Re: any & every

[jorge@PHYAST.PITT.EDU]

la djer cusku di'e

>         1). No ball entered every pocket.
>         2). No ball entered any pocket.
>
>         1') -E(x){ball(x) & All(y)[pocket(y) => entered(x,y)]}
>                                 -------------------------
>         2') All(y){pocket(y) => -E(x)[ball(x) & entered(x,y)]}
>                 -------------------------------------------
>
> Because lojban grammar is based on predicate calculus it is a fairly
> easy matter to translate these into lojban, but I am not going to do it
> here as I doubt that anyone would use these forms. It is like expressing
> the number 5. as s(s(s(s(s(0))))).

You can express both simple forms in Lojban:

        1'') no bolci pu nerkla ro kevna
             No ball entered every pocket.

        2'') ro kevna pu se nerkla no bolci
             Every pocket was entered by zero balls.

The distinction every/any here allows you to reverse the order of quantifiers
in English, without having to reverse the order in which you say the
arguments. In Lojban you have no choice but to reverse the order of the
arguments (or use quantifiers in the prenex). {xe'e} doesn't help you here,
because it is not the right word to translate the "any" of (2).

This is not the problem in the case of the opaque "any". In the opaque
case, rearranging the arguments doesn't solve the problem.


> Shifting the metaphor to the one raised by Jorge, one could say:
>
> 3).  No person needs any box.
> 3'). All(y){box(y) => -E(x)[person(x) & needs(x,y)]}
>
> Now, suppose that 3 was negated by putting "It is not the case that"
> in front of it. I read this as saying , " a person needs any box."

No, the negation of (3) is "at least one person needs at least one box".
Again, the opaque case doesn't appear here. In Lojban, you'd have:

  3'') ro tanxe cu se nitcu no prenu
       Every box is needed by zero persons

Which can be re-expressed as:

       ro da poi tanxe no de poi prenu zo'u de nitcu da
       ro da poi tanxe na su'o de poi prenu zo'u de nitcu da
       na su'o da poi tanxe su'o de poi prenu zo'u de nitcu da
       lo prenu na nitcu lo tanxe

And its negation is:

       lo prenu cu nitcu lo tanxe
       At least one person needs at least one box.

But this is the transparent {lo prenu cu nitcu lo tanxe}. Opaque arguments
are not strictly arguments in the logical sense, but rather they modify the
relationship, so it is not simply a matter of order of quantification.

> Or, suppose that the -E(x) etc. were simply changed to E(x)etc in 3'
> above. Does that say: some person needs any box?  Or can "any" only be
> expressed in the negative with predicate calculus and hence lojban?

The meaning of "any" in a negative sentence is different form its meaning
in an affirmative sentence in English. The meaning of the negative sentence
is clear in predicate calculus. The meaning in affirmative sentences is not
always straightforward.

Actually, even in negative sentences you can have opaque meanings. Compare

        I don't need any box. = I need no box. (transparent)
        mi na nitcu lo tanxe
        It is not the case that there is a box that I need.

        I don't need just any box. (opaque)
        mi na nitcu xe'e lo tanxe
        It is not the case that I need any box whatsoever.


> Why don't we just use xe'e for "any" and be done with it?  Because
> "any" has the meanings of: one indiscriminatly taken; of some; of all;
> and of (one, some, or all). Negation seems to contort it further.

Nobody is asking for a word to cover all the meanings of "any". Many of
those meanings are already covered in Lojban.

Jorge