Philosophical Studies
XXIII
Contents
WOZlllebt <b- MeJlSCh - Einige UberlegwtgeIl ausgehend von Hegels 'fheaie des Begriffs
in <b- 'WtsSenSCbaft <b- Logik"
TAKAYAMA, MamOOl
Does Probability Collapse or Retroact?
IE::IDNOSE, Masaki
Husserls Reflexion auf die lebendige Gegenwart in seiner spateren Zeidehre
SAKAKIBARA, Tetsuya
F.rkmntniswteil und Geschmackswteil bei Kant
110, Mieko

Process Thought and Quantum Theory -in association with the relativistic spatio-tempa-al

perspectives penneating in the former

YAMAZAKI, 0Usat0

Meinongs Gegenstandstheorie

KURATA, Tsuyoshi

Erinnerung und Rekognition

YOSHIDA, Akira

The Relata of Causatioo

YANO, Keisuke

NAKANO, Makoto

Eine Betracbtung liber Hegels Abhandlung : Form und Materie
Sur le temps (m) : observer le ciel

WATANABE, Makoto

Locl<e 00 Money (IT)Remede aux affects clans l'Ethique de Spinoza

IMAMURA, Kenichiro- pc:xtee des notiones communes et etayage de I'acquiescentia

ONISlll, Yoshitomo

Heidegger's Atguments Concerning Death and Conscience in Being and lime-in the Light ofhis Interpretation ofAristotle's ~Locl<e's moral philosophy in the Essay

IKEDA, Takashi
SHIMAMURA, Hisayuki

The general form ofempirical synthetic judgements
What kind of Knowledge Spinoza called the "Third Kind" ?
Detaminisme et Liberte chez Bergson

KANEKO, Yusuke
ASAKURA, Tomomi
110, Toshihiko

Zur Verstiindlichkeit der Stimmung bei Heidegger-im Vergleich mit Aristoteles' Rhetorik
TAKI, Masayuki
La temporalire et la relation entre le Meme et I' Autre chez Uvinas

SAro,Kaori

Cantor's early cooception of number
SAKAMA, Tsuyoshi
Jolm Locke and Private Language
NAGAO, Hidesato
Fitch'sArgument and the Knowability Principle as the Principle Conrerning
Truth of Judgment
OSADA, Ryo
What is Putnam's natural realism?
OIITAN!, Hiroshi
Uber den "Gegenstand" bei Kant
lKEMIZU, Madoka

Depar1ment of Philosophy
Graduate School of Humanities and Sociology
The University ofTokyo
2004

Does Probability Collapse or Retroact?
Masaki ICHINOSE(The University of Tokyo)
1, Probability of the Past Event
It is sometimes questioned, as C. S. Peirce once doubted 1, whether
decision-makings based upon probability can be justified as rational or not.
Even though, for example, we choose the option as the most rational one
whose expected probabilistic utility was the highest, an unexpected event
whose probability was very low might result as a matter of fact. In thatcase, did we make the rational decision? If you say we did, our reason
seems to be useless. 2
I think here is a deep philosophical question concerning rationality,
but what I will focus my attention upon this time by referring to thisquestion is not the question about rationality itself, but the hidden
intuitive idea that happens to appear in the above question. That is tosay, the value of probability must be suddenly fixed as 1 or 0 the moment
the event the probability was attributed has been actualized or failed to be
so. The question above picks up the case that an event was attributed low
probability, nonetheless the event actually happened. Then it was
doubted why the decision from probability can be called 'rational decision',
if we judge backwards after the occurrence of the less probabilistic event.
Obviously, behind this question there must be the idea that, as soon as theless probabilistic event has happened, the event is no longer lessprobabilistic, but has been already fixed with the maximum probability,
namely, probability 1. Otherwise the result of the decision would remain
still pending, so we can't denounce the decision as irrational. As it were,
the situation is suddenly reversed and then settled, which the decision
couldn't follow despite being imposed to follow as far as it professes itself to
Of course, if some model of kinematics of
be 'rational decision'.
probability is adopted like Jeffrey's classic theory3, then the change at thatmoment might not be sudden. Nevertheless, even in that case, thesituation is the same, since there happens a decisive leap from uncertainty
to certainty the moment the event concerned has occurred. Rather, I
suspect that a high probability just before the final moment in such
Peirce (1986), pp.281-2.
David Papineau presented the same question when he talked at the philosophy ofscience speakers series in the University of Oxford on 31 October 2002 under the title of'Decisions in a No-Collapse Universe'. He examined the orthodoxy of decision theory
that appeals to the notion of expected probabilistic utility, compared with Everettian
view of many worlds, and said that the principle of expected probabilistic utility mightlead to an actual unexpected result, notwithstanding that the actual result cruciallymatters in decision theory.
3 Cf., Jeffrey (1983), chapter 11.
1

2

19

kinematics can be assigned retrospectively only after the event concerned
has actually occurred. Anyway, thus, some people suspect that the
principle of maximizing expected probabilistic utility might not be rational,
taking into account the case that an unexpected result has been actualized,
namely, the probability of expected result has been fixed as O. We canparaphrase this idea into the next claim that I want to call the Principle ofProbability of the Past Event (PPE);(PPE) As soon as an event is recognized as just now HAPPENED in a
specified way, the probability of its occurrence must be 1, at leastas far as the probability of single event is concerned4 .
In fact, David Lewis clearly declares this intuitive idea, by saying, 'What's
past is no longer chancy. The past, unlike the future, has no chance ofbeing any other way than the way it actually is'5. Is there anything more
self-evident than the idea that the probability of the event already having
happened is I? It is true that probability 1 is usually introduced asapplying to logical tautology according to Kolmogorov's axiom like, forexample, 'P(t) = 1 if tis a tautology6', but this suggests that being tautology
is only a sufficient condition for probability1. This means that there could
be another case which is assigned probability 1. That is precisely the caseof the past event having just happened. This seems to be an intuitive idea.
Or, in other words, we might say that, when we have the intuitive idea, we
actually committed ourselves to such a sort of tautology as 'what has
happened has happened,' thus the probability 1 of past events could be
subsumed in Kolmogorov's axiom. In any case, the crucial point of PPE isthat, although probability of an event is clearly lower than 1 before
recognition, its probability suddenly rises to 1 as soon as we recognize theevent. PPE is typically illustrated, for instance, by this situation;
probability of the event that the lightening strikes me is lower than 1
before the event occurs, nevertheless its probability leaps to 1 the moment
I am actually struck by the lightening. Who could believe otherwise? I
think that this intuitive idea is universally true of the concept ofprobability, no matter what interpretation of probability is taken, unlessthe notion of the probability of single event is excluded as in the case of
Von Mises' version of the frequency interpretation.
In reality, PPE is clearly confirmed by two major interpretations ofprobability, i.e. the propensity theory (in an original Popper's sense inorder to make sense of single event's probability) and the epistemic(including subjective and interpersonal) interpretation. If we adopt thepropensity theory, there is no doubt that we have to assign probability 1 to
For the time being, I don't search the question of what the bearer of probability is, byroughly taking both the event and the sentence 'the event happens' to be equally thebearer, since my argument would not seem to be influenced by the question.
5 Lewis (1986), p.93 .
6 Howson & Urbach (1993), p.21.
420

an event if we actually just observed it happen. What else could we do? In
reality, this point is conjoined with quantum mechanics that motivatedPopper to propose the propensity theory, which I shall discuss later. Alsothat must be true of the epistemic interpretation, apparently. When wefind something just now actually appear in front of me, we automatically
accept that the sentence, 'the thing has occurred,' is completely true as far
as my epistemic understanding is concerned, apart from philosophicalproblems of illusion or 'theory-laden.' In that sense I said that PPE isuniversally true. As a matter of fact, Bayesian epistemology, which isbased upon epistemic probability, basically presupposes that evidence getsprobability 1 when it is actually observed7 • Of course, I am fully consciousthat there is a crucial difference between the propensity theory and theepistemic interpretation, i.e. the former concerns an event itself and thelatter the degree of belief. But, at least, that difference can be virtuallyignored as regards the case of probability 1 in PPE.
However, what should we think about the past event that is
unrecognized? I think that there are two cases of unrecognized past
events. The first one is the past event that is not recognized at all but
merely supposed to have happened in an unspecified or vague way
regarding space and time, for example, an event that something existed
before the earth was born. PPE avoids mentioning this kind of
unrecognized past events. Certainly, if we accept the propensity theory,
even events that we don't recognized at all but just generally suppose tohave happened might be assigned probability 1. But, on the contrary,since it is supposed in an unspecified way, it seems to be impossible toindividuate a particular event as a bearer of probability. It's hard to see
what probability we are talking about. Then, what if we adopt an
epistemic interpretation? Perhaps, even from such interpretation, we
Here I have to refer to the case in which we can't give probability 1 to a sentence, '8 isP ,' despite actually observing the phenomenon right now . This situation withuncertain evidence was initially noticed by Richard Jeffrey. He gives such an example
that; the agent inspects a piece of cloth by candlelight, and gets the impression that it isgreen, although he concedes that it might be blue or even (but very improbably) violet .
Thus, he proposes a conditionalisation to formulate posterior probability of a hypothesis
in such cases. This is his formulation (which I modify a bit to make it clearer),
7

This is called Jeffrey Condition8lis8tion (in which Ppri means prior probability and Ppos
posterior probability). Jeffrey claims that this is not different from usual B8yesi8n
Condition8iis8tion, since both conditionalisations will be the same when Ppos(e) = 1 canbe affirmed. See Jeffrey (1983), pp.164-172. Well, then, how should we think of PPE inthis case? The event to be considered by PPE in this case is the one that the cloth is lit
by candlelight. When an event is observed, there could be many sentences to describe
the observation, but the fact that the event is observed is determinate. As far as suchdeterminate fact is concerned, probability 1 could be assigned. Actually, Jeffrey
Conditionalisation can't start without probability 1 in this sense, which iscorresponding to probability 1 of (Ppos(e) + ppos("-e) = 1) .

21

might claim that some past events, even if not specifically recognized,
surely happened in an abstract sense, so can be given probability 1. Maybe
those who have personal religious faith might have such claim if it's
particularly concerned with their faith. But, the most honest comment
about the claim is that we are not certain about that. Therefore, I madePPE not touch those cases. The second case is the past event that is notrecognized but represented in a specified way, for instance, an event that anewspaper was on the desk in my room at 4:00 pm yesterday (provided
that it was not recognized at that time). I want PPE to claim that theprobability of such event must be smaller than 1. This must be another
implication of PPE. Otherwise, PPE wouldn't imply the sudden change ofprobability at the moment of recognition. In fact, if we choose theepistemic interpretation, we must give smaller probability than 1 to suchkind of event, because the event is not an actual one but just an uncertain
imaginary, at most inferred, one. But, what if we accept the propensity
theory? It's independent of our belief, so the event has probability 1 if itreally happened despite being not known by us. My idea is that; as far as
the propensity theory comes from quantum physics, it is not completely
unreasonable for us to assign smaller probability than 1 to such kind of
event if we take into account the measurement problem in quantum
mechanics. This idea might sound very weird, but I think that we canmake sense of the idea by reflecting on both Popper's argument ofpropensity and the concept of past in connection with probability, which I
shall try later.
Anyway, here I will examine PPE thus formulated, in order tounderstand the concept of probability as clearly as I could. Perhaps, myargument will be just hypothetical or thought-experimental, but all that I
intend to do is to take a step forward.
2, Mysteries of Probability 1
Honestly speaking, it has been my longtime question, what isprobability I? Because probability 1 seems to be too self-evident, nobody
seriously tried to explain what it is, as long as I know. But, apparently,
probability 1 is very problematic. I can instantly raise at least three basic
difficulties about it, that is to say, MODAL, TEMPORAL, and CAUSAL
difficulties. First, I can pick out the MODAL difficulty about probability 1.
Probability 1 is intuitively applied to the past event that has been
actualized, as in PPE. But, undoubtedly, as I've mentioned, it is also
applied to logical tautology or necessary truth like 'that dog is male or not.'In other words, probability 1 is connected with two fundamental modal
concepts, namely, actuality and necessity. But, a difficulty seems to arise,
particularly from the side of necessity. If the probability of necessary truth
is 1, the probability of non -necessary, contingent sentence must be smaller
than 1. I think that this is an implication of accepting necessary truth as
having probability 1. This is easy to understand if we imagine the case ofprediction. Let's think of those two prediction sentences (suppose that it
is January 2005 now);22

A: Tony Blair either resigns or doesn't resign from British Prime
Minister in April 2005.
B: Tony Blair resigns from British Prime Minister in April 2005.
I adopt the standard notation about probability, i.e. P(A) means theprobability of the occurrence of event-A or the probability of the truth ofsentence- A. Then, obviously, P(A) = 1 since A is a logical necessary truth,
while P(B) < 1 because it is not necessary but contingent. Nevertheless, if
Tony Blair actually resigns from British Prime Minister in April 2005, it
suddenly comes to that P(B) = 1 because it is an actuality, although B still
remains just a contingent truth. But, nevertheless again, even after B is
actualized, we can say that still P(B) < 1 in the sense that we are notnecessarily in the possible world in which B is true if we adopt some simple
possible world semantics to explain modal concepts. Thus, P(B) < 1 &P(B) = 1, which is a manifest contradiction, at least at first sight. Perhaps
a similar difficulty doesn't arise if we start from the side of actuality, sinceactuality might extensionally include necessity. Well, in any case, I thinkthat the contradiction thus produced over necessity is one of the basic
difficulties concerning probability 1. Probably, in order to confront thisdifficulty, the first thing we should do is to consider how modal concepts
are understood if the time, or finally the tense, is taken into account. In
addition, there is another difficulty which is more closely related to PPE.
If we focus upon the problem about the relation between PPE and actuality,
and recognize that the event already actualized in the past is given
probability 1, then we are treating actuality as in the past. So, what
about actuality in the present? How is the past actuality related to thepresent actuality? I believe that this question, which again attempts tocross modal concepts and the tense, is also to be seriously taken inconnection with the problem of probability 18.
Second, we have to mention the TEMPORAL difficulty with respect
to probability 1. Even though we easily accept PPE, namely, the idea thatprobability of the event just having happened must be 1, it can bequestioned whether the probability of the event will remain 1 permanently
hereafter or not. In other words, we have to ask whether probability 1 is
temporally immutable or not after it is established. This question deeply
concerns in a rather perplexed way both the ontological status ofprobability and the metaphysical significance of the past. Is probability asort of natural property inhering in reality, or just a concept functioning in
our thoughts? Does the past eternally persist, or change and vanish as
8 It is often said that probability is contrasted with certainty.
However, exactly
speaking, it is not true, of course, because probability covers both certainty anduncertainty. If P=l , probability is equivalent to certainty, and if P<l , probability canbe take as uncertainty. Well, can we think that the very certainty must be theessential feature of probability 1 in common to actuality and necessity? I believe that
we can't think so easily, since, as far as I understand, actuality and necessity are
typical modal concepts, while certainty is an epistemic concept.

23

time goes by? There is no room here to develop some arguments about
those questions in detail, but, for the time being, I have one thing to affirm ;as a matter of fact, we sometimes take probability to be a natural property,
and sometimes to be just a sort of mental state, and similarly, we
sometimes regard the past as eternally existing in a fixed way, andsometimes as only appearing in our memory. Indeed, if probability isn't
understood in such a flexible manner, it is difficult to explain why
probability is actually working in quantum mechanics as well as our actionof betting on horse racing. This might be also suggested by the fact thatpluralism is often proposed with regard to the interpretation ofprobability9. As to the past, on the one hand, we firmly believe that thepast is already fixed and we can't change that, but, on the other hand, we
think that we can't ascertain the past truly and sometimes doubt the pastby saying that it's shrouded in a mystery. If that is the case, we could
safely say that the event in the past will gradually reduce its probability
from 1 to smaller than I, even though we accept PPE. Why is this?
Because we might forget the past event. As far as there is an epistemic
aspects of probability and there is an idea of the past as memory, it is notinconsistent with our ordinary concepts of probability and past to introduce
the factor of forgetting about probability 1. Timothy Williamson grasps
the situation of PPE as the case of evidential probability I , and, as
correctly as usual, rejects the next claim he calls MONOTONICITY by
taking into account the factor of forgetting;(MONOTONICITY) Once a proposition has evidential probability I , it
keeps it thereafter lO •Williamson reaches the rejection of (MONOTONICITY) through his firmidea of the higher'order probability of the propositions about the evidential
probability. This is an insightful argument certainly, but at the sametime, it clearly indicates an embarrassing situation as regards probability
I , since it somehow compromises PPE, and perhaps challenges theimmutability of the past (maybe with an anti'realistic flavour)ll . In any
For example, Cohen and Gillies clearly propose the pluralism of the interpretation ofprobability. See Cohen (1989) and Gillies (2000). Incidentally, I myself don't sidewith pluralism. It is true that several interpretations of probability are actuallyworking, depending upon contexts, but it's just a fact. We have to question whether
there is some more fundamental moment that penetrates into all of thoseinterpretations, for example, as regards our practical actions in using probability
concepts , or their relationship with the tense concepts. Hopefully, this article is meant
to be one of such attempts as consider the fundamental moment in common to allinterpretations of probability.
10 Williamson (2000) , p .218 ff.
11 I personally asked Timothy Williamson in April 2003 whether his strategy ofrejecting MONOTONICITY might lead to anti-realism about the past or not . He
instantly answered that his standpoint is not an anti-realism in a Dummett's sense.
Of course, it is not . This problem depends upon whether values of probability shouldbe thought to be subsumed in the concept of past as its inseparable component. If it's
924

case, we have to say that this situation is one of the natural implications
about probability 1. Thus, here is also a serious difficulty we must face 12 .
Third, we have to confront the CAUSAL difficulty concerning
probability 1. The situation which strikingly appears when we accept
PPE is that probability is suddenly changed into 1 once the event
concerned becomes past. This is remarkably represented by the case thatsome highly uncertain prediction comes true. For instance, suppose thatthe probability of the proposition, 'I win this lottery,' is estimated at 0.001,
and then I actually win. In this case, the moment the number of mylottery is selected, the probability instantaneously rise from 0.001 to 1.
Isn't there some problem here? Few people have put this situation inquestion so far. But, can we accept this sudden change of probability as aperfectly unproblematic phenomenon that needs no explanation? I don't
think so. The change is often very dramatic and drastic as in the caseabove, so it seems to be very natural to ask why this happens. In fact, we
human being has been developing scientific knowledge by observing
conspicuous changes in physical phenomena and asking why or how theychange. I think that, in asking such questions, we usually look for causal
relations among phenomena. Therefore, I also want to propose thiscausal question in this context, i.e. what causes this sudden change ofprobability from smaller than 1 to I? This is the third basic difficultywith regard to probability 1, which seems to be extremely hard to answer.
3, The Collapse of Probability
Now, then, which is the most noteworthy difficulty we must face firstof those three ones? Obviously the modal difficulty should be discussed
from the metaphysical or logical point of view, and the temporal difficulty
has to be examined from the epistemic or epistemological standpoint.
What about the causal difficulty? I think that the causal problems ingeneral span all fields of philosophical topics including metaphysical andepistemic problems, therefore it seems to be very convenient to focus ourattention first upon the causal difficulty, whose examination might suggest
some promising routes towards some appropriate understanding of othertwo difficulties 13 . Furthermore, it is not only convenient but also crucial
so subsumed, the rejection of MONOTONICITY might reach the idea of changing thepast in a literal sense. But, if probability values are just some numbers externallyattached to the past, this problem has nothing to do with any anti·realism.
12 David Hume once proposed the similar problem to this second difficulty in thechapter entitled 'Of unphilosophical probability' of his Treatise. He points out the factthat all of our knowledge whatsoever must become less probable as the time passes,
because our memory gets fainter and fainter. It is a very interesting argument andcan't be ignored, but I am afraid that this argument would be applied to philosophers'
own assertions in a self-destructive manner. This devastating point might be true of
my second difficulty as regards probability 1. Cf. Hume (1978), pp.143-155.
13 Someone might wonder how the causal difficulty is related to the modal difficulty.
But, is it nonsense to question how necessity or actuality is caused? I don't think so.
In fact, for example, a certain type of the causal theory of time as Michael Tooley
develops explains actuality in terms of causal process. Even logical necessity, as long25

to scrutinize the causal difficulty first, since other two difficulties could
arise only by taking it for granted that something having probability 1
already existed, whereas the causal difficulty brings into question how
probability 1 appears, or what the process of something's getting
probability 1 is. The causal difficulty is the most basic, so it must bequestioned first. That question is my very aim this time.
I must mention one point before examining the causal difficulty.
That is to say, there can be a philosophical problem concerning whichnotion is more basic, probability or causality. It is well known that somepeople try to understand causality by appealing to the notion of probability.
That is called Probabilistic Causality, whose main idea is to define theconcept of cause as raising the probability of its effect14. Obviously, thetheory of probabilistic causality requires the notion of probability to beprior to causality. However, on the contrary, the causal difficulty that I
propose here about probability 1 can make sense only provided thatcausality is more primitive than probability, since the difficulty questions
what the cause of probability 1 is. How can we reconcile this situation? I
want to say; this situation doesn't have to be reconciled in advance. Bothprobability and causality are so extremely fundamental that it is very
natural for them to be entangled with each other.
In fact, even if we find the cause of probability 1, there is still room forthe cause itself of probability to work in a (higher-ordered) probabilistic
way. The similar situation is likewise true of the side of probabilistic
causality. For example, it is well known that Ramsey develops his
elaborate subjective theory of probability based upon the supposed relation
as we take it as one of our (metaphysical) notions, could be questioned as regards its
causal status. I think that metaphysical problems have more to do with causality thanwe usually expect (Cf., Tooley 1997). But, certainly it is difficult to understand therelation between PPE and probability 1 of logical necessity. All that I can do about
this tentatively is to propose another principle parallel to PPE, and then to suggest therelation of both principles. The another principle, which can be called the Principle ofProbability of Necessary Truth (PNT), is like this;(PNT) As soon as a sentence is regarded as NECESSARY truth, its probability mustbe 1.
If we can formulate the case of probability 1 about necessary truth by PNT, we might
say that probability 1 of necessity is subsumed under PPE, because our regarding thesentence as necessary is also temporally made, so it is factually equivalent to having
ALREADY regarded it as necessary. Of course, then, we have to ask those twoquestions, namely, (1) was the probability of the sentence smaller than 1 before ourregarding it as so? (2) is the probability 1 of logical necessary truth changeable in the
future? As to (0, I think that, as soon as we regard the sentence as logically necessary,
we extend the necessity to the past, as it were retrospectively, by taking the necessity toobtain timelessly. Thus, it seems to me that concepts of necessity and past have aclose affinity to each other. But, what about the future? If it's necessary timelessly,
won't it be permanently necessary even in the future? This question leads to (2),which is, as you easily perceive, nothing but the temporal difficulty I called.
14 Cf., Eells (1991), p.1, and MelIor (1995), p.67.

26

between belief/desire and action. And, he regarded the relation as causal15 .
If so, and we propose the theory of probabilistic causality by appealing to
the subjective interpretation of probability, then we must fall into
circularity. Of course, some people believe the effectiveness of the idea ofprobabilistic causality and try to make the idea more acceptable andplausible. Papineau, for instance, attempts to make up a more exquisite
theory of probabilistic causality through replacing the idea of thescreening-off asymmetry, which is usually used to decide the direction of
causation in the theory of probabilistic causality, with another new
requirement. That is, 'the requirement that effects should haveprobabilistically independent causes.l 6 ' But, then, what is 'probabilisticallyindependent'? According to Howson and Urbach, two sentences, hl and h2,
are called probabilistically independent iff P(hl & h2) = P(hl) P(h2)17, whichleads to that P(hl I h2) = P(hl) in terms of the definition of conditional
probability. Is this nothing but that the probability of changing inconnection with other factors is 0, in other words, the probability ofremaining the same despite changes of other factors is I? Thus,
elaborating the theory of probabilistic causality induces us to ask what
probability 1 is, and eventually to run into the causal difficulty aboutprobability 1. Probability and causality are so bewilderingly tangled in an
intrinsic sense. Therefore, I defer the reconciliation of those two basic
concepts until the end of this article, for my article itself is an attempt toconsider the relation between probability and causality.
Well, then, how could we cope with the question, 'what causes thesudden change of probability from smaller than 1 to I?' Perhaps, one ofpossible answers, which can be easily thought of, is that the time itself, inparticular the present itself, causes the sudden change of probability. But,
this is too implausible to be instantly adopted. First, this view represents
the time as a sort of real existence or substance with causal efficacy, which
can be supported only by some peculiar metaphysics, so everyone is notconvinced of it. Second, if we acknowledge that the time causes thesudden change of probability, then there might be a possibility that any
change in our world must be said to be produced by the time, consequently,
this view would have no explanatory power. Then, how should I answer
next? Obviously, the crucial point is how it is established that the event
has happened. A simple answer is that ; it is the very fact that it justhappened and went away into the past, which needn't be elucidated
further more. But, I am afraid this line of thought might come back again
finally to the previous implausible view that the time itself causes thesudden change, because this simple answer eventually appeals only to thechange of the tense in order to explain the sudden change of probability . .
This is just a refusal of investigation, which doesn't seem to bephilosophical.
Ramsey (1990), p.65.
Papineau (2001), p.28.
17 Howson & Urbach (1993), p.4l.

15
16

27

Now, what is another possibility? A slight consideration enables us
to remember an existing theory that explains the sudden change ofprobability into 1. That is what is called 'the collapse (or reduction) of the
wave packet' in quantum mechanics. Mentioning this in this context is notinappropriate, since, as I said earlier, problematic phenomena coming fromquantum mechanics are one of sources that produced a powerfulinterpretation of probability, namely, the propensity theory. Usually thepropensity theory, in particular in an original form by Popper, is classified
as the objective interpretation of probability, but, exactly speaking, suchclassification isn't necessarily true in spite of Popper's own intention if we
consider his way of introducing the propensity theory in connection withquantum mechanics. I will discuss this point later. Anyway, as a matterof fact, apart from its purely mathematical notion, 'probability amplitudes',quantum mechanics itself doesn't always declare anything clear about how
to interpret the concept of probability despite the fact that its main idea
lies in the appeal to probability when describing the reality of nature.
Hence it is sometimes seriously discussed in a philosophical context how
we should interpret probability in quantum mechanics. IS Actually we
could say, independently of Popper's context, that it is doubtful whether
probability in quantum mechanics must be objective in a literal sense,
because, taking into account the so-called measurement problem, we mighteven say that probability concept in quantum mechanics is subjective in acertain sense rather than objective (Popper seemed to recognize thissituation, then propose the propensity theory by explicitly criticizing thesituation). Therefore, I take up 'the collapse of the wave packet' as amaterial that can be universally applicable to the concept of probability,
irrespective of what interpretation of probability to choose, as far as PPE isconcerned l9 . Of course, to consider quantum mechanics in detail is not
my intention here, so my argument in the following provisionally focusesupon the classical Copenhagen interpretation 20. I believe that this18 For instance, Michel Paty recently published his thoughts about it, and asserted that
we should carefully distinguish two meanings of probability in quantum physics, i .e.
theoretical (relational and mathematical) and empirical (in a statistical sense)
meanings. See Paty (2001), pp.235-255.
19 When I considered the relation between subjective interpretation and objective oneof probability along quantum mechanics, I got some hints from Logue (1995). In
addition, I had an opportunity to attend at Logue's lectures at Oxford in the spring of
2003, which was entitled 'Philosophy of Probability and Induction'. I learnt a lot fromhis lecture, too. Here I express my gratitude to him.
20 But, as far as I see, for example, even the manY'worlds approach would be also
involved in another sort of the sudden change, because, if we observe quantum
phenomena in a fixed way and recognize that we are in this particular world, then thereis the sudden moving from one present world to another particular world, even if it
might have nothing to do with probability concept as the Copenhagen interpretation
deals with. This is confirmed by this Polkinghorne's description of the many-world
approach; 'at every act of measurement, physical reality divides into a multiplicity ofseparate universes, in each of which different (cloned) experimenters observe thedifferent possible outcomes of the measurement.' (Cr. Polkinghorne (2002), p.52). This28

strategy wouldn't undermine my points which I'm going to propose as akind of hypothesis or thought-experiment.
Now, then, what is the collapse of wave packet? How does theprobability suddenly change? How is that caused? It is enough here toreconfIrm only the basic points for my purpose. I call for the lucidly
summarized expression of an authority. John Polkinghorne simply
describes like this;Measurements must be occasions of instantaneous and discontinuouschange. If an electron was in a state with probability spread out 'here','there', and, perhaps, 'everywhere', when its position was measured wasmeasured and found to be, on this occasion, 'here', then the probability
distribution had suddenly to change, becoming concentrated solely on
the actually measured position, 'here'. Since the probability distribution
is to calculated from the wavefunction, this too must change
discontinuously ...... This phenomenon of sudden change, called thecollapse of the wavepacket21 ......
In addition, I quote the next Dirac's expression in order to confIrm clearly
what the result of the collapse is;
The state of the system after the observation must be an eigenstate of a,since the result of a measurement of a for this state must be acertain ty22.
This argument is already well known of itself, and also well known for
producing many perplexed difficulties in a philosophical sense as well as ina scientific sense, which are not to be argued here. Now, rather I want toask, what causes the sudden change of probability into certainty? Whatcauses the sudden change of probability from smaller than 1 to I? Well,
as is suggested in the description above, measurement or observation musthave something to do with this.
Polkinghorne expressed that as'occasions ofinstantaneous and discontinuous change.' It sounds to be theway a classical occasionalist, Malebranche, talks. Undoubtedly, here is avery deep philosophical question about causality. According to Barret,
Heisenberg 'must have meant that it was an observer's act of observationthat then forced nature to make a particular choice'23. Yes, it is areasonable and safe opinion to say that nature is the real cause of thesudden change. But, it's too safe to keep our ordinary concept of cause.
approach contrived the way of avoiding using probability concepts crucially, but thestructure is exactly parallel to the standard approach with regard to the sudden change.
Perhaps we may ask likewise, what causes this sudden change? That question isprecisely equivalent to the causal difficulty as I offered even if not using probability
concepts.
21 Polkinghorne (2002), pp.25-26.
22 Dirac (1930), p.49.
23 Barret (1999), p .26.

29

When I give a lethal injection to someone and the person die, it is not a
mistake to say that nature is the cause of her death. Or, when I leave aburning match in the dustbin of a house and then the house burns, it is nota mistake either to say that the burning is caused by nature. Yet, nobody
thinks that such excuses can make sense. Usually, a thing that is anoccasion of other thing and without which the other thing doesn't happen isloosely called a cause of the other thing. As the history of philosophy
proves, an attempt to strictly define the cause is either unsuccessful orvague of itself because of its looseness. In fact, Dirac declares in later
years; 'a measurement always cause the system to jump into an eigenstateof the dynamical value that is being measured. 24' Thus, let's judge,
hypothetically, that our measurement or observation is the very cause of
the collapse of wave packet. I think there is nothing which can reject thisjudgment as entirely ridiculous. Actually I couldn't imagine anotherpossibility at least in this context.
Of course, strictly speaking, those argument concerns the collapse ofwave packet, not the sudden change of probability itself. But, the conceptof cause is actually very loose, intrinsically loose, therefore, if there is theco-extensional phenomena with a cause of some event, we may call thephenomena the cause of the event, too. This is applied to the case of thesudden change of wave packet and of probability, since both of suddenchanges are clearly co-extensional. Thus, if it is our observation thatcauses the collapse of wave packet, then our observation is also regarded as
the cause of the sudden change of probability. As it were, the collapse ofprobability happens here simultaneously with that of wave packet, in thesense that probability distribution collapses and probability becomes
concentrated solely on the actuality position, '1'. My assertion is that; thesame situation is analogously true of the sudden change of probability ingeneral as appears in PPE. Consequently, I propose the next thesis as ahypothesis, which I call 'the Collapse of Probability Hypothesis (CPH);(CPH) If PPE is true, then our observation CAUSES the probability of anevent to collapse into the value 1 as soon as we observe the event
pass away into the past25 •Unless the time itself or the fact itself can be the cause of the suddenchange of probability, I think that our observation is actually the only

Dirac (1958), p.36.
I refer not to the present but only to the past, because I think that the present isalways perishing, so actually impossible to grasp as it is. Presentism, as Zimmermantries to defend, seems to fatally ignore the empirical vanishing character of the present.
Cr. Zimmerman (1998), pp.206-219. But, on the other hand, the concept of the past
has its peculiar difficulty, namely, the past no longer exists, so, exactly speaking, it isfundamentally hard to understand what the truth -maker is about sentences in the pasttense. This is one of traditional puzzles since Aristotle, which is definitely worthwhile
examining further in another occasion.
24
25

30

plausible and explanatory candidate as the cause 26 . There are infinite
numbers of factors that can be theoretically the cause of the sudden changeof probability depending on each case, but what I'm looking for is the causethat equally explains all cases. As far as PPE is accepted, our observation
corresponding to the phrase in PPE , 'is recognized as', seems to be an only
option as the cause. And, the sudden change of probability into 1 in PPE
is remarkably similar to the case of the collapse of probability in quantum
physics as regards the structures of their situations. Thus, I came to CPH.
In fact, rather I want to suggest a bolder claim by reversing the order.
That is to say, probability intrinsically collapses, therefore the collapse ofwave packet and of probability in quantum mechanics is just one of naturalconsequences derived from the concept of probability in general. Probably,
this claim could eliminate the seeming weirdness of my strategy of
appealing to quantum mechanics which is highly controversial in itself,
because my aim is to clarify the nature of probability in general by using
quantum mechanics as just one possible gate to lead to the nature. I don't
intend to at all, and needn't at all, found my argument upon quantum
mechanics.
4, Popper's Legacy
It wouldn't go amiss to turn our eyes to Popper's argument now, or
rather my argument thus far naturally induces us to do that, since CPH
I've just reached is proposed basically through considering the significanceof probability in quantum mechanics, which is precisely what Popper
focused on when constructing his propensity theory of probability.
Actually I have already mentioned Popper's propensity theory a couple oftimes, so I should determine my attitude towards his theory to a certain
extent, which is so remarkable in the history of the philosophy ofprobability as to be called 'Popper's Legacy.' My argument would be
reinforced, and the implication of CPH would be made more explicit, bylooking back a bit over Popper's Legacy.
What I hope to focus on first is that Popper offers thisthought-experiment in the part entitled 'A Random Walk and the'Transition from the Possible to the Actual" in his Quantum Theory andthe Schism in Physics,
A soldier in the desert is given a pocket roulette-wheel and instructed to
set it up, letting the pointer spin and come to rest, and then to march forone minute in the direction indicated by the pointer; and then to repeat
the performance, letting the pointer spin again, and again marching forone minute in the direction of the pointer. It is at once intuitively clear .
26 If we connect the notion of observation with the notion of self or spirit, this viewmight be regarded as one variant of Berkeley's classical standpoint. See, for example,
Berkeley (1951), section 30. Incidentally, Berkeley was highly reassessed by Popper
as one of pioneers of scientific instrumentalism that could be concerned with quantum
mechanics. See Popper (1963), pp.97·119.

31

that, on the basis of this instruction, we obtain a probability distribution
for the position of the soldier which spreads with the soldier's marching
speed from the starting point in all directions -a kind of cloud which isdense in the centre and thins out towards its periphery .....
Now let us observe the soldier, after one hour, when he is justconsulting his pocket roulette-wheel. Then, we may say, the old clouddisappears, and a new cloud is started at the point at which we haveobserved the soldier.
This is precisely the same thing as the 'reduction of the wave
packet,' ...... .It even occurs, though in a slightly more trivial way, in any
such game as tossing pennies: relative to the information that we havetossed heads, the (relative) probability that we have tossed heads from lh
to 1. That this is fundamentally the same problem as that which has
disturbed the quantum theorists 27 .
In short, Popper points out that the sudden change of probability occurs
even in the case of throwing a coin, and that those trivial phenomena are
precisely what the reduction of wave packet means. Obviously this claim
has the same way with my argument of suggesting that the problem ofcollapse of wave packet in quantum mechanics is just one sign ofcharacteristics of probability concept in general. However, Popper
extends his claim to even say that there is nothing problematic in thatsudden change. He develops the same point as this and makes a stronger
claim in other parts of the same book or his The Logic of Scien tine
Discovery in terms of this next thought-experiment that comes fromEinstein and that Heisenberg also mentions28 •Take a semi-transparent mirror, and assume that the probability thatlight will be reflected by it is lh. Thus the probability that light passthrough will also be lh, and we have the next equation (a), if the event'passing through' or 'transmitted' is a, and the experimental arrangementb,(a)P(a,b) = lh = P(-a, b),

where '-a' (that is, 'non-a' ) stands for the event 'reflection.' Provided
that the experiment be carried out with one single photon, the probability
wave packet associated with this photon will spirit, and we shall have thetwo wave packets corresponding to P(a, b) and p( - a, b) in the above
equation. Now let us assume that we find, with the help of photographic
plate, that the photon was reflected, i.e. - a. Then, according toHeisenberg, the probability of finding the photon in the other part of thepacket immediately becomes zercf. 9 . What has happened? We had, andPopper (1982), pp.123-124.
Popper (1982), pp.76-78 and Popper (1959), pp.235-236.
29 Heisenberg (1930), p.39.
27

28

32

still have, the relative probabilities expressed by (a). If we take theinformation - a, then relative to this information we get this (/3),(/3)

Pea, -a) = 0,

PC-a, -a) = 1.

Certainly it is true that pea, - a) = 0, but it is quite wrong to suggest that
it is a kind of changed form of the original packet Pea, b) which'immediately becomes zero.' The original packet pea, b) remains equal to'l2, which is to be interpreted as meaning that if we repeat our original
experiment, the virtual frequency of photons being transmitted will equal
to'l2. And Pea, -a), which is zero, is quite another relative probability. It
refers to an entirely different experiment which, although it begins like thefirst, ends according to its specification only when we find that the photon
has been reflected. No action is exerted upon the wave packet Pea, b),
because pea, b) is the propensity of the state of the photon relative to theoriginal experimental conditions. This has not changed. Thus, thereduction of wave packet clearly has nothing to do with quantum theory: it
is a trivial feature of probability theory that, whatever a may be, pea, a) = 1
and pea, - a) = 0. When we toss a coin, the probability of each of its
possible states (i.e. head or tail) equals 'l2. As long as we do not look at theresult of our toss, we can still say that the probability will be 'l2. If we bend
down and look, it suddenly 'changes.' Was there a quantum jump, owing
to our looking? Was the coin influenced by our observation? Obviously
not. That's Popper's argument, which rather seems to be so strong as totrivialize CPR.
I will raise three points towards the Popper's argument. First, it must
be extremely counterintuitive to accept Popper's argument altogether,because his point would make probability kinematics impossible inprinciple although that is usually recognized as one field of the philosophyof probability. Actually we don't think at all there is somethingproblematic about saying that probability changes, but, Popper's argument
seems to forbid us from saying so, which sounds ridiculous. Additionally
if we follow his argument, we might lead to such an idea that the world is
re-born every moment into a new different one without any continuation.
This idea sounds to be utterly unacceptable. It is true that it's possible toaccept the idea, but doing so makes a lot of sacrifices. In fact, Popper
himself virtually mentions the notion of probability-change when saying;'it is the objective propensity of a which may become zero. 30' This seems
to contradict his criticism against Reisenberg. Second, Popper assertsthat p( - a, - a) = 1 has no problem at all since this is just a trivial
tautology, but it seems to me that he passes thorough a philosophicalquestion without stopping to consider. If we follow his idea and formulate
exactly, what matters here is how P('the photon is reflected', 'the photon
was reflected') should be dealt with, which isn't necessarily treated as a
30

Popper (1982), p.88.

33

self-evident tautology. The second term is the sentence in the past tense,
while the first one is tenseless or in the future tense, and perhaps based on
Undoubtedly here is a delicate
a counterfactual supposition.
philosophical problem hidden like temporal difficulty I mentioned. In any
case, all that Popper can claim is that P('the photon was reflected', 'thephoton was reflected') = 1, which is absolutely trivial, making no sense anduseless here.
Third, as is shown by Popper's declaration that observation has noinfluence on the coin at all, he tacitly makes a presupposition that only
physical phenomena could be accepted as causal influence, upon which heconcludes that observation, knowledge or ignorance has nothing to do withcausal process. The same argument can be found in the part in which hemakes fun of Pauli by saying that Pauli takes our ignorance to increase
entropy31 . However, such argument isn't fair at all. Obviously, it ismerely a dogma that causal influence takes place only about physicalphenomena, as we can draw as many examples as we like concerningcausal influence in a psychological or epistemic phase. For instance, itwould be inevitable that, if I suddenly recognized the person in front of me
to be my boss that I haven't met yet, the recognition will causally influence
my psychological state. Considering this, Popper's argument will notnecessarily work, if the observer in the collapse of wave packet orprobability could be supposed to be something giving causal influence insuch a non-physical level. Actually, as a matter of fact, the causal
understanding of physical phenomena depends upon our network ofknowledge, which would make it not so unreasonable to think that an
observer's action of applying the network to the phaeneomana is the cause
to trigger the whole process. In fact, even Popper himself declares that;'even the 'observation' of a part of the track of the particle in position spaceis ...... actually an interpretation in the light of theories, in spite of its
apparent intuitive obviousness. 32' This line of thought would affect theobjective status of Popper's 'propensity,' which can be actually confirmedby this Popper's definition of the notion of propensity;
This application of probability theory to single cases is precisely what
the propensity interpretation achieves. But it does not achieve it by
speakinfg about particles or photons. Propensities are properties ofneither particles nor photons nor electrons nor pennies. Propensity
statements in physics describe properties of the situation, and are
testable if this situation is typical, that is, if it repeats itself (such as in
the case of the emission of light). They are, therefore, also properties ofrepeatable experimental arrangements: physical and concrete 33 .
If so, we can ask; who sets up the repeatable experimental arrangement?
Popper (1982), pp.109-ll0.
Popper (1982), p.143.
33 Popper (1982), p.79-80.
31

32

34

Who arranges it? Obviously someone must artificially arranges it withthe background of some scientific knowledge. Accordingly, here isdefinitely some person who has a particular knowledge of science and gives
a particular idea, contrary to Popper's official treatment of 'propensity' as
objective. If we mean an independence of observer's knowledge by'objective,' Popper's 'propensity' could not be objective by definition.
Rather it seems that we could say it's epistemic. Anyway, however, it isalso certain there was room in the side of quantum mechanics that
incurred Popper's criticism, since they seemed to describe causal influenceof observation in the collapse of wave packet as if in the same level withphysical phenomena (as if it is light emitted from our eyes). Thus, in any
case we can say that it's possible for our observation to cause the collapseof probability, as CPH claims. At least my argument so far confirmsthat PPE and CPH is concerned with epistemic interpretation ofprobability as well as objective one, because Popper's theory of propensity,
which gives my argument main materials, seems to commit itself to an
epistemic interpretation with professing itself to be an objective theory.
To sum up, Popper's theory supports my argument in that he also
takes what is called the collapse of wave packet to be not restricted tomicrocosmic quantum movements but ordinarily recognized anywhere inour daily life concerning the concept of probability. However, at the sametime, I could show that my argument is applicable to any interpretation ofprobability concept by pointing out that Popper's theory, which gives mymain materials as a representative view of objective interpretations,substantially commits itself to an epistemic interpretation.
5, Argument from Humphreys' Paradox
In the following, I'm going to verify whether CPH is capable ofexplaining anything probabilistic or not, and clarify what kind of
implication CPH eventually brings. In order to do that, I take up the
problem called 'Humphreys' Paradox'. The paradox is not only thought toraise a fundamental difficulty about the concept of probability, but also
seems to be a typical example of puzzles that could be solved by applying
CPH. That is a process of verifying CPH. Mter that, I want to consider
what results CPH produces by scrutinizing why 'Humphreys' Paradox' isthought to be a problem. This is the next step of clarification of CPH's
implication, which will lead to a deeper difficulty.
Now, what is Humphreys' Paradox? This paradox was first proposed
by Paul Humphreys, and then published by Salmon34 , after which it wasnamed 'Humphreys' Paradox' by Fetzer35 . The paradox originally casts
doubt on the propensity interpretation through making inverse conditional
probability in terms of Bayes's theorem. There are some versions of thisparadox, of which I quote the next clearest one that Salmon re-mentioned
in a later year;
34

35

Salmon (1979), pp.183-216.
Fetzer (1981), p.283.

35

(Can Opener Case)
Consider, for example, a factory that produces can opener. There are
only two machines, which we may designate A and B, in the factory.
Machine A is ancient; it produces one thousand can openers per day, and2.5% of these are defective. Machine B is more modern; it produces ten
thousand can openers per day, and only 1% of its products are defective .
Suppose, at the end of the day, that all of the defective can openers(which have been stored by the inspectors) are placed in a box.
Someone randomly picks a can opener out of the box, and ask for the
probability that it was produced by the modern machine B. We caneasily calculate the answer; it is 4/5. Nevertheless, I find it quite
unacceptable to say that this defective can opener has a propensity of 0.8
to have been produced by machine B. It makes good sense to say thatmachine B has a propensity of 0.01 to produce defective can openers, but
not to say of the can opener that it has a certain propensity to have been
produced by that machine36 .
Let D = that the picked can opener is defective. Let A = that it was
produced by machine A. Let B = that it was produced by machine B.
Then, by definition, P(D I A) = 2511000, P(D I B) = 11100, P(A) = 1/11, P(B) =10/11 . The probability questioned here is P(B I D). We can calculate thisby using Bayes' theorem as follows;P(B I D) = P(D I B)P(B) /P(D)= P(D I B)P(B) /P(D IA)P(A) + P(D I B)P(B)= 11100 X 10/11 /25/1000 X 1111 + 11100 X 10/11= 100/11000/

125/11000=4/5
That is perfectly clear, as far as Bayes' theorem is accepted. And I haveno reason to refuse Bayes' theorem at all. Then, what is the problem?
Gillies lucidly describes how this is paradoxical by using the frisbee
example, so I paraphrase his expression using can openers instead offrisbees. That is to say, 'the picking of a defective can opener from the box
in the evening is a partial cause of weight 4/5 of its having been producedby machine B earlier in the day. Such a concept seems to be nonsense,
because by the time the can opener was selected, it would either definitely
have been produced by machine B or definitely not have been produced by
36

Salmon (1984) , p.205.

36

machine B. We can make the point more vivid by supposing that machine
A produces white can openers and machine B orange can openers. If thedefective can opener picked at the end of the day was orange, it would
definitely have been produced by machine B, and it is not clear what would
be the sense of saying that it had a propensity 4/5 to have been produced bymachine B'37. In short, if we understand the can opener example in termsof the propensity theory, we have to adopt the notion of PARTIAL
BACKWARD CAUSATION, which seems to be too absurd to be accepted.
Roughly speaking, three types of reaction to Humphreys' Paradox
have been presented. First, it is claimed that this paradox shows theinappropriateness of the propensity interpretation of probability, therefore
we simply have to give up the propensity theory. This is advocated by
Humphreys himself and Milne. Humphreys is fully aware of the causal
difficulty appearing here, and initially declares; 'the causal nature ofpropensities cannot be adequately represented by standard probability
theory.3s' Milen similarly emphasizes the strangeness of applying Bayes'
theorem of conditional probability to propensity of events, and concludes
that 'conditional probabilities cannot adequately be treated in realist
single-case interpretation of probability,' then suggesting the shift to aninterpretation based upon the long run frequencies 39 . Milen points outthat this paradox consists of three premises, namely, conditional
probability, the link between realist single-case interpretations and theindeterminate, and the claim that the past and the present are
determinate unlike the future. Then he said that anyone rejection ofthese premises is an unhappy choice40 . The second reaction is that weshould clarify which case is and which case is not suitable for us to apply
Bayes' theorem to the propensity of events. This is developed by McCurdy.
He tries to secure the situation in which inverse conditional probabilities
about propensity is perfectly meaningful by taking into account the time
when we calculate the conditional probability. For instance, take up
again the can opener example and let t1 indicate the morning before
machines move, t2 any arbitrary daytime when machines are working, ts
the end of the day when someone picks a can opener from the box, andsuppose that Pti(Dtj ! Btk) is interpreted as "the probability at 1;; for D at tj,
conditional upon B occurring at tk. Let's compare those two probabilities(ignoring the background information now for brevity),(1)

Pt1(Bt2! Dts)

(2)

Pts(Bt2! Dts)

McCurdy's claim is to the effect that (1) perfectly makes sense and is given
Cf. Gillies (2000), p.131.
Humphreys (1985), p.557.
39 Milne (1986), p.131.
40 Milne (1986), p.132.
37

38

37

the value 4/5, while (2) is an absolute nonsense, because once the event Dt3
or "" Dt3 has been realized, there is no indeterminacy about theoccurrence of the event Bt2 - it has either occurred or not41 . In otherwords, on the one hand, he entirely agree that Humphreys' case is involvedin a paradoxical difficulty when we understand it in the sense of (2), but on
the other hand, he maintains that inverse conditional probability through
Bayes' theorem is unproblematically applicable to the propensity
interpretation when we consider Humphreys' case before events concerned
occur, as (1). This is an attempt of defending the propensity theory even if
partially. Then, McCurdy, appealing to Niiniluoto's suggestion42, also
adds the point that (1) is a physical probability, whereas (2) is an epistemic
probability 43 .
Now, the third reaction is simple. Namely, since
Humphreys' paradox only proves that the propensity theory as realist
single-case interpretation of probability is insupportable, we can still
support and refine the long-run propensity theory, which is immune fromthe paradox. That is alluded by Milne as above, and deliberately
proposed by Gillies44 . But, this third reaction is outside the scope of myinterest, because I provisionally want to discuss only the probability ofsingle event as PPE shows.
Certainly, here is a truly serious difficulty about probability. Then,
what reaction should I offer by applying CPH? There are several things
to say, but first of all, I want to make sure this point; as soon as I pick a(either white or orange) can opener from the box and see it, the probabilityof the can opener's being orange collapses into 0 or 1 according to CPH,
therefore there is no room to ask the probability, at least soon after thecollapse. So, if Humphreys' case takes the issue of probability after thecollapse, it is very natural for the case to fall into a paradoxical un solvablesituation because no substantial question exists. It's like the case ofquestioning what the probability of probability 1 is. In this sense, I thinkthat Humphreys' paradox is nonsense from the outset, so it's not
persuasive to conclude the inappropriateness of the propensity theory fromthis paradox. However, I don't intend to say that Humphreys' paradox
doesn't touch any crucial point at all about probability. It seems thatsome hidden significant implication about probability will come into viewby modifying and remaking the paradox and then applying CPH to it.
How do we modify it? Well, suppose the next condition in the can opener
case.
(Modified Version)
The box in which all defective can openers are placed at the end of the
day is made of wood, so its content isn't seen from the outside. Then, I
am told to pick one can opener from the box. I put my hand into the box
McCurdy (1996), p.107 ff.
Niiniluoto (1998), pp.l03-04 note 16.
43 McCurdy (1996) , p .121.
44 Gillies (2000), p .131-136.
41

42

38

and grasp a particular single can opener. At that moment I am asked
what is the probability that it was produced by the modern machine B.
Namely, I am asked before I take it out and see it.
This situation is just arranged in a more detailed way than the original, so
nothing is changed with regard to the calculation of probability.
Therefore, it still come to that P(B I D) =4/5 in terms of Bayes' theorem.
Now, how do we understand the result, p(B I D) = 4/5, in this situation?
Is it paradoxical? According to CPH, the probability hasn't collapsed yet,since we haven't observed it, or in other words along the terminology of
PPE, we haven't recognized whether it's white or orange 45 . Thus, the casemust be that P(B I D) < 1 as far as we accept CPH. We have strong reason
to follow Bayes' theorem to assign 4/5 to P(B I D). Let 13 indicate the time
when I grasp a particular can opener, and I think that even the calculation,
P13(Bt21 D13) =4/5, which was presented along McCurdy's argument above,
The problem isis likewise accepted. Is there something wrong?
undoubtedly our intuitive idea that the can opener grasped by me wasalready actually fixed either white or orange, irrespective of being seen ornot. This idea urges us to refuse to simply apply Bayes' theorem even to
the modified version of Humphreys' paradox as well. Yes, I agree with theidea. I usually live with having such an idea as this, of course. But, I'm
now considering a philosophical problem through a kind ofthought-experiment. From such theoretical point of view, what must be
noticed first is that this idea is supported by determinism about the past,
which insists that the past event either happened or not, wherefore theprobability of its occurrence must be either 1 or O. To put it differently
with Dummett's taste, that idea takes it for granted that the law of
excluded middle is applied to the past. That is presumed. Well, how canit be presumed? By what ground? Honestly speaking, I think there is nodecisive ground on which we have to presume it. We are usually living
having that presumption, and maybe the presumption could be inferredwith A CERTAIN DEGREE OF conviction. But, still there is nothing tocompletely block other possibilities, which means that the conviction is not
perfect, not reaching probability 1. A.J. Ayer once correctly pointed out
about the notion of sense-data that sense-data language is just chosen by
our resolution and nothing more than that46 . I want to say the same thing
here. Our intuitive idea of determinism about the past is always only
45 Someone suspects that the concept of observation or recognition is quite vague here,
for grasping something can be a sort of recognition in a tangible sense, although I'm notsure it can be also called a tangible observation (?). But, this doubt can be easily
dispelled by modifying further, if necessary, the condition into that; each defective canopener placed in the wooden box is randomly numbered by the inspector from 1 to 125,
and then I am told to choose a particular number within those and asked what is theprobability that the particular can opener with the number was produced by themodern machine B. Modifying in this way, we avoid being involved in the problem ofvagueness about the concept of recognition.
46 Ayer (1940), p.28.

39

chosen by our (perhaps instinctive?) resolution. The past event that isn't
recognized yet, although vividly represented in a specified way (e.g. in thisparticular box right now), must be assigned smaller than probability 1, as I
said earlier. As I mentioned before, Milen regarded the determinacy ofthe past as a basic premise incapable ofrejection. But, it's not necessarily
true. This is an implication of CPH. In other words, the difficulty of
Humphreys' paradox directly arises from determinism about the pastrather than from partial backward causation. So, since determinism canbe rejected, Humphreys' paradox can be solved.
Certainly, this argument might sound strange first, but it's always
neither absurd nor inconsistent with our common sense. In reality, what
can you say about the content of the wooden box? We may suppose that
we surely had seen white and orange can openers placed in the box. Even
if so, can we be truly sure that those can openers remains to be with thesame fixed colours in an absolutely perfect sense? You may say, it iscertainly confirmed if we actually check the content. But, I'm asking
about the case before being checked as in the modified version.
Determinism, which is just a result of our resolution, isn't completely
grounded, either. Thus, theoretically speaking, the probability of can
openers' keeping the same situation in the box must be smaller than 1. In
fact, we are ordinarily thinking in this way with regard to historical events.
I t is not strange at all to use the concept of probability in describing history,
even though it was clearly in the past. Suppose the case that a certain
historical figure was confirmed to be in a particular place at a particulartime, but the person was missing for two years after that, then some pluralstories had been told about her behaviours for that missing period. Inthat case, we usually apply probability to each story in such an
unrecognized case, and then assign probability smaller than 1. Actually it
is not rare that Bayesian theory is applied to confirm a hypothesis about
historical events, as McCullagh investigates 47. This kind of argument
might also remind someone of a famous discussion concerning the reality of
miracle that Hume once presented by appealing to the concept ofprobability48. In any case, those arguments regarding history is quite
normal, and is entirely consistent with PPE and CPH. Thus, evenordinarily speaking, we may say that the probability of the can openers'
keeping the same situation in the box is smaller than 1. If so, there isnothing wrong in applying B aye s' theorem to the modified version of
Humphreys' paradox and drawing the result, P(B I D) = 4/5.
Now, someone has already found that what I argued so far isremarkably similar to the measurement problem in quantum mechanics,
at least as regards their structure of argument. Yes, I think so, too.
Actually, there are three reasons for that. First, since even macrocosmic
phenomena are composed of microcosmic quantum phenomena, theproblem deriving from quantum phenomena is theoretically applicable to47
48

McCullagh (1984), pp.57-64 .
Hume (1975), Section X.

40

macrocosmic phenomena, even though by an infinitely subtle amount.
Second, Humphreys' paradox was originally presented in relation to thepropensity theory which had been initially motivated by quantum
mechanics, so it may be natural for the analysis of the paradox to result inthe situation similar to the measurement problem. Third, the collapse ofprobability that arises out of the measurement problem in quantum
mechanics can be rather thought to be one of natural implication ofprobability concept in general, as I said earlier, so the case is inverse, i.e.
the collapse of probability expressed in CPH finally leads to themeasurement problem in quantum mechanics as one result.
However, we mustn't stop here, because the measurement problem isthe very problem which is still controversial. I have said earlier that
determinism isn't completely grounded, but it's just a theoretical remark.
As a matter of fact, we always choose determinism about the past, even if
it's not an only possible option. Therefore, the measurement problem
sounds truly problematic, which was shed light on in an enormously
impressive way by well known Schrodinger's cat. I think that the samething is true of Humphreys' paradox concerning the structure of argument,
although, of course, we mustn't overlook the fact that there is a big
difference between Schrodinger's cat and Humphreys' paradox from theviewpoint of physics. At all events, anyway, we should say that
Humphreys' paradox is theoretically not paradoxical, but practically stillparadoxical. If so, we have to examine further how such a paradox
practically arises and what consequence the paradox brings about.
Before moving to the next stage, I want to notice one basic point.
What I discussed so far concerns mainly the propensity interpretation ofprobability, as is clear from remembering how Humphreys' paradox was
initially presented. However, I want to emphasize that this paradox isapplied to the epistemic interpretation as well. Look at the modified
version of the paradox. In this case, an application of Bayes' theorem
perfectly makes sense, because we are subjectively uncertain of whether
the grasped can opener is white or orange. But, at the same time, we also
accept determinism about the past even from the subjective point of view.
We don't believe that the situation of can openers is changed or unstable
after they are placed in the wooden box, unless something special happens.
Therefore, we subjectively believe that the colour of grasped can opener isalready fixed the moment I grasp. Namely, it is probabilistic in a
subjective or epistemic sense whether I grasp a white can opener or an
orange one, but it is certain even in a subjective sense that the grasped can
opener was already made by either machine A or B independently of mygrasping. The distinction between the physical probability and theepistemic probability as Niiniluoto and McCurdy suggested seems to missthe mode of our subjective belief in determinism about the past.
Humphreys' paradox is the paradox to the epistemic interpretation ofprobability, too. Accordingly, my argument so far is concerned with not
only the propensity interpretation but also the subjective one. Actually
this point was already confirmed by examining Popper's propensity theory41

in the previous section.
6, Retroactive Assignment of Probability 1
Now, let's face the question, why do we accept the determinism about
the past in an objective sense? Why do we believe in the determinism
about the past subjectively as well?
Clearly, this question is deeply
conjoined with a question about how to understand the past. This is so
grand a question that I can't deal with it thoroughly here, of course, but
perhaps, presenting my basic intuition about the past might be forgiven.
It seems to me that the past registers in our minds most realistically when
we regret something we did or feel nostalgic for what we were. Can wethink that such case is an original occasion for us to learn the meaning of
the past, and then we extend the meaning to other cases by generalizing it?
It might be. So, I suppose hypothetically that it is the case. Thus, thesetwo points should be noticed.
(a) When we regret something or feel nostalgic for something, the past
scene in the memory is vividly represented in a specified way,although the memory might be wrong.
(b) The past scene is thought to be inalterable, which characterizes
regret or nostalgia differently from out attitude towards the future,i.e. such an attitude that we can somehow alter or control the future.
According to PPE and CPH, (a) seems to simply imply that the probabilityof the event concerned must be 1, and that our representation causes theprobability 1. However, we must note that PPE and CPH stipulate only
the case of the moment the event has happened (thus I inserted, 'as soon
as', 'just now'), whereas (a) covers the case long after the event happened
as well as the case of the moment it has happened. Therefore, to be exact,
it's not appropriate to apply PPE and CHP to the case of regret or nostalgia.
As a matter of fact, past events regretted or felt nostalgic can be mistaken,
hence I added, 'the memory might be wrong'. In addition, such case mightalso be involved in the temporal difficulty of probability 1 that I mentioned
earlier by appealing to the factor of forgetting. Thus, it's utterly
reasonable to assign probability smaller than 1 to the events regretted or
felt nostalgic, except for the case that we regret or feel nostalgic forsomething which has just now happened. But, nevertheless, the notion ofregret or nostalgia, perhaps grammatically, requires that all past events
regretted and felt nostalgic actually happened and were already fixed interms of (a) and (b). Otherwise, those notions will lose their intrinsic
meanings. If the event concerned could be non-existent or altered, neither
regret nor nostalgia would arise. But, thus, how could we understand thissituation? My understanding might sound Humean; some FICTION as if
we are observing it now is introduced here. We regret or feel nostalgic forsomething past by fictitiously representing those past events as actuallyjust now happened, so as having probability 1. In this case, theprobability of those events is supposed to have probability smaller than 142

before our representation. At least they can't have probability 1 at all
before representation in the sense of PPE and CPR. Therefore, thesudden change of probability appears to occur here. If we thinkanalogously to CPH, we could say that our representing when regretting or
feeling nostalgic, as it were, CAUSES the sudden change of probability
from smaller than 1 to 1. In other words, probability of those past events
collapses in a RETROACTIVE way by our fictitious representation.
The argument thus far is concerned only with such special cases asregret and nostalgia. But, can there be other cases of representing the pastthat are structurally similar to regret and nostalgia? I think there is.
That is the case that we recognize something static as the same as it wassome while before. In this case, we represent the shape of it in the pastand recognize the present one as the same. Rather, we overlap two
shapes. As a result, we represent the past event very vividly. Does thisresemble the case of regret or nostalgia with regard to the way ofrepresentation?
Yes, definitely.
Thus, the retroactive collapse ofprobability by fictitious representation seems to appear, too.
The
probability of the existence of the static thing in the past is estimated as 1
although this case doesn't exactly meet PPE and CPH. This process, ifany, seems to be immensely influential in our conception of the past, since
static things, the most fundamental of which is the earth, forms the mostbasic background of our epistemic activities. Therefore, this process ofretrospectively assigning probability 1 to the past event is naturally
extended to the whole understanding of the past in general, however
fictitious it is. Thus, determinism is born here. We become inclined tothink that past events are already determined and fixed irrespective of ourrecognition, wherefore the probability of the occurrence of those events
must be 1. Perhaps, this way of thinking suits very well our ordinary
language in a practical sense, because the concept of responsibility or
ascription forms a very important part of our practical talk, and suchconcept obviously requires the fixed and determined status of the past,hence I call this context practical rather than theoretical. Anyway, I
propose what I argued thus far as the Retroactive Probability Hypothesis(RPH) in the following;(RPH) Our fictitious representation of past events CAUSES probability
to RETROACT to the past and collapse into value 1, from which
DETERMINISM about the past appears.
RPH might seem to be odd at first sight, but such first impression could be
eliminated by a bit of reflection about our actual notion of the past. I
mean the condition that the past is nowhere by definition of the notion, so
its verification is impossible in nature therefore the past is spoken
necessarily through some fictitious taste. To be fictitious is the intrinsic
character of the past. In this respect, it is quite natural that the narrative
theory of history is often claimed. At least, regarding the past as real in
the literal sense is either a philosophical deception or an intellectual

43negligence.
Two points should be confirmed. First, we could explain why
Humphreys' paradox sounds paradoxical by applying RPH to it. As amatter of fact, we tend to think that can openers in the wooden box has
been already fixed with regard to their colours, i.e. their probabilities ofbeing made by machine B must be either 1 or 0, before we see them, hence
it sounds absurd to apply Bayes's theorem to a particular can opener
grasped by me in the box in order to calculate the probability of having
already been made by machine B. All of this arises only from our
fictitious representation at the time of grasping the particular can opener,
namely, the representation as if we had been observing the can opener inthe box in spite of the fact that it wasn't actually observed. Thus, RPH
elucidates this quite perplexed situation. In this respect, Timothy
Williamson's objection to my modified version of Humphreys' paradox isworth mentioning. He objected that my modified version will not makesense if some videos machine would be set in the box, since in that case thecolour of a particular can opener is known to have been fixed before we seeit49 • To this, first of all, I would like to answer that probability about the
information recorded in video machine would also collapse the moment we
see it, so this case couldn't be an objection to my argument. But, his video
example is very suggestive, because this example will illustrate well thesituation expressed by RPH. Namely, as his example impressively
alludes, we are strongly inclined to make a fictitious representation as if
we were observing the past, thereby virtually causing the retroactive
collapse of probability into 1 through the representation. Thus, RPH might
be strengthened by his video example. Well, incidentally, it seems to me
that the paradoxical impression given by Schrodinger's cat might be partly
derived from our similar fictitious representation as if we were seeing theinside of the box in which a cat is placed. Anyway, determinism about thepast necessarily appears in those situations, where paradoxes stand up.
Second, RPH clearly implies some form of BACKWARD CAUSATION
in the sense that our present representation is the cause of the collapse ofprobability in the past, even though in a fictitious way. As I mentioned,
Milne and Gillies diagnosed Humphreys' paradox as coming from aridiculous notion of (partiaD backward causation. They are right in acertain sense. Some backward causation really matters in Humphreys'
paradox. However, what backward causation is concerned is not in thepartial contribution (by 4/5) to produce the past result, but in the overall
formation of determinism about the past which provides us with thebackground to make the partial backward contribution paradoxical. Then,
how should we treat this backward causation? Should we abandon it
instantly as complete nonsense? Certainly, my argument is consistent
with this strategy of abandoning backward causation, since I take the idea
expressed by RPH to be a fiction. But, what we should notice is that, if we
Williamson raised this objection on 29th May 2003 when I visited his office at New
College of Oxford.

49

44

abandon the notion of backward causation altogether, we simultaneously
have to abolish the conception of determinism about the past, according to
RPH. All are dependent upon how we should estimate the concept offiction. If we must reject everything fictitious, we have to give up thedeterminism about the past. Yet, on the contrary, if we accept that ourepistemic activities are intrinsically involved in something fictitious, as it
were, in the form of metaphysical commitments concerning reality, timeand causation, we should face the fact that we are living with having thedeterminism about the past in a fictitious sense. I think that the latterroute is more natural and reasonable. Perhaps, what we should do after
accepting the latter route is to be very careful to distinguish those fictitious
factors as fictitious when understanding our epistemic activities. In truth,
we can regard the fictitious representation in RPH as a sort ofcounterfactual conditionals such as, 'if we had seen the inside of the box,
we would have found an orange can opener.' And, such sort ofcounterfactual conditionals is quite common for us to use in order tounderstand a dispositional character of usual physical things like, 'fragile','flammable', and so on. Additionally, as is often discussed in thephilosophy of free will, we ordinarily appeal to the conception, 'could havedone otherwise', to judge whether a certain person are responsible or not50 .
Obviously, this ordinary conception is also made through a form ofcounterfactual conditionals51 . If so, the fictitious representation in RPH
shouldn't be instantly refused by reason of being fictitious.
Furthermore, a bit more careful deliberation enables us to notice thateven backward causation is actually supposed in some cases. For
instance, as Suppes points out, 'it is widely known, and often commented
upon, that the basic equation of classical physics remain valid under atransformation from time t to -t. 52' But, of course, this reversibility, or'this invariance under a change of direction of time ...... creates a naturalphilosophical tension about the nature of causal processes, 53' for ourordinary causal concepts work only forwards. In any case, we shouldn't
ignore that the possibility of backward causation is taken seriously at leastin physics. In fact, tachyons or anti-particles are supposed to operate
backwards 54 • Presumably, philosophical discussion about backwardcausation that has become one of indispensable topics in the philosophy of
I discussed this problem in Ichinose (2003), pp .I-18.
As to the character of 'could have done otherwise' condition as counterfactual
conditionals, it was when I joined the lectures of Martha Klein at Oxford in the Trinity
Term of 2003 that I came to realize the necessity to examine it. Her lectures entitled'Free Will, Moral Responsibility and Punishment' was definitely stimulating and full of
insights. I am thankful to her. Also cf. Klein (1990).
52 Suppes (2001), p .203.
Suppes himself firmly believes that causal relation works
forwards , and emphasizes that not strongly reversible systems prevail in nature, which
reinforces our personal experience to support the commonsense view of forwardcausation. See Suppes (2001), p.212.
53 Suppes (2001), p.203.
54 Cf., Horwich (1987), pp.lOl-102.
50
51

45

causation since Dummett might be encouraged by the current stream ofphysics. Moreover, it's even possible to find some examples of backwardcausation in our ordinary experience rather than theoretical physics. I
want to mention an example of apparent motion which Nelson Goodman
once discussed in an impressive way. This is a phenomenon in which we
fictitiously perceive the motion of flashed spot if two spots short distantfrom each other are flashed in a certain interval. This phenomenon is
actually common in the case of neon signs or pictures of TV. Then,
Goodman asks, 'how are we able in the apparent motion to fill in the spotat the intervening place-time along a path running from the first to thesecond flash before the second flash occurs?55' Goodman answers that theapparent motion is constructed only after the second flash occurs althoughseen as running from the first to the second flash. He calls it theretrospective construction theory56. Obviously, this could be interpreted
as an example of backward causation in the sense that our seeing thesecond flash causes the past motion of spot from the first flash to thesecond one. If that is the case, it's reasonable to think that we might beordinarily involved in a certain form of backward causation, even if it's
described as fictitious. Thus, the situation expressed by RPH must be
taken to be more than simply fictitious.
7, Newcomb's Problem and Determinism
Now, how should we estimate the status of determinism that appears
in RPH? I'm closing my argument by briefly verifying this. I want tomention here another paradox that is as perplexed as Humphreys' paradox,
in order only to illustrate my points or make my points stand out. That is,
Newcomb's Problem, which is well known since Nozick published it57.
Here I quote the simplest description that Nozick gave in a later year.

(Newcomb's Problem)
A being in whose power to predict your choices correctly you have great
confidence is going to predict your choice in the following situation.
There are two boxes, Bl and B2. Box Bl contains $1,000; box B2contains either $1,000,000 ($M) or nothing. You have a choice between
two actions: (1) taking what is in both boxes; (2) taking only what is in
the second box. Furthermore, you know, and the being knows you know,
and so on, that if the being predicts you will take what is in both boxes,
he does not put the $M in the second box; if the being predicts you willtake only what is in the second box he does put the $M in the second box.
First the being makes his prediction; then he puts the $M in the secondbox or not, according to his prediction; then you make your choice 58 .

Goodman (1978), p.73.
Goodman (1978),p.8l.
51 Cf. Nozick (1985), pp.107-133.
58 Nozick (1993), p.4l.

55
56

46

It is also well known that a lot of literature on this problem has been piled
up, and many views have been proposed. To examine those views is not
my aim here. I make only summarized comments for the sake ofconcluding my argument. Those views about Newcomb's problem include
so-called one-boxer (people claiming that we should choose the option (2) as
far as we seek as much money as possible), two boxer (people claiming that
we should choose the option (1)59, and other views like no-boxer (people
claiming that this problem is incoherent and underdetermined) 60 orNozick's one (introducing another notion, symbolic utility, to clarify thecomplicatedness of the problem). Roughly speaking, I think that the
viewpoint of no-boxer is a bit too honest to be interesting, since it's quite
easy to argue that this problem is incoherent because of including thesupernatural being that can predict correctly with an extremely highprobability (but probability smaller than 1, otherwise the problem doesn't
make sense). The attitude of no-boxer seems to be obstinately refusing
any fictitious, or metaphysical, factors.
But, such attitude doesn't
harmonize with RPH. In fact, as Mackie describes, this problem might
include a sort of backward causation in the sense that your choice causes
the correct prediction if one-boxer's argument is somehow reasonable.
Mackie himself seems to develop a no-boxer argument by mentioning
various absurdities drawn from Newcomb's problem, one of which comesfrom backward causation 61. However, as I have discussed, it's notnecessarily an acceptable attitude to reject backward causation from theoutset. Also two-boxer arguments seem to share such attitude with
no-boxer in this respect. Those arguments appeal to our intuitive idea
that, once the being makes a prediction and determines what is in thesecond box, the content of the second box is already fixed irrespective of
your later choice. The reason why two-boxers think so is simply that thenotion of backward causation is absurd, although they accept the setting of
Newcomb's problem as possible unlike no-boxers, then offer the choice of
two boxes as rational.
I want to emphasize, I don't intend at all to claim that the setting of
Newcomb's problem is perfectly reasonable and realistic. There is no
doubt at all that this is just a story. But, what I want to pay my attention
to here is the fact that this story is possible to understand. We could
imagine its situation as if it were actually happening. I wish to bring thisunderstandability into focus . From this point of view, it's not appropriate
to estimate Newcomb's problem by presupposing beforehand that notionsof the being with supernatural power of prediction and backward causation
are ridiculous. Evidently, Newcomb's problem is a kind ofthought-experiment to verify our way of making decision, so it's notstrange for it to include somehow unrealistic factors. Furthermore, as I
said, our ordinary understanding of things is tinged with something
Cf. Sainsbury (1995) , p .60.
Cf. Camp bell (1985) , pp.24-27.
61 Mackie (1985), pp.145-158.

59

60

47

,
i

I'
!

fictitious through dispositional words. In any case, we must say thatparticularly two-boxer argument is off the crucial point of this problem,since that argument ignores the role of the supernatural being. Then,
what about the one-boxer argument? My view is that; the one-boxer
argument is the most natural and rational as long as we accept the settingof Newcomb's problem as understandable, although it sounds weird in thesense of implying a sort of backward causation. But, how relevant is such
my view to the present context?
What we should notice here is that there is an obvious parallelism
between Newcomb's problem and Humphreys' paradox. In the case of theoriginal version of Humphreys' paradox, to apply B aye s' theorem andcalculate the probability of a particular can opener's having been made by
the machine B at 4/5 is nonsense, since the probability has alreadycollapsed after your seeing the orange can opener. The same is true of
Newcomb's paradox. To ascribe some proQability of the being's having
made the correct prediction after your choosing either (1) or (2) must be
nonsense. But, as to the modified version of Humphreys' paradox,
theoretically speaking, it's perfectly reasonable to apply Bayes' theorem tothe grasped can opener to calculate the probability of its having been madeby machine B, since the probability hasn't yet collapsed. For the same
reason, it's not problematic at all to talk about the probability of the correct
prediction before your actually choosing either (1) or (2), particularly if twoboxes are not to be seen from the outside as in the modified version of
Humphreys' paradox. Contents of box B can't be regarded as having
probability 1, in principle. Of course, there is a difference between two
puzzles. In Humphreys' paradox, there is mixture of white and orange
can openers, so actually nobody knows which can opener you grasp in thebox. On the other hand, Newcomb's case presupposes the special being
who himself puts nothing or $M in box B, so at least the being definitely(i.e. assigning probability 1) know what is in the box B. But, even though
it is the case, the content of box B mightn't be always said to haveprobability 1, as far as we consider the temporal difficulty of probability 1.
Strictly speaking, the same argument is applied to box A if the box A is not
to be seen from the outside. At all events, there is nothing irrational toapply the concept of probability to the correctness of the being's prediction
before your actual choice.
However, those arguments so far concern only the theoretical aspects.
If we practically think, the situation is changed and determinism about thepast revives strongly. As I saw along Humphreys' paradox, we are, as itwere fictitiously, inclined to think that the colour of the particular can
opener has been already fixed and determined when you grasp it before
seeing it, and in that stage determinism appears through the retroaction ofprobability in terms of backward causation according to RPH. A similar
point will be true of Newcomb's problem. As a two-boxer argument
unintentionally shows, we have a strong tendency to believe that, once thebeing made his prediction about your choice, the content of box B has been
fixed and determined, so your choice has nothing to do with what is in the48

box B. If we accept RPH, this belief could be explained by your imaginary
representation as if you had been observing the being's prediction and thecontent of box B until now. In other words, at the time of your choice or
your resolution to choose either (1) or (2), your fictitious representation as
if you had seen the prediction and the content of box B CAUSES theprobability of the correctness of the being's prediction to RETROACT to thepast and collapse into the value 1 or O. Certainly, backward causation
matters here, even if fictitiously. But, as I've already pointed out
concerning Humphreys' paradox, backward causation in this context works
in the overall formation of determinism about the past, rather than in thepartial backward contribution of your choice to the past prediction. Thus,
the close parallelism obtains between Humphreys' paradox and Newcomb's
problem. In fact, if we suppose that the probability of the correctness ofthe being's prediction is 4/5 and try to choose (2) expecting $M, conditionsof both puzzles would be exactly similar.
Nevertheless, we mustn't overlook a big difference between them.
That is to say, Newcomb's problem includes the existence of the being withsupernatural power of prediction, while Humphreys' paradox doesn't
involve such a purely fictitious element. Thus, if we consider that even
Humphreys' paradox is actually eroded by fictitious representations about
the past and that there is a parallelism between two puzzles, we should saythat Newcomb's problem only makes the core of the relation between
probability 1 and the past come to the surface more manifestly. But,
strictly speaking, Newcomb's case sheds light on a hidden implication of
RPH, which Humphreys' case couldn't explicitly show, by introducing thenotion of highly probabilistic prediction of the being. What's that?
First of all, introducing such prediction makes it possible for the
retroactive collapse of probability through backward causation to stand out
vividly, because the concept of prediction grammatically requires a strong(probably causal) connection between an action of predicting and a
predicted thing. As far as the structure of Newcomb's problem is exactly
similar to Humphreys' paradox, a character of Newcomb's case must be
helpful to understand better Humphreys' case and then the problem about
the relation between probability 1 and the past in general. Additionally,
there is another, more noteworthy point. If we took the viewpoint of thebeing at the moment he makes a prediction and determines the content ofbox B (i.e. we needn't consider the temporal difficulty of probability 1), his
prediction would have two implications. The first is that the prediction
will be true with an extremely high probability, and the second is that thecontent of box B has been just fixed and determined and he would intend
the content to remain the same until your choice, otherwise the setting of
the problem wouldn't obtain. I want to focus my attention on the second
point. This is nothing but taking the viewpoint to look forward back after
looking backward. In other words, first we fictitiously causes theretroactive collapse of probability by looking backward, and then (of course,
fictitiously) take the standpoint of looking forward from that past point,
which can be drawn from the setting of Newcomb's problem through the49notion of prediction. It is the merit of Newcomb's problem, as far as myquestion is concerned, that such a perspective as considering the aspect oflooking forward back is clearly introduced to RPH.
Well, what does this perspective bring out? This perspective enables
us to understand how determinism is extended to the future as well as thepast. In fact, I have intentionally avoided so far mentioning thedeterminism about the future, since it sounds extremely offensive to myears, although I know that the determinism is often supposed to virtually
cover the future by being used tenselessly or timelessly. But, at least tome, it sounds totally implausible to use determinism without considering
the temporal factor, for determinism usually concerns natural phenomenain space and time. And, more crucially, I can't imagine at all how we aresure of determinism applicable to any time including the future. Suchoverall determinism might be unproblematic as our ordinary belief or faith,
but to claim it as true is absolutely beyond our capacity by definition.
Thus, if we make the determinism about the future associated with such
pure tenseless determinism, the situation must be completely hopeless.
But, I think that there is another kind of the determinism about the future.
That is the determinism which is established through the process of RPH,
namely, such the process as we fictitiously cause the retroactive collapse ofprobability through looking backward and then look forward from thatpast point to the later time. This process is initially occasioned by thepresent representation about the past event and then retroacts to the pastby regarding that past event as what determined this present situation,
I think that this
therefore a sort of determinism appears here.
determinism can be said to be concerned with the future from theviewpoint of that past which probability retroacts to and collapses at. I
call this determinism boomerang determinism in contrast to tenselessdeterminism.
I attempt to illustrate those situations around thedeterminism about the future in the following diagram, which may besomehow helpful to see my point.
(Determinism about the Future)
past

presenttenselessdeterminism

boomerang
determinism

future---------><--------------->

50

In short, I claim that boomerang determinism is the only meaningful
option as the determinism about the future , and this is clarified byexamining Newcomb's problem. Boomerang determinism starts from asort of prediction, and, as it were a prior, it is qualified to be called
determinism, since that is defined to have been already verified by thepresent representation. In other words, the prediction is confirmed atpresent and then the prediction became the determinism in a retroactive
way. That might sound bizarre. Yet, isn't it natural to characterize thedeterminism about the future as a sort of prediction? I think it is. And,
probably, we also explain why people firmly believe the determinism about
the future. That's because boomerang determinism has been always
already verified by definition.
However, finally, I have to strongly emphasize that those processes tointroduce all-out determinism is thoroughly fictitious, and involved inbackward causation which is quite unacceptable without special
qualifications. Probability 1 that is intuitively assigned to the past event
as expressed by PPE eventually leads to those tricky paradoxicalsituations. Of course, we always have to make commitment to thosefictitious states.
But we must also recognize the fact that thosecommitments make us involved in perplexity about the concept ofprobability. As a result, we learn a lesson. Let's apply the concept ofprobability only to the pure future without using the notion of probability 1
and 0 at all (perhaps except the case of metaphor), and we will be exempt
from weird complication about probability. Perhaps, on the contrary, theconcept of causality is in principle the most appropriate tool for us to deal
with past things, since what mostly matters concerning the past seems to
be an ascription of origins of things and the language of causation is
supposed to stem from our talk of attributing responsibility, whether theconnection between cause and effect is perfectly certain or not. In anutshell, it seems to me that causality leans to the past and probability
works towards the future. But maybe, in any case, we can't actually
restrict the concept of probability to the future . If so, all that we should
do is to let that intrinsically paradoxical nature of probability be engravedon our minds when we use the concept of probability.
*My grateful thanks are due to Professor Timothy Williamson,
Professor Hugh MelIor, and Professor Donald Gillies for their mosthelpful comments about this article that I received when I stayedin Oxford, England, from 2002 to 2003 by using my sabbatical. .

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