1. A puzzle about justification
Consider the following principle:
If I know that a coin is fair and, for all I know, it is going to be flipped then, for all I know, it is going to land tails.
Dorr, Goodman and Hawthorne (Reference Dorr, Goodman and Hawthorne2014) call this principle FAIR COINS. What this principle says, in effect, is that the only way to know that a fair coin won’t land tails is to know that it won’t be flipped at all. If we put aside possibilities such as clairvoyance, time travel and ‘cheesy’ modes of presentationFootnote 1 then FAIR COINS seems a very appealing principle. Now suppose there are 1000 fair coins arrayed before me which I’m about to start flipping. If the first coin lands tails, I’ll flip the second. If the second coin also lands tails, I’ll flip the third and so on. As soon as a coin lands heads, I stop flipping and the remaining coins are destroyed. Suppose that, in actual fact, the second coin is going to land heads and that will be the end of it – but imagine that I’m just about to start the process. If I know everything about the set-up, what can I know, at this point, about the outcome?
Before I start flipping, it is extremely objectively likely that coin 1000 will not be flipped. After all, this would require throwing 999 tails in a row, and the objective probability of this is 0.5999 – (far) less than 1 divided by the number of atoms in the known universe.Footnote 2 But can I know that coin 1000 won’t be flipped? Given my knowledge of the set-up, if I know that coin 1000 won’t be flipped, then I know that coin 999 won’t land tails. Given FAIR COINS, it follows that I know that coin 999 won’t be flipped. But then, given my knowledge of the set up, I know that coin 998 won’t land tails and so on. After 1000 iterations we derive the absurdity that I know that coin 1 won’t be flipped. If FAIR COINS holds then I cannot know that coin 1000 won’t be flipped, even though this is true and extremely objectively likely.
As Dorr, Goodman and Hawthorne point out, a similar principle to FAIR COINS can be used to derive the result that I can’t even have justification for believing that coin 1000 won’t be flipped.
If I know that a coin is fair and I lack justification for believing that it isn’t going to be flipped, then I lack justification for believing that it isn’t going to land tails.
Dorr, Goodman and Hawthorne call this principle JUSTIFIED FAIR COINS.Footnote 3 What this principle says, in effect, is that the only way to have justification for believing that a fair coin won’t land tails is to have justification for believing that it won’t be flipped. Once again, if we set aside clairvoyance, time travel, cheesy modes of presentation etc., JUSTIFIED FAIR COINS seems a very appealing principle. Using this principle, Dorr, Goodman and Hawthorne’s argument can be run as before, but with justification in place of knowledge: Given what I know about the set-up, if I have justification for believing that coin 1000 won’t be flipped, then I have justification for believing that coin 999 won’t land tails. Given JUSTIFIED FAIR COINS, it follows that I have justification for believing that coin 999 won’t be flipped. But then, given my knowledge of the set up, I have justification for believing that coin 998 won’t land tails and so on. After 1000 iterations we derive the absurdity that I have justification for believing that coin 1 won’t be flipped. If JUSTIFIED FAIR COINS holds, then I cannot have justification for believing that coin 1000 won’t be flipped, even though it is true and extremely objectively likely.
Although JUSTIFIED FAIR COINS seems an appealing principle, surely this puzzle forces us to give it up – surely we can’t deny that I have justification for believing that coin 1000 won’t be flipped. And, even if we can bring ourselves to deny this, things may get even worse. Following Dorr, Goodman and Hawthorne (Section 3), suppose I’m looking at a particular leaf on a particular maple tree at the beginning of autumn. I know that the tree will soon start to shed its leaves and by the end of winter is almost certain to be bare. Consider the proposition that, by the end of winter, the leaf I’m looking at will have fallen from the tree. This is not a proposition about an artificial or esoteric set up but is, rather, the kind of thing we could easily imagine ourselves believing about the future.
But the period between now and the end of winter can be divided up into hour-long intervals and I know that, for any given hour, if the leaf is still on the tree at the beginning of the hour then there is a high objective probability that it will still be on the tree at the end of the hour. With this in mind, consider the following principle, which we might call JUSTIFIED AUTUMN LEAF:
If I know that leaf shedding is a chancy process then, for any number n, if I lack justification for believing that the leaf will have fallen by the beginning of the nth hour, then I lack justification for believing that the leaf will have fallen by the end of the nth hour.
If I have justification for believing that the leaf will have fallen by the end of winter then, by this principle, I must have justification for believing that the leaf will have fallen by an hour before the end of winter and, by applying the principle again, I must have justification for believing that the leaf will have fallen by an hour before that and so on. By applying the principle enough times we can derive the absurdity that I have justification for believing that the leaf will fall from the tree in the next instant.
If we accept JUSTIFIED FAIR COINS, we are under some pressure to accept JUSTIFIED AUTUMN LEAF and, if we do, we are forced to deny that I have justification for believing that the leaf will have fallen by the end of winter.Footnote 4 But if we deny this, then we really do seem to be inviting a more pervasive scepticism about the future – a view on which most, if not all, of the things we believe about the future are unjustified. Furthermore, we could hardly prevent such scepticism from infecting some beliefs about the present and past as well. It seems, then, that we have a powerful reason for rejecting JUSTIFIED FAIR COINS.
In addition, as Dorr, Goodman and Hawthorne point out (n8), there is a widespread conception of epistemic justification that provides further, independent reason for rejecting JUSTIFIED FAIR COINS. On this conception, one has justification for believing a proposition P just in case, given one’s evidence, it would be unlikely for P to be false. Elsewhere I have termed this the risk minimisation theory of justification – it makes justification a matter of minimising the risk of error. To make this idea a little more formal, suppose we set a threshold t greater than, but close to, 0 to represent the maximum tolerable risk of error. According to the risk minimisation theory, I have justification for believing a proposition P just in case the probability that P is false, given my evidence, is less than t.
If the risk minimisation theory is correct, then JUSTIFIED FAIR COINS is false – the risk of falsity to which I expose myself in believing that a fair coin won’t land tails is lower than the risk of falsity to which I expose myself in believing that it won’t be flipped. By plugging the risk minimisation theory into JUSTIFIED FAIR COINS we derive the following principle:
If I know that a coin is fair and the probability, given my evidence, that it will be flipped is greater than t, then the probability, given my evidence, that it will land tails is greater than t.
It is obvious that this principle is false – the probability of a fair coin in the sequence landing tails is lower than the probability of it being flipped and, as such, the two values could easily sit either side of a numerical threshold.
Suppose we fix the threshold t as equal to 0.1 – so I have justification for believing anything which, given my evidence, has less than a 0.1 probability of being false. In Dorr, Goodman and Hawthorne’s scenario, the probability, given my evidence, that coin 2 will be flipped is 0.5, that coin 3 will be flipped is 0.25 and that coin 4 will be flipped is 0.125. The probability, given my evidence, that coin 2 will land tails is 0.25, that coin 3 will land tails is 0.125 and that coin 4 will land tails is 0.0625. The risk minimisation theory straightforwardly predicts, then, that I have justification for believing that coin 4 won’t land tails, but I lack justification for believing that coin 4 won’t be flipped.Footnote 5
Here are the probabilities of each coin being flipped and of landing tails, respectively:
Evidently, the error threshold t at crossed at coin 4.
At this point, the case against JUSTIFIED FAIR COINS may seem decisive – even overdetermined. Not only does the principle appear to lead to scepticism about the future, it clashes with a widely held, independently motivated, theory of epistemic justification. And yet, as powerful as these considerations seem, there are still reasons for being somewhat uneasy. JUSTIFIED FAIR COINS does seem like an appealing principle, and, in a way, we are still no closer to understanding why this should be. Indeed, the fact that the risk minimisation theory converts this principle into such a patent falsehood is, perhaps, something that should give us pause. It is, of course, right and proper that a theory of justification should prompt us to revise certain pretheoretic judgments – but the risk minimisation theory makes it difficult to see why this principle would have ever seemed attractive in the first place.
Further, consider the following, which we might call JUSTIFIED BIASED COINS:
If I know that a coin is heavily biased in favour of tails and, I don’t have justification for believing that it isn’t going to be flipped then I don’t have justification for believing that it isn’t going to land tails.
This principle might seem even more appealing than JUSTIFIED FAIR COINS. If I know that a coin is very heavily biased towards tails, it would be very odd to believe that it won’t land tails, while remaining agnostic about whether it will be flipped – and it’s difficult to accept that such a combination of attitudes could ever be justified. But the risk minimisation theory will also translate this into an obvious falsehood. Even if a coin is heavily biased in favour of tails, the probability of it landing tails will still be slightly lower than the probability of it being flipped and these two values could easily sit either side of a threshold.
2. Two theories of justification
In previous work (Smith Reference Smith2010, Reference Smith2016), I’ve argued against the risk minimisation theory of justification, and in favour of an alternative that I’ve termed the normic theory. According to the risk minimisation theory, one has justification for believing a proposition P just in case, given one’s evidence, it would be unlikely for P to be false. According to the normic theory, one has justification for believing a proposition P just in case, given one’s evidence, it would be abnormal for P to be false.
Sometimes when we describe a situation as ‘abnormal’ we mean only that it is relatively rare or infrequent. If this is how we are understanding the notion of normalcy, then the normic theory looks very similar to the risk minimisation theory, and is not obviously an alternative at all. Other times, though, when we describe a situation as abnormal, we are not making a claim about frequencies. Rather, we are flagging the situation as an exception to a pattern or regularity – as something that would need special explanation if it were to obtain. If the lights in my house suddenly start to flicker, or my car fails to start when I turn my key in the ignition and I remark ‘that’s not normal’, I’m not simply pointing out that this is something rare or infrequent – what I’m saying is that there needs to be some explanation for what is occurring. This is the understanding of normalcy that I have in mind in putting forward the normic theory – an understanding that will make it into something recognisably different from the risk minimisation theory.
Suppose I’m wondering whether the milk has gone off. I sniff it and I can’t smell anything unusual, and I check the use by date and see that it’s still a few days away. Clearly this evidence makes it very unlikely that the milk is off – but this is not its only effect. If the milk were off, in spite of the evidence, then some special explanation would be needed – I’ve lost my sense of smell or the odour has been masked and the wrong date was stamped on the milk or the milk has not been properly refrigerated etc. Given my evidence it would be unlikely and abnormal for the milk to be off. Both the risk minimisation and normic theories predict, in this case, that I have justification for believing that the milk is not off – but they offer different diagnoses for why this is.
While the risk minimisation and normic theories will often agree in their predictions, they do not always do so. The normic theory seems, for instance, to offer a very different perspective on JUSTIFIED FAIR COINS. By plugging the normic theory into this principle we derive the following:
If I know that a coin is fair and it is normal, given my evidence, for it to be flipped then it is normal, given my evidence, for it to land tails.
Call this NORMIC FAIR COINS. Given the intended reading of ‘normal’ this principle seems clearly true. If I know that a coin is fair and the situation in which it is flipped requires no special explanation, given my evidence, then the situation in which it lands tails requires no special explanation, given my evidence.
It may even be possible to motivate NORMIC FAIR COINS without appealing to any substantial theory of normalcy. Given that a coin is fair, it would be equally normal for it to be flipped and land heads and for it to be flipped and land tails. Arguably, this is part of the analysis of what it means for a coin to be ‘fair’ – if it is more normal for a coin to be flipped and land heads than for it to be flipped and land tails or vice versa, then it is not a fair coin. But, given a few principles about the formal properties of normalcy, NORMIC FAIR COINS can be ‘proved’ from this trivial starting point.Footnote 6 In the Appendix I describe a simple formal model of normalcy in which this derivation can be set out – but I leave the discussion informal here.
Since the normic theory predicts that JUSTIFIED FAIR COINS is true, it will play into Dorr, Goodman and Hawthorne’s hands and predict that, in the scenario they describe, I lack justification for believing that coin 1000 won’t be flipped. But this result should come as no particular surprise – given my evidence, it may be extremely unlikely for me to throw 999 tails in a row, but this outcome is not abnormal in the sense of requiring special explanation.Footnote 7 Put aside Dorr, Goodman and Hawthorne’s particular set-up for a moment and imagine a simpler case in which I’m about to throw 999 fair coins and note the result. Of the 2999 possible sequences that I could get, it’s clear that none of these is abnormal, or requires any special explanation. The sequences are all on a par. When it comes to a ‘patterned’ sequence like 999 tails, we might be inclined to think that some special explanation would be needed for this – but, in reality, this sequence could come up just as easily as any other (Smith Reference Smithforthcoming).Footnote 8 In a case like this, the normic theory predicts that I should be open minded about how the coins will land – I don’t have justification for ruling out any outcomes in advance.
In Dorr, Goodman and Hawthorne’s set-up, in which the sequences terminate with the first head, there are far fewer possible outcomes and the sequence of 999 tails is the least likely of them. But this sequence is no less normal, and requires no more explanation, than it does in the simpler case.Footnote 9 While I may be justified in believing that this outcome is extremely unlikely to obtain, according to the normic theory, I lack justification for believing that it won’t obtain.
What then of Dorr, Goodman and Hawthorne’s claim that this verdict threatens a slide into scepticism about the future? Does the normic theory carry such a threat? What does it predict, for instance, about the autumn leaf example? I will build up to answering this question by considering two related cases: Suppose I’ve set up my computer to sound an alarm at some randomly determined moment within the next two hours. Presumably, I’m justified in believing that my computer will sound an alarm some time during the next two hours. This case has a number of structural features in common with the autumn leaf example. After all, the next two hours can be divided up into, say, minute-long intervals and I know that, for any given minute, if the alarm has not sounded by the beginning of the minute, then there is a high objective chance that it will not have sounded by the end of the minute. Consider, then, the following principle:
JUSTIFIED ALARM
If I know that the alarm is determined by a chancy process then, for any number n, if I lack justification for believing that the alarm will have sounded by the beginning of the nth minute, then I lack justification for believing that the alarm will have sounded by the end of the nth minute.
While this principle seems very solid for the first minute, the second minute, the third minute … the 120th minute is clearly special. It’s consistent with my computer functioning properly that the alarm has not sounded by the beginning of the 120th minute. It is not consistent with my computer functioning properly that the alarm has not sounded by the end of the 120th minute. It’s plausible at least that this minute serves as a counterexample to JUSTIFIED ALARM – I lack justification for believing that the alarm will have sounded by the beginning of the 120th minute, but I have justification for believing that it will have sounded by the end of the 120th minute.
This impression is further cemented by considering the normic translation of JUSTIFIED ALARM:
NORMIC ALARM
If I know that the alarm is determined by a chancy process then, for any number n, if it is normal, given my evidence, for the alarm not to have sounded by the beginning of the nth minute, then it is normal, given my evidence, for the alarm not to have sounded by the end of the nth minute.
Given the way I set up my computer, the instant at which we reach the end of the 120th minute is the instant at which a failure of the alarm to sound becomes abnormal and requires explanation of some kind – the computer has lost power, someone has taken to it with a hammer etc.
What this example shows, at the very least, is that the normic theory will not lead to an all-encompassing scepticism about the future. My justification for believing that the alarm will sound within the next two hours seems to be safe from a Dorr, Goodman and Hawthorne-style argument. One might complain, though, that this case is rather artificial and contrived – it’s rare to have such precise information about the time period during which an event might occur. Here is a more realistic, but closely related, case: Suppose my roommate calls and tells me that he’ll be home within the next two hours. Consider the following principle:
NORMIC ARRIVAL TIME
If I know that arriving home is a chancy process then, for any number n, if it is normal, given my evidence, for my roommate to have not arrived home by the beginning of the nth minute, then it is normal, given my evidence, for my roommate to have not arrived home by the end of the nth minute.
NORMIC ARRIVAL TIME looks a lot more secure that NORMIC ALARM. It doesn’t seem right that, at the end of the 120th minute, my roommate’s failure to arrive home would suddenly become abnormal and require explanation. In fact, it’s strange to think that there’s any minute during which this happens.
Unlike NORMIC ALARM, we can’t find a clear counterexample to NORMIC ARRIVAL TIME – but this is not to say that we should accept it. If my roommate has not arrived within 20 min or within 1 h, then there may be nothing abnormal about that. But if my roommate has still not arrived within 10 or 24 h then that clearly would be abnormal and some explanation would be needed. NORMIC ARRIVAL TIME is, however, inconsistent with these judgments. If the situation in which my roommate has not arrived by the end of the first minute is not abnormal then, by repeatedly applying NORMIC ARRIVAL TIME, it follows that the situation in which my roommate has not arrived within 24 h is not abnormal. This, of course, is nothing other than an instance of the sorites paradox.
In the classic presentation of the paradox, we begin with a base case – a man with 0 hairs on his head is bald (it is normal for my roommate to have not arrived home by the end of minute 1/beginning of minute 2). We then introduce a soritical principle – for any number n, if a man with n hairs on his head is bald, then a man with n + 1 hairs on his head is bald (if it is normal for my roommate to have not arrived home by the beginning of the nth minute, it is normal for my roommate to have not arrived home by the end of the nth minute/beginning of the n + 1th minute). From these two premises, an absurd conclusion follows – a man with 1, 000, 000, 000 hairs on his head is bald (it’s normal for my roommate to have not arrived home within 24 h).Footnote 10
At first, NORMIC ARRIVAL TIME might look more like NORMIC FAIR COINS than NORMIC ALARM. After all, there is a clear counterexample to NORMIC ALARM, while NORMIC ARRIVAL TIME and NORMIC FAIR COINS seem to be counterexample-proof. On closer inspection, though, these two principles turn out to be very different. NORMIC FAIR COINS can be given a sound motivation – and perhaps even be motivated by an analysis of what it means for a coin to be ‘fair’. NORMIC ARRIVAL TIME is not motivated by an analysis of anything – it’s appeal lies in that which makes all soritical principles seductive – our reluctance to impose a sharp boundary on vague expressions. Almost all contemporary treatments of the sorites paradox involve denying that soritical principles are perfectly true, in spite of their undeniable appeal, and lack of clear counterexamples.Footnote 11 I am inclined to follow this approach, but whatever view we take of soritical principles, the fact that they lead to trouble when combined with the normic theory is evidently no reflection on the theory – they lead to trouble on their own.
In the case described, the normic theory predicts that I have justification for believing that my roommate will have arrived home within 24 h – and our only reasons for thinking otherwise are disreputable soritical reasons. What this example shows is that the normic theory will not lead to a widespread scepticism about the future. My justification for believing that my roommate will be home within 24 h – typical of much of what we have justification for believing about the future – is safe from a Dorr, Goodman and Hawthorne-style argument.
Return, finally, to the autumn leaf example. If we plug the normic theory into JUSTIFIED AUTUMN LEAF we obtain the following:
NORMIC AUTUMN LEAF
If I know that leaf shedding is a chancy process then, for any number n, if it’s normal, given my evidence, for the leaf to have not fallen by the beginning of the nth hour, then it’s normal, given my evidence, for the leaf to have not fallen by the end of the nth hour.
Here is one compelling thought: The leaf shedding process serves a function or purpose for the tree and, absent external interference, will unfold in a particular, predetermined way. If the leaf I’m looking at were still on the tree at the end of winter, then this would represent a breakdown in the process and, given my evidence, would require explanation in terms of some anomaly with the season, the immediate environment, the tree or the leaf. If we’re struck by this thought then, given the normic theory, we should take ourselves to have justification for believing that the leaf will not be on the tree at the end of winter. On this way of thinking, we should regard NORMIC AUTUMN LEAF as nothing more than a soritical principle, with the associated deceptive allure.Footnote 12
Here is an alternative thought: The leaf shedding process is essentially random, in the same sense as a series of coin flips, and each of the possible outcomes of the process, including those in which the leaf remains on the tree at the end of winter, is equally normal. If we’re struck by this thought then, given the normic theory, we should not take ourselves to have justification for believing that the leaf won’t be on the tree at the end of winter – the most we would have justification for believing is that it is highly likely that the leaf won’t be on the tree at the end of winter. On this way of thinking, NORMIC AUTUMN LEAF will be trivially true.Footnote 13
Each thought can seem appealing, in a certain frame of mind – but the crucial point, for present purposes, is just this: If the circumstance in which the leaf is still on the tree at the end of winter would be abnormal, given my evidence, then the normic theory predicts that I have justification for believing that the leaf will have fallen by the end of winter. If the circumstance in which the leaf is still on the tree at the end of winter would be normal, given my evidence, then the normic theory predicts that I lack justification for believing that this won’t happen. But whatever the case, it was obvious from the get-go that the predictions of the normic theory would hinge on this very issue – the NORMIC AUTUMN LEAF principle is a wheel turning idly. That is, whatever sceptical liabilities there are in the normic theory, it wears them on its sleeve – there is no latent sceptical threat in the theory that a Dorr, Goodman and Hawthorne-style argument can draw out.
3. Conclusion
JUSTIFIED FAIR COINS is an appealing principle. And yet, Dorr, Goodman and Hawthorne argue that the principle is troublesome, ultimately leading us to scepticism about the future. The principle is also invalidated by a widespread and well-motivated theory of justification – the risk minimisation theory. This observation might be thought to clinch the case against JUSTIFIED FAIR COINS and to be a point in favour of the risk minimisation theory. In this paper I have attempted to subvert this picture of the dialectical situation. I have argued that JUSTIFIED FAIR COINS does not inevitably lead to scepticism about the future. I have outlined a theory of justification – the normic theory – which turns out to validate JUSTIFIED FAIR COINS, and on which scepticism about the future is not inevitable. In light of this, the fact that the risk minimisation theory invalidates JUSTIFIED FAIR COINS, and does so in a way that seems to make its appeal inexplicable, might even be seen as a reason to question the theory.
Some of what I say here has implications for the original FAIR COINS principle on which Dorr, Goodman and Hawthorne focus most of their attention. In the scenario that Dorr, Goodman and Hawthorne envisage, FAIR COINS will lead us to the result that I cannot know that coin 1000 won’t be flipped, in spite of the fact that this is true and extremely objectively and evidentially likely. From the perspective of the normic theory, though, this provides no decisive reason to reject the principle. The idea that we are unable to know certain highly likely truths about the future is one that a normic theorist should already be quite accustomed to. After all, some highly likely truths about the future are such that, given our evidence, their falsity would involve no departure from normal circumstances. These will be truths that we cannot justifiably believe and, assuming that knowledge requires justification, cannot know either. The truth that coin 1000 won’t be flipped is one such.
Funding
Work on this paper was supported by the Arts and Humanities Research Council [grant number AH/M009610/1].
Acknowledgements
This paper was presented at the University of St Andrews in October 2016. Thanks to all of those who participated on this occasion. Thanks also to two anonymous referees for this journal.
A model of normalcy
This model is based upon the idea that claims about what is normal, and about how events normally unfold, can be treated as involving quantification over possible worlds that stand in relations of comparative normalcy. I introduce a simple formal language L consisting of countably many sentential constants (A 1, A 2, A 3 …, ) the truth functional operators (∧, ∨, ~, ⊃, ≡), two special operators (♦, ▪→) and punctuation. The sentences of L are built up using the standard recursive clauses for operators and punctuation. The semantics for L is provided by models of the form 〈W, ≤, I〉 where W is a set of possible worlds, ≤ is a relation on W reflecting the comparative normalcy of worlds, given background evidence, and I is an interpretation function mapping pairs of sentences of L and members of W to the truth values (1, 0). The relation ≤ is assumed to be reflexive, transitive and connected (a total preorder on W). The function I is subject to the following constraints:
For any sentences φ, ψ ∈ L and world w ∈ W:
I w (φ ∧ ψ) = 1 iff I w(φ) = 1 and I w(ψ) = 1
I w(φ ∨ ψ) = 1 iff I w(φ) = 1 or I w(ψ) = 1
I w(~φ) = 1 iff I w(φ) = 0
I w(φ ⊃ ψ) = 1 iff I w(φ) = 0 or I w(ψ) = 1
I w(φ ≡ ψ) = 1 iff I w(φ) = 1 and I w(ψ) = 1 or I w(φ) = 0 and I w(ψ) = 0
I w(♦φ) = 1 iff for every world x ∈ W, if I x(φ) = 0 there is a world y ∈ W such that x ≤ y and I y(φ) = 1
I w(φ ▪→ ψ) = 1 iff there is a world x ∈ W such that I x(φ) = 1 and I x(ψ) = 1 and there is no world y ∈ W such that x ≤ y and I y(φ) = 1 and I y(ψ) = 0
Put less formally, ♦φ is true iff for every world at which φ is false, there is an equally or more normal world at which φ is true, and φ ▪→ ψ is true iff there is a world at which φ and ψ are true and which is more normal than any world at which φ is true and ψ is false. If we assume that there are maximally normal worlds (worlds in W that are maximal with respect to ≤) then ♦φ is true iff φ is true at some maximally normal worlds. If we assume that, for any sentence φ that is true at some possible worlds, there are maximally normal worlds at which it is true (worlds in I(φ) that are maximal with respect to ≤), φ ▪→ ψ is true iff ψ is true at all maximally normal worlds at which φ is true. ♦ might be read ‘It is normal that …’ and ▪→ might be read ‘If … it would normally be the case that __’.
As can be checked, the following are valid inferences:
φ ▪→ ψ ♦(φ ∨ ψ) m
♦φ ⊃ ♦ψ ♦φ ∨ ♦ψ
Let φ be the sentence ‘Coin c is flipped’, ψ be the sentence ‘Coin c is flipped and lands heads’ and χ be the sentence ‘Coin c is flipped and lands tails’. Let the background evidence include the fact that coin c is fair and no further relevant facts. Given this evidence, if coin c is flipped, it would normally land heads or tails – φ ▪→ (ψ ∨ χ). This entails ♦φ ⊃ ♦(ψ ∨ χ) – if it is normal for coin c to be flipped then it is normal for coin c to land heads or tails. But ♦(ψ ∨ χ) entails ♦ψ ∨ ♦χ from which we derive ♦φ ⊃ (♦ψ ∨ ♦χ). Given that the background evidence includes the fact that the coin is fair, we also have it that ♦ψ ≡ ♦χ – it is normal for the coin to be flipped and land heads just in case it is normal for the coin to be flipped and land tails. This is a special case of the conjecture flagged in the above discussion of NORMIC FAIR COINS: Given that a coin is fair, it would be equally normal for it to be flipped and land heads and for it to be flipped and land tails. These premises together entail ♦φ ⊃ ♦χ – if it is normal for coin c to be flipped, it is normal for coin c to land tails. This gives us the NORMIC FAIR COINS principle. QED
This model can be applied to Dorr, Goodman and Hawthorne’s set-up as follows: For any n between 1 and 1000, let φn be the sentence ‘Coin number n is flipped’, ψn be the sentence ‘Coin number n is flipped and lands heads’ and χ n be the sentence ‘Coin number n is flipped and lands tails’. Let the background evidence include the coin flipping rules, the facts that coins 1 through to 1000 are all fair and no further relevant facts. Given the rules we have ♦φ1 and, for any n, ♦χn ⊃ ♦φn+1 – it’s normal for the first coin to be flipped and, for any coin, if it is normal for the coin to be flipped and land tails, it’s normal for the next coin in the sequence to be flipped. By the above result we have, for any n, ♦φn ⊃ ♦χn – for any coin, if it is normal for the coin to be flipped then it is normal for the coin to be flipped and land tails. From this we can derive ♦φn ⊃ ♦φn+1 and, using the base case and this inductive step, we can prove ♦φ1000.
Goodman and Salow (Reference Goodman and Salowforthcoming, Section 5) apply a similar model to this set-up, but with the starting assumption that the outcomes vary according to their normalcy, with the abnormality of an outcome being a function of the number of consecutive tails that it involves. This could perhaps be underwritten by a picture on which the normalcy of a sequence of coin throws is associated with its ‘randomness’ or ‘paternlessness’ (see Dutant Reference Dutant2016, Section 2.3). From this starting point, we have it that, for some n, ♦φn, ♦χ n and ♦φn+1 but ~♦χ n+1 – NORMIC FAIR COINS is violated. Since the model still validates ♦φn+1 ⊃ (♦ψn+1 ∨ ♦χn+1), we have ♦ψn+1. Although coin n + 1 is fair, it is normal for it to be flipped and land heads, but it is not normal for it to be flipped and land tails. As such, the model also violates the claim that, whenever a coin if fair, it would be equally normal for it to be flipped and land heads and for it to be flipped and land tails. One might treat this as a reductio of Goodman and Salow’s starting assumption or, alternately, as an argument to the effect that the outcomes of a fair coin flip can differ with respect to their normalcy. In the latter case, NORMIC FAIR COINS would need to rest, after all, on certain substantial assumptions about normalcy. I won’t pursue this further here.
Open access
