Sort of an axiom or theorem in Modal Logic.

MathematicalPhysicist October 26, 2019 at 13:03 2150 views 4 comments

To me the following sentence sounds perfectly valid:"
What is good for you is not necessarily good for others"

But how would you formalize it in Modal predicate logic?
And in which system would it be an axiom or a derived theorem?

Cheers!

Comments (4)

Pfhorrest October 26, 2019 at 17:03 #345746 0 likes

for all x, p, and q, it is not obligatory that if x(p) then x(q)

or equivalently

for all x, p, and q, it is permissible that x(p) and not x(q)

where obligation and permission are the equivalent of necessity and possibility in deontic modal logic

MathematicalPhysicist October 27, 2019 at 02:23 #345877 0 likes

Well, I am not sure if my formalization is correct, let me know.

If we denote by Gx the predicate that says: Good for x.
And let L - denote the necessary operator.
Then I would write it as:
\forall x \forall y (Gx \rightarrow (~(x=y)^ ~LGy)

I think.

MathematicalPhysicist October 27, 2019 at 02:24 #345878 0 likes

where ^ stands for the conjunction connective.

Nicholas Ferreira October 28, 2019 at 04:07 #346301 0 likes

I think a more adequate formalization would be ?x?y(?(Gyx??z(¬Gyz?z?x))), that it, for any x and y, is possible that (y is good for x and there exists a z different from x such that y is not good for z). This means that for anything that is good for you can be person for which it's not good.