jeudi, février 16, 2006

Question of the Day


Necessitation, Collapse, Banalisation and other Issues

One of the curious properties of the Modal Semantics of Human Languages is that they do have a T principle and a Necessitation rule, and even Aristotle’s law, but lack other principles, like banalisation and collapse.
The T principle may be stated as follows(1):
T: A⊃A

In Logic the Necessitation Rule requires that if A is a thesis of a certain modal system S, then A is also a thesis of the S. In the study of Natural Languages one may, for instance, think of a variation of Necessitation by simply saying that if a string or sentence that expresses a proposition is a sentence of a language, then the sentence that expresses that the same proposition is necessary must also be a sentence of the same language. But that would be a very basic and elementary principle of natural language.
However, neither Necessitation nor T must be confused with the Banality and Collapse, which are quite different laws. Banality is a principle that claims that if a proposition is true than it is necessarily true:
(Ban) A⊃A

Collapse is a stronger law that establishes equivalence between necessity and truth:
(Coll) A≡A

Banality is a fallacy in Natural Languages. There are abundant examples that show this, such as the nonsense sentences below:
(1) #If Harper is the Prime Minister then he is necessarily the Prime Minister.
(2) #If it is raining outside, then it must be raining.
(3) #If it is true that your car is broken, then it is necessary that your car be broken.
(4) #If Governor General Michaela Jean came from Haiti, then she needs to have come from Haiti.

Of course, the same can be said about Collapse.
These facts indicate that the Modal Semantics of Natural Languages includes systems that, in the Logician's jargon, are not degenerated. Thus, the interesting and relevant question turns to be why is that so? There are several possible answers. One tentative manner to tackle this issue would be assuming that, inasmuch as the semantics of Natural Languages has a displacement property (the discourse is not limited to the here and now), they are modal languages, while banal systems are quasi-limits of modal systems. This kind of explication probably requires further elaboration.
The issue gets more complicated when we consider that instances of the P law below are odd statements:
P: A⊃◊A

Although it is very intuitive to assert that if A is true then A is possible, sentences with equivalent meaning have bizarre effects:
(5) #If Harper is the Prime Minister then he may be the Prime Minister.
(6) #If it is raining outside, then it might be raining.
(7) #If it is true that your car is broken, then it is possible that your car be broken.
(8) #If Governor General Michaela Jean came from Haiti, then she possibly has come from Haiti.
(9) #If today is Valentine's Day, then maybe it is Valentine's Day.

What is the problem with the sentences above? Does it have to do with truth-conditions? Or is it simply the case that these sentences are infelicitous?

(1) There are apparent exceptions though, in the cases where the lexical item that should correspond to the operator  have a deontic reading or actually corresponds to ◊. The (somewhat redundant) sentences below are instances where T works without doubt:
i. If it is necessarily true that oranges do not grow on apple trees, then it is true that they do not.
ii. If natural catastrophes necessarily happen sooner or later, then they (do/ will) happen sooner or later.
iii. Young males of necessity seek girls’ company and vice versa, which means that they seek each other.