I am puzzled about these efforts, for two reasons.

First, as Lewis and Kratzer pointed out in the 1970s and 80s,
if-clauses often (according to Kratzer, always) function as
restrictors of quantificational and modal operators. So when we see an
if-clause in the vicinity of a modal like 'the probability that', the
*first* thing we should consider is whether the if-clause
restricts the modal.

How would an if-clause restrict a probability modal? Well, what is the probability of B restricted by A? An obvious answer is that it's the probability of B given A. So if the if-clause in 'the probability that if A then B' restricts the probability modal, then the expression denotes the conditional probability of B given A. Which is just what we find.

In other words, there is independent evidence about if-clauses suggesting that 'the probability that if A then B' should be analysed as 'P(B/A)' rather than 'P(if A then B)'. If that's correct, then what's expressed by

(*) the probability that if A then B equals the conditional probability of B given A

is the trivial identity 'P(B/A) = P(B/A)'. There's no need to make a big effort trying to make (*) true.

The case of subjunctive conditionals is parallel. We have the intuition that

(**) the probability that if A were the case then B would be the case equals the conditional probability of B on the subjunctive supposition that A.

Again, the first thing we should check is whether the if-clause restricts the modal. And, plausibly, subjunctive if-clauses restrict probability modals by subjunctive supposition (aka imaging). And then (**) expresses the trivial 'P(B//A) = P(B//A)'.

When people try to give a semantics motivated by (*) or (**), they practically never explain what's wrong with the simple and obvious explanation of (*) and (**) that I've just given.

That's one reason why I'm puzzled by these efforts. Here's a second reason.

For concreteness, let's look at subjunctive conditionals, which I'll write 'A > B'. As Lewis shows towards the end of "Probabilities of Conditionals and Conditional Probabilities", if you want to validate 'P(A > B) = P(B//A)', you have to assume a Stalnaker-type semantics for '>' on which, for any world w and any proposition A, there is a unique A-world that is "closest" to w; 'A > B' is true at w iff B is true at the closest A-world.

But if we assume a Stalnaker-type semantics of would counterfactuals 'A > B', then what should we say about might counterfactuals, 'if A were the case then B might be the case' -- for short, 'A *> B'?

Clearly, 'A *> B' can't be the dual of 'A > B', otherwise the two would be equivalent. The only option I can think of is to say, with Stalnaker, that 'A *> B' must be analysed as Might(A > B).

But that's unappealing, especially in the present context.

For one, the idea that 'A *> B' means 'Might(A > B)' is incompatible with a broadly Kratzerian treatment of 'if' and 'might'.

Moreover, syntactically, 'would' and 'might' seem to play similar roles in 'if A then would B' and 'if A then might B'. One would at least like to see some more evidence that 'might' scopes over the conditional and 'would' does not. Relatedly, (as I mentioned in an earlier post), it seems to me that

What if A were the case? It might be that B

is equivalent to 'if A were the case then it might be that B'. But surely 'might' in the second sentence doesn't somehow scope over 'if' in the first.

Moreover, let's look at the probability of might counterfactuals. Assuming that 'Might' in 'Might(A > B)' is epistemic, 'Might(A > B)' is true relative to an information state s iff s is compatible with A > B. What is the probability that s is compatible with A > B, relative to s? Unless the information state is unsure about itself, it will be either 0 or 1. Specifically, we get the prediction that P(A *> B)) = 1 if P(A > B) > 0 and P(A *> B) = 0 if P(A > B) = 0. But intuitively, 'the probability that if A then might B' is not always 1 or 0.

So what you have to say, if you want to analyse 'A *> B' as 'Might(A > B)', is that despite surface appearance, the expression 'the probability that if A then might B' does not denote the probability of the embedded might counterfactual 'if A then might B'. Perhaps the two epistemic modals merge and the expression denotes the probability of 'if A then would B'. Or whatever. But in the present context, it's funny that you have to say such a thing, given that your whole approach is motivated by your commitment to the idea that 'the probability that if A then would B' denotes the probability of the embedded would counterfactual.

]]>But it's hard to spell out how exactly the argument is meant to go. In fact, I'm not aware of any satisfactory statement. Here's my attempt.

For concreteness, I'll focus on the argument for probabilism, but the case of conditionalization is similar.

The argument begins with an uncontroversial mathematical fact, the
*Dutch Book Theorem*:

Let aunit beton a proposition A be a deal that pays $1 if A is true and otherwise $0. Suppose for any proposition A, an agent is prepared to buy a unit bet on A for up to $Cr(A) – the dollar value corresponding to her credence in A – and she is prepared to sell a unit bet on A for $Cr(A) or more. If her credences do not satisfy the axioms of non-negativity, normalization, and finite additivity, she will then be prepared to buy and/or sell unit bets in such a way that if she makes all these transactions she incurs a guaranteed loss.

How do we get from here to an argument that rational credences should conform to the probability axioms? A few problems immediately stand out. (See "Hajek 2008.)

First, the theorem seems to say nothing about people who aren't prepared to trade bets in accordance with their expected monetary payoff. Surely epistemic rationality does not require having a utility function that is linear with respect to monetary payoff. An epistemically rational agent need not care about money at all.

Worse, even if an agent does care only about money, and her utility function is linear with respect to monetary payoff, she ought not to be prepared to buy a unit bet on any proposition A for up to the dollar value corresponding to her credence in A. For example, let A be the proposition that the agent will not buy any bets today. An agent's credence in A may well be high, yet she ought not to pay much for the corresponding bet, since doing so would render A false.

Even if the relevant propositions are unaffected by the considered bets, there can be interference effects between different bets. For example, what if our agent has earlier bought a high-stakes bet on the proposition that she will not buy any more bets today? Then she may not be prepared to buy a unit bet on any proposition whatsoever. Relatedly, the Dutch Book argument for finite additivity involves at least three bets; if a probabilistically incoherent agent cares about the net outcome of all her transactions, rather than myopically about the isolated outcome of whatever transaction she currently considers, it is not clear why she ought to make all three transactions. (This is the "package principle objection". Interestingly, it seems not to arise for the case of conditionalisation, as "Skyrms 1993 shows.)

Stepping back, why is the mere possibility of making a sure loss normatively relevant? After all, as Lewis said, "there aren't so many sneaky Dutchmen around".

Finally, why is the possibility of financial loss a sign of
*epistemic*, rather than *practical* irrationality?

A neat way to get around most of these problems, which I haven't seen in the literature, is to invoke some broadly Humean principles about the independence of belief and desire. In outline, the idea is that for any probabilistically incoherent agent X there is a possible agent Y who (1) has the same credences as X, (2) only cares about the monetary payoff of whatever transaction they presently consider, and (3) is offered the relevant bets that make up a Dutch Book. Y then makes a sure (and avoidable) loss, despite trying to get as much money as possible. Something has gone wrong. But the fault must lie in Y's beliefs, for neither her utilities nor her decision process is faulty. So Y's beliefs are irrational. But Y's beliefs are identical to X's. So X's beliefs are irrational.

That's the outline. Let's fill in the details.

Let X be an arbitrary agent whose credences violate one of the probability axioms. Our aim is to show that X is epistemically irrational.

Let Y be a possible counterpart of X with the same (centred) credences. But Y has strange desires. Whenever Y is offered a monetary gamble, she only myopically cares about the net amount of money she will make through the present transaction. Specifically, if Y has the option to buy a unit bet on some proposition A for some amount $x, then the only thing she cares about is whether she will eventually (i) win $1 after having paid $x, or (ii) not win after having paid $x, or (iii) not win after not having bought the bet; the utility she assigns to these outcomes are, respectively, (i) $1-$x, (ii) -$x, (iii) $0. Similarly, mutatis mutandis, if Y has the opportunity to sell a unit bet.

I'll also stipulate that when faced with a choice, Y always chooses an option with maximal expected utility.

All this still doesn't ensure that Y is prepared to pay up to $Cr(A) for a unit bet on A because her credence in A may be affected by getting an offer to buy the bet or even by the act of buying (as when A is the proposition that she won't buy any bets today). If we want Y to accept a Dutch Book that involves several transactions, we must also ensure that, say, buying the first bet does not affect the expected utility of buying the second.

I'm not sure how best to get around these problems. Here's a brute force response.

Let's say that X's (and Y's) credence function is *stable* with
respect to some propositions A,B,...,N iff Y regards $Cr(A) as the
fair price for a unit bet on A, $Cr(B) as the fair price for a unit
bet on B conditional on having bought/sold a unit bet on A, and so
on. That is, a credence function is stable with respect to a list of
propositions if the credence in each proposition on the list is not
affected by whether a bet on that proposition or a proposition earlier
in the list has been bought or sold.

If we assume that there is a list of propositions for which X's credences are stable and violate the probability axioms, we can stipulate that Y is made the relevant offers and gets caught in a Dutch Book.

So we need to assume that X's probabilistic incoherence isn't restricted to unstable parts of her credence function. To get a general argument for probabilism, we'll need the following premise.

Premise 1. If any restriction of an agent's credence function to stable propositions should satisfy the probability axioms, then so should her entire credence function.

(Here and throughout, a violation of the probability axioms means that either (i) some proposition has negative probability, or (ii) the tautology does not have probability 1, or (iii) there are disjoint propositions whose disjunction has a probability that is not the sum of the probability of the disjuncts. Boolean closure is not treated as an axiom.)

The motivation for Premise 1 is that the probability axioms are supposed to be general consistency requirements on rational belief. They are meant to hold for beliefs or any kind, not just for beliefs with a specific content.

Jeffrey makes a similar move in *Subjective Probability: The Real
Thing* (pp.4f.):

If the truth [of a proposition about distant planets] is not known in my lifetime, I cannot cash the ticket even if it is really a winner. But some probabilities are plausibly represented by prices, e.g., probabilities of the hypotheses about athletic contests and lotteries that people commonly bet on. And it is plausible to think that the general laws of probability ought to be the same for all hypotheses – about planets no less than about ball games. If that is so, we can justify laws of probability if we can prove all betting policies that violate them to be inconsistent.

Jeffrey here assumes that there's a reasonably wide set of propositions for which our credences match our betting prices. I assume something much weaker. But I'd still like to know how to do better.

(One reassuring fact to keep in mind is that probabilistic incoherence is infectious: if, for example, your credence in an exclusive disjunction A v B is not the sum of your credences in A and B, there will be lots of other propositions for which you'll violate additivity. So it requires some fine-tuning to limit incoherence to unstable fragments of a credence function.)

Moving on, recall that X was an arbitrary agent whose credences fail to satisfy the probability axioms. We want to show that X is epistemically irrational. By Premise 1, we can assume without loss of generality that there are some propositions with respect to which X's credence function is stable but fails to satisfy the probability axioms.

The next premise is that all the differences between X and Y are irrelevant to whether the credence function shared by X and Y is epistemically rational. More precisely:

Premise 2. If X is epistemically rational, then so is Y.

The basic idea is that whether someone's beliefs are epistemically rational does not depend on her goals or desires. If we want to know whether it is epistemically rational (as opposed to practically useful) for an agent to have such-and-such beliefs, we don't need to know anything about her goals or desires.

I've also assumed that Y is an expected utility maximizer, which X may not be. But again, arguably the epistemic rationality of someone's beliefs would not be undermined by finding that they are an expected utility maximizer.

Finally, Premise 2 implies that it does not affect the epistemic rationality of an agent's beliefs if they are about to be offered a series of bets. (That's the final difference between X and Y.)

Premise 2 looks fairly good to me.

Now the Dutch Book theorem tells us that there are certain transactions that Y is prepared to make that would amount to a guaranteed loss. Let's stipulate that Y is made the relevant offers and thus really does make a sure loss.

The next premise states that something has then gone wrong.

Premise 3. It is irrational of Y to make choices that together amount to a sure loss (a loss she could have avoided by making different choices).

Here the guiding idea is that it is irrational for an agent whose sole aim is to maximize monetary profit to knowingly and avoidably enter transactions that are logically guaranteed to cost her money.

Premise 3 relies on the Converse Dutch Book Theorem: that probabilistically coherent agents cannot be Dutch Booked.

I'm not entirely happy with Premise 3. The problem is that, by
assumption, Y does not care about her net wealth. When offered a
series of choices, she only cares about the net outcome of the
*present* choice. It would be nicer if we could stipulate that Y
cares about the net payoff of all the choices she's about to
make. This would make Premise 3 quite compelling, I think. But then
we'd need to explain why Y accepts the individual deals that together
constitute a Dutch Book. (Skyrms offers such an explanation in his
1993 paper on conditionalization, but the argument sadly doesn't
generalize to the case of finite additivity.)

The rest of the argument is simple.

Premise 4. If an agent makes irrational choices, then either she is epistemically irrational or her desires are irrational or her acts don't maximize expected utility.

Premise 5. Y's desires are not irrational.

Y's myopic desires are admittedly weird, but on a suitably weak notion of rationality, I think they should pass. Since Y maximizes expected utility, we can conclude that Y's credences are irrational. Intuitively, Y misjudges the profitability of the relevant bets.

So Y is epistemically irrational. By Premise 2, it follows that X is epistemically irrational. QED.

]]>Many of the reviewers seemed to be either mathematicians or cognitive scientists who (1) didn't appreciate any of the wider philosophical content and (2) complained about minor details that are completely irrelevant to my main points. For example, one reviewer commented on my use of the term "doxastic space": "Ordinary mathematical use of ‘space’ suggests that the au wants to have some structure on the extended doxastic space beyond what comes from it being based on a Cartesian product (most likely a topology); but we are told nothing about that structure, leaving the au’s use of ‘space’ mysterious."

]]>

The question I address is simple: how should we model the impact of perceptual experience on rational belief? That is, consider a particular type of experience – individuated either by its phenomenology (what it's like to have the experience) or by its physical features (excitation of receptor cells, or whatever). How should an agent's beliefs change in response to this type of experience?

Why care? A few reasons:

First, the question is closely related to several traditional issues in epistemology. Intuitively, many of our beliefs are justified because they are suitably connected to relevant perceptual experiences. But what is that connection? How does a "nonconceptual" experience support a "conceptual" belief? How do we accommodate the holism of confirmation? How can an experience justify a belief about an external world if one could have the same experience even if there were no external world? A good model of how perceptual experiences should affect belief would help to make progress on these issues.

Second, the question is important for the "interpretationist" (formerly known as functionalist) approach to belief. On this approach, what makes a physical state a belief state with such-and-such content is that it plays a certain causal role – the role characteristic of the relevant beliefs. Part of this role links the beliefs (and desires) to choice behaviour. Another part of the role links them to perceptual experience: it specifies how beliefs tend to change through perceptual experience. In the paper, I'm effectively trying to spell out that part of the role.

Third, the question plays an import role in cognitive science. People in artificial intelligence have models of how incoming perceptual stimuli should affect a belief system. Similar models have proved fruitful in the neuroscience of perception. The model I propose looks a lot like these models from neuroscience and artificial intelligence. But the models have strange features that call for philosophical comment. In particular, they seem to imply a form of sense datum theory: perceptual experiences are supposed to provide infallible information about a special realm of sense data. How should we understand these sense data? Aren't there decisive philosophical arguments against sense datum theories?

Fourth, the question might provide the key to the hard problem of consciousness. In the paper I suggest that the appearance of irreducibly non-physical properties in perceptual experience is a predictable artefact of the way our brain processes sensory information.

Fifth, the question is interesting because it's really hard to answer. A common idea in the philosophy of perception seems to be that (1) perceptual experiences represent the world as being a certain way, and that (2) in the absence of defeaters, having the experience makes it rational to believe that the world is that way. But that's hardly a full answer. For one, how does an experience – individuated, say, by its physical features – come to represent the world as being a certain way? Moreover, how should the rest of an agent's belief system change if there is no defeater? How should the agent's beliefs change if there is a defeater? (Surely it should still change in some way.)

In the paper, I assume a Bayesian framework. So the question becomes: how should a given type of experience affect an agent's subjective probabilities? The classical Bayesian answer assumes that perceptual experiences make the agent certain of a particular proposition, so that her probabilities can be updated by conditionalization. But that doesn't seem right. Richard Jeffrey proposed an alternative which allows experiences to convey less-than-certain information. But the relevant less-than-certain information in Jeffrey's model is not just a function of the experience; it also depends on the agent's prior probabilities. So how do the prior probabilities together with an experience determine the input to a Jeffrey update? No-one knows. In fact, I argue that it is impossible to know, because the effect an experience should have on a belief system is not fixed by the experience and the (prior) belief system at all. It depends on a further aspect of the agent's cognitive state.

]]>Van Fraassen's cube factory nicely illustrates the problem. A factory produces cubes with side lengths between 0 and 2 cm, and consequently with volumes between 0 and 8 cm^3. Given this information, what is the probability that the next cube that will be produced has a side length between 0 and 1 cm? Is it 1/2, because the interval from 0 to 1 is half of the interval from 0 to 2? Or is it 1/8, because a side length of 1 cm means a volume of 1 cm^3, which is 1/8 of the range from 0 to 8?

Some try to dodge the problem by saying that the alternative
propositions that should be given equal probability, by the Principle
of Indifference, must be "equally supported by the evidence". The
problem now re-arises as the question what it takes for alternative
propositions to be equally supported. I think it is best anyway to
express the Principle as a constraint on *prior* probabilities,
which don't factor in any relevant evidence.

Many philosophers seem to have given up on the Principle of Indifference, suggesting that there simply is no such norm on rational credence.

That seems wrong to me. Suppose a murder has been committed, and exactly one of the gardener, the butler, and the cook could have done it. In the absence of further evidence, surely one should give roughly equal probability to the three possibilities. What norm of rationality do you violate if you are confident that the gardener did it, if not the Principle of Indifference?

So we're stuck with the problem of saying when two propositions
should count as "similar" so as to deserve equal prior
probability. Here's an answer that looks promising to me and that I
haven't seen in the literature: propositions should get equal prior
probability if they are *equally simple* and *equally
specific*, in a sense I am going to explain.

This criterion is motivated by a certain approach to the problem of induction.

Suppose we know that there are 1000 ravens in the world. We've seen
100 ravens, all black. Without further relevant information, we should then
be reasonably confident that all the ravens are black. Why so? Why
should the hypothesis that all ravens are black be preferred over the
hypothesis that the first 100 ravens are blacked and all others white?
An attractive answer is that the first hypothesis is
*simpler*.

Let's note one fairly obvious connection between induction and Indifference. Consider the class of all hypotheses about the colour of the 1000 ravens that are compatible with our observation of 100 black ravens. That class is huge. If we gave equal probability to every colour distribution compatible with our data, we would be practically certain that some ravens are non-black. (Even with just two possible colours, white and black, the probability that all ravens are black given our data would be roughly 0.000000–here come 270 zeros–8.)

The lesson is that a simple-minded application of the Principle of Indifference is incompatible with inductive reasoning. If want to spell out a plausible Principle of Indifference, we should make sure it doesn't get in the way of induction.

So perhaps it makes sense to start with models of induction.
Return to the attractive idea that induction is based on a preference
for simpler hypotheses. Roy Solomonoff found a way to render this more
precise. Let's think of scientific hypotheses as algorithms for
producing data. Any algorithm can be specified by a string of binary
symbols fed into a Universal Turing Machine. Define the
*complexity* of a hypothesis as the length of the shortest input
string to a Universal Turing Machine that computes the algorithm. Now
we can understand simplicity as the reciprocal of complexity. That is,
if we want to privilege simpler hypothesis, we can give higher prior
probability to hypotheses with lower complexity. Under some further
modelling assumptions, it turns out that there is only one natural
probability measure that achieves this: Solomonoff's *universal
prior*. (See "Rathmanner & Hutter
2011 for more details and pointers to even more details.)

The assumption that simpler theories should get greater probability is meaningless without some criteria for simplicity. Any theory and any data whatsoever can be expressed by a single letter in a suitably cooked-up language. So we either have to make the rational prior language-relative or we have to assume that there is a privileged language in terms of which simplicity is measured. The second option is more daunting, but I think it's clearly the way to go. I don't know how the privileged language should be defined. It is tempting to stipulate that all non-logical terms in the language must express what Lewis calls "perfectly natural" properties and relations, but I'm not sure.

Of course, this is a problem for everyone. If seeing lots of green emeralds makes it reasonable to believe that all emeralds are green and not that they are grue, and if we think that this is not a language-relative fact – what makes it irrational to conclude that all emeralds are grue is not a fact about our language or psychology – then there must be something objective that favours hypotheses expressed with 'green' over hypotheses expressed with 'grue'. There is no special problem here for Solomonoff.

I do have a few other reservations about Solomonoff's
approach. Some of the modelling assumptions used to derive the measure
look problematic to me. I'm also not sure that we should always favour
simpler hypotheses. For example, I think the Copenhagen interpretation
of quantum mechanics deserves practically zero credence, but not
because it is so complicated. (Although perhaps it *does* come
out complicated if we tried to translate the concept of an observation
that figures in the theory into the language of perfectly natural
properties?)

So I'm not convinced that Solomonoff's prior is the uniquely ideal
prior probability. But we don't need to go all the way with
Solomonoff. It seems plausible to me that simpler theories should
*generally* be given greater prior probability, and that this is
what vindicates inductive reasoning, however exactly the idea is
spelled out.

Now notice that if the probability of any hypothesis with complexity k is greater than the probability of any hypothesis with complexity k+1, and smaller than the probability of any hypothesis with complexity k-1, then there can't be much variability in the probability of hypotheses with complexity k. Solomonoff's universal prior, for example, assigns exactly the same probability to hypotheses with equal complexity.

That may seem odd. Intuitively, the hypothesis that *it is windy
and sunny* is equally simple as the hypothesis that *it is windy
or sunny*, but surely the two should not be given equal prior
probability.

In response, recall that the relevant "hypotheses" in Solomonoff's account are algorithms that generate present and future data. The idea is that the entire world can be modelled as a big stream of data. Only recursive streams deserve positive probability, and their probability is supposed to be determined by the length of the shortest computer program that produces the stream. (There is a way to allow for stochastic programs, as Hutter discusses, but let's ignore that.)

So Solomonoff's account entails a Principle of Indifference for
*maximally specific* hypotheses – hypotheses that determine
a unique stream of data. The Principle says that maximally specific
hypotheses that are equally simple should get equal prior
probability. Let's call this *Solomonoff's Principle of
Indifference*.

Can we generalise the Principle to less specific hypotheses? Yes,
but the generalisation isn't obvious. The problem is that some
disjunctions of maximally specific hypotheses are equivalent to
simpler sentences in the privileged language, while others are
not. But if the disjunctions have the same number of disjuncts, they
will will have equal (universal prior) probability. So it looks like
we'll need an unusual measure of simplicity or complexity on which the
complexity of an unspecific proposition is defined as the sum of the
complexity of the maximally specific hypotheses that entail the
proposition. If we then stipulate that two propositions are *equally
specific* iff they are entailed by the same number of maximally
specific hypotheses with positive probability, it follows from
Solomonoff's account that equally specific propositions that are
equally simple should have equal prior probability.

Again, we don't need to go all the way with Solomonoff. The general
point is that a promising approach to induction implies *something
like* Solomonoff's Principle of Indifference.

What does that Principle say about the murder example? It's hard to prove, but it seems plausible that the gardener, butler, and cook hypotheses are equally specific and equally simple in the relevant sense. So they should get equal prior probability.

What about the cube factory? No surprises: it's completely unclear what Solomonoff's Principle will say here. It might be instructive to work through the details to see if, for example, the answer depends on the choice of the "privileged" language.

Anyway, here's the upshot: if Solomonoff's approach to induction is on the right track, then anyone who isn't a radical subjectivist about induction should endorse a Principle of Indifference.

]]>**Example 1**. My left arm is paralysed. 'I can't lift my (left)
arm any more', I tell my doctor. In fact, though, I *can* lift
the arm, in the way I can lift a cup: by grabbing it with the other
arm. When I say that I can't lift my left arm, I mean that I can't
lift the arm *actively*, using the muscles in the arm. I said
that I can't do P, but what I meant is that I can't do Q, where Q is
logically stronger than P.

**Example 2**. I have bought a piano and just took my first
lesson. So I can play a few basic tunes. Can I play the piano? In most
contexts, the answer would be no. Normally, when we say that someone
can play the piano, we mean that they can *play reasonably well*,
which is logically stronger than *play*.

(Without context, the standards vary widely, as "this forum thread illustrates, where someone asked, 'at what point can you say that you can play the piano?'. Answers range from 'when you can play all of Chopin's etudes in one sitting' to 'anyone can play the piano'.)

**Example 3**. I'm standing in front of a safe, but I don't know
the combination. I only have a minute. 'I can't open the safe', I
say. But whatever the right combination is, I *can* dial that
combination. So I can open the safe, but only by luck. What I can't do
is *open the safe deliberately* or *at will*, which is
logically stronger than *open the safe*.

**Example 4**. You don't know the way to the train station. I
could walk you there, but due to a disability I can only walk very
slowly, so that you would miss your train. 'I can't walk you to the
station', I say, meaning that I can't walk you there *in
time*.

It is not obvious that these are examples of a single phenomenon. But they all have in common that a 'can P' statement is interpreted as 'can Q', where Q is logically stronger than P.

The effects arguably also arise for deontic 'can' and 'must'. For
example, 'you must (or must not) raise your arm' is naturally
understood as conveying an obligation to *actively* raise the
arm.

What might explain these effects? I can think of five explanations, none of them very good.

**Explanation 1**. The appearance of a strengthened prejacent
comes about through a contextual restriction on the domain over which
the modal quantifies. I don't see how this could work, if we want to
retain the idea that the domain of the modal is given by contextually
salient worlds compatible with relevant circumstances (or the most
"ideal" of these worlds relative to some salient ordering). Consider
example 1. The circumstances surely allow me to lift my left arm with
my right arm. We might try to say that worlds in which I don't
"actively" perform an act are ignored. But in worlds where I lift my
left arm with my right arm, I *am* actively performing that
act. And it is unclear what ideal the act might float. (The best
candidate is perhaps an ideal of normalcy, but that doesn't
generalise.)

**Explanation 2**. The prejacents are ambiguous. 'Lift an arm'
is ambiguous between actively lifting an arm and passively lifting an
arm; 'opening the safe' is ambiguous between opening the safe
deliberately and opening it by luck. But this doesn't seem right. If I
opened the safe by luck, the claim that I did not open the safe is
unambiguously false.

**Explanation 3**. The strengthened prejacent is an
implicature. After all, it can be cancelled: `I can open the safe, but
only by luck'; `I can play the piano, but only poorly'; `I can lift my
arm, but only with the other arm'. But I can't think of a Gricean
explanation for the supposed inference. In the three examples, a
sentence is uttered that is literally false (on the present
explanation). What kind of reasoning leads us from there to the
conclusion that the speaker wanted to convey an alternative
proposition that is true? Paradigm examples of implicatures
strengthen the content of an assertion; here it weakens the
content. Also, the supposed implicatures are equally present in
questions (`can you raise your arm?'), which makes it hard to explain
them in terms of norms of assertion.

**Explanation 4**: The prejacent is strengthened by a process of
"free enrichment". Recanati, Bach, Carston and other have argued that
when we process utterances, we often supplement the uttered sentence
by further, unarticulated constituents that don't have to be
pronounced because they can be taken for granted in the relevant
context. Perhaps this happens in the prejacent of our modals. For
example, `I can't raise my arm' is understood as `I can't raise my arm
actively' – with the adverb `actively' supplemented in the
contextual processing. This would explain the observed effects, but
the whole idea of free enrichment is controversial, and there is (to
my knowledge) no precise model that would predict when an enrichment
occurs, and what kind of enrichments can occur.

**Explanation 5**: Modals have a hidden parameter of adverbial
type that can either be left empty or supplied by conversational
context. If the parameter is supplied, it restricts the interpretation
of the prejacent. In the context of example 1, "active" actions are
relevant, so an unarticulated `actively' modifier is passed to `can',
which restricts not the accessible worlds but the interpretation of
the prejacent. This makes the right predictions, but one would like to
have some independent evidence for the postulated mechanism.

1. I haven't looked at Nodelman and Zalta carefully enough (I should!), but the view doesn't look like the classical kind of structuralism I had in mind. The structuralism I had in mind postulates only concrete individuals and abstract properties. Since a "place in a structure" can hardly be a concrete individual, it would have to be a property. Nodelman and Zalta don't seem to think of structures as properties; in any case, they also say that 'i' and '-i' are not referring terms at all, but have to be quantified away. So they agree that e.g. 'i ≠ -i' doesn't express the non-identity of two definite things.

2. I wasn't assuming any indiscernibility principle. Instead, I was assuming that complex numbers are properties, defined by the conjunction of all structural predicates which they satisfy in the complex field. It then follows immediately that i = -i. You're right that if we have an ontology in which property-like "structures" are in some mysterious sense related to individual-like platonic "places", then we could say that i and -i are primitively distinct "places" even though they share their structural properties.

(And yes, Button's hybrid solution amounts to a kind of eliminative structuralism for constructed structures.)

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Interesting read! I'm certainly no expert and not necessarily a friend of structures myself, but here are some loose thoughts:

1. One thing that perplexes me a bit is why do you presume that a stucturalist would want (or need) to identify the number 2 with some property. Isn't it quite customary for a structuralist to say instead that numbers are not properties but objects of a special kind - namely, positions within the structure? For this to be convincing, of course, you'd have to take seriously the structuralist's talk of positions and it looks like you dismiss it as being too contrived and metaphorical. What would convince you otherwise? (In particular, I wonder what's your take on an axiomatic approach of the sort provided by Nodelman and Zalta.)

2. The argument leading to $latex i = -i$ seems to hinge on an indiscernibility principle, and one strong enough to guarantee that two elements (or places/positions) of a given structure are identical whenever they are related by an automorphism of the structure in question (cf. Keranen's paper). But why think that a structuralist must commit herself to such principle in the first place? Why can't she, say, treat the identity of positions as primitive and proceed from there, thus avoiding the need for an explicit identity criterion? I suppose that some epistemological considerations might be at work here, but as I said, I'm not overly familiar with this.

P.S. Oh, this could be a red herring, but what you say about $latex i \neq -i$ being a universally quantified statement bears a prima facie resemblance to Tim Button's hybrid solution and his distinction between basic and constructed structures. ]]>

Re 2: my idea wasn't that 'might have p' is true iff there was a time in the past at which p was compossible with the then available evidence. That doesn't look promising to me. (After all, it might have been that there are no sentient beings, and so nobody had any evidence for anything.) Rather, the idea was that 'might have p' evaluates p relative to our actual information state, but shifting all worlds in the state into the past. This would be analogous to my proposal in the post about might counterfactuals. The problem is that if all worlds in the current state are (say) worlds where some coin landed tails, then all temporal predecessors of these worlds are worlds where the coin is going to land tails. So this idea probably doesn't work either. In any case, I don't have any real view about 'might have'.

Re 1: Bare 'might's do always seem epistemic to me. In your example ('if you had tossed it, it might have landed tails'), I think the 'might have' is naturally interpreted as an epistemic 'might' with a past-tense/subjunctive prejacent, in which case the switch would be unsurprising.

Anyway, thanks for your thoughts!

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