Tuesday, February 17, 2009

On the negation of modal verbs

I recently learned (it goes to show how diligently I've been practicing my German) that Germans use the phrase "must not" in the opposite sense of English speakers to mean "need not." Thinking about this, I realized that there is an implicit "or" in any statement involving modal verbs and the sense of the negative depends upon which part of the logical proposition is negated.

For example, I would translate the phrase:

"You must do A"

into a logical proposition as follows:

^A -> P

where P is some form of punishment. Or:

A or P

We could negate the phrase either by negating the whole thing:

^A and ^P

Or we could negate only one part:

A or ^P = P -> A
(e.g., "You (must not) play in the street.")

^A or P = A -> P
(e.g. "You must (not play in the street.)")

The first example would seem to be how the Germans use the phrase since whether we do A or not, we will not get punished for it, while the third form is more in line with how we use the phrase, that is, A implies punishment. The second example seems rather more ambiguous and in fact inverts the construct: now, getting punished implies that we have done A.

It has rather deep implications, since all of ethics, law and morality is related to the use of modal verbs. Can we use this idea to justify breaking the Ten Commandments?

I translate,

"I shall go to the store,"


F (uture) -> S (going to the store) = ^F or S

The sixth commandment becomes:

You shall (not kill.) = F -> ^ K = ^F or ^K = ^(F and K)

You (shall not) kill. = ^F -> K = F or K

or perhaps,

^(F -> K) = F and ^K

The second says that if there's a future, there may or may not be killing while the first and third say what we want them to say: if F is true, K must be false.