Newcomb's Paradox provides an illuminating non-theological illustration of the problem of divine foreknowledge and human freedom. We are to imagine a being with great predictive powers and to suppose we are confronted with two boxes, B1 and B2. B1 contains $1,000; B2 contains either $1,000,000 or nothing. We may choose either B2 alone or B1 and B2 together. If the being predicts that you choose both boxes, he does not put anything in B2; if he predicts that you choose B2 only, he puts $1,000,000 in B2. What should you choose? A proper construction of the pay-off matrix for the decision vindicates the one-box choice. If this is correct, then those who claim that God's knowledge is counterfactually dependent on future contingents foreknown by Him are likewise vindicated.
Let's analyze this problem.
I. Initial Analysis
As an initial analysis, let's assume that (a) we want more money than less money, and (b) the predictive ability of the being is perfect, not just very great.
In that case, the solution is obvious, pick B2. Result %100 chance of receiving $1M.
II. Return to the Presented Question
When we take away (b), such that the predictive power is great, but not perfect, we are faced with options that are somewhat more complicated. We'd like to increase our amount from $1M to $1.001M, but we don't want to receive only $0.001M or, even worse, nill.
There are four possible scenarios:
1. You pick both, and the being predicts that correctly.
2. You pick both, but the being predicts you pick only B2.
3. You pick B2, and the being correctly predicts that.
4. You pick B2, and the being predicts that you pick both.
If each were to occur, the results (payout) would be as follows.
Looking only at your payouts, there is not a monotonic preference between picking B2 or both. In other words, there is no one choice that is always better than the other.
A. Strategies that do not involve assessing the being's predictive ability
There are a couple game theory strategies that you could employ at this stage. If you were aiming to minimize your worst outcome, you would pick both. This is called the "maxi-min" strategy. Employing that strategy, you are guaranteed at least $0.001M, and you could get lucky and strike it rich. A person whom we would consider very risk averse would choose both to avoid getting nothing.
Likewise, a very risk loving person would (using a different game theory strategy) choose both, becuase doing so provides the possibility of the maximum payout. This is called the "maxi-max" strategy. Employing this strategy, there is some chance (however remote) that you will obtain the big prize of $1.001M. Such a person would be willing to take the risk that he could get the low payoff in the hope of obtaining the big prize.
Notice that neither of these strategies requires any knowledge about how good the being is at predicting, as long as the being does not have perfect predictive ability.
B. Strategies that requiring assessing the predictor
i. Expected Value
The most analytical strategy for making a selection is based on assessing the ability of the predictor. This is the expected value strategy.
In the expected value strategy, we multiply the payout for each outcome of each of the scenarios by the chance of the scenario occurring. Thus, we would have the following, in which P is the probability that the predictor predicts correctly, expressed as a decimal.
1. $0.001M x P
2. $1.001M x (1-P)
3. $1.000M x P
Item number 4 remains zero, because zero times anything is zero.
Notice that there are two pairs A (1,2) and B (3,4). The expected value strategy cannot pick just one of the items, it must pick one of the pairs. The value of each pair is the sum of the outcomes within the pair. The expected values of the pairs is:
A. ($0.001M x P) + ($1.001M x (1-P))
B. $1.000M x P
B is equal to 3, because adding zero adds nothing.
Let's examine a few sample values for P.
If P = 0, A = $1.001M and B = Nada.
Such a scenario is the scenario where the predictor always picks wrong.
In contrast, if P = 1, A = $0.001M and B = 1.000
Such a scenario is the scenario where the predictor always picks correctly.
In the first scenario, the Expected Value strategy would be to pick A, and in the second scenario, the Expected Value strategy would be to pick B. As you might expect, as P increases from zero chance to a 100% chance, A loses favor to B.
The precise point at which the Expected Value stategy says both A and B are equally good, is:
($0.001M x P) + ($1.001M x (1-P)) = $1.000M x P
which simplifies to
$1.001M = $2M x P
or P = 0.5005
When P = 0.5005, A = $0.5005M and B= $0.5005M
When P is less then 0.5005, A is going to be higher than B, and when P is greater than 0.5005, B is going to be higher than A.
Thus, if we believe that the being's predictive power is greater than 0.5005, and we use the expected value strategy, we will pick B, and if we believe that the being's predictive power is less than 0.5005, we will pick A.
If we had no information about the predictive being's guessing abilities, we might default to the view that the predictive being guessed correctly 50% of the time. If we had that view, and we employed the expected value strategy, we would pick A.
But the problem stipulates that predictive being does not just have average guessing ability, but that the predictive being has great predictive ability. Accordingly, we would be inclined to pick B using the expected value strategy, as long as we believed that 0.5005 is not >> 0.5.
C. Second Guess Strategies
In addition to the strategies that evaluate risk or payoff alone, and the strategy of using expected value, there is an additional level of strategy that is intuitively attractive. This intuitively attractive strategy is to try to predict the prediction of the predictor.
In other words, we ask the question, what would the predictor expect me to pick? If we predict that the predictor will predict our selection of B (i.e. that we pick B2), then we will actually pick A (i..e. both boxes) because then we will have $1.001M instead of $1.000M. If we predict that the predictor will predict our selection of A (i.e. that we will pick both boxes), then we will pick A again because that way we will have $0.001M instead of nada.
If we employ this second guess strategy, we will always pick A, because (regardless of what we think that the predictor will predict) the result is that A will be better.
III. Comparison of the strategies
There is really no one strategy that is better than the other strategies. One might argue that the Expected Value strategy is the best strategy, but such a strategy assumes that one is willing to accept the risk of a zero payout, and willing to lose the thrill of the maximum payoff. Likewise, such a strategy assumes that one cannot second guess the predictor and try to choose based on predicting the predictor. The strategy also assumes that the predictor has a predictive ability that exceeds (or at least equals) 0.5005.
IV. Comparison to W. L. Craig's claims
Recall that W.L. Craig had claimed:
A proper construction of the pay-off matrix for the decision vindicates the one-box choice. If this is correct, then those who claim that God's knowledge is counterfactually dependent on future contingents foreknown by Him are likewise vindicated.
As has been shown above, a pay-off matrix "vindicates" the one-box choice if and only if the other available game strategies are ignored, and the predictive ability of the being is assumed to be greater than a certain threshold. Such a qualified vindication cannot be said to vinidicate those who say that "God's knowledge is counterfactually dependent on future contingents foreknown by Him."
Additionally, there is no apparent link between this game and the point to be proved. But let us briefly review and critique W. L. Craig's own analysis detailed analysis of the problem.
V. A Common-Sense Strategy
The common sense strategy is simply to deny that what one does now will have any effect on what is in the second box. That is to say, either the predictive being placed the money in the box already or he did not. If he did, picking both boxes won't hurt, and if he did not, picking only the second box is foolish, because it automatically discards the $0.001M in the first box, leaving you with nada.
VI. A critique of W. L. Craig's analysis (Link to his analysis, here.)
In W. L. Craig's article, he favors the expected value strategy, while noticing various objections, including the common sense objection noted above.
W. L. Craig's main point appears to be that God already knows what you are going to pick, i.e. P = 1. As we have shown above, if P = 1, we should choose B2, in order to maximize our wealth.
The problem is that such a state of perfect predictability requires, as a mechanism, backwards causation. In other words, the only reason to favor the one-box strategy over the common-sense strategy, is if one believes that one's choice will actually affect the contents of the box. But the contents of the box have already been determined. Accordingly, one's choice must have a causal effect in the past, in order for one's choice to matter in whether the box is empty or full.
WLC attempts to interact with this rebuttal in a manner that reminds me greatly of Godismyjudge's comments in previous posts. Here goes:
This analysis, however, seems to rest upon a misunderstanding in which the causal relation between an event or thing and its effect is conflated with the semantic relation between a true proposition and its corresponding state of affairs. For if at tn I choose B2 alone, then the proposition "W chooses B2 alone" is true at tn because of the semantic relation which obtains between a true proposition and the corresponding state of affairs which makes it true; by the same token " W will choose B2 alone" is true prior to tn. "W chose B2 alone" is true subsequent to tn, and " W chooses B2 alone at tn" is omnitemporally true. The relation obtaining between a true proposition and its corresponding state of affairs is semantic, not causal. Now God, knowing all true propositions, therefore knows the true future contingent proposition concerning my choice of the boxes. Again no causal relation obtains here. Hence, the charge of backward causation seems entirely misconceived: we have simply the semantic relation between true propositions and their corresponding states of affairs and the divine property of knowing all true propositions.
Here WLC makes much the same error that we have seen previously in Godismyjudge's posts. There is confusion between the proposition and the event. This confusion is somewhat subtle. Let's see if we can make it evident in a line-by-line analysis.
WLC writes: "This analysis, however, seems to rest upon a misunderstanding in which the causal relation between an event or thing and its effect is conflated with the semantic relation between a true proposition and its corresponding state of affairs. "
I respond: This is WLC's thesis, that cause-and-effect relationship has been confused for "the semantic relation between a true proposition and its corresponding state of affairs." Before we rush to judgment regading this thesis, let's see what WLC tenders as support.
WLC writes: "For if at tn I choose B2 alone, then the proposition "W chooses B2 alone" is true at tn because of the semantic relation which obtains between a true proposition and the corresponding state of affairs which makes it true; by the same token " W will choose B2 alone" is true prior to tn."
I respond: Let's leave aside the portion after the semi-colon for now. Looking at the first part of WLC's assertion, you will notice the following terminology, "... if ... then ... true ... because ... which makes it true." This is the language of causation, with the effect being the truth of the proposition. If the word "broken" were used instead of "true," no one would have the least doubt that the omitted portion of the quotation explained how the object came to be broken.
Calling this relationship between the cause of the proposition being true, and the truth effect of the cause a "semantic" relationship, appears intentionally obfuscatory. The term "semantic" usually conveys the sense of "relating to meaning," and derives from the Greek word semantikos, which indicates signification. Yet, WLC's explanation is one of cause and effect, not signification or meaning.
In other words, the correspondence to reality is the reason that a statement about reality is either true or false. That relationship is a causal relationship. If reality is as the statement says, then the statement has a "true" truth value. If reality is contrary to the statement, then the statement has a "false" truth value. In other words, the state of reality is the cause of the truth/falsehood value of propositions that describe reality.
WLC appears to attempt to bolster his view by the following example pair: ""W chose B2 alone" is true subsequent to tn, and " W chooses B2 alone at tn" is omnitemporally true. "
I respond: This difference between the two statements is something semantic, it has to do with the relation between the action verb and time. The first statement employs the past tense, and thus is making a claim regarding the past. The second statement employs language that simply states an event at a particular point in time (without a relative reference) and thus states a general, or gnomic, truth. As such, it avoids the semantic problem of the first statement.
Nevertheless, the same causal relationship exists between the second statement and the event. The second, gnomic, statement is true or false as result of its correspondence or lack thereof to reality. The fact that there are semantic differences between different descriptions of an event does not mean that the proposition's truth is causally independent of the event. Such a conclusion cannot possible follow from the demonstration provided.
WLC then simply asserts: "The relation obtaining between a true proposition and its corresponding state of affairs is semantic, not causal."
I respond: But we have shown that the truth value is causal, and even WLC's own explanation testifies to the causal relationship.
WLC continues: "Now God, knowing all true propositions, therefore knows the true future contingent proposition concerning my choice of the boxes."
I respond: Since WLC acknowledges that God knows all true propositions, it would appear that WLC is acknowledging that God not only knows them, but knows their truth or falsehood value. If so, then there is no way to exclude from God's knowledge the gnomic absolute proposition concerning the act of choosing the boxes. For if God knows the proposition, and if God knows that the proposition is true, then God knows the reality of the act/event, because God surely recognizes that the only proposition to be true is for it to have a positive correspondence to reality. If not, (i.e. if God does not know which propositions are true, and which are not true), then knowing the "the true future contingent proposition concerning my choice of the boxes," cannot help Him.
Either way, WLC's argument again collapses. As shown above, God knows the absolute gnomic truth of the proposition "A picks B2 at tn," regardless of whether God additionally knows the reason why the event occurs (either "contingently" or "determinately").
WLC again simply states: "Again no causal relation obtains here."
I respond: And of course, WLC's saying three times does not make it so. We have shown above, that there is a cause and effect relationship between the proposition's truth and reality. That's enough to show backward causation IF God has advance knowledge of the truth of the proposition.
WLC then concludes: "Hence, the charge of backward causation seems entirely misconceived: we have simply the semantic relation between true propositions and their corresponding states of affairs and the divine property of knowing all true propositions."
I respond: I have shown above why this conclusion is incorrect. There is more than just a semantic relationship between gnomic propositions and events - if the gnomic proposition corresponds to reality, then (and as a result of that correspondence) it is true. That is a causal relationship, and even WLC's own explanation (as demonstrated above) shows that it is a causal relationship.
Since WLC acknowledges that backward causation is a problem for his view of what he calls God's "foreknowledge," we are in a position to refuse to accept WLC's view of God's omniscience.
WLC has the opportunity to make one more attempt to address the shortcomings of his argument. WLC provides the following discussion and objection.
Choosing B2 alone is the right strategy, but one must live with the "uncomfortable knowledge" that at the time of choosing B2 alone God's belief is "unalterably tucked away in the past" and there is really $1,001,000 in the boxes. After choosing B2 alone one must be prepared to say, "If I had chosen both boxes, I would not have gotten the $1,001,000. " But an opponent might retort, "Of course you would have, since it was there! Therefore, you must not have been free to choose both."
Interestingly, WLC never finds time to the opponent's retort described above.
VII. A Reformed alternative
The Reformed alternative is to deny that God acts based on what he foresees us doing. The Reformed alternative indicates that God has, out of his infinite wisdom, foreordained all that will occur, and that includes both His and our actions. God is immutable and impassive. Acordingly, nothing that a creature does can cause God to do anything.
Thus, the Reformed critic can look at the problem two ways:
1) as absurd, because God knows the future as a logical result of knowing His eternal decree, not as a result of prediction,
2) as absurd, because whatever God decided to put in the box is already in the box, and will not come out, or
3) with slight modification, as a confusing revelation: i.e. God has revealed the consequence of picking one or two boxes. If two boxes are picked, the person will receive $0.001M and if one box is picked, the person will receive $1.000M; but how that would be implemented is simply a mystery, because it would appear that the only way that prophecy could be true with respect to the one-box-choice is if there is already the cash in the box, and likewise that for the two-box-choice if there is already no cash in the box. The solution being that God is behind both the cash placement and your choice. If God has ordained that you will choose both, he has already prepared no cash in box 2, and likewise if God has ordained that you will chose box 2, he has placed in that box. This solves the problem, solves the paradox, and leaves the reader wondering why WLC holds to an autonomous view of man in the first place.
WLC leaves a couple of gems in his own conclusion, and it is worth touching on them here.
A proper understanding of the counterfactual conditionals involved enables us to see that the pastness of God's knowledge serves neither to make God's beliefs counterfactually closed nor to rob us of genuine freedom. It is evident that our decisions determine God's past beliefs about those decisions and do so without invoking an objectionable backward causation.
Neither of these propositions have been established in WLC's article, nor can they be. Furthermore, the straw man is the "counterfactual conditionals" in contrast to the gnomic absolutes. God both knows the truth of hypothetical scenarios, but also the reality of the future. Finally, the second sentence is shocking in obvious self-contradictory nature. "... determine ... past ... without invoking an objectionable backward causation." The only possible hope for WLC to extricate himself from the self-contradiction would appear to be to rely on the "an objectionable" to qualify (rather than describe) the kind of backward causation that the present determination of a past event implies.
And of course, that's the penultimate problem with WLC's position: it is internally inconsistent and self-contradictory. The ultimate problem is that it is unscriptural. That it is internally inconsistent and self-contradictory is an outworking of its failure to begin from the presuppositions contained in the Word of God.
In short, in this critique of WLC, we see the practical impact that a flawed epistemology has.