A Problem with Probability

A Problem with Probabili ty: Decision-justification and Carnap’s Principle of Total Evidence

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Chris Lang, University of Wisconsin-Madison 5185 Helen C. White Hall , 600 N. Park Street, Madison, WI 53706

(608) 264-0817, email: CCL6@cornell .edu

Contents

The “ Problem of Application” of Induction Page 1

Carnap’s Pr inciple of Total Evidence Page 2

Objections to Carnap’s Pr inciple of Total Evidence Page 3

Carnap’s principle is counterintuitive Page 4

Carnap’s principle itself lacks justifi cation Page 4

Carnap’s principle is impracticable Page 5

First proof that justifi cation requires conclusive evidence Page 7

Second proof that it requires conclusive evidence Page 8

Common Objections to the Demand for Conclusive Evidence Page 9

Explaining the behavior of gamblers and casinos Page 9

Explaining the success of science Page 11

Explaining “ random sampling” and the role of

probabilit y in quantum mechanics Page 12

Conclusions Page 13

References Page 14

Appendix Page 15

1 For feedback on earlier drafts, I thank the participants of the 2nd Annual Graduate Conference in the Philosophy of Logic, Math and Physics, Elliott Sober, Branden Fitelson, Zach Ernst, Joel Velasco and especially Ellery Eells.

A Problem with Probabili ty

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The “ Problem of Application” of Induction When philosophers of science ask, “What good can come from science when

its models of the world are so often inaccurate?” we may answer that science produces decision-justification, that it can tell us how we should act, even if it lacks some precision and/or accuracy in describing that upon which we are acting. For example, when we see a meteor headed towards our vicinity, science indicates a trajectory from which we should try to move away, and it does this accurately and precisely even if we don’ t realize that true laws of mechanics must account for relativistic effects and that the meteor is actually composed of many subatomic particles, each with their own trajectories, and that our perception of the meteor could be an ill usion (etc.). This resolution of the Realism debate, the recognition that science’s production of decision-justification shields its value from any epistemological problems there may be with knowledge about the external world,2 raises the new question: “Does decision-justification have its own potential limitations comparable to those attributed to knowledge about the external world?”

There are at least two reasons to separate the concept of decision-justification from that of knowledge about the external world. First, of course, is the aforementioned potential to circumvent the worries raised by Hume and other skeptics about such knowledge. The second reason is that scientists and those who fund science care about decision-justification independent of their quest for knowledge about the external world. That is, even if science succeeds in discovering the truth about the external world, it cannot be considered a complete success unless it also produces decision-justification. The FDA must justify their decision to permit the sale of a given drug; courts must justify their decisions to convict defendants; sellers must justify their decision to raise or lower their prices; builders must justify their decision about what materials to use; publishers must justify their decision about what to print; and governments/companies must justify their decisions about what artistic and scientific projects to fund. If science cannot facilit ate such decision-justification, then it is, as Carnap put it, “ inapplicable”.3

Justified decisions are (1) above criticism—they are proven to be “the best one could do given the circumstances” . In addition, it is often thought that they are (2) praiseworthy—they are based on something like “suff icient evidence”. For example, if the ball of dung is removed from the grasp of a dung beetle en route to plug its nest, it will pantomime the plugging ritual as though the ball had not been removed. Although the beetle would be doing the best it can, we would hardly call such actions the results of justified decisions. The problem is that, although the beetle

2 Note that neither the Realist nor the Anti-realist defeats the other in this resolution. Rather, their dispute is simply treated as uninteresting. 3 Carnap (1947)


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may base its decision on evidence that it holds dung (i.e. it collected it), that evidence is not considered “sufficient” to justify the decision to engage in the plugging ritual. If the FDA (etc.) similarly did “ the best they could” but acted on insuff icient evidence, we would be similarly inclined to mock them and take littl e or no precautions against accidentally squashing them.4 Thus, it is the praiseworthy sort of decision-justification that people usually demand from science.

Regarding the problem of application of inductive logic, Carnap presented the issue of suff icient evidence by describing a researcher interested in the prediction that a given Chicagoan will be found to have red hair.5 The researcher shows Carnap a list of 40 Chicagoans, only 20 of which have red hair, and asks whether he could (justifiably) apply a probabili ty estimate of 0.5 (i.e. 20/40) to the prediction. When Carnap asks whether the list of 40 Chicagoans represents all of the researcher’s observations of Chicagoans, the researcher replies that he had actually observed 400, but wasn’ t interested in the other 360 because all of them had previously been found not to have red hair. To this, Carnap responds that the 0.5 probabili ty estimate is not applicable (and I should think we’d all agree) because it would be based on insuff icient evidence. The purpose of the current essay will be to determine what constitutes suff icient evidence in the general case.

Carnap’s Pr inciple of Total Evidence To formally express the requirement for suff icient evidence, Carnap needed a

principle beyond the standard axioms of probabilit y calculus developed by Kolmogorov: (K1) No probability is less than zero nor greater than one. (K2) If a proposition necessaril y must be true, then its probability is one. (K3) The probability of the disjunction of two incompatible propositions is equal to

the sum of their probabilities.

Much follows from these axioms, but one thing that does not follow from them is any sort of claim that probabili ty calculations justify decisions. Let’s me offer another example to demonstrate why. Suppose we had a machine that flips coins, that it had been flipping coins (one per minute) for the last 100 years, amazingly and consistently alternating between heads and tails, and that, if the machine continued to alternate, the next flip would yield heads. Suppose someone offered 10-to-1 odds that the next flip will produce tails. How would the probabili ty calculus tell us whether we would be justified in refusing their offer? On the one hand, if one supposed (as one usually does) that the probabili ty of tails on each flip were the relative frequency of tails on all previous flips, then the probabili ty of winning the bet would be 0.5, so the expected value of taking the

4 I mean the organization, rather than its employees, of course! 5 Carnap, R. (1947), page 139.


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bet (according to neoclassical economics) would be vastly greater than that of not taking it. In contrast, if one supposed that the probabili ty of tails on any even flip were the relative frequency of tails on all previous even flips (and analogously for odd flips), then both the probabili ty and expected value of taking the bet would be very low. Thus, in order to “apply” the probabili ty calculus we need a means to choose between such probabili ty calculations.

Let’s call the various potential methods for calculating probabili ty “accounts” of probabili ty. Some accounts, such as the naïve blanket policy of assigning a probabili ty of 0.5 to all events, can be dismissed on the grounds that they violate Kolmogorov’s axioms.6 Not all accounts can be dismissed on such grounds, however. For example, neither of the accounts named above contradict Kolmogorov.7 Therefore, we need an additional axiom to make the probabili ty calculus applicable. Adding this fourth axiom to Kolmogorov’s three would yield a new calculus—one of “decision-justifying probabili ty” . Carnap proposed the following principle to serve this purpose:8

Carnap’s Pr inciple of Total Evidence: If [evidence] e expresses the total knowledge of X at time t, that is to say, his total knowledge of the results of his observations, then X is justified at this time to believe [hypothesis] h to the degree r [where r is the result of applying inductive logic to e and h], and hence to bet on h with a betting quotient not higher than r.

To this claim that the application of induction on total evidence does justify beliefs (and presumably the decision-making that follows from them), Carnap goes on to add that less than total evidence cannot justify beliefs, except in the application of deduction. In the case of the coin-flipping machine, this principle of total evidence would allow us to see that we mustn’ t apply the first account of probabili ty, since the second account employs more evidence, namely, the additional fact that the machine had been alternating between heads and tails. Note that total evidence involves both the amount of data (i.e. number of coin flips) and the number of potential patterns for which we check in that data.

Objections to Carnap’s Pr inciple of Total Evidence

Carnap seems to be right that the incorporation of total evidence is necessary for decision-justification in induction but wrong that it is sufficient. It seems possible for someone to lack access to the evidence required for justification, to find themselves in a situation like that of the dung beetle with no possibili ty of justification. I will present five

6 According to this account, Pr(a)=Pr(~a)=Pr(a&~a)=0.5, which contradicts K3: Pr(a&~a)=Pr(a)+Pr(~a). 7 Since it is logically possible for the machine to deviate from its previous pattern, we cannot appeal to K2 to dismiss the first account. 8 Carrnap (1950), page 211. Actually, he credits Bernoulli, Keynes and Peirce with recognizing the principle earlier (albeit less formally).


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arguments to this effect: (1) Carnap’s principle is counterintuitive, (2) Carnap’s principle itself lacks justification, (3) combined with practical matters, Carnap’s principle entails that induction is impossible, (4) from the premise that justification rests on evidence, we can prove that suff icient evidence must be conclusive, and (5) from the premise that our goals are about things yielding consequences that we can experience, we can prove that all decision-justifying probabiliti es are either 0 or 1.9

Carnap’s pr inciple is counterintuitive. While, on the one hand, it is certainly intuitive (or traditional) to suppose that one should not use less than the total relevant evidence at one’s disposal, the claim that total evidence is suff icient has counterintuitive consequences. Specificall y, no matter how much evidence is available, Carnap’s principle entails that it yields a decision-justifying probabili ty. Thus, it entails that decisions can be justified with no evidence at all , if that’s what we had. But a decision based on no evidence at all could hardly be praiseworthy! Carnap’s principle seems even more counterintuitive when we imagine a situation in which the justifiable action is to try to collect more evidence before making the decision at hand. If someone in that situation were to subsequently learn that a powerful demon will prevent them from gathering further evidence, then they would cease to be justified in trying to collect additional evidence, so they would become justified in making their decision based on the evidence at hand. Whereas we would intuitively suppose such demons to be an enemy to our quest for justification, Carnap’s principle would entail that they would aid us by elevating the justificatory powers of our evidence!

Carnap’s pr inciple itself lacks justification. Justification cannot be achieved

through any old set of axioms of probabili ty one cares to propose, of course; the axioms themselves must be formally justified (i.e. intuitions don’ t cut it). The search for such justification of Carnap’s principle was an open problem in philosophy of science for twenty years. One diff iculty was that we tend to face conflicting evidence—in which case, although some may be truth-pointing, some must be misleading. If the majority of the evidence we face happens to be misleading,10 then our total evidence would be more misleading than the average subset of it, so Carnap’s principle would tend to lead us astray (on average). To know that misleading evidence is rare would seem to require a spectacular measurement of the peculiarities of our universe. The currently popular

9 It follows from these arguments (many times over) that the bounded optimality problem of artificial intell igence is intractable (i.e. one cannot rationally cut corners in decision-justification). 10 One need not resort to stories of trickster demons to conceive this possibility. For the majority of our own history, humanity has been led away from recognizing the atomic, relativistic and quantum realiti es of our universe. One might even doubt that the average modern non-physicist really sees past the ill usion of independent solid entities. Thus, history shows that humans have faced a lot of misleading evidence.


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justification, first proposed by I. J. Good (1967),11 tries to use mathematical slight of hand to avoid the need for such measurement.

Good produced an elaborate mathematical proof concluding that the incorporation of additional inconclusive evidence must usually lead to higher maximum “expected utiliti es” in rational decision-making. He then (mistakenly) claimed that this conclusion entails that inconclusive evidence usually will not lead us astray. The slight of hand is achieved by confusing the following two behaviors:

(1) selecting actions so as to maximize expected utili ty when probability estimates

are held fixed, and (2) selecting evidence so as to maximize expected utilities (via altering our

probabili ty estimates). The first of these behaviors is the very definition of rational decision-making, but the second is not rational at all—it amounts to inflating our expectations.

Rational agents seek to maximize actual utility, but want their expectations to be accurate (which isn’ t necessarily maximum). For example, if my brother told me that a particular lottery number I can purchase for $1 has a 50% chance of winning $1M, then believing him would raise the maximum expected utili ty on my lottery ticket purchase decision from less than that of $1 to about that of $500K. Whereas I would previously have expected to lose money on average, believing my brother would make me expect to win big on average by playing the number he recommended. However, the increase in expectation that would result from including my brother’s claim among the evidence I incorporate is not a rational reason to include it. My decision about how much to trust or distrust my brother should not be influenced by the pleasantness of the picture he paints—it should be based entirely on other factors. Agents who select evidence so as to maximize expected utili ty as Good proposed are not “ rational”—they’ re “optimistically gulli ble”!

Thus, although Good deserves credit for recognizing the importance of justifying Carnap’s principle, his attempt to accomplish that justification failed. Since different possible universes might have different proportions of misleading evidence, the justification of Carnap’s principle cannot be had a priori. Thus it remains unjustified.

Carnap’s pr inciple is impracticable. The practical problem with Carnap’s

principle derives from the fact that one’s total evidence generally cannot be enumerated and thus cannot be what scientist and other statisticians incorporate in their calculations. As an example of a failure to use total evidence, Carnap’s cited Laplace’s calculation of the probabili ty that the sun will rise tomorrow. Laplace noted that history had recorded 1,826,213 previous sunrises and then applied his rule of succession to determine that the

11 Skyrms (2000), page 155, makes essentially the same argument.


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probabili ty of the next sunrise is 99.9999452% (i.e. 1 – 1/1,826,215). Carnap says that Laplace violated the principle of total evidence by faili ng to account for other observations that function as confirming (or disconfirming) instances for the laws of mechanics. These additional observations are relevant evidence because tomorrow’s sunrise would be an instance of those laws. But here we see that it would be impractical to incorporate into our formal calculations every observation we have ever had of mechanical phenomena. ”Thus,” wrote Carnap, “examples of the application of inductive logic must necessarily show certain fictitious features and deviate…from situations that can actually occur...” 12

Carnap believed that increasing rigor would allow us to reduce the impracticali ty of inductive logic and thus converge on applicable induction much as scientific advancement allows us to converge on scientific truth. The obstacles to such convergence are much worse than Carnap originally thought, however. Consider the case of the coin flipping machine—since the application of total evidence requires not only that we account for every flip but also that we look for every possible pattern in the data,13 we must in general check not only for patters of alternation but also for all other periodic patterns (i.e. triplet patterns, quadruplet patterns, etc), asymptotic patterns (such as heads on all prime numbered tosses) and chaotic patterns. But chaotic patterns can be indistinguishable from randomness, so they cannot be checked. Thus, even computers (which may very well have reasonably small data sets at their disposal) cannot abide Carnap’s principle.

On top of that, we face the diff iculty that arises from the subconscious nature of observation. The very best scientists use intuition,14 and that entails using evidence of which they aren’ t even aware. As Albert Einstein described it,

Only something which did not in similar fashion seem to be ' evident' [i.e. in the fashion that geometric facts seem ' evident' ] appeared to me to be in need of any proof at all ...thus it appeared that it was possible to get certain knowledge of the objects of experience by means of pure thinking...15

Dr. Einstein was careful not to deny that sense-data figured into his process, but it is quite clear that he was unaware of how it figured in, if at all . Thus, contrary to what modern

12 Ibid, page 213. 13 Carnap acknowledges this as well on page 215. 14 Thomas Kuhn wrote of “ flashes of intuition through which a new paradigm is born” (The Structure of Scientific Revolutions, 1962, p. 123), and even Willi am Whewell denied the possibil ity of fully explicating scientific convictions, ”…this conviction, that the inductive inference is…necessary, finds its place in the mind gradually…It is scarcely possible for the student at once to satisfy himself...” (Novum Organon Renovatum, II .iv.16). 15 Albert Einstein: Philosopher-Scientist, pg 11.


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textbooks on scientific methodology suggest, the best scientific conclusions are based on far more evidence than scientists report or even could report.

Might it then be the case that scientists actually do use their total evidence, only not formally? It is possible, but we can never prove that any subconscious probabili ty calculation actually follows the probabili ty calculus, so informal induction cannot yield decision-justification. Thus, Carnap’s principle cannot be abided in practice, and it follows from the requirement of total evidence that the only practicable way one could justify a decision would be deductively.

From the premise that justification rests on evidence, we can prove that sufficient evidence must be conclusive. Although the following proof confirms what Carnap’s principle reduces to in practice, it contradicts Carnap’s principle in theory, since one’s total evidence could, in theory, be inconclusive. The proof goes like this:

Suppose evidence, E, would justify (to non-zero degree) decision, D, for subject, F. Take fD to be the proposition that “F would be justified (to non-zero degree) in

choosing D,” and ef to be the proposition that “F has evidence E” . Then (ef ⇒ fD).16 If E were inconclusive evidence in the context of F’s other evidence, then it would have to be

possible to devise some claim, c, for which (c ⇒ ef) and yet ~(c ⇒ fD). For example, if D were the decision to look for milk in the refrigerator, and E were F’s earlier observation of milk in the refrigerator, then the claim that “F has E and knows that she has removed any milk that has been in the refrigerator since she observed E” would constitute a c that strictly entails ef but not fD. Now, by the transitivity of strict implication, it follows from

(c ⇒ ef) and (ef ⇒ fD) that (c ⇒ fD), so, if we maintained that E is inconclusive in the

context of F’s other evidence, then we would have the contradiction (c ⇒ fD) and ~(c ⇒

fD). Therefore, E cannot both be inconclusive and also justify F’s decision to D. It is important to note that this argument is sound and valid even if c happens to

be false. If c were true (i.e. if F knew that she had removed any milk that had been in the refrigerator since observing E), then E could not be all of the evidence F had, so Carnap’s own principle would entail that E cannot justify F’s decision (and the proof above would not contradict Carnap’s principle). However, the contradiction in the reductio above lies in what c strictly implies, and c can have strict implications, even if it is false. Thus, E could be F’s total evidence (in which case c would be false), and we would still have proven that E fails to justify any decision to any non-zero degree. If we were to suppose that justification does not rest on anything like evidence, then, although it would then cease to be clear how one could be better justified than a dung beetle, we could deny (ef

⇒ fD). This strikes me as an insane way of clinging to the possibili ty of applying

16 The double arrow indicates strict implication (i.e. (a⇒b) means ~M(a&~b), where M is the modal operator for possibili ty).


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induction, but let me offer a second proof from independent premises for anyone still i n doubt.

From the premise that our goals are about things yielding consequences we can experience, we can prove that all decision-justifying probabilities are either 0 or 1. I consider this proof to be the most compelli ng because denial of its premise (i.e. claiming that you have a goal that certain things occur in epistemologically remote “other possible worlds”) could cause one to be shipped off to an institution (which would limit your abili ty to participate in the debate). As we will see, the premise allows us to justify the following axiom of logic:

(K2+) If any proposition, a, is proven in a derivation or subderivation, it is valid to append, in that same derivation or subderivation, the line DPr(a)=1 (where DPr(a) signifies a probabili ty that can justify decision-making contingent on a).

This axiom is a stronger version of Kolmogorov’s second axiom (K2), which was equivalent to the claim that it is valid to append “Pr(a)=1” if one has proven that a is necessarily true. Whereas K2 demands that the proposition be true in all possible worlds, K2+ merely requires that it be true in our own world. If we suppose a proposition to be true in our world, then we are supposing that any evidence we will ever experience to the contrary must be misleading. Thus, we are supposing that our experience will never be able to refute the claim that its probabili ty is one. But it follows from our premise about the nature of the goals our decision-making is supposed to satisfy that if a result of probabili stic calculus justifies one of our decisions, then it cannot contradict experientially irrefutable claims. Thus, to suppose that a proposition is true in our world is to suppose that the decision-relevant probabili ty of that proposition is one. This proves the soundness of K2+.17

K2+ turns out to be even stronger than Carnap’s principle, for it allows us to derive the following theorems (see appendix for derivations):

(T1) (~a≡(DPr(a)=0))

(T2) ((DPr(a)��⊃(DPr(a)=0))

(T3) ((a⊃b)⊃~(DPr(a)>DPr(b)))

(T4) (b⊃(DPr(a)=DPr(a|b))) The last of these theorems is a stronger version of Carnap’s principle—it states that the decision-relevant probabili ty is conditional not just on all of the evidence we

17 Logicians typically prove the soundness of an axiom relative to a set of semantics, but here we have done better: we have proven soundness relative to our inter-subjective goals.


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have, but on all of the evidence that exists. The second theorem is equivalent to the conclusion of the previous argument—it states that any decision-relevant probabili ty that is not one must be zero. Thus, any evidence that entails a decision-relevant probabili ty will be conclusive (if not conclusively for, then conclusively against).

Common Objections to the Demand for Conclusive Evidence The justification of the K2+ axiom and the derivations of the theorems that follow

from it answer the question of what constitutes “suff icient evidence” for decision-justification, but they also raise new questions: If decision-justification requires conclusive evidence, how shall we explain the behavior of gamblers and casinos? If intermediate probabili ty calculations cannot justify decisions, how shall we explain the success of science? How can we explain the scientific practice of “ random sampling” and the role of probabili ty in quantum mechanics? Although I think that the earlier sections of this essay stand on their own and that these new questions deserve more consideration than I can give them here, I feel obliged to outline some answers lest the demand for conclusive evidence be dismissed as unviable.

The behavior of gamblers and casinos. Regarding the question of how to

account for the behavior of gamblers and casinos, it must first be recognized that all previous accounts of probabili stic reasoning have failed in this regard. Thus, if the demand for conclusive evidence facilit ates a good explanation, we will have an additional reason to accept the demand. Previous accounts18 have run into trouble, for example, in accounting for why people buy just a few lottery tickets instead spending all (or none) of their money on them. If the Powerball j ackpot reached one trilli on dollars, for example, and there were 0.86 trilli on Powerball sequences to play and it costs one trilli onth the utili ty of the jackpot to play each sequence (i.e. $1),19 then previous accounts would suggest that the expected utili ty of each play would be 1.16 times its cost, and that rational agents are therefore obliged to buy as many as they possibly can. This would seem to be a wise decision for those who can buy-out the entire lottery (and make a guaranteed profit), but if we had only enough money to buy 200,000 tickets, it would be surely be irrational to spend it all on the Powerball—there would be a severe risk of losing everything!20

A proper account of rational gambling behavior must be able to explain why the rationali ty of spending on the inflated Powerball Lottery depends upon how much money

18 Neoclassical Economics. See Von Neuman, J. and Morgenstern, O. (1944): 19 I say “one trilli onth the utili ty of the jackpot” rather than $1 because some economists have doubted that one trilli on dollars has one trillion times as much util ity as one (see Keeney and Raiffa, 1976, p.210). 20 According to previous accounts of probabilit y, we would have a 99.99998% chance of losing everything in this scenario, but would be obliged to take the risk because of the massive potential pay-off .


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one has. The obvious answer that comes from K2+ is that a gambler who buys-out the entire lottery is justified in doing so because she would have conclusive evidence that she would profit, but a gambler than can only buy a few tickets would lack such evidence. But what about the gambler who can buy all but one ticket—wouldn’ t it be rational for her to do so as well? To explain how her purchase could be justified by conclusive evidence, we must consider the complexities that arise from the potential to engage in multiple gambles and to inherit/will gambling legacies in a family. The following thought experiment may help to explain the issues:

Consider a war in which 52 soldiers on one side are each dealt a card (face down) from a complete deck and told that they must either bet their li ves that the card is an ace or bet their li ves that it is not an ace. Each soldier that loses their bet will die, but for each that bets on the ace and wins, 1000 enemy soldiers will die. The general would be justified in instructing them all to bet on the ace, since she would have conclusive evidence that this would yield a net profit of 3,952 soldiers.21 Soldiers that share the general’s goal would likewise be justified in following that gambling policy. Since they would be considering themselves part of family that inherits a gambling legacy and engages in multiple gambles, they would not care who the casualties happen to be. Soldiers with individual goals, such as to save their individual skins, however, would not have the luxury of conclusive evidence; if they lose, they would not have any opportunity to recoup their losses. Betting on the ace might be their best move or it might not, and with no way of justifying any claim about how “ randomly” the cards have been dealt, we should not be surprised if the soldier begins to wonder whether “higher powers” might have dealt her a special card and what such “powers” would expect her to choose (etc.).

The arguments in this essay explain why the individualistic soldier’s second-guessing should not surprise us. Her justificatory powers would have been reduced to that of a dung beetle or infant, so she would be unable to act with as much confidence as the general. These arguments likewise explain why people spend only a portion of their money on lotteries instead of all -or-nothing. For such gamblers, the utili ty of twice their money is more than twice that of the money they have now, so, from their perspective, even normal lotteries can have inflated jackpots as in the Powerball example above. They do not spend all of their money in such lotteries, however, because they lack conclusive evidence that they will win. They do, however, have conclusive evidence that a family policy of abiding a balanced plan of regular small i nvestments in the lottery combined with additional investments in other income sources must eventually pay-off f or their family.

To be justified in believing that gambling will eventually pay-off , one must be justified in believing (a) that the ratio of wins to losses in the set of gambles they are

21 That is, 48 of her soldiers would be guaranteed to die, but so would 4000 of the enemy.


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considering is fixed for suff iciently many trials (much as the number of aces per deck is fixed) and (b) that they can engage in suff iciently many trials. Casinos provide the clearest example of seeking justification along these lines. To take care of (a) they have explicit policies of not gambling with anyone who wins a lot (regardless of whether they can prove that the gambler cheats), and to take care of (b) they make it a policy not to operate a casino without a large quantity of capital to cover losses. Regular gamblers do something similar—to take care of (a) they avoid games that seem to be “fixed” , and to take care of (b) they devote a portion of their time to an alternate income source that will permit them to keep playing (indefinitely). Thus, lotteries need not be explained as a “tax on those bad at math” ; they can be seen as a rational reflection of the fact that large sums of money bring us more utili ty than their dollar figures would suggest, and that certain gambling policies can therefore conclusively be shown to be profitable in the long run. I submit that the demand for conclusive evidence thus provides the best explanation for gambling behavior.

The success of science. The more serious objection that will be raised by my

fellow philosophers of science is the concern that the demand for conclusive evidence completely undermines the received explanation of the “success” of science. They are right—if we are loathed to suppose that scientists are, li ke dung beetles, merely lucky that the world has so far provided a situation favorable to their style of reckoning, then the only alternate explanation that the demand for conclusive evidence leaves available is the claim that successful science springs from deduction (rather than induction). But it is supposedly impossible to deduce anything non-trivial, so that class of explanations has long been abandoned.

I have three reactions to this objection: (1) It would be circular to dismiss the possibili ty of lucky science on inductive grounds and, although human ego may tempt us to dismiss it out of hand (lest our epistemic practices be considered no more praiseworthy than those of dung beetles), such encouragement does not constitute justification for such dismissal. (2) It would seem self-contradictory to conclusively prove that it is impossible to conclusively prove anything non-trivial. (3) Even if it happened to be impossible to deduce claims about the external world, deductive process may yet explain the success of science, for I have elsewhere shown that one can deduce the decision-justifiably of fundamental scientific laws in the same way Aristotle deduced the decision-justifiabili ty of the laws of non-contradiction and excluded middle.22

Personally, I suspect that all good science is deductive. As we saw earlier, great science comes from intuition, so the descriptions that scientists have given of their own process are most assuredly inaccurate. Considering the disparity between

22 See my forthcoming Decision-Justifying Logic and Generalized Evolution.


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modern neuroscience and folk psychology, it would hardly be the first time intelli gent people misrepresented their own mental processes. Given the value of modesty in our culture, it would not be surprising if scientists were biased towards masking the conclusiveness of their process to avoid offense. Neither would it be surprising if they translated their conclusions from ones about decision-justifiabili ty to ones about the external world so as to avoid being labeled “moralists” (i.e. in direct conflict with the teachings of contemporary religious leaders). At the very least, it seems quite plausible to me that the account of science as inductive is nonsense invented to support an egocentric conception of freewill , and I hope the current essay will encourage others to join my efforts to explore an alternative philosophy of science (worst case: we’ ll map-out more dead-ends).

“ Random sampling” and the role of probability in quantum mechanics. If

justification requires conclusive evidence, then scientists cannot be justified in claiming that any phenomenon is less then completely deterministic (metaphysically). How then shall we explain their talk of “ randomness”? This talk (which was controversial long before this essay was written) 23 comes up regarding random sampling, genetics and quantum mechanics. Regarding genetics, the “randomness” of mating (and so forth) could be purely epistemological24, and we will now show that this is usually the case for sampling as well . In most experimental set-ups, li ke, for example, those involving the distribution of a placebo to human subjects, selection is a function of the order in which the subjects arrive at the laboratory, which in turn is a function of such variables as the order in which they were contacted, which may be a function of where their names appear in the telephone book (etc.), ultimately boili ng down to a complex function of the properties that make each human unique. Since human beings are invariably unique, experimenters cannot control all of the variables that influence sampling of human societies. Thus, the apparent randomness of placebo allocation may be an epistemic ill usion caused by the fact that so many influences went uncontrolled. Most importantly, if there are sufficiently many such influences, ill usory randomness may effectively dilute what would otherwise be systematic sampling error.

Quantum mechanics provides an apparently separate but actually related problem. Actually, this problem is restricted to a small class of experiments, namely Bell correlation experiments and quantum interference experiments. These are the only two kinds of experiments in all of science that seem to provide any evidence at all that our universe is metaphysically indeterminate. I hasten to add that the physics community is 23 See the debate inspired by Kolmogorov, Chaitin and Solmonov. 24 Sober (2000), page 66. He uses the term “subjective” for what I am call ing “epistemological” and “ irreducible and objective” for what I am call ing “metaphysical” ; I use different terms because


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split on the interpretation of these experiments (Einstein25 and even Bell ,26 himself, presented deterministic interpretations) and that none of the great successes of quantum mechanics (e.g. the transistor, etc.) have been shown to necessitate a belief in metaphysical non-determinism. These two kinds of experiments also happen to be the ones in which experimentalists have achieved the greatest experimental control. Each of the particles involved in them have identical mass, charge, trajectory, spin, size, shape, color, intelli gence, you name it. Ironically, this undermines the power of the experiments because it leaves very littl e room for external factors to create an epistemological randomness that would dilute systematic sampling error. In short, for these experiments to prove that our world contains metaphysical indeterminism would be circular, since their validity relies on metaphysically random sampling and we have no reason to suppose such sampling exists without first accepting the results of the experiments.

My responses to the above common objections to the demand for conclusive

evidence do not, of course, stand as proof that gamblers use policies that rest on conclusive evidence, that great scientific intuition has rested on purely deductive reasoning, or that our universe is metaphysically completely determinate. They merely show that all of this is plausible, and therefore, that the common objections to the demand for conclusive evidence lack justification. It may follow from the arguments presented earlier that we would be justified in denying that there is metaphysical indeterminacy and/or that science holds a proper place for induction (etc.), but being justified in denying something doesn’ t necessarily entail that it is actually false.27 Thus, if we are to engage in decision-justification, it seems that we need a modest philosophy that does not pretend to establish truth about anything other than just that.

Likewise, if science is to produce decision-justification, it needs some changes. Gerd Gigerenzer (1993) has shown that, over the last century, many areas of science have developed a tradition of rejecting any work lacking inductive support. We have shown this to be a bad policy. Scientists attempting to produce decision-justification must never assign intermediate probabiliti es to propositions (as in “There is a 40% chance that the gene will mutate in the first hour” or “Patient A is twice as likely to recover as Patient B”). They should feel free to assign relative frequencies to ensembles (as in “Given enough coin flips, 50% will be heads”), but it should be clear that such assignments do not justify decision-making regarding non-ensembles (e.g. 99% of the integers are not multiples of 100, but this does not justify the decision to bet that the integer following 99

epistemological randomness can be inter-subjective and I worry about distinguishing the inter-subjective from the objective, but my terms probably have their problems as well . 25 Fine (1986), page 41. 26 Albert (1992), page 134. 27 For example, I would think that patients are justified in denying that the placebos they receive from doctors are placebos, since such denial is supposedly what makes the placebo effective.


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is not a multiple of 100). Inconclusive evidence should be reported, since other scientists may augment it to form a conclusive set, but it should be made clear that such augmentation will be required before the evidence justifies anything. Scientists should be encouraged to report their intuitions, to report inconclusive evidence without drawing conclusions from it, and to report purely theoretical work (i.e. deductive arguments proceeding from premises that the reader may be willi ng to assume).28 These are the only ways they could possibly fulfill t heir decision-justification mission.

References Albert, D. Z. (1992): Quantum Mechanics and Experience. Cambridge: Harvard University Press. Ayer, A. J. (1957): “The Conception of Probabilit y as a Logical Relation.” In Observation and Interpretation, ed. by S. Korner. London: Butterworths. Pages 12-30. Carnap, R. (1947): “On the Application of Inductive Logic”, Philosophy and Phenomenological Research 8: 133-148. Carnap, R. (1950): Logical Foundations of Probabilit y. Chicago: University of Chicago Press. Pages 211-213. Fine, A. (1986): The Shaky Game. Chicago: University of Chicago Press. Gigerenzer, G. (1993): “The Super Ego, the Ego, and the Id in Statistical Reasoning.” In G. Keren and C. Lewis (eds.) A Handbook for Data Analysis in the Behavior Sciences: Methodological Issues. Hill sdale, NJ: Erlbaum. Good, I. J. (1967): “On the Principle of Total Evidence.” British Journal for the Philosophy of Science 17: 319-321. Keeney, R. L. and Raiffa, H. (1976): Decisions with Multiple Objectives: Preference and Value Trade-offs. New York: Wiley. Royall , R. (1997): Statistical Evidence—a Likelihood Paradigm. London: Chapman and Hall . Skyrms, B. (2000): Choice and Chance (fourth edition). Belmont, CA: Wadsworth. Sober, E. (2000): Philosophy of Biology (second edition). Boulder, CO: Westview Press. Von Neuman, J. and Morgenstern, O. (1944): Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.

28 I have made some progress in developing a one standard procedure for this sort of thing that produces valuable results (see http//philosophy.wisc.edu/lang/pd.htm).


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Appendix

The following derivations are given in a graphical format developed by Frederick B. Fitch. I will use the notation “DPr(a)” to signify “ the decision-relevant probabili ty of a” . Since probabili ty can be confusing, I have endeavored to make the validity of these derivations more transparent by avoiding use of the law of excluded middle and using the following rule to avoid reiteration of probabili ty claims into subderivations:

&-elim*: Given a and ~(a & b), it is valid to append the line: ~b.

1. ~a (hypothesis) 2. (DPr(~a)=1) (1, K2+) 3. (a or ~a) (1, or int.) 4. (DPr(a or ~a)=1) (3, K2+) 5. (DPr(a or ~a)=DPr(a) + DPr(~a)) (K3) 6. (DPr(a)=0) (2, 4, 5, math) 7. (~a⊃(DPr(a)=0)) (1-6, ⊃ int.) 8. (DPr(a)=0) (hypothesis) 9. (a & (DPr(a)=0)) (hypothesis) 10. a (9, & elim.) 11. (DPr(a)=1) (10, K2+) 12. (DPr(a)=0) (9, & elim.) 13. (1=0) (11, 12, = elim.) 14. ~(a & (DPr(a)=0)) (9-13, ~ int.) 15. ~a (8, 14, & elim*) 16. ((DPr(a)=0) ⊃~a) (8-15, ⊃ int.) T1. (~a≡≡(DPr(a)=0)) (7, 16, ≡≡ int.)

1. (DPr(a)≠1) (hypothesis) 2. (a & (DPr(a)≠1)) (hypothesis) 3. a (2, & elim.) 4. (DPr(a)=1) (3, K2+) 5. (DPr(a)≠1) (2, & elim.) 6. ~(a & (DPr(a)≠1)) (2-5, ~ int.) 7. ~a (1, 6, & elim*) 8. (DPr(a)=0) (7, T1) T2. ((DPr(a)≠≠1) ⊃ (DPr(a)=0)) (1-8, ⊃ int.) 1. (a ⊃b) (hypothesis) 2. (DPr(a)>DPr(b)) (hypothesis) 3. ((DPr(b)=1 & (DPr(a)>DPr(b))) (hypothesis.) 4. (DPr(b)=1) (3, & elim.) 5. (DPr(a)>DPr(b)) (3, & elim.) 6. (DPr(a)>1) (4, 5, = elim.) 7. ~(DPr(a)>1) (K1) 8. ~(DPr(b)=1 & (DPr(a)>DPr(b))) (3-7, ~ int.) 9. (DPr(b)≠1) (2, 8, & elim*) 10. (DPr(b)=0) (9, T2) 11. ~b (10, T1) 12. ~a (1, 11, modus tollens) 13. (DPr(a)=0) (12, T1) 14. (DPr(a)=DPr(b)) (10, 13, = elim.) 15. ((DPr(a)=DPr(b))&(DPr(a)>DPr(b))) (2, 14, & int.) 16. ~(DPr(a)>DPr(b)) (2-15, ~ int.) T3. ((a⊃⊃b) ⊃⊃ ~(DPr(a)>DPr(b)) (1-15, ⊃ int.)

A Problem with Probability

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1. b (hypothesis) 2. (b & (DPr(a)

��a|b))) (hypothesis)

3. (a & b & (DPr(a)��

a|b))) (hypothesis) 4. a (3, & elim.) 5. b (3, & elim.) 6. (a & b) (4, 5, & int.) 7. (DPr(b)=1) (5, K2+) 8. (DPr(a)=1) (4, K2+) 9. (DPr(a & b)=1) (6, K2+) 10. (DPr(a|b)=DPr(a & b)/DPr(b) (definition) 11. (DPr(a)=DPr(a|b)) (7, 8, 9, 10, math) 12. (DPr(a)

��a|b)) (3, & elim.)

13. ~(a & b & (DPr(a)��

a|b))) (3-12, ~ int.) 14. ~a (2, 13, & elim*) 17. (a & b) (hypothesis) 18. a (15, & elim.) 19. ~a (14, reiterated) 20. ~(a & b) (15-17, ~ int.) 21. b (2, & elim) 22. (DPr(a)=0) (14, T1) 23. (DPr(a & b)=0) (18, T1) 24. (DPr(b)=1) (19, K2+) 25. (DPr(a|b)=DPr(a & b)/DPr(b) (definition) 26. (DPr(a)=DPr(a|b)) (20, 21, 22, 23, math) 27. (DPr(a)

��a|b)) (2, & elim.)

28. ~(b & (DPr(a)��

a|b))) (2-25, ~ int.) 29. (DPr(a)=DPr(a|b)) (1, 26, & elim*) T4. (b⊃⊃ (DPr(a)=DPr(a|b))) (1-27, ⊃ int.)

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A Problem with Probability