any & every
Gerald Koenig
into the universal quantifier when expressed in predicate calculus.
Any subtleties they may have can only be expressed in predicate
calculus by sequencing the universal quantifier, or altering its scope.
I am sharing some examples from a logic text below which deal with the
question of "any" vs. "every". I have changed the metaphor to a pool
game as first used by PC, but the logical form of examples (1-2') is
from a logic text.
1). No ball entered every pocket.
2). No ball entered any pocket.
It is pretty clear from these that "every" \= "any"
Here is the textbook translation of these into predicate calculus:
1') -E(x){ball(x) & All(y)[pocket(y) => entered(x,y)]}
-------------------------
2') All(y){pocket(y) => -E(x)[ball(x) & entered(x,y)]}
-------------------------------------------
The lines indicate the scope of the universal quantifier. It is longer
for the "any" example. Apparently it has something to do with the fact
that these statements are negated, but I can't say that I understand
this.
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))))).
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."
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?
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.
Context determines which is meant and hence the word is not parseable.
In short this is one of those places where we have an opportunity to
vastly improve English, if we can just sort it all out.
As I posted previously, there are at least three "anys", I now believe
there are the 4 mentioned above. I call them alpha, sigma, zeta, and
rho. They are all quantifiers. Does anyone want to go for 5?
mi nitcu rho danfu
djer
jo...@phyast.pitt.edu
> 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
Gerald Koenig
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).
-----------------------------------------------------
GK> Its not so easy as you think. Consider this scenario: The white
ball is marked with an X. During the course of a game it happens to get
hit into pockets one through six. So we can say as above in 1.), except
for the negation:
E(x)( ball(x) & All(y)( pocket(y) => entered(x,y)))
Notice that the other balls went into the pockets. When this sentence
is negated as in 1), it just denies the existence of the white ball
with the X on it, or any that behaved similarly. It doesn't say all the
pockets are empty.
Your sentence, 1'' says that "0 balls entered every pocket." You can't
translate the lojban word "no", which means the number 0, into the
English word "no" which is a logical connective, and make sense.
If you are inclined to argue about the meaning of the above sentence, I
suggest you look at page 215 of the book Logic and Prolog, Cambridge
University Press, which is where I learned about it.
I havn't digested your other comments to my any & every post, I
hope others will comment.
djer
jo...@phyast.pitt.edu
> > 1). No ball entered every pocket.
> > 2). No ball entered any pocket.
with result:
> 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.
This means that the effect of reversing the universal and negated existencial
quantifiers, that is achieved in English by changing from every to any, can
only be achieved in Lojban by actually reversing the arguments.
(This "any" is not {xe'e}.)
djer disagrees:
> GK> Its not so easy as you think. Consider this scenario: The white
> ball is marked with an X. During the course of a game it happens to get
> hit into pockets one through six.
Then you can say: su'opa bolci pu nerkla ro kevna
At least one ball (to wit, the white one)
entered every pocket
> So we can say as above in 1.), except
> for the negation:
>
> E(x)( ball(x) & All(y)( pocket(y) => entered(x,y)))
Yes.
> Notice that the other balls went into the pockets. When this sentence
> is negated as in 1), it just denies the existence of the white ball
> with the X on it, or any that behaved similarly. It doesn't say all the
> pockets are empty.
Of course not. 1'') doesn't say that either.
2) = 2'') both say that all pockets are empty.
> Your sentence, 1'' says that "0 balls entered every pocket."
Maybe the English translation is confusing, but 1'') does not say that
every pocket is empty. It simply says that the number of balls that
entered all of the six pockets is zero. If each pocket was entered by
one different ball, it is still true that {no bolci pu nerkla ro kevna}.
If you say {pa bolci pu nerkla ro kevna} you claim that one ball entered
each and every pocket, the same ball. If each pocket has one ball, but
not the same one, then it is false that {pa bolci pu nerkla ro kevna},
but true that {ro kevna pu se nerkla pa bolci}.
> You can't
> translate the lojban word "no", which means the number 0, into the
> English word "no" which is a logical connective, and make sense.
Yes you can, in most cases. {noda} can always be replaced by {naku su'oda}.
If we do this in this case:
no bolci pu nerkla ro kevna
naku su'o bolci pu nerkla ro kevna
It is false that at least one ball entered every single pocket.
No ball entered every pocket.
> If you are inclined to argue about the meaning of the above sentence, I
> suggest you look at page 215 of the book Logic and Prolog, Cambridge
> University Press, which is where I learned about it.
I don't understand your comment. Sentences 1) and 2) mean two different
things. I don't see that we have any disagreement on what the English
sentences mean, so what can I check in that book? What we seem to disagree
on is the meaning of the Lojban sentence 1'').
Jorge