# Negation of a E quantifier in a Modal Logic T System?

In System T its possible to conclude:

(1) □Ex->Ex

Therefore its possible to conclude:

(2) Ex->◊Ex

But whats the result, if I negate the existence quantifier? Is it:

(3) ~Ex->~◊Ex

or is it:

(4) ~Ex->◊~Ex

I found nothing in the usual introduction textbooks, but Im not a (modal) logic expert. Im thankful for all your answers

• Not clear; you are trying to derive the negated formula from what ? Commented Jan 7, 2018 at 13:43
• You have to negate formulas, and not quantifiers. Commented Jan 7, 2018 at 13:44
• My question is: If nothing existists (~Ex), is it inpossible (~◊) that something existists (Ex)? That is what I was going for with (3). And can i show (3) in the T system? Commented Jan 7, 2018 at 13:49
• Obviously, (4) follows from (2). Commented Jan 7, 2018 at 13:56
• From (1), by contrapostion, we get: ~∃x → ~□∃x i.e. ~∃x → ◊~∃x. Commented Jan 7, 2018 at 13:58

## 1 Answer

As @MauroALLEGRANZA mentions, you do get

(4) ~Ex->◊~Ex

I'd say it just follows from System T in general:

System T makes the accessibility relation reflexive, so that if there is no x with a certain property P in a world w, then there is a possible world relative to w, namely w itself, where there is no x with property P.