A sole sufficient quantification function: for()
Following up on my previous discussions about a sole sufficient modal function and the meaning of the existential quantifier, I propose a sole sufficient quantification function:
The for() function that takes three arguments, the first being a set of values that some variable can take to satisfy some formula, the second being that variable, and the third being that formula. (This would then be read as "for [these values of] [this variable], [this statement involving that variable] (is true)").
This replicates some of functionality of another function frequently used together with the traditional quantification operators, ?, which properly indicates that whatever is on the left of it is a member of the set on the right of it, but together with the existential operators is often used to write things like
And once again, like with the single operator I proposed for all modal logic earlier, if the first argument is the empty set, we are left with a special case of this function meaning "for no...", which we can then easily turn into "for some..." by negation, and then turn those two into "for all..." and "for not all..." by applying them to the negations of the formula in the third argument. E.g.
negating "for no m, if m is a man then m is mortal",
meaning "no men are mortal",
gives us "for some m, if m is a man then m is mortal",
meaning "some men are mortal";
while "for no m is it not the case that if m is a man then m is mortal"
means "for all m, if m is a man then m is mortal",
or in other words "all men are mortal";
and "for some m, it is not the case that if m is a man then m is mortal",
in other words "some men are not mortal",
of course means the same thing as "for not all m, if m is a man then m is mortal",
or in other words, "not all men are mortal".
for()The for() function that takes three arguments, the first being a set of values that some variable can take to satisfy some formula, the second being that variable, and the third being that formula. (This would then be read as "for [these values of] [this variable], [this statement involving that variable] (is true)").
This replicates some of functionality of another function frequently used together with the traditional quantification operators, ?, which properly indicates that whatever is on the left of it is a member of the set on the right of it, but together with the existential operators is often used to write things like
?x?S...?x?S...And once again, like with the single operator I proposed for all modal logic earlier, if the first argument is the empty set, we are left with a special case of this function meaning "for no...", which we can then easily turn into "for some..." by negation, and then turn those two into "for all..." and "for not all..." by applying them to the negations of the formula in the third argument. E.g.
negating "for no m, if m is a man then m is mortal",
meaning "no men are mortal",
gives us "for some m, if m is a man then m is mortal",
meaning "some men are mortal";
while "for no m is it not the case that if m is a man then m is mortal"
means "for all m, if m is a man then m is mortal",
or in other words "all men are mortal";
and "for some m, it is not the case that if m is a man then m is mortal",
in other words "some men are not mortal",
of course means the same thing as "for not all m, if m is a man then m is mortal",
or in other words, "not all men are mortal".
Comments (3)
Same for modality.
The sole sufficient operator for modality was really a lot more interesting (see the other thread). I admit that this topic is boring in comparison.