# How to denote physical (not logical) impossibility in modal logic?

Say I want to note that in this world, it's physically impossible to jump over the Eiffel Tower. I can just write ¬◇x, but this seems to say that in all possible worlds x is impossible, and I want to specify THIS world.

Say I want to note that in this world, it's physically impossible to jump over the Eiffel Tower. I can just write ¬◇x, but this seems to say that in all possible worlds x is impossible, and I want to specify THIS world.

Well, in the rather popular Kripke semantics, ¬◇x means there are no accessible worlds where x is true. Now, depending on your accessibility relation, this might not prohibit x from being true in the current world (by default the "actual world").

So, if x is the predicate "One can jump over the Eiffel Tower," then just write ¬x , or to be clear that you specifically mean the actual world, w*, use the forcing operator. (aka the validation operator, or satisfaction operator).

`````` w* ╟─ ¬x
``````

¬◇x actually says more or less what you want. It doesn't explicitly specify THIS world, but at least it doesn't say that in all possible worlds x is impossible. Instead, it says something about an arbitrary but fixed world. It need not be completely arbitrary, the meta-context may have restricted it a bit, for example to a world reachable from another previously given arbitrary world, or to an arbitrary world with some additional properties (like being a transitive model of ZFC). In theory, the meta-context may even say that it is actually THIS world, but I am not aware that this is actually done in practice.

You may wonder why those additional properties of the arbitrary world are not also written down in modal logic, but only specified at the meta-context. Some of those properties (like being reachable from another previously given arbitrary world) could be expressed in modal logic, for some it is unclear whether they could even be expressed (like being a transitive model of ZFC), and some (maybe meaningless) properties (like actually being THIS world) can definitely only be stated as meta-context. But you basically always need the meta-level, even for those properties of the arbitrary world which can be expressed in modal logic. This can be hard to accept, even for experienced logicians, at least when it comes to free variables. (See http://math.andrej.com/2012/12/25/free-variables-are-not-implicitly-universally-quantified/, or perhaps rather https://chat.stackexchange.com/transcript/message/29883983#29883983 to see the reaction if the experienced logician is not talking to somebody famous like Andrej Bauer, but to a "student" having some perceived misconceptions.) You may look at the worlds in modal logic as similar to free variables in predicate logic. They somehow seem to be universally quantified, while they are actually just (more or less) arbitrary but fixed.

• It actually does not. Negation of a weak operator or quantifier makes it a strong operator or quantifier. For instance, just like ¬∃xP(x) is equivalent to ∀x ¬P(x). ¬◇x is equivalent to □¬x. Which, in essence, translates in the semantics of possible world: x is false in every possible world. Oct 24, 2018 at 0:14
• @BertrandWittgenstein'sGhost Maybe it gets clearer if we give the name w to the world under discussion. Then the translation becomes "x is false in every possible world which is reachable from world w". What I try to make clear with my answer is that even so w is basically arbitrary, it is not necessary to assume an implicit universal quantification over w. (The case with the free variables is easier, since there the name of the free variables at least occurs in the formula under discussion.) Oct 24, 2018 at 10:48
• I see what you are saying, but the problem is not semantical. That is, it is not because of how we define our worlds it is syntactical. Regardless of how you define it, negation of a weak operator will give you a strong operator which will necessarily have to be instantiated in every subcontext accessible to the main context. Oct 24, 2018 at 18:45
• Also, keep in mind in quantified modal logic there can not be a free variable. It must be bound in order for the formalism to make sense. "Modal Logic for Philosophers"/systems of Quantified Modal Logic. Oct 24, 2018 at 18:49

Let P: jump over the Eiffel Tower. Then, ¬◇P implies ¬∃\$w_k\$, k∈[1,n) such that P is true in it. So, in essence, when you negate a weak modal operator, you get a strong operator just like it is the case for Universal and Existential. For example, ¬∃xQ(x)≡∀x¬Q(x). Keeping that in mind, we can not speak of contingent impossibilities within possible-world or Kripkean semantics.

We can only speak of possibilities and necessities such that they hold in all possible worlds, or in all worlds accessible to the actual world.

That said, once you start quantifying over possible worlds then you can meaning full speak of contingent impossibilities. There are system which do that.

The Eiffel example doesn't quite fit the idea of "physical impossibility". It may be biology, not physics, that prevents the jump, and with different physical laws one would have to ensure the Eiffel Tower can still exist and not collapse due to stronger gravity etc. But let's grant for the sake of argument that jumping over the Eiffel tower is logically but not physically possible.

Let the accessibility relation between possible worlds x and y be

y is physically possible relative to x

In other words, x and y obey the same physical laws (so this will be an S5 modal logic).

Then "Nobody jumps over the Eiffel Tower" is not logically valid, because there are some worlds with different physical laws where somebody does jump over the Eiffel Tower. But it necessarily true, at our current world.