von Mises: Probability, Statistics and Truth (1951), ch. 1

Richard von Mises; from Wikipedia.

Richard von Mises was an important proponent of the frequentist philosophy of probability.

In his book Probability, Statistics and Truth, he militates against the use of the word "probability" for anything other than indefinitely repeatable experiments with converging relative frequencies (pp. 10–12). He also compares probabilities to physical constants like the velocity of a molecule (p. 21) and asserts that the law of large numbers is an empirical generalization comparable to physical laws like the conservation of energy (pp. 16, 22, and 26).

Reference Class Relativism

A consequence of this frequentist notion of probability is that specific events do not have probabilities. Only infinite classes of comparable events can have probabilities.

For instance, when your coin comes up heads at 10 o'clock, that's a different event from the coin coming up heads at 11 o'clock in infinitely many ways. Only because you choose which properties from the situation to select can you identify the two events as equivalent.

As a kind of argument for this reference class relativism, von Mises asserts that a specific person has a different probability of dying depending on the reference class (e.g., people over 40, men over 40, male smokers over 40, etc). We thus have to explicitly select the reference class before we can talk about "the" probability.

He comments:

One might suggest that a correct value of the probability of death for Mr. X may be obtained by restricting the collective to which he belongs as far as possible, by taking into consideration more and more of his individual characteristics. There is, however, no end to this process, and if we go further and further into the selection of the members of the collective, we shall be left finally with this individual alone. (p. 18)

Even from a frequentist perspective, I'm not sure this makes sense. The fact that we have narrowed down our reference class so much that there is only a single real person left in it should not change the fact that we still have an intensional definition of the class. In so far we do, we should be able to apply that definition to the outcome of any sequence of candidates, like an infinite sequence of people or experiments. In reality, it is only data sparsity that keeps use from going "further and further."

So I think von Mises has a theoretical choice to make: Either, he must require that reference classes be actually infinite, or he must merely require that they be potentially infinite.

"Randomness" and Insensitivity to Subsequence Selection

Von Mises spends a large part of the lecture elaborating a notion of "randomness" which is intended to capture the difference between asymptotically i.d.d. sequences and not asymptotically i.d.d. sequences with the same limiting frequencies. He does so by adding the requirement that the limiting frequencies are independent of subsequence selection.

A possibly more intuitive way of stating that definition would be in terms of a Topsøe-style game: A structure-finding player is tasked to pick infinitely many places in a sequence based on past data and is rewarded when the empirical frequencies fails to converge to a given distribution; a structure-hiding player is tasked to select the sequence and is rewarded when the frequencies do converge to the given distribution.

If the structure-hider then introduces any systematic dependence between the experiments, the structure-finder can exploit these regularities to outgamble the structure-hider. Thus, only asymptotically i.d.d. sequences are part of an equilibrium.

I haven't checked the details, but this game seems to be the same as that suggested by Shafer and Vovk, although (if I remember correctly), they only consider fair (that is, maximum-entropy) i.i.d. coins, not arbitrary biases. But at any rate, coin flipping is, like distributions on a finite set, one of the cases in which there is a maximum entropy distribution even in the absence of an externally given mean.

Barbara Dancygier: "Mental space embeddings, counterfactuality, and the use of unless" (2002)

According to many native English speakers, the word unless has a tendency to work less well in hypothetical contexts:

• You're not clinically depressed unless you're tired.
• ?You wouldn't be clinically depressed unless you were tired.

In Barbara Dancygier's paper on the use of unless, she tries to explain this fact by means of some rather extravagant cognitive assumptions about people's use of hypothetical mental spaces. This explanation mainly amounts to describing the different distributions of unless and except if.

Unacceptability: Counterexamples

Although unless is indeed not appropriate in many hypothetical constructions, there are couterexamples. Dancygier gives a number of quite nice corpus examples, including

• 'Unless I was naked, you'd still worry I was wearing a gun or a wire.' (p. 365)
• 'I have pulled my tail off,' replied the younger Mouse, 'but as I should still be on the sorcerer's table unless I had, I do not regret it.' (p. 368)
• [I]f Miss Catherine had the misfortune to marry him, he would not be beyond her control, unless she were extremely and foolishly indulgent (p. 369)

So apparently, some hypothetical uses are OK, whatever the reason is.

Some Acceptability Judgments

Zooming a bit out from this observation, we have, according to Dancygier's intuitions, the following pattern (p. 369; I've abbreviated the sentences a bit):

• If she married him, he would be under control unless she were extremely indulgent.
• If she married him, he would be under control except if she were extremely indulgent.

Further, we have the following distribution in the mouse example, again according to Dancygier's intuitions (p. 372):

• As I should be on the table unless I had pulled my tail off, I do not regret it.
• *As I should be on the table *except if I had pulled my tail off, I do not regret it.

I suppose it's fair to assume the following acceptability judgments as well, even though Dancygier does not explicitly say so:

• *I wouldn't have finished unless you had helped me.
• *I wouldn't have finished except if you had helped me.

If these three cases are representative, then unless is more lax than except if: Whenever except if fits in a frame, unless does, too. Whatever story we tell about these distributions, it thus better be one in which except if requires some kind of higher standard of well-formedness.

Acceptability Patterns: A Possible Explanation

Dancygier's theory has something to do with contrast.

She proposes that except if needs to introduce a condition that stands in contrast with the whole hypothetical context (I think?) all the way up to the actual situation. Unless, on the other hand, only needs to introduce a condition that stands in a contrastive relationship with the immediate intensional context.

What this means is not quite clear, and it requires some further (and quite strong) assumptions about the introduction of new layers of intensional context. Such layers are introduced quite often in Dancygier's theory; not only modal verbs like would and should trigger them, but also future tenses and conditional items like if.

Beyond Control

So, and example: Let's look at the marriage example and show how this example allegedly builds up layers of context according to Dancygier's analysis. The relevant embeddings are:

This should be read as follows: In the base layer, Miss Christine and the male character are not married. However, in the first conditional scenario (if…), they are imagined to be, and from this hypothetical situation (would…), it is judged that "he would not be beyond her control."

This imaginary situation is then equipped with an exceptional condition (unless…), namely that "she were extremely and foolishly indulgent." This condition is taken to imply to lack of control in a future scenario (not lexically expressed). (Cf. pp. 369–70.)

Pancake Upon Pancake

Another way to visualize the same thing is by keeping track of when different assumptions are introduced, inherited, and negated in the embedded structure:

Not marriage (assumed)
(No assumption about control)
(No assumption about indulgence)
Hypothetical scenario:
        Marriage (overridden)
        (No assumption about control)
        (No assumption about indulgence)
        Future scenario:
                Marriage (inherited)
                Control (assumed)
                (No assumption about indulgence)
                Exceptional scenario:
                        Marriage (inherited)
                        Control (inherited)
                        Indulgence (assumed)
                        Future scenario:
                                Indulgent (inherited)
                                No control (overridden)
                                Marriage (inherited)

The important grammatical fact is here which of its ancestor layers the exceptional condition (indulgence) is inconsistent with. The ancestor layers are here the base layer, the hypothetical scenario, and the future scenario.

Since the indulgence is neither assumed nor denied in any of these layers, the exceptional assumption stands in a relationship of contrasts with all of them. Both unless and except if are thus grammatical in this context.

Note that this conclusion crucially depends on the fact that the base layer assumed to be undecided about control. If we suppose that Miss Christine has no control over the male character in the base layer, except if should be ungrammatical

Tailless Escapes Captivity

This analysis should be compared to the example I should be on the table unless I had pulled my tail off (cf. p. 371):

Pulled tail (assumed)
Not on table (assumed)
Hypothetical scenario:
        (Assumptions about the tail deleted)
        On table (overridden)
        Exceptional scenario:
                Pulled tail (assumed)
                On table (inherited)
                Future consequence:
                        Pulled tail (inherited)
                        Not on table (overridden)

In this case, the exceptional condition (I pulled my tail off) is in contrast with the immediately preceding scenario, which has no assumptions about the tail. The exception is, however, not in contrast with the base layer. Consequently, except if is ungrammatical, and unless is grammatical.

But note again the assumptions going into this conclusion: Without the assumption of "forgetfulness" in the hypothetical scenario, the conclusion would not follow. It is thus crucial that the hypothetical If I were still on the table… "deletes" a particular one of our assumptions in a seemingly rather arbitrarily fashion.

And All the Other Cases…?

Let's look at the case I wouldn't have finished, which I assumed above was ungrammatical with both unless and except if. This sentence should presumably be analyzed as follows:

I have finished (assumed)
You helped me (assumed)
Hypothetical scenario:
        I have not finished (overridden)
        You helped me (inherited)
        Exceptional scenario:
                I have not finished (inherited)
                You did not help me (overridden)
                Future consequence:
                        I have not finished (inherited)
                        You did not help me (inherited?)

In this case, the conditional exception (you did not help me) contrasts with both of the two layers above. So why not a grammatical use of unless and except if?

I think questions like these point to the fact that the cognitive assumptions behind Dancygier's theory are pretty vague and very, very speculative. It is by no means clear when assumptions are "forgotten," or exactly which or how many assumption that have to contrast with the intensional context.

In fact, it is not even quite clear to me whether except if requires contrast to all ancestor levels, and whether it requires contrast on a particular, high-focus parameter, or just any single parameter. Without an answer to these questions, it will be very difficult to assess the theory to any interesting degree.