ~p)." So, timeless causation entails a contradiction and hence cannot occur.

At the end, note that Craig's definition of "begins to exist" entails that time itself is caused; which entails either simultaneous or timeless causation. But, since both are impossible (and time itself cannot be caused), it follows that the CAP is actually false and the KCA is not a cogent argument.

What would actually be a causal principle for a coherent theistic creation scenario, and a less objectionable principle, would be the notion that whatever "comes into existence" has a cause. For in "begins to exist," it entails that time itself is caused, because there are no prior moments of time at the beginning of time. But, in "coming into existence" it explicitly involves the notion of there existing prior moments at which the object did not exist i.e. it moved from potentiality to actuality. So, then, all things that "come into existence" could be caused (in a modal sense), and the theist might argue that they actually are caused.

Unfortunately, the universe did not come into existence, under Craig's relativistic cosmology, because time had a beginning at the beginning of the universe. So, if Craig's cosmology is correct, the universe is uncaused and God could not have created the universe. Since the definition of "God" involves the notion of "creating the universe," it follows that that the KCA strictly disproves the existence of God.

A timeless entity cannot interact causally for causal interaction explicitly involves change, which a timeless entity cannot do. For instance, a timeless entity must move from a state of not acting to acting. Even if we suppose that it was always acting, this raises the same problem as mentioned earlier. For its action brings about an effect, and when the effect is brought about, it ceases to bring about the effect. So, in effect, any causal entity must move necessarily from acting to not acting when producing its effect. So, when the effect is brought about, there is a proposition expressing its existence such that "For t, it is true that the cause is not acting"; but, if the act is timeless, then it follows that "For all t, it is true that the cause is acting." This entails that "For t, it is true that the cause is acting." But, this conjunction of facts entails that " For t, it is true that the cause is acting and not acting." So, timeless action entails a contradiction identical to the one we examined earlier. Clearly, causal structures change in interaction, as this illustrates.

This also demonstrates the untenability of Craig's strange idea of "relative" timelessness. Craig believes that God was timeless sans the universe; ontically prior to the universe, God was timeless. But, upon the universe's creation, God became temporal. Craig offers this strange doctrine in an attempt to reconcile his views on the KCA and relativistic Big Bang cosmology (which require timeless causation) and his views on time, particularly A-theoretic presentist frameworks (which require God to be a changing entity; for instance, God's knowledge of tensed facts continually changes). Of course, this is nonsensical. A timeless entity cannot change. If God became temporal, that means He changed from being timeless to temporal. But, a changeless entity cannot change; the moment we have a proposition of the form "it is true that x is timeless" it follows that "For all t, it is true x is timeless." A timeless entity always exists, so propositions expressing its existence and timelessness are omnitemporally true. The point is that we have object that has this property timelessly. Obviously enough, a timeless entity is timeless timelessly. Therefore, this entails the proposition that "For all t, it is true that x is timeless." So, God can never be temporal if He is ever timeless. (Of course, God can't be timeless either for the reasons stated above). Since timelessness is the lack of event-sequences and hence, the lack of change, it follows that timeless entities cannot change; so a timeless entity can never be temporal or place themselves in time. Craig's doctrine of relative timelessness is incoherent.

Of course, perhaps our interlocutor may object that it is not the case that God may never become temporal, for "never" is used in an explicitly temporal sense. Yet, God was initially timeless, so perhaps this move is unwarranted.

This objection is multiply confused. First, to state propositions of the form "For t, p" is to not state the truthmaker of the proposition is temporal. For instance, the following proposition is true: "For t, it is true that numbers exist." And yet, the truthmaker of the proposition is timeless (if we accept Platonism). What the proposition expresses is not that the object is temporal, but when a proposition is true. So, such explications are talking about the truth-value of a proposition. Moreover, recall that timelessness entails changelessness. Our interlocutor has failed to sufficiently answer that objection. Moreover, if our interlocutor insists nonetheless, the above objection can be restated in terms of ontic sequencing. And, in any case, an ontic sequence is a temporal sequence (because of the arguments above). Propositions about timeless entities are going to be true or false at a time, and indeed, for all times.

So, we may conclude that the intersection of the causal principle as interpreted by Craig (which involves the notion of "beginning to exist" rather than "coming into existence") and relativistic cosmology (which is necessary to conclude that this section of space-time had a first interval of time) entails timeless causation. Since timeless causation is impossible and the conjunction of the CAP and relativistic cosmology is necessary for the cogency of the KCA, it follows that the KCA is not a cogent argument. It is unfortunate that this line of reasoning is not found in critiques of the KCA typically speaking.

1.2. Support for the causal principleAlthough we have already shown the KCA to not be cogent, I wish to go for "overkill", so to speak, to show just how many egregious errors Craig has made.

So, let us examine the three lines of evidence Craig supplies for the CAP.

1.2.1. Empirical confirmation

Craig claims that the CAP is continually confirmed in our experience. But, is this really so? Let us examine this claim.

First, what actually appears to be confirmed macroscopically in our experience is that all things that come into existence have a cause of their existence. But, recall that the beginning of time does not come into existence. Therefore, our intuitions regarding causation cannot apply to the beginning of time, since time does not come into existence.

Second, all caused events we observe occur within the natural order. This provides an inherent context for occurrence. This is why, for instance, we would disbelieve the idea of "tigers popping into existence." But, there is no background context for the natural order itself. Causation, empirically, has been confirmed within the natural order and in its background context; but the natural order lacks a background context. So, we cannot transfer our intuitions with respect to causation within the natural realm to the natural order itself.

Third, our experience confirms that caused events occurs in time and involve the rearrangement of energy from one form to another. But, this is clearly not the case of a timeless entity bringing about the universe, so, our intuitions cannot carry over to the conjunction of the CAP and GTR-cosmology.

Fourth, this could hardly indicate that causal principle (either in its incoherent or in its improved formulation involving "coming into existence") is true for all possible worlds i.e. it is a necessary truth. Empirical generalizations, at best, establish a physical necessity, not any sort of logical necessity. Of course, Craig has a ready reply to this. He insists that the causal principle is metaphysically necessary and points out Kripkean/Putnamian strong a posteriori necessities i.e. "Water is H2O."

Of course, this move is highly objectionable. First of all, Craig cannot simply handwave in the direction of a posteriori necessities but he must actually defend them and show them to be an acceptable metaphysics of modality. A posteriori necessities are highly controversial in modal philosophy in contemporary times. Moreover, it seems unclear that (a) that there is such a class of modality entitled "metaphysical possibility" and (b) that a posteriori necessities express anything more than semantic manipulation.

First, what is metaphysical necessity in the first place? Typically speaking, a modality is defined with respect to some standard. For instance, physical possibility is construed relative to the actual laws of nature. But, metaphysical possibility is not construed with respect to any standard. So, one is struck by the obscurity of the claim that something is "metaphysically necessary." Just what does it mean for something to be metaphysically possible?

Typically, an attempt at ostensive definition is attempted. For instance, Plantinga has written that "all bachelors are married" and "2 + 1 = 7" are metaphysically impossible, yet not logically impossible. Apparently, first-order logic does not allow one to conclude the impossibility of either. And yet, each is really impossible. I believe this analysis is mistaken. "2 + 1 = 7" and "all bachelors are married" are logically impossible. The reason is that both involve a definitive contradiction. In the former, to be a bachelor means to be unmarried. Therefore, to assign the property "married" to a bachelor is to speak of a man who is both married and not married. The same applies with the latter: given the axioms of Peano Arithmetic, 2 + 1 = 3, per the successor function. To deny this is to invoke a contradiction under Peano Arithmetic. In each case, we observe an actual contradiction, once we insert actual premises under a logical apparatus. So, these sorts of propositions are logically impossible, which invites the question as to how metaphysical possibility is defined. Indeed, all of the ostensive examples offered involve contradiction; but, the case of the causal principle does not invoke a contradiction, so its negation cannot be taken as being impossible.

Second, it is doubtful whether a posteriori necessities represent a meaningful modal metaphysics. The intuition behind a posteriori necessities is that there are true expressions that are synthetic, in which the predicate is not simply contained in the subject, and yet cannot be false. A popular example is that "Water is H2O." Now, I contest that this is really necessary. For here, we either have a trivial proposition or semantic manipulation. First, how do we establish that this is necessary? What Kripke has us do is first to rigidly designate water i.e. we take the stuff we call "water" and refer to that and that alone. Now that we are referring only to that stuff, we inquire as to its nature. But, its nature has to be discovered empirically, so we can establish necessary truths a posteriori. This is flawed. What we are really doing here is taking the referent we have for "water" and assigning as its properties "whatever properties it actually has." Of course, under this, it is true that "water" is H2O. But, note the triviality of it all. The referent that is referenced is only the one of the actual world; so, it is trivially true that whatever actual properties the referent has it actually has. A more credible example can be taken in the instance of origin essentialism. Consider a person. Now, suppose we define that the person according to whatever properties they actually have. One such property is that they are evolution of the union of a particular egg and sperm. So, per origin essentialism, anyone that is not the union of those particular sex cells is not that person. This is flawed because necessity here scopes over all possible worlds. But, the actuality operator is indexical; in different possible worlds, that is the actual world (not to say that modal realism; rather that if a different possible world existed, then it would actual i.e. actuality is indexical). It shifts reference from world to world. It hardly means that the person in question is the result of said particular sex cells in all worlds, but that the person in the actual world is the union of said particular sex cells. In other words, the modal scope of a posteriori necessities is none other than the actual world! Clearly, whatever properties x actually has, it actually has, but that does not say anything about any such properties being essential to x, unless we trivially redefine x as referring to these actual properties. In that instance, we have mere semantic manipulation. For instance, consider the origin of the universe. The actual universe had a particular origin, but this origin is hardly necessary, unless we redefine "universe" to include the property-relation. So, a posteriori necessities are not profound, but trivialities. And if one cannot say anything profound save without semantic manipulation. Or more relevantly, consider things that "begin to exist" or "come into exist." Suppose we find that all of them have causes. We could take this to say that "all things that begin to exist have a cause" as being necessary; but given the indexical actuality operator, all this indicates is that all things that begin to exist in this actual world actually have a cause. So, any sort of profound metaphysics to be gained from a posteriori necessities is an illusion. It is little more than semantic manipulation.
Furthermore, Kripkean a posteriori necessities invariably involve the question of naming and reference: if “watery stuff” exists at world w and is composed of XYZ, not two hydrogen atoms and one oxygen, is it water? Nay, says Kripke. It is necessary, according to Kripke, that water is composed of two hydrogen atoms and one oxygen atom, but that doesn’t mean that it is necessary that water exists. Kripke’s statements are intended to apply to names, using a baptismal/causal-theory of reference. We call some stuff a particular name and hold it fixed; so our semantics apply across the entire modal landscape. Craig has not shown such a similarity with respect to the causal principle.

Fifth, it might be pointed out that causal principle is actually not metaphysically necessary. It is constantly disconfirmed in the realm of quantum mechanics. Most interpretations show quantum phenomena to be undetermined and uncaused. For instance, radioactive decay and spontaneous virtual particle formation. Given violations of Bell's inequality and given the no-communication theorem, it follows that such events are uncaused. Nonetheless, there are deterministic interpretations of quantum mechanics, but it remains that the most accepted notion of quantum mechanics involves indeterminism. This, at least, provides evidence that the causal principle is not necessary.

Craig has an interesting reply with respect to quantum mechanical events. Discussing virtual particle formation, he states that although there are no sufficient conditions for their existence and formation, it remains that there are a number of necessary conditions and hence, cannot be considered uncaused. This view, is in my opinion, false. First, this contradicts Craig's view of "mechanical causation" in which the cause is sufficient for the effect. Secondly, it is unclear that effects without sufficient conditions and yet necessary conditions are "caused." Note that crucial to the notion of "causation" is that the cause actualizes the effect, it brings it about. Yet, what could be said to bring about the virtual particle? If the prior conditions are absolutely compatible with or without the existence of the virtual particle (that is, it is not a matter of epistemic failure, but that ontically, either is compatible), then what brought it about such that it existed? Clearly, the state of affairs consisting of the virtual particle's forming has no cause of its existence, regardless of what necessary conditions there may be. So, it is insufficient to point out necessary conditions.

1.2.2. Axiomatic confirmationAccording to Craig, the axiom ex nihilo nihil fit confirms the causal principle's necessity. In English, this translates to "From nothing, nothing comes." Presumably this is taken to support the causal principle.

But, this is contestable. First of all, it is obscure as to how the axiom supports the CAP. How are we to interpret the axiom in a manner such that a denial entails an impossibility?

Ultimately, this line of support largely amounts to handwaving. Apparently, the reasoning goes as thus: There was nothing prior to the universe. Originally, there was absolutely nothing: no states of affairs whatsoever. Afterwards, the universe came about. Clearly, this is absurd, so we are justified in foregoing an elaborate defense of the causal principle.

This is confused. Note the terminology used here. Namely, that there was some original state in which nothing at all existed. Afterwards, the Big Bang occurred. So, we have an event-sequence of nothing followed by the Big Bang. This is mistaken. First of all, to say that there was nothing prior to the Big Bang does not mean that there was some esoteric state of absolute nothingness prior to the Big Bang, but that the Big Bang is the first moment of time; there were literally no earlier moments of time or anything at all. Moreover, absolutely nothing does not even contain time; so how could it originally be a state and then change? Furthermore, absolutely nothing is not a state of affairs and hence there could never have been an ontic point or a time at which nothing existed. So, the entire scenario Craig draws simply does not exist and does not support the causal principle. This doesn't even apply to temporally embedded events, for such events, by definition, are preceded by other events. So, prior to such uncaused events, there is not "absolute nothingness" from which to come from. Uncaused events simply lack sufficient causal conditions for their existence, not a state of prior "nothingness." Even if we take "nothing" to signify "no concrete objects whatsoever", "nothing" could still not be a moment of time, since temporal states of affairs are concrete, insofar as they involve change. But, if no concrete objects exist, then there are no states of affairs to be sequenced; hence, "nothing" under this definition could not ever be a moment in time. In any case, while "nothing" under this definition is possible, "absolute nothingness" is clearly impossible, for there would not even be Platonic abstracta, true or false propositions, or anything at all.

And clearly, this interpretation of ex nihilo nihil fit is what Craig takes to be correct. And yet, this does not support the CAP. Now, suppose we interpret the axiom tenselessly, even though Craig does not make this move. We could express it such that "If nothing existed, then nothing could exist." However, "could" expresses a modality, presuming a necessity. But, what is the scope of necessity operator? On the wide scope interpretation, the resulting proposition is that "Necessarily, if nothing exists, then nothing exists." But, this is a trivial analytic truth and would be true even if uncaused events existed. On the narrow scope interpretation: "If nothing exists, then necessarily nothing exists." But, this is hardly true. If nothing actually exists, then it is not the case that nothing exists in all possible worlds. Furthermore, if we take "nothing" to signify "absolutely nothing," it is arguable that "absolutely nothing" is impossible (as per above); but if so, any conditional such that "nothing" is the antecedent is a counterpossible, and from an impossibility, anything at all is derived. (This is due to the principle of explosion). Also, "nothing" is not a condition on anything; it is not some strange sort of something that can prevent things or make things occur.

So, what interpretation of the axiom could promote the CAP? Perhaps this: "If nothing has a cause, then nothing begins to exist." This does entail the CAP, but that is because this is the mere contrapositive of the CP and trivially entails it. So, it is a mere assertion of the CAP. So, we find that the axiom does not actually support the CAP at all.

1.2.3. Metaphysical confirmationCraig also offers a metaphysical line of reasoning to support the necessity of the causal principle. He quotes Jonathan Edwards, who argues that if uncaused events can exist, then how is it that anything and everything does not come into existence uncaused? It seems inexplicable.

There seems to be a pretty simple answer for this, although the issues run much deeper. Namely, that there are physical laws, that is regularities which explain and establish causal connections between specific phenomena, and that there are regularities that establish boundary conditions on uncaused phenomena. For instance, your releasing a ball, in conjunction with gravitational acceleration, entails that the ball falls. So, natural law entails that there are sufficient conditions for various effects, and these sufficient conditions exist in the universe. Moreover, uncaused phenomena are constrained by boundary conditions and natural law. For instance, virtual particle formation has particular rules governing it, but that does not mean that the particular formation of a particular virtual particle is caused.

Now, the deeper issue is that this seems to be inexplicable. Namely, why is it that the laws of nature constrain uncaused phenomena as such? But, that would seem to be simply asking Why theses laws of nature? I contend that there is no answer and that the laws of nature are brute. The point is that we are trying to explain something here. But, we explain by pointing out regularities, how something is sufficient for something else. However, our explanandum here is all regularities themselves. But, this is surely mistaken. We can explain regularities with deeper regularities, but ultimately, we cannot explain the existence of regularities at all. To explain the existence of regularities at all, the explanans would be explanatorily prior to the explanandum. Hence, the explanans of any regularity at all cannot be a regularity. But, if there is no regularity at all, then the explanans is not an explanation, since all explanations employ regularities! Hence, the existence of any regularity at all is a brute fact. More importantly, this "explanation" performs none of the marks of a true explanation: that is simplifies our data set and predicts new phenomena. Ultimately, this "explanation" doesn't really explain why things are thus and not some other way. The sum of contingents facts is actually contingent; in other words, it is otherwise in alternate possible worlds, and unless we invite modal collapse, there is no explanation for why some possible world is actual rather than another. That's just the way it is.

What this issue really delves into is the validity of the Principle of Sufficient Reason. But, the KCA's supposed strength is that it does not rely on the Principle of Sufficient Reason, a rather esoteric principle that yields controversial results. Indeed, we have reasons to believe that such a principle is incoherent. I argue as much in the following under my alternate handle "rayndeon":
Allow me to attempt to strengthen Peter van Inwagen's fatal argument. I was originally going post the following on Victor Reppert's blog, but, in any case,

Allow me to summarize what I construe as being the strongest objection to any version of the principle of sufficient reason (which holds that necessary facts can explain contingent facts), which Peter van Inwagen raises in An Essay on Free Will (Oxford: Oxford University Press, 1983) pp. 202-4 and can also be found in Metaphysics: Second Edition (USA: Westview Press, 2002) pp. 119-22. Consider the cosmos, which is the set of all contingent state of affairs (and is itself a contingent state of affairs). Now, the explanation of the cosmos is going to be either necessary or contingent. If it is necessary, since the explanation entails the explanandum (recall that it is the Principle of Sufficient Reason, moreover that if the explanation does not entail the explanandum, then what makes the explanandum actual?), it follows that the explanandum is necessary. But, the cosmos is contingent, for the very definition of the cosmos is the set of all contingent states of affairs, which is itself contingent. So, if the explanation entails the explanandum, and the explanation is necessary, then the explanandum is necessary. But, if the explanandum (the cosmos) is contingent, in which case it is not-necessary, it follows by modus tollens that the explanantion is not necessary. But, the explanations is necessary, which is a contradiction. Furthermore, necessary states of affairs are self-explanatory: they must be actual, hence, they are actual. It is difficult to see how necessary SOAs can be explained since nothing makes necessary SOAs actual. This is especially true if construe non-self-explanatory explanations as being plausibly interpreted as being causal ( it seems that explanation consists of figuring out what *makes* a state of affairs actual, or what is the truthmaker of a proposition. The same definition seems to be clear of the word "cause."); however, necessary SOAs cannot stand in causal relations, due to causal asymmetry, for given causal priority, if we construe events in some chain of causation, at the "placeholder" wherein cause exists, the effect does not exist. But, necessary SOAs can never fail to be actual, hence, necessary SOAs cannot stand in causal relation, and hence, cannot be caused. So, we have two good reasons to deny a necessary explanation of the cosmos.

What if the explanation is contingent? The problem is that any contingent explanandum whatsoever must appeal to contingent explanations that are neither parts of the explanandum nor entailed by the explanandum and it cannot be the explanandum itself. The explanation cannot be a part of the explanandum nor entailed by the explanandum (if the explanation is a part of the explanandum, then the explanandum entails the explanation, for the actuality of the explanandum entails its parts, which therefore, if the explanation is a part, entails the explanation). First, that is not what is meant by "explanation" since to say that something is explained is to say that there is some SOA such that given a different SOA, the latter *makes* the former actual. We cannot, for instance, consider the fact that a car blew up last Saturday explains the fact that the car blew up last Saturday and the smoldering remains fell into a lake. Secondly, if the explanandum entails the explanation, then the explanations explains itself since the explanandum entails the explanation for that the explanation is a conjunct of the explanandum, and explaining a conjunction explains the conjuncts, therefore, the explanation explains the explanation, which I address in below regarding where or not the explanation can be the explanandum. Moreover, the explanation is prior to the explanandum; but, if the explanandum entails the explanation, then the explanandum must exist in the same "placeholder" as the explanation; but if the explanandum is actual, then the explanantion cannot explain the explanandum for the explanandum is already actual then. Therefore, the explanandum cannot entail the explanation. If the contingent explanation is the explanandum, then it follows that explanation and explanandum are one the same; but this cannot be true, for no contingent SOA can be self-explanatory, since nothing makes self-explanatory SOAs actual (for they must be actual), but contingent SOAs do not have to be actual. Furthermore, explanations are prior to the explanandum. But, if the explanation is the explanandum, it follows that the explanation is prior to itself; that in the "placeholder" wherein the explanation is actual, it is not actual, which is a logical contradiction, and therefore, the explanation cannot be the explanandum.

So, no contingent explanation of the cosmos can be the cosmos nor a part of it. But, then, it follows that the cosmos necessarily has no explanation at all for there cannot be any necessary explanations at all, and any contingent explanandum is to be explained by a contingent explanation that is neither entailed by the explanandum nor identical to the explanandum; but the cosmos entails any contingent SOA whatsoever since it is the set of all contingent SOAs and there is nothing contingent outside of the cosmos; therefore, the cosmos is necessarily a brute fact. In any case though, it seems pretty clear even without van Inwagen's objections that it is not entirely clear why precisely we should hold that there cannot be brute facts. First of all, empirical generalizations leading to the PSR are generalizations within the cosmos, so it is difficult to see how this entails application on the cosmos itself. And if we seriously take the Copenhagen interpretation of quantum mechanics, then there are brute facts; for instance, that a specific atom decays at such and such time is a brute fact, under the Copenhagen interpretation. Moreover, it is difficult to see how the PSR could apply to eternal SOAs, since explanation is prior to the explanandum; but there is nothing prior to eternal explanandums, so, it is plausible to regard eternal explanandums as necessarily being brute facts, and the cosmos as also necessarily being a brute fact.

Typically speaking, there are two types of objections to van Inwagen's thesis. The first is a denial that explanations entail their explanandum. The second is an appeal to libertarian free will. What is to be made of the first objection? Well, first of all, it does not seem that it is entirely consistent with the PSR, since the PSR is a principle of sufficient reason, and moreover, that the principle consists of finding the complete explanations of things. Futhermore, it is difficult to see how explanation cannot entail the explanandum, for if it does not, then what is the explanation explaining? For the explanation makes the explanandum actual, so, saying that the explanation does not entail the explanandum is to say that the explanandum is in some sense unexplained. For if in some logically possible world we have the explanation C1 and the explanandum E1 such that C1 is actual and E1 is non-actual, then what makes the explanandum actual? It seems that the actuality or non-actuality of E1 turns out to be a brute fact, which is incompatible with the PSR. Furthermore, suppose we insist that there is a partial explanation for every contingent SOA. First of all, what counts as a partial explanation to begin with? This immediate restriction looks too ad hoc, but I suppose that is merely my perspective. Moreover, to say that an explanation is partial seems to be saying that given some contingent SOA that corresponds to some conjunction [(p & q) & r], one partially explains it by explaining p. But, plainly, we have the same problem as above; for then, what makes (q & r) true at all? Also, doesn't seem clear that explaining p entails p to begin with? Perhaps we wish to insist that other explanations explain (q & r); but, wouldn't those explanations in turn entail (q & r)? Moreover, if we take all those explanations together as one conjunction, surely that explains the original conjunction. It also can't be that we appeal to that there must be some explanation for [p & (q & r)], since while it is true no particular explanation would entail that conjunction, each particular explanation would also be contingent, which is not what the PSR is to show in the first place... Moreover, perhaps we can construe complete explanations such that it is true that [(p → q) & p]; plainly, in all possible worlds wherein this holds, q is true. So, this entails q, so, it is necessary that given [(p → q) & p], q is true.

The second objection is an objection from libertarian free will. It insists that there is some necessary being such that the necessary being created the cosmos, but that necessary being is insufficient for the cosmos' existence, for agent-causes do not entail their effects. I think this is a confused thesis. First of all, the mere existence of the necessary being is not the explanation of the cosmos. Indeed, when we cite that an agent was responsible for some SOA, we do not cite their mere existence, but that the agent did certain things that then in turn brought about the SOA in question. In this case, the explanation in question is in reality the necessary being willing the cosmos into existence. Perhaps we can picture this necessary being saying "Be!" or something like that, and the cosmos comes into being. But plainly, the necessary being's willing the cosmos is sufficient for the existence of cosmos' existence. But, that is the explanation of the cosmos, not the mere existence of the necessary being. And showing that the cosmos has a necessary explanation is to appeal to an explanation like that. So, it seems that one must admit to that the necessary being wills this cosmos is a contingent SOA (this makes it susceptible to the above arguments against any contingent explanation of the cosmos). However, this explanation is going to be a brute fact of the world since per libertarian free will, that any agent does such and such thing is causally undetermined, and hence, that a necessary being willed the cosmos into actuality is a brute fact. But, this seems to be at odds with the PSR, which would eliminate brute facts. Moreover, why find this brute fact preferable to the brute nature of the cosmos? We have derived some speculative explanation such that that explanation has no explanation at all. But, how is this better than the cosmos has no explanation at all? This is of course all prior to whether or not libertarian free will is a true theory of freedom, which I do not think that it is.

So then, given Peter van Inwagen's objections (with a few extra additions), I find the PSR to be wholly false and construe it as necessary that this cosmos is a brute fact."

Moreover, there is the questionable contention that there can be any concrete necessary being at all. I briefly argue for this at IIDB - For Dean, the Modal Ontological Argument:

"Furthermore, God's necessity seems to indicate a contradiction. Given a certain type of compatabilism, [4] given a certain circumstance, in all possible worlds wherein that circumstance obtains, a person does the exact same thing every single time. So, we have conditional propositions such that

(18) Given a circumstance C and abilities A, a person P will actualize a state of affairs S.

Moreover, these conditionals are grounded in who P is, which is to say that the maximal set of these conditionals (as I detail in the fourth footnote) is individuative; it marks a distinct person. Now, in God's case, the problem is that the exact same C obtains in all worlds in which He exists: He is prior to non-eternal entities. If we include the compatibilist account and cite that the type of God in question is a God who would create a particular cosmos in response to the situation, it follows that in all worlds in which God exists, the exact same cosmos obtains; but if God is necessary, then the cosmos is necessary. But, this means that there is exactly one possible world. But, there can't be one possible world because other worlds are possible. Therefore, the necessity of God (and compatibilism) entails modal fatalism. Moreover, it seems that necessary states of affairs cannot be caused to exist, and furthermore since cause entails effect, it seems that there cannot be any necessary causes either. [5]

Finally, I would offer the same objection that Jade raised, that the ontological argument does not establish the existence of an individual but of a class of entities. From a previous post:

I do not think you grasped my point entirely. I apologize for my ambiguity. Allow me to be more explicit. In order to show that God the concrete entity is necessary, you have to establish all individuating aspects of God as necessary, otherwise God, the individual will be contingent.

This situation is pretty similar to the contingency of the actual world. [Clearly, an actual world must be exist, but no particular actual world must be exist] The necessary parts of the actual world exist in all possible worlds, so that shall not concern us. What makes the actual world contingent is the contingent states of affairs present. No particular contingent states of affairs had to be present, but some contingent states of affairs had to be present. Look: for each contingent SOA, there is some proposition p. Qua contingent states of affairs: ◊p & ◊~p. Now, either p v ~p. Exactly one conjunct is going to be true and it is necessary that exactly one such conjunct be true. But, no particular conjunct need be true, hence, the SOA in question remains contingent even though there must be some actual SOA. So then, the actual world in question is contingent, even though, there must be some actual world since (a) the necessary facts obtain in all possible worlds and hence obtain in an actual world and (b) it is necessary that for every contingent SOA, it is either actual or non-actual, per the above.

Now, we have the same sort of case with God for at best we have established (if the ontological, cosmological, and conceptualist arguments work) necessary conditions for His individuality, but not sufficient conditions. So, at best, the ontological, cosmological, and conceptualist arguments establish that of the universal "God," it is necessary in all possible worlds that there be some particular that instantiates this universal, but there is no necessity of that there be an individual that is necessary; therefore, this arguments fail to establish the necessity of God the individual or even God the person.

So, I do not regard the ontological argument a good argument at all. And, if anything, it is impossible for there to be any necessary concrete entity whatsoever."

...

A brief edition: I accidentally made the statement "it seems that explanation consists of figuring out what *makes* a state of affairs actual, or what is the truthmaker of a proposition." What I actually meant is that explanation (of a non-necessary entity) consists of figuring out what makes the SOA or the truthmaker of the proposition actual i.e. one does not explain a contingent proposition by pointing out the truthmaker of the proposition, but by explaining the truthmaker of that proposition.
So, we must regard the Edwardian argument as failing to support the CAP.

1.2.4. What is left for the causal principle?Given that the three lines of evidence Craig employs have failed, we now come to the question of whether or not it is true. Now, under the "begins to exist" formulation, it is false because it is incoherent, as I pointed out in 1.1. The "comes to exist" formulation is more coherent, but it is not metaphysically necessary for the reasons already stated and it is likely false given most interpretations of quantum mechanics. Hence, given that Craig has a false premise in his argument, the KCA is not a cogent argument and fails to deduct its conclusion. Indeed, what is interesting to note is that only under the "begins to exist" formulation would the universe have a cause. But, that formulation is incoherent. But, the "comes to exist" formulation does not apply to the universe, because the universe did not come to exist, as I explicated above. Hence, since the former formulation is incoherent and the latter formulation is irrelevant, we must assuredly come to the conclusion that under relativistic cosmology, the universe is actually uncaused. If anything, the arguments against infinities are irrelevant. The universe, since it exists for all times, is eternal.

Actually, Craig objects to this sort of move, saying that the universe began to exist, although he accepts that the universe existed for all times. Here, one is puzzled by such a claim. It is true that the universe had a beginning, but what is crucial to note is that it did not come into existence; but this entails that the universe is actually uncaused, if it began to exist yet did not come into existence, being the beginning of time itself. Since universe is uncaused and exists for all times, this certainly qualifies as being eternal, wherein something is eternal iff it has always existed.

Consider a neutral formulation of "always existed" such that x always has existed iff the proposition "x exists" is true for all times t. This covers objects with infinite durations in both past and present, timeless objects (for any timeless object x, any proposition that expresses its existence is true for all times), and, of course, our present situation. So, we reasonably may conclude that the universe has always existed, particularly under Craig's cosmology. For the point is that throughout the entire event-sequence (i.e. time itself), the universe has existed. It does not matter as to the topology of this event-sequence, but merely that the universe has existed throughout its entirety. How could the topology of time affect eternity here?

2. Infinity
Craig has also given two lines of argumentation against the existence of an infinite past. There is also an interesting story from Bertrand Russell that Craig employs. Of course, a foray into this will be a largely mental exercise, since even a finite universe can be eternal and fail to come into existence, as I pointed out earlier. So, his arguments to this regard are largely irrelevant. Nonetheless, given the preponderance of appalling mathematics, I feel compelled to point out the errors.

2.1. The general argument against actual infinityThe first line of argumentation against an infinite past is a denial of the existence of an actual infinite, that is, the existence of any concrete object or state of affairs that exemplifies the property of being infinite. Craig offers a general line of argumentation against any actual infinity and he argues that such infinities are, supposedly, metaphysically impossible. What is interesting is that Craig accepts Cantorian set theory, which employs the very notions of infinity that Craig derides, and yet he denies that there could be a concrete state of affairs have some infinite property. So, to Craig, there is a clear demarcation between mathematics and reality here.

2.1.1. Endless librariesThe first three arguments Craig offers deals with his example of an endless library. Imagine a shelf that extends infinitely in one direction, filled with books. Now, Craig draws forth three "absurdities" from this.

First, Craig asks to further posit that the books in question have only two colors: black and red. Now, here's the upshot of his first argument: Being an actually infinite collection of books, the number of black books and red books is equivalent to the number of red books. This is absurd, Craig would have us believe, and hence, the actual infinite cannot exist.

This is deeply mistaken. First, before we adduce the problems in this argument, a brief understanding of what is going on is required. Craig here is discussing Cantorian set theory. Now, one of the features of it is the existence of sets with transfinite cardinality i.e. sets with an infinite size. Now, what's interesting about infinite sets is that they can be put into a bijection (a one-to-one correspondence) with their proper subsets. Consider the set of natural numbers {1,2,3,...}. Now, consider the set of positive even numbers {2,4,6...}. Clearly, although the even numbers are a proper subset of the natural numbers, both can be put into a bijection with each other, demonstrating that each has the same cardinality. In other words, the set of even numbers has the same size as the size of the even and odd numbers together, namely, the first transfinite cardinal aleph-null.

Now, Craig would have us believe that the situation with the library is absurd. But, why? After all, Craig accepts Cantorian set theory, and hence accepts the consequence that infinite sets can have a bijection with their proper subsets. Yet, he denies that this applies to actual infinites. But, the mathematical facts map directly onto the physical facts here; one cannot accept one and deny the other. The point is that Craig misunderstands the nature of the relationship between mathematical facts and physical facts. For particular physical facts, one takes a mathematical schema to describe such physical facts. So, to talk about a number of books or a set of books is to detail entailments and theorems in the relevant mathematics. For example, consider what one gets from adding two apples. In reality, one is not adding the apples, but taking the union of the sets that we take to correspond to the apples in question. Addition, subtraction, and sets are, after all, mathematical. So, in dealing with the question as to whether or not one can make bijections between proper subsets of an infinite set and the superset, one must necessarily refer to the mathematical facts in question. So, all infinities are abstract since they are mathematical entities; to say that actual infinites exist is to say that there are concrete states of affairs that can be described by and correspond to infinity. So, it is nonsensical for Craig to accept Cantorian set theory and yet espouse the "absurdity" of actual infinites in which proper subsets have the same cardinality as their supersets. After all, the notion of a "set" is a mathematical one. If a physical fact can correspond to a particular mathematical fact, any manipulations in the mathematics is, necessarily, derivations and entailments within the mathematical facts themselves, not the physical facts. So, one cannot, out of pain of contradiction, reject the possibility of an actual infinite and yet accept Cantorian set theory.

Moreover, one wonders why one should be surprised by this result. After all, since the set of black books and red books is infinite, and the set of red books is infinite, there seems to be an intuitive appeal to that they have the same number of books, namely, an infinity.

The second related "absurdity" that Craig outlines once more involves our infinite library. Suppose that each book has written on its spine a natural number, so as to create a bijection from the books to the natural numbers. Since the collection is actually infinite, it would be impossible to add another book to the library, for what would its number be? And yet, one can clearly add books to an infinite library.

This too is deeply flawed. First, the natural numbers here consist of a set of labels that we arbitrarily affix onto each book. And it is hardly any feat to add a book to the library without any number at all. So, if anything, this illustrates the impossibility of labeling the book with a natural number. We could easily use different numbers, different labels, or not label at all. This does not illustrate any inability to add the book, at best, it might illustrate an inability to label the book according to some particular labeling scheme.

But, of course, one can label the new book. One such scheme would be to simply reuse a number. Here, one has a surjection from the books onto the natural numbers. After all, the natural numbers here are merely labels. Another scheme would be the use of the ordinal ω + 1. Craig has objected to this particular labeling scheme because ω + 1 has the same cardinal number as ω, and we need a new cardinal number. This is mistaken of course as we do not need a new cardinal number because the cardinality of ω has not increased. The ordinal ω + 1 suffices.

Moreover, we can re-label the books according to the particular labeling scheme Craig sets out. Suppose we label the new book "1" and label the previous books with a number 1 greater than their original value. Craig has a few replies to this. First, he argues that this violates the initial conditions set down by the example. But, of course, the conditions are hardly initial anymore; a new book has been added, so one is puzzled by the appeal to the initial conditions. Craig also makes the claim that although this could be done in the mathematical realm, this could not be done in the physical realm for there exists books which completely exhaust the natural number system i.e. we could not call Book 1 Book 2, or Book 2 Book 3, and so on. So, an actual infinite constitutes a determinate whole and a re-count necessitates the creation of a new number.

This is very much confused. First, the natural numbers are determinate and complete because they are the set of all natural numbers by definition; but an infinite library of books should not be confused with the set of all books. More importantly, this view is entirely tackled by the fact that we are simply dealing with labels here, and we can freely employ other labels.

An example that parallels this case can be given. Suppose we have the positive integers Z+. Now, there is a function f that maps a bijection from Z+ to the natural numbers N. Is it impossible to add new elements to Z+, because given f, there are no numbers left to correspond in a bijection with? Hardly, since we can add new elements to Z+ (for instance, {0}) and there will simply be a function f' mapping a bijection from Z+ union {0} to N. Now, f "exhausts the natural numbers" just as the books; but this hardly disproves other functions f' from existing to map a bijection from Z+ union {0} to N. What prevents us from having such a function f' to the library?

As Craig accepts that f' bijects Z+ union {0} to N, then why is it that he cannot accept that N could biject the books labeled by N union the new book? It is true that each book is already labeled, but the modal force of Craig's argument is important; that the books could not be numbered. But, of course, as we've seen, they can be, so there is no gap in the possibility of mapping a transfinite superset onto a proper subset anymore than an actually infinite superset onto its proper subset. Given the nature of how mathematical facts and physical facts correlate, such a disparity would be absurd, if not contradictory.

The final absurdity Craig asks us to contemplate is as follows. Suppose we check out books 1, 2, and 3. Here, we have not decreased the number of books because an infinity still remains. Now, suppose we instead check out 1, 3, 5,...; even with the subtraction of an infinite number of books, we remain, strangely enough, with an infinity of books. Finally, suppose we instead checked out books 4, 5, 6...; lo and behold! in one stroke we have once more checked out an infinity of books and reduced the infinite library to a finite number of 3 books. We remove the same quantity and yet, we remain with different amounts, illustrating that inverse operations with transfinite numbers cannot be done. But, this only applies to the mathematical realm, not the physical world. So, while we may correct the mathematician from subtracting from a transfinite number, we cannot prevent people from checking out books.

This is perhaps the most egregious of Craig's mathematical mistakes. Presumably there is some important mismatch here between the mathematical facts and the physical facts. Certainly, people may check out what books they will. But, so the mathematician can also subtract elements from a set. There is no real mismatch here. Any difficulty in physical exemplifications of this entails difficulty in the mathematics of this, due to the relationship between the world and mathematics. But, no such problem arises.

To say that transfinite subtraction is undefined is not to say that it is impossible or absurd to subtract elements from a set. Rather, the subtractive process of removing such elements does not entail or derive a formally correct subtraction operation. Consider for example how division is often defined; as an inverse operation, x/y = r iff there there is one and only one r that satisfies x = yr. For example, 1/0 is standardly undefined because there is no r that satisfies r x 0 = 1. However, 0/0 is undefined because any r satisfies r x 0 = 0. So, the point of a mathematical definition of an operation to obtain a process that determines a uniform solution; that it derives the same way in any instance. Or, more relevantly, consider how cardinal addition is defined: the sum of the cardinality of A and the cardinality of B is defined as the cardinality of the disjoint union of A and B. So, cardinal addition is a derivative operation such that it derives the same way in all instances, so no matter which two sets we choose, the sum of their cardinalities is equal to the cardinality of their disjoint union. But, no such derivation or theorem exists for cardinal subtraction, indicating, not that it is absurd, but that there is no formal operative process that derives a uniform, determinate result for all instances. We can have sets with the same cardinality and subtract the same cardinality yet be left with differing cardinalities. So, cardinal subtraction is not the uniform subtraction of sets with like cardinality.

So, although cardinal subtraction is undefined, one can still remove elements, differing in no respect from a person being able to remove books. So, there is no real mismatch here, hence, it is false that one cannot subtract from the infinite.

Now, as we have seen, all of these absurdities are little more than mathematical error and a lack of understanding of how mathematical facts relate to the physical. Most importantly, we've seen that the argument is intuitive; there is no actual contradiction or anything of the sort here, but that the results are presumably absurd.

Yet, if one accepts the mathematics, then the results are not absurd, given the relationship between the mathematical and the physical. It remains a question of whether or not there are actually exemplifications of the actual infinite, but there is no legitimate a priori argument that ensures their impossibility. More importantly, it is unclear how mere intuition could guarantee metaphysical impossibility. Indeed, considering that metaphysical possibility is not a meaningful modal space, Craig's arguments to this regard seem largely irrelevant.

2.1.2. Euclid's maximCraig correctly believes that the root of these "absurdities" lies in the Principle of Correspondence. According to the Principle of Correspondence (POC), two sets which can be placed into a bijection with each other have equivalent sizes. Craig notes that whatever intuitive appeal that the POC has is matched by the Euclid's Maxim (EM), which says that the whole is always greater than the part. While both the POC and the EM apply perfectly well to finite sets, they are, supposedly, in disagreement with respect to infinite sets. According to Craig, the POC allows for the possibility of the whole being equal to the part. So, both cannot be true. So, what should we do? Craig believes that we should accept both principles and deny the possibility of the actual infinite.

This procedure seems mistaken to me. First of all, in the mathematical analysis, we cannot simply deny the actual infinite but it seems that we must choose between these two principles. And given that we would describe physical reality with this mathematical analysis, it appears that the principle chosen would carry over to the physical realm, removing the argument against the physical possibility of the actual infinite.

More importantly, it can be successfully argued that the EM does not actually contradict the POC, let alone that it does not apply consistently to infinite sets. We distinguish between "more" in terms of cardinality and "more" in terms of supersets. Whereas the POC deals with "more" in terms of cardinality, the EM deals with "more" in terms of supersets. For instance, the natural numbers are greater than the even numbers in the second sense, in that the natural numbers contain the even numbers and are a superset of it. Nonetheless, it does not have a greater cardinality; hence, these two senses of "more" are not coextensive.

Craig continually equivocates between these two senses of being greater. Craig asks us to suppose that we could add to the infinite library. But, as he states, we have added to the library, and yet our collection has not increased by a single book. We remove the same book, and the amount has not lessened by a single book.

It can be easily pointed out that Craig equivocates between the two senses of greater here. While the collection has certainly not increased in the sense of there being a greater cardinality, the collection has certainly increased by one book in that the collection containing the book is the superset of the original collection by one book. So, in a very reasonable sense, EM applies perfectly well to infinite sets.

One wonders how an addition to an infinite physical collection could increase cardinality, when there is no increase in the mathematical realm? One analyzes such claims in the mathematical realm as I elucidated earlier; cardinality and sets are, after all, mathematical notions. If anything, such a notion appears to be wholly unsupported. One finds these "absurdities" to actually be absurdities because one is already committed to the notion that adding an element necessarily increases cardinality. So, these absurdities illustrate this notion; they do not establish it. Unless we automatically beg the question by assuming that only finite sets could be exemplified, such a notion seems to be largely unsupported.

Moreover, what makes the EM plausible is the derivation it naturally receives from the POC. Why do we suppose that the whole is greater than the parts? Well, suppose we take the objects in a line. Now, consider some particular part of it; the initial segment of the whole line is congruent to the entire part. So, there's a segment left over at the end. This intuitive way of attempting to create correspondence between parts and wholes, namely a mapping of identities, necessarily fails to find one-to-one correspondences between parts and wholes. And yet, while the POC drives the EM, it does not properly motivate for transfinite cardinalities, since the failure of one method of mapping to derive a correspondence doesn't entail failure of other methods.

This particular feature is among the reasons why cardinal equivalence is defined according to the POC. Namely, that because we can alter the ordering and hence alter the part-whole relata, these relations per part to whole are varying. For instance, checking out an infinity of books sometimes leaves an infinity of books left and other times leaves it nearly barren. So, we cannot use such part-whole relations in determining an equivalency as they are varying. However, one-to-one correspondences do not vary with ordering and part-whole relationships. So, if anything, we've got excellent reason to prefer the POC as being correct here.

This is well-established. No one contests, for example, that the natural numbers can be injected or bijected upon the integers. So, we aren't really left with room to doubt that such varying collections of physical items is not possible. This variance is a derivative result of infinity.

2.1.3. God and infinityIn denying the infinite in general, one wonders if Craig leaves himself open to the charge that he denies the God of traditional theism, the very God the argument is supposed to prove. For, certainly, God's qualities, such as power, knowledge, goodness, and so on are typically construed as being infinite. The area of contention rests around whether or not God does have properties exemplifying an actual infinity.

Allow me to use a singular, simple example to illustrate one such property. God is said to be omnipotent: He can do whatever is logically possible. Now, consider some of the implications of this. God could create 1 human. He could also create 2. And 3. And 4, and so and so forth. So, if we take the set of what number of humans God can create, His power scopes over an infinity of possible actions corresponding to the natural numbers. Or, consider for instance that God could create any possible world whatsoever. Yet, there are infinitely many possible worlds, so God's power is infinite in extent as well.

As we can see in this singular example, God, as typically construed, contains infinite properties, properties that contradict the assured conviction of the KCA regarding the impossibility of the actual infinite.

2.2. The additive argument against actual infinity
The second line of argumentation that Craig draws upon is the argument from successive addition. What distinguishes this line of argumentation from the first is that Craig argues that there is an actual impossibility here, one that many are willing to grant, an improvement over mere assertions of "absurdity" that marked the first line of argumentation.

2.2.1. The formation of an actual infinite
The core argument Craig puts down is an argument from successive addition. According to Craig, if there had actually been an infinite past, then that would mean that an actual infinite has been traversed, one interval of time at a time. Yet, it is impossible to traverse an actual infinite. One cannot form an actual infinite by adding successively, for no matter how much one adds, one will always remain with a finite amount. Hence, it is impossible for there to be an infinite past.

This argument, although better than the general argument against infinity, is also deeply flawed. Note that the argument states that it is impossible to form an actual infinite. Namely, we cannot start with some member and continue adding members until the collection becomes infinite. That will never occur.

But, surely, this is irrelevant to an infinite past? For in an infinite past, it is hardly the case that it is formed. To suggest that the infinite past were formed would be to say that the infinite past came into being, that there was some time that the past became infinite. But, this is entirely false on the view of an infinite past; for there is no interval of time at which there is not an infinity of prior intervals. In other words, an actually infinite past is not formed, but is in effect always there. So, Craig's argument to this effect is deeply mistaken, as it cites an irrelevancy to argue against an actually infinite past.

But, perhaps Craig might insist that since every interval of an infinite past has been traversed, so has the entire infinite past. Well, this would depend on what Craig means when he says that the infinite past was "traversed." Craig's comments strongly indicate that any "traversal" as defined by him has a beginning and an end. But, surely, then, we cannot say that the infinite past has ever been traversed, at least according to Craig's definition. So, what is the actual problem here?

Consider for instance the negative integers. While each integer is "traversed" the whole is hardly traversed. Moreover, the negative integers with an ordinal of *ω lacks a beginning and hence cannot be formed; moreover, any particular integer is only a finite distance away from another integer. So, there is no real indication of there being, absurdly enough, a beginning to the beginningless, that the passing of an infinite past indicates that it can be or was "traversed" under Craig's definitions, or that there was ever a moment of time of ordinal type 1 + *ω. If anything, these arguments commit the fallacy of composition.

2.2.2. Counting down from infinity
Craig has an interesting reply to some of the above. He argues that if indeed if it is impossible to complete an infinite count of the natural numbers, that is, to count to infinity, then how could one count all the negative numbers, that is, count from infinity? If one cannot pass the infinite in one direction, how can one do so in the other direction?

This objection is entirely mistaken. Craig is, moreover, equivocating the reversals of the counts. Craig grants, supposedly, that the order type corresponding of an infinite past is that of *ω, namely {...-3,-2,-1}. Note that aleph-null is not a member of this set, so there is no question of counting particularly from beginning at infinity, but that there is no beginning at all.

Now, Craig asks to contemplate the case of the ordinal ω + 1, where there is an interval infinitely distant from the beginning, the termination of this sequence. Of course, it is impossible to count this, but this is irrelevant because the reversal of *ω is not ω + 1, but ω. So, Craig's appeals are once more irrelevant. The point here is that Craig realizes what while in *ω, an infinity of events have passed, what he fails to realize is that the analog of the reversal, namely ω, is the future. And plainly enough, in an infinite future, an infinity of events will pass. But, just as there was no beginning to the passing of past events, so there is no end to the passing of future events. So, Craig's comments here are motivated by a severe misunderstanding of the reversal of the ordinal *ω.

2.2.3. The eternal counter
The second response to our refutation of the additive argument is another argument from example. He asks us to imagine that there is a man who has been counting from eternity and has just finished counting the non-positive integers. Craig asks why is it that the man did not finish yesterday or the year before. For an infinite time would elapse. Indeed, at no point in time will one find the man counting as he will already have finished.

This objection is largely confused. It is true that yesterday our eternal counter would have counted an infinite number of integers. But, from this, it does not follow that our counter would necessarily finish his count. For he was counting to 0 and yesterday he counted to -1 and the day before to -2 and so forth. So, there is not any sort of reason to suppose that our counter would have always finished his count. If anything, Craig's argument confuses counting an infinity of numbers with counting all the negative numbers. All of these proper subsets we are considering here have the same requisite transfinite cardinality, so pointing out that an infinite amount of time has elapsed does not entail the completion of a particular count.

Craig has an interesting response. Supposedly, the opponent of the anti-infinitist must argue that the counter would reach 0 because there was enough time to do so, according to the Principle of Correspondence, allowing Craig's initial objection that there was always enough time.

If anything, Craig's imaginary opponent is distinct from our actual response. The POC does not entail that the non-positive integers would have been counted by now, but that they could have been counted by now. The proper response is that our infinite counter could have finished at any particular moment of time, the actual time of which being indeterminate from the mere assumption that he has been counting since eternity. Perhaps Craig is appealing to the Principle of Sufficient Reason here, namely, what is the reason as to why the counter finishes today rather than yesterday or what not.

It is doubtful whether the Principle of Sufficient Reason could be applied here. Craig admits, after all, that infinite pasts do not come to begin to exist, for all of his philosophical confusion concerning the additive argument. So, any reason as to why the count finished today would necessarily have to be prior to the count itself; but, since the count never begins to exist, it cannot be brought about by some prior cause or reason, hence, infinite pasts of this sort are reasonably immune to the PSR. Indeed, recall that the PSR is an incoherent principle, no less.

2.3. Tristram Shandy

The final argument Craig offers is an example from Bertrand Russell. Russell had an interesting story about the character Tristam Shandy in a novel by Sterne. Tristam Shandy is writing his own autobiography, but writes so slowly that it takes him precisely 1 year to describe the events of 1 day. Russell noted this, but pointed out that if Shandy lives for an infinitely long time, then all of his life will be written in his autobiography.

According to Craig, this indicates that Shandy will eventually finish his autobiography, given an infinite amount of time. This interpretation is inaccurate. What Russell actually said was that Shandy will write about each event in his life, but not that there will be some point in time at which he will finish writing about his entire life. For Shandy will be able to write about each day, eventually, but there will never be a point at which he ever finishes writing.

Russell was saying that if we took the function f(x) = int(x/365) such that the domain is the natural numbers, then the range is also the natural numbers. If Shandy writes for every day for an infinitely long time, there will be no day of his life he will not eventually write about. This isn't to say that he'll ever finish writing; it's not as if the function in question need be extended to a set with an ordinal of ω + 1; there are no questions of a termination of Shandy's writing, in effect, no mention of a ωth day, in effect, a day aleph-null days away from the day Shandy began writing.

So, with this confusion in hand, Craig takes an "inverse" Tristram Shandy case. Suppose that Shandy has been writing since eternity i.e. for an infinity of past times. He has just finished his autobiography today. Given that had an infinite amount of time, he should be finished. But, Shandy could hardly have written about today's events. If anything, each day of writing indicates another year of writing about that day. Hence, we observe the bankruptcy of the POC on the real world.

Presumably, this notion is mathematically coherent whereas it is physically impossible. So, there is, supposedly, some important disparity between the mathematical and the real here. This, I contest. We can take the mathematical case of the inverse Shandy case. Let us consider a function defined on the non-positive integers, such that the function f(n) provides the day on which that Shandy is writing on day n. Now, 0 is the present day whereas the negative integers as naturally ordered are the past days we are to consider. Now, we need a couple of restrictions, first, if n - m = 365 (the difference between two separate days is 365 days), then f(n) - f(m) = 1, moreover, that for any n, n > f(n). But, clearly, no such function is definable. Consider the present day. f(0) = p, for some p. But, it follows that f(365p) = 0, which contradicts the second axiom. Therefore, the inverse Shandy case is impossible, but it is mathematically impossible.

There is no important disparity between the real and the mathematical here. If the inverse Shandy case shows anything, it is not the impossibility of an actual infinite or the "bankruptcy of the Principle of Correspondence," but that the type of task Shandy is engaged in is simply impossible.

3. PersonalityIn the conclusion that the universe has a cause, Craig attempts to deduce a number of qualities about this cause. If his first two premises and cosmology were correct, then he could deduce that the cause of the universe was a non-physical entity that is timeless and hence uncaused and eternal. Craig also believes that there is an argument to show that the cause is personal as well.3.1. A personal cause?The basic argument that Craig sets out is as such. He argues that since the cause of the universe is timelessly eternal and the universe itself is temporal, it raises a puzzle as to how the universe is temporal. For, as he says, mechanical causes are sufficient for their effects and if a mechanical cause existed eternally, its effect would exist eternally. He then posits that the only way to have a temporal effect from an eternal cause is to have a free libertarian agent. A man sitting from eternity may choose to freely stand up, hence, an eternal cause can produce a temporal effect in this manner and there is no change in the agent to be conceived. So, we don't merely have a cause to the universe, but a personal cause.

I believe this line of reasoning is wildly flawed.3.2. Sufficiency and EternityFirst, a few prefatory remarks. Note that his view on mechanical causes being sufficient for their effects contradicts his view that quantum phenomena are caused and yet lack sufficient conditions. Furthermore, it is well known that the determinism entailed by the causal principle contradicts libertarian free will, which argues that only uncaused actions are truly free. So, we have contradictions in Craig's views here. And finally, this is largely a mental exercise, since timeless causes, as indicated earlier, are simply impossible. In any case, let us proceed.

Craig is under the presumption that libertarian free will accords freedom from this "dilemma." And yet, the only relevant aspect that Craig cites here is the indeterminism of libertarian free will, namely, that the conditions antecedent to a free choice do not determine a choice. Hence, a man may freely choose to stand up. But, it is much to easy to point out that quantum mechanical phenomena are likewise indeterministic; indeed, one can quickly point out that there are a number of vacuum fluctuation scenarios regarding the origin of the universe that rely on this indeterminism.

It is all too easy to create a parallel non-theistic creation scenario here. The universe is eternal and was timelessly eternal as an atemporal singularity. However, due to indeterministic quantum phenomena, the universe entered time itself and began its life in time as we know it.

This is parallel to Craig's scenario: God is eternal and was timelessly eternal. However, due to indeterministic libertarian free actions, He entered time itself and began His life in time as we know it.

That isn't to say that either of these scenarios are reasonable (indeed, I find them to be nonsensical), but that there isn't any real difference in positing either of these creation scenarios.

Secondly, one questions whether or not God's eternal act to create the universe is really insufficient for the universe's creation. Craig holds that God eternally had the intention to create time. Of course, it is all too easy to show that this is sufficient for the effect, namely, the universe. God's will is sufficient for his effect; after all, if an entity actualizes something, then it is sufficient for that thing being actual. So, if God eternally wills the universe, then it is sufficient for the universe existing, which, by Craig's principles, entails that the universe should be timelessly eternal. Moreover, given the nature of intention, it is sufficient for the universe, since if God had the eternal intention of creating the universe, that would mean that God would create the universe if He had the power to do so. But, clearly, God (being omnipotent) had the power to create the universe. Given this conjunction of power and intention, since these antecedent conditions are sufficient for God actualizing the universe, which is sufficient for the universe being actual, it follows that the universe's existence is entailed. Hence, this eternal divine cause is sufficient for the temporal physical effect. The point is that it is not as one can explain a state of affairs brought on by an agent by pointing out the agent's existence, but the agent's action; the agent did something, presumably for some reason. These facts explain and are sufficient for the state of affairs in question. Moreover, as I pointed out earlier, God's act of creation must be timeless, and since God's actions are sufficient for their effects, God's timeless act of creation is sufficient for the universe existing.

But, doesn't God have the eternal will to create a universe with a beginning? Perhaps our interlocutor will argue that this is sufficient for the universe having a beginning? Perhaps it does, but if so, then this entails that Craig is committed to an inconsistent conjunction of propositions. Consider (a) the universe had a beginning (b) God's intention of, ability of, and act of creating are eternal (c) God's intention, ability, and act of creating the universe is sufficient for the universe's existence and (d) if an eternal cause is sufficient for its effect, then the effect is eternal. Now, if (a) is true, then the universe is presumably non-eternal. Craig certainly agrees. Now, if so, then (a) entails that the universe had a beginning, whereas (b) to (d) entail that it did not. So, this set of propositions entail a contradiction.

So, what must go? Certainly not (a); that is the entire basis of Craig's argument. And not (c), under standard views of God's power. Perhaps (d)? Craig might argue that God's willing to create the universe with a beginning makes the proposition "there is a universe with a beginning" eternally true, without the universe itself being eternal. I'm not sure there is much merit to this idea. For the proposition in question would be true eternally; yet, the proposition expresses the existence of the universe as well. So, the proposition must entail that the universe is indeed eternal. But, since the universe has a beginning, it follows that it does not have a beginning. Hence, we are left with the same contradiction.

But, even if our interlocutor insists, then Craig's argument disappears, for (d) is the crucial mechanism by which he concludes a personal cause. For if an eternal personal cause can be sufficient for temporal effects, then why not a non-personal cause?

What has happened to Craig's argument? First, Craig seems to move back and forth between considering eternity as an everlasting, infinite duration and as a timelessness. When Craig says that God is causally prior to the universe, he means that God is timeless sans the universe. Yet, when he speaks of God willing from eternity, a man choosing to stand up after sitting from eternity, and mechanical causes from eternity, he is considering eternity as a beginningless, infinite, endless duration.

The point is that a personal agent in time can plan for the future. If God is temporally prior to the universe (where the universe is not actually the beginning of time; in this scenario, there is no beginning of time). Presuming that libertarian free will is coherent, just as someone can sit for a long time, having the intention of standing up at a particular time, so can God wait from eternity, always intending to create at a particular time. But, one wonders what prevents mechanical causes from acting on a distance, through time at least. This is precisely what is typical of vacuum fluctuation models, for instance.

What if we switch over to timelessness? But, if so, the supposed difference between a personal cause and a non-personal one vanishes, for a timeless agent either wills timelessly for a world with a beginning or not. There cannot be any temporal lacuna between the willing and the time it occurs. In this sense, both personal and non-personal causes are identical in this aspect, for they both are immediately sufficient for their effects.

Given the arguments above, I conclude that the argument for a personal creator fails.

3.3. Agency and CausationBesides the relevant concerns of eternity and sufficiency, another important issue is raised, namely, the question of free will. Craig explicitly draws on libertarian free will in supporting his argument. This move, if anything, is surprising. Craig makes no attempt to defend libertarian free will, something one would expect given that the majority of modern philosophers deny the coherence of libertarian free will. Indeed, I shall briefly outline the reason why I believe libertarian free will to be incoherent and why compatibilism is correct. The following is from a previous post on this very subject.

Freedom of the will is one of the most largely debated subjects in both philosophy, theology, and even among the broader populace. The main question is thus:

(1) What does the term "free will" mean?

The formulation of this question has a few ramifications. First of all, we're trying to ascertain a referent onto the term that matches the conventional usage of it. Terms are meaningless with referents, and referents are conventionally attached to terms. As Wittgenstein might have said, the use of a word is its meaning. So, it matters little if there are obscure, esoteric definitions of "free will" that do not satisfy the conventional requirements, otherwise, one is not talking about "free will", but instead changing the subject. So, let's open the floor to that question.

Most people would offer something like the following to describe "free will": "'Free will' seems to require the presence of a choice, in which a person decides upon a course of action and has the power to undertake or not undertake that action. For example, when I freely choose to eat a sandwich, I have in me the power to undertake upon or not undertake upon that action. I merely choose to do so."

This understanding of "free will" has led philosophers to opine the thesis of libertarianism. The doctrine of libertarianism holds that a person freely makes a choice if and only if their actions are undetermined and uncaused [1]. Now, why would such a radical consequence be apparent to them? Most of them reason from a seemingly acceptable principle, called the Principle of Alternate Possibilities. It goes as follows

(2) PAP: A person S is free with respect to an action A if and only if S could undertake A and could refrain from A.

Now, this seems like a very reasonable doctrine. In fact, it seems to be a rigorous formulation of what the common person labelled "free will." But, here comes the interesting part. Libertarians have a particular interpretation of the PAP. This interpretation centers around the phrase "S could undertake A and could refrain from A."

Now, "could" tends to signify a modal space. Libertarians have argued back and forth over the space this tends to be, and most have agreed it to be local possibility; given the prior conditions of the world that are antecedent to the choice, it must be possible that the action could be undertaken or refrained from in the context of those conditions.

This seems reasonable as well. After all, the modal space could hardly be logical or physical possibility. Both leave themselves open to extremely unfeasible situations that are nonetheless possible in their respective modal spaces i.e. teleporting from place to place or pseudo-telekinesis due to extreme magnetism. The former is logically possible and the latter is physically possible. They are hardly locally possible and given the antecedent conditions of the actual world as it is, both are locally impossible.

Now, how does this entail that radical thesis we talked about in the first place? A very simple deduction shows this. Recall that libertarians interpret the "could" as signifying local possibility such that

(3) Expanded Libertarian PAP: A person S is free with respect to an action A if and only if S could undertake A such that given the antecedent conditions of the actual world as they are, it is locally possible that S undertakes A and S could refrain from A such that given the antecedent conditions of the actual world as they are, it is locally possible that S refrains from A.

Now, this means that given the antecedent conditions of the world prior to the choice, they cannot determine the choice. Why not? Because if they determined the choice, then it would be impossible for S to refrain from A since the antecedent conditions as they are would entail that S does A. This is similar to the impossibility of the set {p -> q, p, ~q}. So, if we consider the notion of an initial world segment (abbreviated as IWS), and consider it to scope over all events prior to the choice. So, if the choice occurs at time t, the IWS scopes over the set of times t' such that t' < t.

Now, the IWS cannot entail that S does A or S is not free with respect to A. If it entails that S does A, then given the antecedent conditions of the world as they are, it is no longer possible that S refrains from A. It is logically impossible (the broadest form of possibility and the necessary condition of all other spaces of possibility); if there is an entailment relation such that p -> q; and the antecedent condition p holds, then it must be the case that q, per logical possibility. So, ~q is impossible given the antecedent and the entailment relation. So, if S is to be free with respect to A, the IWS cannot be determinative. And, this applies to all free choices, hence, all free choices cannot be determined. Therefore, all free choices are in fact uncaused.

This is the libertarian perspective and this is a brief outline of why they believe as they do. I agree with the previous posters who have argued that omniscience deprives "free will." But, it deprives "free will" only in the sense of libertarian freedom, because omniscience sets up a determinative relation such that God knowing that S does A entails that S does A; this is a determinative relation, therefore, S is not free with respect to A.

That said however, I believe libertarian freedom to be false and not what is meant by the term "free will."

I hold to compatibilism, which, as its name suggests, offers that causation (as well as determination) and free will are compatible. I hold a specific sort of compatibilism that is becoming extremely popular nowadays which rejects the PAP, holding that determination is required for free will. Well, that's not entirely true. This version of compatibilism rejects the libertarian interpretation of the PAP. The compatibilist has a differing and far more defensible interpretation.

The compatibilist argues that "could" signifies as having the ability to do A or refrain from A. There is, however, an active difference between having the ability to do A and the conditional local possibility (I call it conditional local possibility since it is local possibility conditional on the antecedent conditions) of doing A or refraining from A. The former identifies the abilitites or the powers of S. This concerns what S could do. The latter identifies the possibility of S will do.

Consider the fact that I have the power to smash my computer to bits. Now, given the conditional local possibility as it is, I think that given my nature and who I am, I will never smash my computer to bits given these specific conditions. There will never be the possibility of me doing that. But, this does not restrict my freedom. Note that all that is required to have compatibilist freedom is to have the power to do or refrain from A. It does not require the possibility of me actually refraining from it. It simply means that every time I am confronted by those exact conditions, I make the same choice every time. It describes who I am, my nature, and the fact that there is no conditional local possibility of my actually refraining from it does not decrease this.

Compatibilists tend to support their point in three ways. First of all, they point out that this easily reconciles with what people mean by "free will." The second way builds upon the first by arguing that libertarian freedom does not actually exist and yet the term "free will" actually has meaning. Finally, compatibilists argue that libertarian freedom cannot be the meaning of the term "free will"; that is to say that libertarian freedom is incoherent.

I shall not argue for the first since it seems to have already been argued for above.

The second relies on an inductive argument. All around us, we find the world to be a maze of causation. Now, quantum mechanics aside, macroscopically, causal, determinative relations hold between states of affairs. So, for virtually all of our choices, causal, determinative relations hold. This means that none of our choices are free in the libertarian sense. Nonetheless, we still use the term "free will" and the term still has meaning, the most prominent of which is the compatibilist term. Therefore, libertarian free agents do not actually exist and the best definition of "free will" out there continues to be the compatibilist definition.

Finally, even though the term "free will" has a compatibilist definition, it is impossible that the libertarian definition could ever be a true definition of it. In a different forum, I argue as thus
Libertarian free will considers that one has the ability to have done otherwise, and by that, philosophers do not mean that given different circumstances, a person acts differently, but given the exact same circumstances, that same person acts differently in some logically possible world. In order to hold to this thesis, defenders of libertarian free will hold that persons are causally undetermined: there are absolutely no causal antecedents of one's actions whatsoever, not even the prior mental states of the agent. Now, here's the problem: if that the agent does some action is uncaused, then how is that action the agent's action? For if that action is uncaused, then the agent exercises no causal influence over the action since the action is uncaused. Most defenders of libertarian free will reply that the agent causes the action. But, what about the event "the agent causes the action?" Is that caused or uncaused? If uncaused, the agent continues to lack freedom. If caused, then either the agent was caused by something else (and hence lacks libertarian free will), or the agent caused the action. And so, we continue down an infinite regress with the agent causing that they cause that they cause that... But, this is absurd; no agent can control an infinite regress. And moreover, what about this infinite regress? Nothing can be prior to such an infinite regress, so the agent exercises no causal influence over this infinite regress, and so, that the agent acts as they do consists of a brute fact; it's just something that sort of happens to the agent. And such is not a free action. Since it is not the agent's action, it is not anyone's action, which means, that it was not an action to begin with. Therefore, libertarian free will is incoherent. Moreover, plainly, agents act according to their mental states, desires, and so on. To deny this is to divorce the agent's action from the agent itself. So, needless to say, libertarian free will is not the theory of freedom we should adopt.

What should we adopt? First of all, it has been shown that indeterminism precludes free will. Therefore, we have the following argument:

(1) It is logically possible that agents exist.
(2) If anything is an agent, then that thing is free.
(3) Either the agent's actions is caused or uncaused.
(4) If the agent's actions are uncaused, then the agent is unfree.
(5) No agent can be unfree.
(6) Therefore, the agent's actions are caused.

(1) is necessarily true since there is no contradiction in the actuality of agents, and moreover, agents are actual; we are agents, and we have infallible phenomenal knowledge as such. (2) is necessarily true by definition. (3) is true analytically, as it exhausts range of logical possibilities. (4) follows by our above analysis, and (5) and (6) follow by modus tollens.

So then, what have we shown? We have shown compatibilism to be true: that our actions are caused, and that such causation is compatible with freedom. However, it must be made clear that this type of freedom precludes the ability to do otherwise. And the ability to do otherwise does not support freedom, it destroys freedom (see above). Per the ability to do otherwise, there is some logically possible world in which given the exact same circumstances, I act differently. But why? Given who I am, why would I act differently at all? The ability to do otherwise is disguised as freedom, when all it really consists of is the antithesis of freedom: for I am no longer free to act in accordance to my desires and personality, in other words, who I am. And given that I am the same person in all logically possible worlds, it is necessary that given the same circumstances, I do not act differently. I lack the ability to do otherwise not because of external circumstances, but given who I am. This type of compatibilism is necessarily true (per the above) and accords with our understanding of freedom. It should be made clear that the only thing that really causes our actions besides ourselves is that fact of our creation: that we are brought into being in this world. And from that point on, we act in accordance to who we are, and this does not impinge on freedom. It seems that this form of compatibilism is immune to many of the challenges given by defenders of libertarian free will, such as the Consequence Argument. The best statement of the Consequence Argument is espoused by Peter van Inwagen. He argues from the notion of an untouchable fact, a fact being untouchable if and only if we could have causally influenced such a fact to be other than what it is. According to van Inwagen, if it is an untouchable fact that (p → q) and an untouchable fact that p, it follows that it is an untouchable fact that q. So, van Inwagen continues, it is an untouchable fact that given that the laws of nature and determinism hold, I do some action. It is an untouchable fact that given the laws of nature and determinism, I do some action. And therefore, it is untouchable that I do some action. But, van Inwagen is missing some parts of the causal chain. First, part of the causal chain is the fact that I am born (it should be clear that "I" is not completely born until around the age of seven, when we have pretty much developed our personalities) and then after that, I act according to who I am. And there is no lack of freedom in this, per the above.

So, I think we can conclusively agree that libertarian freedom is not and cannot be what is meant by the term "free will" as conventionally used and that compatibilist freedom is meant and must be meant by the term.

Given the incoherence of libertarian free will, Craig's argument for a personal creator is not cogent, as it relies on the coherence and existence of libertarian free will. An "agent" with libertarian freedom is not an agent at all.4. Cosmology

We now come to the relevant issue regarding cosmology. Much of Craig's argument revolves around his view of cosmology. I wish to engage these views.

4.1. The universe had a beginning?

Craig often points out that the Big Bang theory under general relativity points out the existence of a singularity 13.7 billion years ago. This, he argues, shows that time and space and the universe itself had an absolute beginning. But, is it really so that the universe had an absolute beginning? I wish to contest this.

First, we must point out that the Big Bang theory is often regarding as a theory of cosmic evolution, not cosmic origin. Many physicists often speak of a "cause of the Big Bang" or a "time before the Big Bang" without contradicting Big Bang theory. The point is that Big Bang theory does not entail that there was an earliest time for all physical reality. Indeed, a number of proposals regarding what caused the Big Bang have been posited, such as Neil Turok and Paul Steinhardt's cyclic ekpyrosis, Lee Smolin's cosmic natural selection, Sean Carrol and Jennifer Chen's infinite-entropy inflationary scheme, and various other models. This is not to say that there is not actually a beginning (that is not the point of this argument), but that the Big Bang theory does not actually imply one.

Indeed, many physicists are uncomfortable with the idea of a beginning of time. It is not so much a difficulty with the uncaused beginning of time, but there being a beginning at all. And yet, these same physicists easily accept Big Bang theory, so we cannot regard the theory as entailing a beginning of time. That is not to say that these intuitions are correct however; intuition can only take us so far in matters sufficiently beyond our experience. There is certainly nothing incoherent with a beginning to time. But, in any case, it is clearly indicated that Big Bang cosmology does not entail a beginning of time.

More importantly, the Big Bang theory, at best, addresses the evolution of this particular space-time continuum. But, it hardly addresses the existence of other space-time continua. The existence of such entities remains an open question, one that the Big Bang does not rule. And given the proliferation and indication of other such space-time continua in modern cosmology, that the KCA explicitly assumes that this space-time continuum composes all of physical reality is illegitimate. Even if one succeeded in showing a cause for this space-time continuum, that would say nothing at all about the existence of other space-time continua or other physical entities that gave rise to this space-time continuum. Such a hypothesis is, at least, more parsimonious than the hypothesis of a disembodied, timeless mind. At least we have actual, coherent, empirical evidence of the former.

The only reasonable cosmological model in which time has an absolute beginning tends to be classically relativistic models, in which a GTR-framework is sufficient for the existence of singularities, and most importantly, the beginning of time, as shown in the Hawking-Penrose theorems. However, there is, in modern times, a large bit of tension between two new paradigms of cosmology, relativistic cosmology and quantum-mechanical cosmology. Relativity and quantum mechanics disagree on several key issues, which may make one wonder if relativity is once more something of a deeper regularity, perhaps ones found in quantum mechanics. For instance, relativity expresses gravity in terms of the curvature of space-time; in quantum mechanics, gravity is a force with discrete messenger particles, namely, gravitons. And, given the proliferation of various quantum-mechanical schemes in modern cosmology, such as cosmic inflation for a celebrated example, it remains doubtful whether or not there was really a beginning of time. Indeed, in the evolution of this universe, our knowledge extends only up to the Planck epoch, as general relativity breaks down. Prior to that, we cannot directly ascertain the existence of an initial singularity. There, one must resort to quantum mechanics, which tends to eliminate singularities.

Moreover, given the problems that singularities tend to entail, namely, the breaking down of the laws of physics among other things, most physicists deny that there was ever really a singularity. If anything, it represents the limit of the universe's origin, akin to the half-open interval (t, t']. Of course, this doesn't do away with the idea of a first interval, but given all of the above, it remains an open question whether or not the universe had a beginning. That question is not answered by the Big Bang theory, contrary to Craig.

4.2. The universe came from nothing?Craig also argues that the universe literally came from nothing. He offers two arguments to this effect. First, he argues that, under standard GTR-cosmology, the universe was initially shrunk down to a single point, an infinitely dense point of spacetime; the singularity. According to Craig, there cannot ever be an object with infinite density, for any object, if it has a size at all, has a non-zero density. Therefore, an object with infinite density is synonymous with nothing, hence, the earliest state of the universe was nothing at all; therefore, the universe came from nothing.

This is very confused. First of all, the singularity is not an object in the sense of being a particle or any of that. Rather, a singularity is a state that occurs when the curvature of spacetime is infinitely dense. Secondly, Craig is equivocating here when he says "nothing." Earlier, and throughout, Craig refers to "nothing" to mean "absolutely nothing at all"; no space, no time, nothing whatsoever. This sort of nothingness is incoherent and cannot have any properties. And yet, a singularity possesses a number of properties, namely, infinite density, being a curvature of spacetime, and so and so forth. So, by no means is the singularity "nothing at all." Moreover, although a respectable number of physicists nowadays would accept the idea of there being no object with infinite density, Craig misunderstands what this tends to entail. What this entails is that, since there can be no such object, there was never such a state in the universe's history. For if there can be no such object, then the singularity never really existed, but is at best a limit of the universe's past history. The universe, therefore, would strictly lack an earliest time. In any instance, as I said above, few physicists nowadays believe in such initial singularities.

Craig's second argument involves the notion of there being "nothing prior to the singularity," a notion I answered in 1.2.2.
4.3. The universe is finite? Craig believes that the actual infinite is to be found nowhere in modern physics. He goes on to flatly deny, echoing Hilbert, that the actual infinite is found nowhere in the world. But, is this really so? Does modern physics deride the infinite? I disagree.

Consider Craig's own interpretation of cosmology, namely, a relativistic Big Bang theory. Now, the thing about GTR-cosmology is that they explicitly involve singularities and continuous time. This has consequences for Craig's denial of the actual infinite. First, singularities are states such that there is an infinitely dense curvature of space. So, singularities signify an actual infinite. Secondly, continuous time corresponds to the real-numbers via bijection, hence, set of moments has a transfinite cardinality, namely, aleph-one. There is an infinity of moments and intervals in a continuous time.
Craig has a strange reply to the latter objection. He argues that time is only "potentially" infinitely divisible. He says that we can continue dividing indefinitely, but that we will never complete an infinity of divisions. However, this is highly confused. It is not as if the actual sub-regions do not exist prior to one's conceptual division as such, but that the sub-regions already exist. It is not as if, when moving through a continuous line, one has only passed through a "potentially" infinitely divisible line, but that the line already had an infinity of sub-regions. Either it is the case that between each moment there is another moment and has a bijection with the reals or not. If not, then time is discrete; i.e. there are moments such that there is no moment between them. And if the universe began to exist, then such a topology of time has its set of moments with a finite cardinality. However, if between any two moments there is another moment and there is a bijection with the reals, it follows that time is continuous and is actually infinite. Craig's mistake here is similar to his mistakes in his argument against general infinities, namely, a misunderstanding of how mathematics applies to physical theories. We take the physical theory and a mathematical structure maps on to it such that entailments in that structure itself are entailments for the physical theory. I also provided a few examples of how this procedure works. The claim that between any two moments there is another moment maps onto to the proposition that between any two points there is another point. But, this mathematically entails an infinity of points and sub-regions and likewise entails an infinity of moments and intervals. In any instance, there is no problem of "infinitely division", since we can simply define the continuity of time at one stroke i.e. the rule that between any two moments, there is another moment.

This is sufficient for showing that Craig cannot reasonably hold to the general argument against infinity and at the same time hold to the relativistic cosmology he espouses.5. Intuition
It may be worth pointing out that Craig's general argument against the actual infinite and his support of the CAP is largely intuitive. I wish to examine the role of intuitive support in the KCA.

5.1. The reliability of intuitionIntuitions about abstract matters, or more accurately, matters sufficiently removed from our immediate human perspective, more often than not, tend to be wildly wrong. Consider the various intuitions that humanity has shown wrong: (a) velocities are additive (b) there is an "absolute" time (c) space is Euclidean (d) time and space are separate entities (e) matter and energy are separate entities and so on. All of these intuitions, which seem very reasonable to us and counter-intuitive, if not "absurd" to deny, are entirely wrong. Special and general relativity alone have falsified all of the above intuitions. So, it may well be insufficient to cite intuitive generalization of this type.

The point is that, in our upbringing, we have found various "rules of thumb" that work in the cosmos of our collective human experience. They include intuitions such as (a) through (e), that all things have a cause, that all things have a beginning, and so on. These certainly work in the macroscopic realm we occupy on this Earth. But, this hardly indicates that these generalizations are really more than crude, pragmatic "rules of thumb", the reality being far stranger and far more wonderful than we would merit. The entire history of quantum mechanics is an interplay of such tensions and, if anything, this indicates how cautiously we must apply our intuitions to matters sufficiently removed from our experience. So, the KCA, in using such intuition, appears to make the same methodological mistake, and we cannot grant much evidentiary value to them, insofar as they portend to express uniform principles of the universe that are also necessary.

5.2. The conflicts of intuitionIt is also clear that our intuitions often conflict, which raises even more doubt as to their total evidentiary value in matters as abstract as these.

Consider the intuition that we have that every event we find is preceded by another event that brings it about. This is effectively determinism or the totality of the CAP. Now, we also have strong intuitions regarding libertarian free will. The reasoning goes as thus: we have a Principle of Alternate Possibilities, where we argue that someone cannot be free with respect to an action unless they could have done otherwise. What libertarians argue is that, given the antecedent conditions, I still could have done otherwise. In other words, the antecedent conditions do not determine my actions. But, that is to say that the state of affairs consisting of me acting has no cause. So, libertarian free will and universal causation are mutually incompatible. So, here, we have two strong intuitions that nonetheless conflict.

Or, consider Euclidean space versus the counter-intuitive nature of actual infinites. Now, non-Euclidean space is very counter-intuitive; for instance, in a Riemannian space, parallel lines always meet, whereas in a hyperbolic space, parallel lines get further and further apart. So, our intuitions favor a Euclidean universe. And yet, a Euclidean universe is infinite in extent. So, it appears that our intuitions regarding Euclidean space and actual infinites are conflicting. Moreover, we have a pretty clear intuition that beyond any point of space, there is another. But, it seems absurd to think of an "end" to space, because that would invite the question what lay beyond it. Well, nothing at all; it is the end of all space. And yet, if we deny an end to space, we accept an infinite extent. But, even with an infinite extent, we tend to think of the universe as a contained "whole"; and yet, if the universe is infinite in extent, it would appear that the universe is not really contained. It wouldn't really make sense to discuss an infinite universe in that aspect. And yet, if we deny that, we move back to a finite space.

Or, consider discrete time versus continuous time. On one hand, discrete time seems very counter-intuitive, because it seem absurd that time "jumps" and does not actually flow naturally from past to present in a continuous inexorable fashion. So, between any two moments, there would seem to be another moment. And yet, this implies an actual infinity, since an infinity of moments would be between each moment. And yet, a continuous time defies the notion of there being a "next" state. For example, consider causality. If each event brings about the "next" event, since time is continuous, then it follows that there really is no next event, but that there is a continuous progression of events. But, for there to not be a "next" moment like there being a next segment would also seem counter-intuitive as well. And yet, to deny that would be to accept a discrete time.

Or, consider an infinite duration of time versus a beginning of time. The former seems incoherent because it invites the question of how time ever arrived at the present. So, it would appear that we would have to accept a past with a beginning. And yet, it would seem plausible to believe in the inexorable flow of time, where each moment comes into existence and then passes, each moment of time preceded by another. And yet, this view of time entails an infinite past. So, the idea of a past with a beginning is repugnant to us. And, yet so it is without one!

This doesn't mean that any of the above intuitions in this section are really reasonable. Indeed, I wince at the egregious lack of understanding in many of them, some of which we have already detailed as being confused. But, the point is that these are natural intuitions, intuitions that we strongly believe in, and yet intuitions that violently conflict with each other. If our intuitions are all over the place and contradict one another in the abstract, how can one reasonably assign strong evidentiary value for such intuitions for abstract metaphysical principles? The point is that you cannot, and hence, Craig's appeals to intuitions are largely unfounded.

What's interesting to note is Craig's selectivity of what intuitions to accept. He freely dismisses uncaused events and infinities as "absurd", and yet he has no problem with timeless entities, when our intuitions (and logic) tell us that time is required for causality, and disembodied minds, when our intuitions tell us that minds that interact have material brains and that the nonphysical cannot affect the physical. This sort of egregious selectivity amounts to little more than special pleading on Craig's part. If strong intuitions about causality and time indicate a caused origin of the universe, then what about strong intuitions regarding timeless, disembodied minds?6. Conclusion
Well, what we have learned after all this? Given the above preponderance of errors, we find the Kalam Cosmological Argument to be a comedy (or a tragedy) of mistakes, of mathematical errata, of astronomical misconceptions, and of metaphysical egregiousness. It has not shown that everything that begins to exist has a cause, it has not shown the universe to come into existence or even begin to exist, and it has not shown that the cause of the universe would be personal. Does it establish the existence of God?

Not by a long shot.7. Bibliography
Craig, William Lane. "Prof. Grünbaum on Creation." Erkenntnis 40 (1994): 325-341.

Craig, William Lane. "Must the Beginning of the Universe Have a Personal Cause?: A Rejoinder." Faith and Philosophy 19 (2002): 94-105.

Craig, William Lane. "The Existence of God and the Beginning of the Universe." Truth: A Journal of Modern Thought 3 (1991): 85-96.

Craig, William Lane. "The Origin and Creation of the Universe: a Reply to Adolf Grünbaum." British Journal for the Philosophy of Science 43 (1992): 233-240.

Dever, Josh. “Worlds Apart: On the Possibility of an Actual Infinity.” Taiwanese Journal For Philosophy and History of Science 10 (1998): 95-116.

Grünbaum, Adolf. “Creation as a Pseudo-Explanation in Current Physical Cosmology.” Erkenntnis 35 (1991): 233–254.

Grünbaum, Adolf.“The Pseudo-Problem of Creation in Physical Cosmology.” Philosophy of Science 56.3 (1989): 373–394.

Morriston, Wesley. "Must the Past Have a Beginning?" Philo 2.1 (1999).

Morriston, Wesley. "Must the Beginning of the Universe Have a Personal Cause?" Faith and Philosophy 17.2 (2000): 149-169.

Morriston, Wesley. "Causes and Beginnings in the Kalam Argument: Reply to Craig." Faith and Philosophy 19.2 (April 2002): 233-244.

Morriston, Wesley. "Creation ex Nihilo and the Big Bang." Philo 5.9 (Spring-Summer 2002).

Morriston, Wesley. "Craig on the Actual Infinite." Religious Studies 38.2 (June 2002).

Morriston, Wesley. "Must Metaphysical Time Have a Beginning." Faith and Philosophy 20.3 (July 2003): 288-306.