In this post I’ll look at some cases which might motivate a mushy credence view. Which particular mushy credence we use to account for any particular case will depend on the details of the interpretation we place on mushiness – on which kind of indeterminacy, in particular, we take to be represented by the provision of a set of credence functions rather than a single function. For a survey of the different possible interpretations of set-valued credal states, see Bradley [Synthese, 2009]. In part 3 of this post I’ll consider these interpretations in more detail, and raise some problems for the use of sets of credence functions based on the sheer multiplicity of suggested interpretations. But let me anticipate the results of that discussion by restricting my attention in this post to what Bradley calls the ignorance interpretation of mushiness, and seeing how that interpretation is most naturally applied to our problem cases.

Here’s what Bradley says about the ignorance interpretation:

“The agent may be unable to arrive at a judgement because she lacks the informational basis for doing so. This seems to be the kind of situation in which subjects find themselves, for instance, when placed in an Ellsberg paradox set-up in which the consequences of their decisions depend on the colour of a ball drawn from an urn containing an unknown proportion of balls of different colours. Many authors argue that in these kinds of situation the agent is not merely in a state of uncertainty in the sense that they don’t know for sure which colour ball will be drawn but can assign a probability to the prospect of each colour, but are rather in a state of ignorance in the sense that, such are the limits on what they know and can find out, that they have no non-arbitrary basis for assigning such a probability.”

So how can we apply this to the first of our cases?

[Percy] Someone you’ve never met emails to tells you that they use the name ‘Percy’ for some particular proposition; they don’t tell you anything about what Percy says. What is your credence that Percy is contingent? What is your credence that Percy is true?

Here the relevant ignorance is ignorance of which proposition Percy is. Therefore, applying the ignorance interpretation involves assigning a different credence function for each candidate for Percy’s identity. In particular, each candidate for Percy’s identity carries with it a credence distribution over contingency and non-contingency. For example, we are very confident indeed that ‘there are no round squares’ is noncontingent, while being equally confident that ‘there are no round windows in the White House’ is contingent. So it would seem that if we are to use a set of precise credences to represent our belief state concerning Percy’s contingency, then at least some of the credences in the set should be 1 or close to 1, and others 0 or close to 0. Perhaps there are some sentences whose status as contingent is controversial. ‘Other times exist’, ‘energy is conserved’, ‘ghosts exist’ might all fall into this category, and (if so) would contribute middling credences to the set. By aggregating the credences associated with all candidates of the relevant sort, we might hope to end up with a set spanning the [1,0] interval, or at least a set which comes very close to spanning that interval.

What about my state of belief in Percy’s truth? Again, different Percy-candidates come with a different credence distribution over truth and falsity. If Percy is ‘1=1′, then we might assign credence 1 to it; if Percy is 1=2, we might assign credence 0. And there are plenty of propositions to which we assign middling credence. So again, our state of belief in Percy’s truth will plausibly be represented by a set of credence functions spanning the [1,0] interval.

The case of Percy is relatively suitable for mushy treatment, as it’s plausible that there do exist continuum many propositions; this makes it at least a mathematical possibility that for any real number N between 0 and 1 we could find a proposition such that our credence in that proposition’s contingency or truth is N. But notice that even this is not guaranteed; it seems unlikely to be a requirement of rationality, for example, that an agent should have a credence in some proposition’s contingency equal to N, for any N in any range between 0 and 1. That is, it doesn’t seem out of the question that for no proposition whatsoever do I have a credence of 0.55551 in that proposition’s being contingent. But if so, then my state of belief in Percy’s contingency cannot be represented by an ignorance-interpreted mushy credence over an interval including 0.55551; the set which represents my state of belief in Percy’s  contingency would have to be ‘gappy’.

Another worry emerges when we notice that sets spanning the [1,0] interval are being used to represent our state of belief both in Percy’s contingency and in Percy’s truth. It follows straight away from this that our state of belief that Percy is contingently true must also be represented by a set spanning the [1,0] interval. This issue will be revisited later on.

[Quiz] In a tie-breaker round of the pub quiz you are asked how many times Australia have won the Ashes. You never watch sports or read the sports pages, and don’t know what the Ashes are or how often they are competed for. What is your credence that the number is larger than 10? What is your credence that the number is even?

In this case, the relevant ignorance concerns the properties of the Ashes competitions. ‘I don’t know enough about what the Ashes competitions are like’, the mushy credence lover might reason, ‘to assign a credence in their having been won any particular number of times by Australia’.  Suppose first that the Ashes are the prize in a daily card game between Australian and British airline crews, and that the card game is strictly a game of chance with a 50% likelihood of either side winning on any given occasion, and that they have been competed for on 10,000 occasions, and that neither side has ever cheated. Then the expectation value for Australian wins is 5,000; your credence that the number exceeds 10 is close to 1; and your credence that the number is even approaches 0.5. But now suppose that the Ashes refers to a single cricket game, played in 1882, and the rules were such that only one ball would be bowled, rendering it impossible for either side to win. Then the expectation value for Australian wins is 0, your credence that the number exceeds 10 is zero; and your credence that the number is even is close to 1.

It is easy to see that there are hypotheses about the rules and frequency of Ashes competitions which will result in any credence between 0 and 1 being ascribed to Australia winning more than 10 times, or to the number of wins being even. As in the case of [Percy], it seems that we must assign a set of credences spanning the [1,0] interval both to the proposition that the number is larger than 10, and to it being even.

Here a further question may occur to us. Surely some of the candidates for the nature of Ashes competition are less plausible than others; if the rules were as given in either of my toy examples, then the Ashes competitions would be highly unlikely to feature in a pub quiz, or even to have been held at all. With this in mind, it would be natural to weight some of the candidates more strongly than the others; but this is not part of the standard mushy credence machinery. Without weighting, the only way that we can restore some plausibility to the application of ignorance-interpreted mushy credences in this case is if there is some natural partition of possible rules-candidates, such that each is equi-probable. But such a partition looks like it would be highly problematic. I will return to this issue in part 3 of this post.

[Constant] Fundamental physics reveals that the value of a  certain ‘fundamental constant’ Q of nature is around 75. It turns out that a value of 74 or lower for Q would have resulted in a failure for stars to form; a value of 76 or higher would have resulted in a supergiant black hole sucking in the entire universe. Whether or not we take these results to be evidence for eg a benevolent God or multiple universes will depend on how antecedently unlikely we take a value of 75 for Q to be. What prior credence should we have had in Q being between 74 and 76?

In this example, it is less clear how to apply ignorance-interpreted mushy credences. My best stab at characterizing the ignorance involved is ignorance of the range of metaphysically possible worlds in which Q takes any value at all. If all metaphysically possible worlds that instantiate Q have a Q-value of around 75, then the appropriate prior credence to have in Q being around 75 is 1; if there are metaphysically possible worlds featuring all values of Q from 0 to 100, then the appropriate prior credence to have in Q being around 75 is about 0.01; if there are metaphysically possible worlds featuring all real-numbered Q-values, then our prior credence in Q being around 75 is 0. Once again, then, we are led to ascribe a set of credence functions covering the [1,0] interval as the correct representation of a rational belief state in Q being between 74 and 76.

[Cube] Bas van Fraassen locks you in a mystery cube factory. You discover that no cubes produced have edges longer than 2 metres.  You have no other evidence about the distribution of cube size. What is your credence that the next cube produced will have a volume larger than a cubic metre?

Here the relevant ignorance is ignorance of the distribution of cube size. One hypothesis has it that all cubes produced by the factory have edge lengths of 10cm. This would result in a credence of 0 in the next cube produced having a volume larger than a cubic metre. Similarly, the hypothesis that all cubes have edge lengths of 1.5m would result in a credence of 1 in the next cube having a volume larger than a cubic metre. And it’s easy to see that various hypotheses about edge length distribution could make appropriate any real-valued credence in the next cube having a volume larger than a cubic metre.  So once again, we find a set spanning the [1,0] interval mandated to represent our belief state that the next cube produced will have a volume larger than a cubic metre. Anyone spot a theme developing here?

[Ignorant tennis] You sit down to watch a tennis match. You have never heard of either player, they appear evenly-matched in fitness and physique. What is your credence that player A will win?

Our ignorance concerns the capacities of the two players. As part of this ignorance, we don’t want to rule out the hypothesis that player A has a heart pacemaker with a faulty battery which is about to expire, and which will force him to retire on medical grounds, handing victory to B. Nor do we want to rule out that player B is similarly afflicted. On the former hypothesis, your credence in A winning should be 0; on the latter, it should be 1. And there are plenty of intermediate hypotheses about the relative ability of the two players according to which the credence in A winning can take any value between zero and one.  So, once again, your credence that player A will win ought to be represented by a set spanning the [1,0] interval.

[Knowledgeable tennis] You sit down to watch a tennis match. You have coached both players, and have an detailed knowledge of their abilities and playing style. You consider them exactly evenly-matched. What is your credence that player A will win?

We presume that the ‘abilities’ referred to here includes all eventualities, such as faulty pacemakers. There is no ignorance involved in the case ex hypothesi – therefore, the correct credence in A’s winning is 0.5. The friend of mushy credence will presumably point to the difference between 0.5 and [0,1] as representing the intuitive difference between Ignorant tennis and Knowledgeable tennis.

From the above treatment, every single case which seems amenable to mushy treatment (under the ignorance interpretation of mushy credence) seems to land us with a set of credences spanning the [1,0] interval. This does not lead to any immediate contradiction, just as we can have credence 0.5 in many distinct propositions without contradiction. But it does lead to a number of worries, which I will list briefly here and examine in more detail in part 3 of this post:

• Sets of credences spanning the [0,1] interval are immovable, assuming that we update by conditionalization of each function within the set. If we start with a set spanning [1,0] and conditionalize, no further evidence will shift us from this position.
• Representing belief states with sets spanning [0,1] intervals seems to wash out relevant epistemic differences. For example, we are presumably more confident that Percy is true than that Percy is contingently true, as the latter claim is logically stronger. But it our belief state in Percy’s truth and our belief state in Percy’s contingent truth must both be represented by a set spanning the [0,1] interval.
• Although the ignorance-interpreted mushy credence machinery has the scope to represent partial suspension of judgement (for example, by a set spanning [0.4-0.6]), it seems that in a range of straightforward cases (all those surveyed above, in any case) this partial suspension of judgement is inapplicable, and we are stuck with maximal suspension of judgement.
• It is not clear that the sets resulting from an ignorance interpretation of mushiness will not be ‘gappy’ – that is, it is not clear that there will always be a function in the set corresponding to every real number in some interval. Gappiness significantly detracts from the intuitive appeal of mushy credence functions, and complicates the mathematics needed to apply them.
• Applying the ignorance-interpreted mushy credence machinery in the way described above seems to wash out differences between the plausibility of the various hypotheses (the Ashes resting on the results of a normal cricket match vs a 1-ball cricket match). To avoid this, we would require either a) weighting of different credence functions within the set or b) a natural partition of hypotheses into equiprobable hypotheses. Applying either of these solutions would significantly undermine the simplicity and intuitive appeal of the mushy credence approach.

In part 3, I will reassess motivations for using mushy credences, and tie various loose threads into a concerted case against the use of mushy credences in a probabilist epistemology.

[To be continued]

Recently I’ve been puzzling over the ‘coin puzzle’ recently spotlighted by Roger White. White uses it to raise trouble for a view of personal probabilities (variously called ‘mushy credences’, ‘fuzzy credences’, ‘imprecise probabilities’, ‘vague probabilities’, ‘thick confidences’) which characteristically represents personal probability states using sets of credence functions, such that each of the functions in the set individually conforms to the probability calculus and updates by conditionalization. The mushy credence view is often motivated by the desire to account for the difference between judgements of equiprobability and suspension of judgement.

I think White’s coin puzzle does serious damage to the mushy credence view. Properly understood, it is a way of making vivid the phenomenon of dilation, which results when mushy credences interact with sharp credences. Dilation isn’t news to the proponents of mushy credences, but as the coin puzzle shows, in combination with objective chance it becomes problematic. A known correlation of some p in which we have mushy credence with the result of some coin toss for which we know the sharp chance of heads leads to a mushy credence in heads, even when we knew the pre-toss chance. This mushification of our credence in chancy outcomes is repugnant and provides us with a reason to reject one of the premises that led us to it. But the only controversial premise was the mushy credence view.

This is bad news for proponents of mushy credences. Some I have spoken to would bite the bullet and accept the counter-intuitive consequences of known chance dilation, seeking to soften the impact by emphasizing that the coin case is unrealistic. Others (here I am thinking of Scott Sturgeon’s forthcoming paper in OSE) take this as reason to abandon the sets-of-credence-functions-individually-updated model. Others, like White, take it as reason to abandon mushy credences altogether and explain suspension of judgement in a different sort of way. Which route is the most promising?

I suspect that biting the bullet would prove too painful, and that once we abandon the formal model there wouldn’t be much left of the mushy credences view; but I won’t defend these claims here. Rather, my plan is to support the case against mushy credences by examining some cases which might have seemed amenable to mushy treatment, and providing an alternative non-mushy explanation of what is going on in these cases. So here are some example cases to ponder over. In the next part of this post, I’ll apply the mushy treatment to these cases and ask how well it fares with them.

[Percy] Someone you’ve never met emails to tells you that they use the name ‘Percy’ for some particular proposition; they don’t tell you anything about what Percy says. What is your credence that Percy is contingent? What is your credence that Percy is true?

[Quiz] In a tie-breaker round of the pub quiz you are asked how many times Australia have won the Ashes. You never watch sports or read the sports pages, and don’t know what the Ashes are or how often they are competed for. What is your credence that the number is larger than 10? What is your credence that the number is even?

[Constant] Fundamental physics reveals that the value of a  certain ‘fundamental constant’ Q of nature is around 75. It turns out that a value of 74 or lower for Q would have resulted in a failure for stars to form; a value of 76 or higher would have resulted in a supergiant black hole sucking in the entire universe. Whether or not we take these results to be evidence for eg a benevolent God or multiple universes will depend on how antecedently unlikely we take a value of 75 for Q to be. What prior credence should we have had in Q being between 74 and 76?

[Cube] Bas van Fraassen locks you in a mystery cube factory. You discover that no cubes produced have edges longer than 2 metres.  You have no other evidence about the distribution of cube size. What is your credence that the next cube produced will have a volume larger than a cubic metre?

[Ignorant tennis] You sit down to watch a tennis match. You have never heard of either player, they appear evenly-matched in fitness and physique. What is your credence that player A will win?

[Knowledgeable tennis] You sit down to watch a tennis match. You have coached both players, and have an detailed knowledge of their abilities and playing style. You consider them exactly evenly-matched. What is your credence that player A will win?

(thanks to John Cusbert for this last pair of examples.)

[To be continued]

Stalnaker and Moss have given the following argument for the claim that ‘will’ and ‘might not’ are not contraries:

1.    ‘Arsenal might not win, but I believe that Arsenal will win’ is assertible.
2.    If ‘will’ and ‘might not’ are contraries then this is an instance of a Moorean paradox: p and I believe that not p.
3.    Moorean paradoxes are not assertible.
Therefore:
4.     ‘will’ and ‘might not’ are not contraries, and a fortiori ‘will’ and ‘might’ are not duals.

A similar argument appears to show that ‘might not have’ and ‘would have’ are not contraries:

5.    ‘If the game had been played, Arsenal might not have won; but I believe that if the game had been played, Arsenal would have won.’ is assertible.
6.    But if ‘will’ and ‘might not’ are contraries then this is an instance of a Moorean paradox: p and I believe that not p.
7.    Moorean paradoxes are not assertible.
Therefore:
8.    ‘would have’ and ‘might not have’ are not contraries, and a fortiori []→ and ◊→ are not duals.

These argument might lead us to reject the duality of ‘will’ and ‘might’, and of the []→ and ◊→ operators, which is enshrined in the Lewisian semantics for counterfactuals. But these dualities are motivated by the following sort of argument:

9.    ‘Arsenal might not win, but Arsenal will win’ is not assertible.
10.     ‘Arsenal might win, but Arsenal will not win’ is not assertible.
11.    There is no good explanation in pragmatic terms for the non-assertibility of 9 and 10.
Therefore
12.    The best explanation for their non-assertibility is that it derives from semantic inconsistency.
Therefore
13.    ‘might’ and ‘will’ are duals.

And for the counterfactual case:

14.     ‘If the game had been played, Arsenal would have won; but if the game had been played, Arsenal might not have won’ is not assertible.
15.    ‘If the game had been played, Arsenal would not have won; but if the game had been played, Arsenal might have won’ is not assertible.
16.    There is no good explanation in pragmatic terms for the non-assertibility of 14 and 15.
Therefore
17.    The best explanation for their non-assertibility is that it derives from semantic inconsistency.
Therefore
18.    []→ and ◊→ are duals.

We see that these two kinds of arguments lead to opposite conclusions. How should we respond to this state of affairs?

One common response in the non-counterfactual case is to reject 11). And although this response is less common in the counterfactual case, it seems equally open to us to reject premise 16).

How might rejecting these premises be motivated? One approach, suggested to me by John Hawthorne, goes as follows (take 9) as an example)

a) For ‘Arsenal might not win’ to be assertible at t, it must be the case that Arsenal not winning is compatible with my knowledge at t. (by the rules for deploying the epistemic modal ‘might’.)

b) For ‘Arsenal will win’ to be assertible at t, I must know at t that Arsenal will win. (by the knowledge norm of assertion.)

c) It cannot be the case at t both that I know that Arsenal will win, and that it be compatible with my knowledge that they not win. So the conjuncts are never simultaneously assertible, and the conjunct is never assertible.

This does provide a neat pragmatic explanation of the non-assertibility of 9), which obviously generalizes to 10). How do we generalize it to 14) and 15)?

The natural route is to interpret ”if the game had been played, Arsenal might have won’ as ‘it might be the case that, if the game had been played, Arsenal would have won’. Eagle here calls this the ‘epistemic’ reading of ‘might’ counterfactuals. No doubt this is the right reading for some ‘might’ counterfactuals – ‘if I were to look in the fridge, I might find some beer’ for example. It’s either the case that I would find some, or that I wouldn’t find some, I just don’t know which. But it doesn’t seem to be the right reading for ‘ontic’ readings of ‘might’ counterfactuals, which ascribe some genuine possibility of the consequent given the antecedent. We want to say that if the game had been played, it’s neither the case that Arsenal would have won nor that they wouldn’t have won. Either might have happened; ‘might’ here gets the same kind of reading as ‘could’.

A similar complaint can be made about the pragmatic explanation for the non-assertibility of 9) and 10). We might not want to accept that it’s either the  case that Arsenal will win, or that they won’t win, and we just don’t know which – we could insist that the ‘might’ and ‘will’ are given a ‘ontic’ or ‘objective’ readings whereby’might p’ requires a genuine possibility of p, and ‘will p’ requires no genuine possibility of ¬p. If this is so, how should we respond to the clash of arguments discussed above?

A different type of response to this clash would reject 2) and 6). Perhaps, despite appearances, ‘Arsenal might not win, but I believe that Arsenal will win’ and ‘if the match had been played, Arsenal might not have won; but I believe that if the match had been played, Arsenal would have won’ do not express propositions of the form ‘p, and I believe that not p’. If so, then which propositions do these sentences express?

An interesting point to note is that the following sentences do not seem assertible, unlike the original Stalnaker/Moss sentences:

19.    Arsenal will win, but I believe that Arsenal might not win.
20.    If the match had been played, Arsenal would have won; but I believe that if the match had been played, Arsenal might not have won.

These two sentences do seem like instances of Moorean paradoxes, as the duality theses would suggest. And the explanation of non-assertibility in terms of the knowledge norm of assertion does not seem to apply to them; it’s possible that I know Arsenal will win, but also that I know that I believe it to be compatible with my knowledge that Arsenal not win.

This suggests that there is something unusual about the way that ‘will’ and ‘would’ embed into propositional attitude contexts. Here is a proposal: perhaps we should read ‘I believe that Arsenal will win’ as expressing a high credence in Arsenal’s winning, and read ‘I believe that if the match had been played, Arsenal would have won’ as expressing a high conditional credence in Arsenal winning on the match played.

If this is right, then it predicts that the following two sentences should be inequivalent:

21.    I believe that Arsenal will win.
22.    I believe that ‘Arsenal will win’ is true.

And so should the following two sentences:

23.    I believe that if the match had been played, Arsenal would have won.
24.    I believe that ‘if the match had been played, Arsenal would have won’ is true.

We can test for this by reinserting 22) and 24) into the original Stalnaker/Moss sentences, as follows:

25.    Arsenal might not win, but I believe that ‘Arsenal will win’ is true.
26.    If the match had been played, Arsenal might not have won, but I believe that ‘if the match had been played, Arsenal would have won’ is true.

I submit that neither of these sentences are assertible. There is a tension between the two conjuncts, which is absent from the original Stalnaker/Moss sentences. If this is correct, then we have reason to reject 2) and 6), and with them the argument against the might/would and the []→ / ◊→ dualities. We can then adopt the simplest explanation for 9), 10), 14), and 15): the failure of assertibility derives from straightforward semantic inconsistency.

We can generalize the proposal to other epistemic propositional attitudes, such as suspecting. ‘I suspect that p will be the case’ plausibly also expresses a high credence in p, though probably not as high a credence as that expressed by ‘I believe that p’. We cannot, of course, generalize to the case of knowledge; ‘p might not be the case, but I know that p will be the case’ is not assertible. What about hoping and fearing? These are interesting cases, because they seem to have both epistemic and non-epistemic uses. To get a grip on the difference I have in mind, compare ‘I hope that p will happen’ to ‘I hope that p doesn’t happen’ – the former, but not the latter, appears to express optimism about p, while the latter remains neutral. I propose, then, that ‘I hope that p will be the case’ can be taken to express a credence in p which is satisfactorily high – similarly, ‘I fear that p will be the case’ can be taken to express a credence in p which is unsatisfactorily high.

Why should ‘will’, and ‘would have’ embed in this non-standard way into epistemic propositional attitude contexts? I suggest that it is because of the practical utility of being able to express our credences and conditional credences in a way which does not sound clumsy. But whether or not we have a genetic explanation for this non-standard behaviour, it seems to be the best explanation of our responses to the simple sentences involving ‘might’, ‘will’ and ‘would have’ which I have discussed.

Can we apply a fictionalist strategy to the metaphysics of property-boundaries? The following thoughts are still very sketchy, and I’d be glad to be put right if I’ve missed something obvious. The aim is to clarify somewhat the motivation for supervaluationism.

Consider first modal fictionalism, which is to a good approximation the doctrine that ‘possibly p’ is true iff ‘according to the modal realist fiction of concrete possible worlds, p is true at some world.’ The fictionalist claims that the possible-world ontology is firmly embedded in our thinking about modality, but nevertheless functions as a useful fiction; fundamentally speaking, there are no possible worlds of the sort that that the modal realist acknowledges, but their existence is a presupposition of our practice of modalizing.

The reason that modal fictionalism is unconvincing is that it fails to explain why modal thinking is so useful, despite involving a presupposition which is literally false.  Compare the no-miracles argument in philosophy of science – it would be a miracle if modal thinking worked as well as it does, despite having a false existential presupposition about concrete worlds, just as it would be a miracle if quark science were as successful as it is, despite having a false existential presupposition about quarks.

Now consider an application of this strategy to vagueness. The view I am thinking of is that, to a good approximation, ‘x is F’ is true iff, according to the fiction of precise unknowable boundaries, x falls within the extension of F.’ The semantic fictionalist claims that existence of precise unknowable boundaries is firmly embedded in our thinking about vagueness, but nevertheless functions as a useful fiction; fundamentally speaking, there are no precise boundaries of the sort that the epistemicist acknowledges, but their existence is a presupposition of our practice of modalizing. The commitment to the precise and autonomous extension-fixing mechanism envisaged by epistemicism is merely a fictional one, and semantic theory ought to be treated purely instrumentally.

The thought is that the demand for explanation here is easier to resist than the demand for explanation in the case of modal fictionalism. We want to be able to use the simplest possible logic for our language – taking for granted the existence of sharp boundaries allows us to use classical logic in full generality. This could be taken as an independent explanation of why predicative language is a much more powerful tool if we make the presupposition of sharp boundaries. And the explanation which is analogous to the scientific realist’s explanation for the success of science – that is, that there is indeed a landscape of semantic facts out there determining sharp boundaries – seems much less plausible in the current case. So the objection which in my opinion cripples modal fictionalism leaves semantic fictionalism untouched.

However, another objection looms large. Which fiction should we use? There are myriad precise-boundary fictions available, which draw the lines slightly differently from one another. It seems that a small difference in the details of the fiction wouldn’t make a significant difference in the practical utility of the fiction – classical logic holds whichever fiction we choose, and the differences under consideration can be made small enough so as to not clash significantly with use. The objection is that any choice of fiction would be arbitrary.

This arbitrariness, I think, is fatal to semantic fictionalism. It doesn’t get such a good grip against modal fictionalism, because arguably the Lewisian modal realist ontology is to at least a significant degree non-arbitrary. But the arbitrariness of the sharp boundaries, which is so intuitively problematic for epistemicism, recurs for semantic fictionalism as the arbitrariness of which is the ‘correct’ fiction.

The thought that any particular placing of the sharp boundaries is somewhat arbitrary is the main motivation for supervaluationist approaches to vagueness. According to supervaluationism, it is indeterminate which of some set of precise fictions is the correct one – there is simply no fact of the matter. Nonetheless, when evaluating logical reasoning, logically valid arguments will be valid whichever fiction we choose – which is what matters for the theoretical utility of language-use.

How supervaluationism should be developed is a further question, and a difficult one. My preference is for views which reject the truth-supertruth identification – I’ll try to say more about this in future posts.

In MLE today we discussed Boris Kment’s ‘Counterfactuals and Explanation’. See here for the details of the discussion, handout, etc. I really liked the view of this paper, and found myself wondering how to carry it over to my own view.

According to Everettian Modal Realism, all worlds share the same fundamental laws. So criterion 1) of Kment’s theory of closeness gets explained by EMR, rather than stipulated.

Of course, EMR can’t appeal to the notion of impossible worlds to explain counterfactuals with necessarily false antecedents. So we have two options here – either take all such counterfactuals to be trivially true, and explain their assertibility or lack thereof in pragmatic terms, or give an alternative metalinguistic account of their truth-conditions. I prefer the former option, as explained here. This might look like a disadvantage of EMR, but only if you’re comfortable with ersatz linguistic possible worlds – for those of us persuaded by Lewis’ criticisms of such worlds, impossible worlds just make no sense at all.

Kment’s account also involves worlds featuring violations of laws of nature, which looks incompatible with EMR. But if we restrict the exceptions used in the analysis to apply only to non-fundamental laws, like the laws of statistical mechanics or economics, then Kment’s account of closeness can be retained for all ordinary counterfactuals. The only problematic cases will be counterfactuals explicitly about fundamental physics.

Consider two cases:

1) If the electron was now here rather than there, the entire history of the world would have been different.

2) If there were now no electrons within a million miles of here, the entire history of the world would be different.

Even on EMR, 1) comes out as false, because the closest worlds with an electron in a slightly different position are worlds which diverged quite recently, due to indeterministic evolution. Since on EMR determinism is not even a metaphysical possibility, the burden on Kment’s account to deal with deterministic worlds is lifted, and cases like 1) present no real problem.

Cases like 2) are a bit harder. The indeterministic evolution required to get rid of all electrons within a million miles from here in the relatively recent past is a phenomenally unlikely event – so unlikely, I think, that it counts as a ‘big violation of law’ in Kment’s terms (although the only laws violated are special-scientific laws, like classical electrodynamical laws.) What seems to matter is that very low-probability events would have to occur to get rid of the electrons in the recent past, while courses of events starting much longer ago which would have led to a lack of electrons round here are much higher-probability.

So I think a defender of EMR should adopt the following criteria of closeness:

- avoid very low-probability events.

- achieve match in matters of particular fact, where the facts have the same explanation in each world.

These are the only two criteria we need; and the trade-off between them will be non-trivial. However, we have a good enough intuitive grasp on the trade-off – it seems plausible that the spontaneous disappearance of all the electrons within a million miles is low-enough probability to cancel out billions of years of match in matters of particular fact, whereas the spontaneous movement of one electron across a micrometre or so of space is not low-enough probability to cancel out more than a few seconds of match in matters of particular fact.

The term ‘current best scientific theory’ gets thrown around a fair bit. Quineans about ontology say that we ought to believe in the existence of exactly those entities which are the values of bound variables in our current best scientific theory; and metaphysical naturalists argue that metaphysics should be informed by current best scientific theory. In general, epistemological naturalism proposes that our beliefs should be in some way constrained by by current best scientific theory. But what is ‘current best scientific theory’, and why should it play these kinds of normative role?

Start with the obvious. ‘Our current best scientific theory’ is a definite description. Lets take theories to be timeless abstract objects, understood as sets of propositions, and take it that ‘our’ refers back to some non-vague epistemic community. Assume also that ’scientific theory’ refers to some total theory which is the result of conjoining theories from all of the special sciences. Then ‘our current best scientific theory’ could be disambiguated as follows (I’m not suggesting there aren’t other ways you could read it):

1) Of all the total scientific theories in the public domain, the best one.

2) The total scientific theory which, currently, is considered by some informed majority to be the best in the public domain.

3) The totality of our scientific knowledge.

It is fairly clear that 1) can’t play any very interesting methodological role for grounding ontological commitment. It is consistent with a theory meeting condition 1) that it also be disbelieved and discounted as a serious contender by everybody. So an account of ontological commitment which appealed to best scientific theory understood in the sense of 1) might have us committed to entities in whose existence we flatly disbelieve.

But perhaps 1) can help in explaining the normative role of best scientific theory in guiding philosophical theorizing. If some theory T is in the public domain, and really is the best theory in that domain, then it would be better in some sense to be guided by T than by any other theory in the public domain. But presumably it would be better still to be guided by a true total theory T* which is not in the public domain. The norm ‘be guided by T’ seems just to be a special case of the norm ‘believe the truth’.

If we understand ‘current best scientific theory’ according to 2), then it becomes much more plausible that the norm ‘conform your credences to current best scientific theory’ is one we can in fact follow. Through polls, perhaps, we could collect opinions of experts in various scientific fields, and conjoin them to construct a total theory which informed consensus picks out as the best. Of course, this procedure is highly idealized, but it doesn’t seem entirely impossible for us to build an accurate picture of current best scientific theory, if we understand it according to 2). And if you are persuaded by scientific realism, you will be able to account for the expected good epistemic outcome of following this norm. I think 2) is the standard way of thinking about ‘current best scientific theory’, and underlines much of the use of this phrase in sloganeering.

3) also seems to hold some interest, though. Take ‘current best scientific theory’ to be the common knowledge fragment of the theory given by 2). Since knowledge is factive and sensitive, following correctly the norm ‘conform your credences to the current best scientific theory’ understood according to 3) cannot possibly lead us to false or unreliable beliefs, a risk which is still present with the version of the norm which appeals to 2). Taking ‘current best scientific theory’ in the sense given by 3) yields a norm which seems no harder in principle to follow than Williamson’s conception of knowledge as the norm of assertion.

Conclusions? None really, except that we should try and be precise about what we are talking about in methodological appeals to ‘current best scientific theory’. Depending on how we take that term to pick out a particular set of propositions, we end up with different strengths of epistemic norm. 1) leads to a norm which we cannot in general follow, and 3) leads to a norm which we cannot in general know we are following.

One thing I haven’t considered at all here is exactly how some given total scientific theory should constrain our beliefs. Presumably we don’t want to conditionalize on best current theory, for reasons related to the preface paradox and the pessimistic meta-induction. But saying that we should Jeffrey-conditionalize on it seems to lose all the distinctive force of the naturalist methodology. Perhaps the best way of representing the naturalist credo in Bayesian terms is that there should be some input from current best scientific theory into selecting correct prior conditional credences. But this hasn’t really been thought through at all.

Caveat – I’m still on painkillers after a paragliding incident so the above is probably a little garbled!

This post was inspired by a talk Barbara Vetter gave to the Ockham society last night. The view of dispositions as not necessarily needing a stimulus is hers – the extra stuff I’ve added shouldn’t be blamed on her!

Folklore has it that dispositions are individuated both by their stimulus conditions and their manifestations. Fragility, for example, is said to be roughly the disposition to break when struck. There is an active element (breaking) and a passive element (being struck) in this characterization. But could we have purely active or purely passive dispositions?

Consider the purely active case first. This would be a disposition to behave a certain way, independently of external stimuli. There’s one particularly clear candidate; radioactivity. An atom is radioactive if it is disposed to decay spontaneously.

Purely passive cases are harder. But maybe edibility is an example. What matters to whether something is edible is whether or not some external entity can come along and eat it; what happens next (the manifestation) seems to be somewhat irrelevant. Of course, in some sense of ‘edibility’, being edible requires not poisoning the eater when eaten. But leave these cases aside. Or perhaps something like ’strikability’ is a better example.

If these cases are as I’ve described them, then a kind of pattern emerges. There are similar kinds of property which may be characterized by an active condition (manifestation), by a passive condition (stimulus), or by both, as follows:

A question which now arises is – which of these kinds of properties count as dispositional? I suggest they all do, even though we don’t normally think of properties like ’strikability’ when giving examples of dispositions. Perhaps this is partly because strikability, if it is a disposition, will be an extrinsic disposition.

The question mark after fragility is because I’m not sure that fragility really is the disposition to break when struck – perhaps it’s just the disposition to break, or the property of breaking easily (this is actually how the OED defines it.) Evidence for this claim might be that ‘fragile’ and ‘breakable’ are used pretty much interchangeably on packaging.

Provisional conclusions -

• Some dispositions lack a (non-trivial) stimulus condition (radioactivity is a clear example).
• We can perhaps make sense of properties which lack a (non-trivial) manifestation condition (edibility and strikability are not-so-clear examples).
• We can lay out a spectrum of possible properties ranging from ‘purely active’ to ‘purely passive’, depending on the role of active and passive conditions in characterizing those properties.
• Which properties count as dispositional is an interesting and non-trivial question.

Warning – this might be somewhat half-baked.

I got to wondering about the vagueness of the term ‘our linguistic community’ and its consequences for epistemicism. On a natural Williamson-style view, the exact meaning of a term is determined by the global pattern of use of that term amongst our linguistic community. But what about the term ‘our linguistic community’? Presumably this term itself is vague – there can be borderline cases of entities which may or may not be people, or may or may not be speaking Old English as opposed to modern English, and it can be unclear whether they are part of our linguistic community. But how is an epistemicist to cash out the vagueness in the size of our own linguistic community?

If the meaning of the term ‘our linguistic community’ has its meaning fixed in the same sort of way as other terms, then it seems like what we’re saying is that the meaning of the term ‘our linguistic community’ gets fixed by its global pattern of use among our linguistic community. Is there a threat of circularity here?

Consider various precise linguistic communities which are candidates to be the referent of the term ‘our linguistic community’ as in fact used by us. For consistency, on the assumption that community A is the best candidate, the meaning of ‘our linguistic community’ must be determined by the use of that term by community A. If this is true of all candidate linguistic communities, everything is hunky-dory for Williamson. But can we be sure that things will always work out this nicely? If there is an argument that they will, I’d love to see it.

Could there be, for example, a linguistic community P whose use of the term  ‘our linguistic community’ is such that in their mouths it always refers to some distinct linguistic community Q? In this case, the reference of P’s terms is fixed by the global usage of community P, even though no term actually used by any of the members of P in fact refers to P. There is still no inconsistency: we can simply think of this setup as involving a linguistic community who have no term within their own language for their own linguistic community, and hence no way of delineating the supervenience base for the meaning of their own terms. Of course, members of community P can refer to the supervenience base for their terms descriptively: ‘the supervenience base for the meanings of our terms’. But they may have no other way of referring to that supervenience base: in which case it would be true for a member of P to assert ‘the meaning of our terms is not fixed by the global pattern of their use amongst our linguistic community’.

The question that interests me is – how do we know that we’re not in the situation of community P? If a community like P is a close possibility, then we can’t know that we’re not in that situation – in which case we can’t know that Williamson’s view about how meanings are determined by global use is correct. But perhaps it can be argued that P is not a close possibility, or even a possibility at all.

But even if communities like P are not possibilities, or not close possibilities, we don’t seem to be out of the woods yet. Another way to worry about the potential circularity is to worry whether there might be various (overlapping) candidate linguistic communities within the actual world all of which include you, all of which are consistent in the sense that members of them refer to their own community by the term ‘our linguistic community’. Since the communities are different sizes, they have different global patterns of use, so the meanings they determine for other terms may systematically differ; we only require that each community self-refer by the term ‘our linguistic community’.

Now: what could possibly determine which of these various candidate linguistic communities is in fact our own? Presumably it would be the degree of knowledge each community has about non-linguistic matters. A speculation; perhaps this knowledge is maximized by selecting the largest possible linguistic community which still has the required consistency property of correctly referring to itself by the term ‘our linguistic community’.

Recall the Lewisian definitions of branching and of divergence; two objects branch if they share a common initial part, and they diverge if they have intrinsically exactly similar but numerically distinct initial parts. Saunders and Wallace [2008] are keen to stress that their semantics is compatible with both a branching metaphysics and a diverging metaphysics:

The salient distinction, between similar and identical initial segments, is invisible at the level of syntax and logical form.

Saunders and Wallace [2008], p.297

However, as Saunders [forthcoming] recognises, the distinction between branching and divergence is not invisible at the level of metaphysics. He concludes that in the Lewisian sense, Macroscopic Pairing is a theory of diverging worlds.

I think that this is incorrect; Macroscopic Pairing and Space-time Point Pairing are strictly speaking neither theories of branching worlds nor theories of diverging worlds. In the context of Everettian quantum mechanics, a more helpful distinction is that between overlapping and non-overlapping worlds. I will argue that both Macroscopic Pairing and Space-time Point Pairing tell in favour of taking Everettian worlds as non-overlapping.

Take Macroscopic Pairing first, and consider the situation depicted in Figure 1. According to Macroscopic Pairing, there are two worlds present: <A, A> and <B, B>. There are also two distinct material objects present, <A, C> and <B, C>. So what do we have here? Is <A, C> identical to <B, C>, or are they intrinsically exactly similar? The answer appears to be ‘neither’. <A, C> cannot be identical to <B, C>, for the obvious reasons that the former includes A as a element while the latter does not, and that the latter includes A as a element and the former does not. But equally, and for the same reasons, it appears that <A, C> cannot be intrinsically exactly similar to <B, C>. So, according to the Lewisian definitions of branching and diverging, worlds <A, A> and <B, B> neither branch nor diverge.

However, there is a natural alternative sense of ‘exactly similar’ according to which <A, C> and <B, C> are exactly similar. Everything which is knowable about <A, C>at some time when it is present is also knowable about <B, C>: no possible observation can distinguish whether we are observing <A, C> or <B, C> up until the interaction that separates the worlds <A, A> and <B, B>. This sense of ‘exactly similar’ is an epistemic one: it could be explained as ‘observationally exactly similar’. If we appeal to this notion of exact similarity, rather than the notion of intrinsic exact similarity which is part of the Lewisian definition of divergence, then we can recover a sense in which worlds <A, A> and <B, B> diverge.

Another sense of ‘exactly similar’ which could recover divergence of worlds could be defined as follows, along the lines of the parthood* relation discussed above: two world-continuants x and y are exactly-similar* iff they share a second element. <A, C> and <B, C> are exactly-similar*: so if we define divergence by a having exactly-similar* initial segments, then worlds A and B do diverge. If you are prepared to countenance parthood* as a replacement for parthood in the context of world-continuants, then you might equally be prepared to countenance exact-similarity* as a replacement for exact similarity in the context of world-continuants. Such a replacement would once again render Everettian worlds as diverging.

Similar comments apply to the case of Space-time Point Pairing. Fusions of pairs <A, pC> (where pC means ‘a point-like part of C’) cannot be either identical or (strictly speaking) intrinsically exactly similar to fusions of pairs <B, pC>. However, if we are prepared to extend the notion of exact similarity to that of observational exact similarity, or to exact-similarity*, then we can recover senses in which the world which is a maximal fusion of pairs <A, pC> and the world which is a maximal fusion of pairs <B, pC> diverge from one another.

Despite the failure of Lewis’ distinction between branching and divergence to apply directly to the worlds delivered by Macroscopic Pairing and Space-time Point Pairing, it remains the case that these worlds are non-overlapping. The distinction between diverging (in the strict Lewisian sense) and branching worlds is not obviously more important than the distinction between overlapping and non-overlapping worlds. And in terms of this latter distinction, it seems clear that the Everettian should think of worlds as non-overlapping rather than as overlapping. This conclusion, somewhat unexpectedly, vindicates the application of the popular phrase ‘parallel worlds’ to Everettian quantum mechanics.

What is left of the intuitive thought that Everettian quantum mechanics involves branching worlds? The Space-time Point Pairing account which I have been urging involves no branching at the level of worlds; however, it does still incorporate branching at an underlying level, the level of Metalanguage 2. The structure of branches, which provide the ‘raw material’ for the elements of the pairs identified with space-time points, is a branching structure; different branches genuinely do have different parts in common. The difference between the view I am defending, and the straightforward picture, is that I would not identify branches directly with worlds. Building worlds up out of branches through the Space-time Point Pairing procedure gives us the resources to explain ignorance of the future and to explain probability. The straightforward view lacks these resources.

The picture we are left with has much in common with the Modal Realism of David Lewis (with the exception, of course, that the laws of quantum mechanics hold in every Everettian world). Both Everettian quantum mechanics and Modal Realism give us distinct disconnected non-overlapping space-times populated with distinct disconnected non-overlapping macroscopic objects and events.

It’s a consequence of my favoured version of Everettian quantum mechanics that there is no determinate ontology of macroscopic objects and events. Here’s how I see this working.

At the fundamental level, the ontology of Everettian QM is monistic – there is just one single highly-structured object, the universal state. Call the language in which this claim is true Metalanguage 1. This language trades off minimal ontological commitment for maximal ideological commitment.

By use of decoherence theory to pick out privileged structure from the universal state, we can construct a language in which we can quantify over structures. In this language, Metalanguage 2, there are objects such as branches, branch segments, and so on – all of the entities identified directly with objects on the ‘Literal Fission’ picture I criticize in this post. Because decoherence does not pick out exactly a decomposition basis for the universal state, the translation scheme between Metalanguage 1 and Metalanguage 2 will be vague.

Metalanguage 2 can be thought of as the working language of metaphysical theorizing on this picture. It is flexible enough to be able to refer to everything we want to speak about, whether in or out of the metaphysics room. But it is still a metalanguage; it is unsuitable as an interpretation of our ordinary thought and talk about the macroscopic world. This role is reserved for the object language, which can be explained in terms of Metalanguage 2. My preferred account of the relationship between Metalanguage 2 and the object language is that set out below, in my previous post. According to this view, our ordinary language quantifiers are generally restricted to range over a very special range of entities – fusions of space-time point pairs.

According to this framework, the connection between Metalanguage 1 and Metalanguage 2 is vague, but the connection between Metalanguage 2 and the object language need not be vague. There is still an ineliminable vagueness in ordinary language, but this results from the inexactness of decoherence, rather than from any inadequacy of the translation schema between Metalanguage 2 and the object language. It may be objected that the ineliminable vagueness of the object language is a fault of Everettian QM; but this simply amounts to a complaint that the decoherence-based solution to the measurement problem is faulty. In the absence of any better solution to the measurement problem, we may simply have to bite the bullet of ineliminable non-epistemic vagueness in ordinary thought and talk.

This metaphysical framework is inimical to the epistemicist conception of vagueness. Quantum mechanics allows for no physical facts at the level of Metalanguage 1, not even necessarily unknowable ones, which could determine a precise value for, for example, numbers of branches in Metalanguage 2. If there is no determinate fact of the matter about macroscopic ontology, vagueness in ordinary talk about macroscopic objects and events cannot be a purely epistemic phenomenon. Our concepts cannot have sharp boundaries of application if there are no sharp boundaries in the world for them to latch on to.

If this brand of Everettian QM is even a metaphysical possibility, then it seems to present problems for epistemicism. Presumably epistemicism is intended as an account of vagueness in any language structurally like ours – and if it fails in application to vagueness in Everettian multiverses, do we really have good reason to hold to it even if we’re not ourselves living in such a universe?