I have the following sentence which I want to formalize in predicate modal logic. The sentence is:

What is good for you is not necessarily good for others"

My attempt at formalizing this is as follows with Gx be the predicate - Good for x:

∀x ∀y (Gx → (~(x=y) ∧ ~□Gy))


I see two problems with the current formulation. First, 'good for' (in the sense that you mean) is a 2-place predicate, as in 'tea is good for Alice' and 'coffee is good for Bob'. Second, your formulation entails that if something is good for x, then ~(x=y). But of course x=y in some cases, since those variables range over the same individuals.

So I would formalize "What is good for you is not necessarily good for others" as follows:

∀x∀y∀z((Gxy & ~(y=z)) → ~□Gxz)

The following should be equivalent (though I'm not 100% sure). It formalizes "it's not necessary that: everything that is good for you is good for others".

~□∀x∀y∀z((Gxy & ~(y=z)) → Gxz)

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  • I think your last comment in your answer depends on the particular PML you are using. – MathematicalPhysicist Oct 29 '19 at 5:42

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