## Wednesday, November 7, 2012

### Vendler: "Each and Every, Any and All" (1962)

This paper was first published in Mind in 1962, by just like everybody else, I read the version reprinted in the Linguistics in Philosophy (1967). It discusses the meaning of the words in the title and is famous for having described the meaning of any in terms of a certain "freedom of choice" (p. 80).

### Each, Every, and All

Vendler describes the differences between each, every, and all in terms of collective reference vs. individual reference. His theory is that all is collective, while each and every are distributive.

We thus have differences like
• You can buy each of these items for $5 (distributive) • You can buy all of these items for$5 (collective)
Every, on the other hand, can be seen as a quantification over all the distributive attributions so that "every is between each and all" in meaning (p. 77). We thus get — according to my intuitions — slightly more ambiguous examples with every:
• You can buy every one of these items for $5 According to my intuition, this could lean towards both a collective ($5 in total) and a distributive reading (\$5 per item).

### The Blank Check

Vendler describes his ideas about any nicely in this quote:
To say
Any doctor will tell you …
is to issue a blank warranty for conditional predictions: you fill in the names. You choose Dr. Jones; well, then he will tell you if you ask him. You pick twenty-five others; then, I say, they will tell you if you consult them. (p. 85)
This means that
… the any-proposition is an unrestricted warranty for conditional statements or forecasts and, we may add, for contrary-to-fact conditionals. In other words, to draw an obvious conclusion, it is an open hypothetical, a lawlike assertion. (p. 89)
I like the phrase "open hypothetical." It both highlights why any can be used in couterfactuals and other modals, and why it does not have existential import.

Vendler also notes that every single time any is used, it issues this blank warranty anew:
… I can certainly not say
*He took any one
even if you acted on my words: Take any one. […] Any calls for a choice, but after it has been made any loses its point. (p. 81)
In other words, once all the facts are settled, you cannot use any to make a report, since "facts are not free" (p. 84).

### Any and The Pragmatics of Preferences

One more quote from his explanation:
With Take any one, it is up to you to do the determining; here it does not make sense to ask back, Which one? Thus while in the former case [Take one] I merely fail to determine, in the latter case [Take any one] I call upon you to determine, in other words, I grant you unrestricted liberty of individual choice. (p. 79–80)
He notes that this also explains why a command like You must take any seems odd. Interestingly, though, the British National Corpus does contain examples like the following:
• You must report any losses immediately.
It is probably fair to paraphrase this sentence as
• If you have any losses, you must report them immediately.
So it appears that you can in fact order people to take any apple, but only if they are placed in an environment in which they are exposed to apples, and they be tempted to not take all of them (so to speak).

### A Probabilistic Interpretation of Any

The last thing Vendler does in the article is to informally sketch a way that the difference between any, every, and all could be implemented in a compositional probabilistic semantics:
A bag contains a hundred marbles. We inspect ten at random and all ten are red. Then the probability that any one marble we care to pick out of the hundred will be red is quite high. Yet the probability of every one's being red is much lower. (p. 94)
I interpret this the following way: When you evaluate the formula
• All the marbles are red.
you are really asking for the posterior probability that the relevant parameter is 1. When you ask
• Some of the marbles are red.
you are asking for the posterior probability that the parameter is larger than 0. However, when you evaluate the formula
• Any marble we draw will be red.
you are looking for the posterior probability that one randomly drawn marble will be red, given your evidence. This amounts to summing up the probabilities of the statements
• The bag contains 100 red marbles, and if I draw one at random, it will be red.
• The bag contains 99 red marbles, and if I draw one at random, it will be red.
• The bag contains 98 red marbles, and if I draw one at random, it will be red.
To take an example with slightly lower numbers, suppose I have drawn a marble twice from a bag of 10 marbles, and that in both cases, I drew a red marble. Then the posterior probability of the different parameter settings are shown in the graph below:

With these numbers, we get the probabilities
• P("All marbles are red") = P(p = 1 | k = 2, n = 2) = 26%
• P("Some marble is red") = 1 – P(p = 0 | k = 2, n = 2) = 1 – 0% = 100%
• P("Any marble is red") = Σi P(p = pi | k = 2) * P(k = 1 | p = pi, n = 1) = 79%
The sum in the last line then ranges over all the parameters values p = 0, 0.1, 0.2, 0.3, …, 0.9, 1. As stated by Vendler, the probability of the any-sentence is substantially higher than the probability of the all-sentence.