## Tuesday, May 15, 2012

### Hintikka: "What is elementary logic?" (1995)

The claim of paper is that independence-friendly logic is more "natural" that ordinary first-order logic. That is, the restriction to quantifiers with nested scopes is unnecessary and unfounded.
In this article, as in everything he has written, there are some serious linguistic issues with the examples he uses, and it is by no means clear that his own semantic intuitions are generalizable.

The paper is reprinted in an 1998 anthology of Hintikka's work, but Hintikka referred to the paper as "forthcoming" in 1991, and it was published for the first time in 1995.

### The Old Example: Villagers and Townsmen

His old natural-language argument for the usefulness of independence-friendly logic comes from his introspective intuitions about the following sentence:
• Some relative of each villager hates some relative of each townsman
This sentence has two readings, a classical and an independence-friendly. These readings can be distinguished by the following model: Suppose that there is one villager and one townsman, and that they are related to themselves and to each other; suppose further that they hate each other, but do not hate themselves.

The Verifier then instantiates some relative of each (= the only) villager by picking either the villager or the townsman, since everyone is related. The same goes for some relative of each (= the only) villager. The sentence is true exactly when the two choices are different, and not true when they are the same (since no one, per assumption, hate themselves).

When the two choices are independent, the Verifier has no winning strategy, and the sentence is thus not true. The Falsifier, on the other hand, also doesn't have a winning strategy, since some combinations of Verifier choices do in fact make the sentence true, and others don't. In the independent-friendly reading, the sentence is thus neither true nor false. In the classical reading, it's true.

Now Hintikka's claim is that the independence-friendly reading of this English sentence is a plausible reading (or the most plausible?). He does not give any empirical arguments for the claim.

Note that the same logical structure can be replicated with a slightly less far-fetched example:
• A north-going driver and a south-going driver can choose to drive in a side of the road so that they avoid a collision.
If you think that this sentence is true, you have read it in the classical way. If you think it is false, you have read it in the independence-friendly way.

### The New Example: The Boy that Loved the Girl that Loved Him

In support of his claim, he provides the following "perfectly understandable English sentence" as evidence (p. 10):
• The boy who was fooling her kissed the girl who loved him.
He then claims that this sentence cannot be expressed in first-order logic "no matter how you analyze the definite descriptions" (p. 10).

So how can we analyze the definite descriptions? I guess we have at least the following options:
• The boy1 who was fooling her2 kissed the girl2 who loved him1.
• The boy1 who was fooling her2 kissed the girl3 who loved him1.
• The boy1 who was fooling her2 kissed the girl2 who loved him4.
• The boy1 who was fooling her2 kissed the girl3 who loved him4.
I suppose that the problematic case that he is thinking about is the first one. That's the one where the sentence implicitly states that the identity of the girl uniquely identifies a beloved boy, while the identity of the boy also uniquely identifies a fooled girl.

This is obviously a circular dependence, but can still meaningfully apply (or not apply) to various cases. For instance, if x fools y and y loves x, then it applies. If x fools y, and y loves z, or loves both x and y, then it doesn't.

But unlike the villagers-sentence, I can't see how this is not expressible in terms of first-order logic, given the usual legitimate moves in sentence formalization. But perhaps Hintikka has some strange and far-fetched "natrual" reading of the sentence in mind?