Double Modal Operators and Modal Equivalences
There are many strange things about modal logic. One of them is how we can define one modal operator by using another. This situation is not unlike saying (in propositional logic):
p = df. ¬¬ p ⊃ p
Or we can have:
i) (p ∧ q) = df. ¬(p ∨ q)
ii) (p ∧ q) = df. ¬(p ∧ ¬q)
That is, the logical constants reappear in our definitions of them.
The other interesting aspect of modal logic is the joint use, before a predicate or variable, of two different - or two of the same - modal operators. Such as:
p
or
◊p
The former is the more counterintuitive of the two above because, prima facie, it seems to ‘multiply entities beyond necessity’, as it were. Take another case of double modal operators:
Axiom S5 ◊p ⊃ ◊p
To translate: If p is possible, then it is necessarily possible.
What does that actually mean? How is a possibility necessary? If Tony Blair could possibly be a serial killer, then that possibility is necessarily possible. The possibility of Blair’s being a serial killer is necessary. What is necessary, then, is that it is possible, not impossible, for Blair to be a serial killer. Clearly such a thing is indeed possible! How could it be impossible? So now we can say that it is impossible that Blair could not possibly be a serial killer:
1) ¬¬◊p [= is false]
2) ¬¬◊p [= is true]
3) ¬¬◊p ⊃◊p [= is true]
4) p [= is false]
5) ¬((p ⊃◊p) ⊃(◊p ⊃◊p))[= is true]
To translate the final formula: If it is not the case that if it is necessarily the case that Blair is a serial killer then it is possibly the case that he is, then it is the case the if it is possible that Blair is a serial killer, then it is necessarily the case that it is possible that he is a serial killer. Blair’s being a killer is not necessary! If that’s the case, i.e., that necessity has been ruled out, then the possibility of Blair’s being a killer is necessarily possible because it is not necessary that he is a killer. If we are of the essentialist persuasion, we can now say that if p were necessary, then being a killer would be an essential property of Blair. It is not. Thus that property is contingent in that at least one possible world Blair is a serial killer, but he is not a killer at every possible world. So can we have
◊p
instead of
◊p [?]
Of course it is possible that Blair is essentially a serial killer or he has this property at all possible worlds at which he exists. Why couldn’t Blair’s being a killer be necessary?
Axiom S4: p⊃ p
Trans: If p is necessary, then p’s being necessary is itself necessary. The necessary status of p is itself necessary.
Of course, if p is necessary, then p’s being necessary will also be necessary - precisely because necessity is so strong. That is why p is necessary. The status of being necessary must also be necessary. It is the nature of a necessity that it must be the case at all possible world at which p applies. In other words, contingent necessity is self-contradictory. Thus:
p ⊃◊p [= is false]
How could p’s necessity be only possible? If p’s necessity were only possible, then it wouldn’t be necessary in the first place. If 2 + 2 = 4 is a necessary truth, and then we say that 2 + 2 = 4’s necessary truth is only possible, then it wouldn’t be a necessary truth at all. It would be contingent. It is not the case that it is only possible that 2 + 2 = 4 is a necessary truth; it must be necessarily necessary the case that 2 + 2 = 4 is a necessary truth. Thus
p ⊃p [= is true]
p ⊃◊p [= is false]
This is the inverse of:
◊p⊃◊p [= is true]
With a possible truth it could be the case that it is a necessary truth. That is, because p’s possibility is precisely that, a possibility, then this epistemic or modal restriction of knowledge to mere ‘possible’ means that ◊p could be ◊p. If we have a complex and difficult mathematical truth that has the status of only being possibly true, then that means that it could be true, but we have not yet established its truth, as with Golbach’s theorem. But if p’s truth is only at present possible, then it not only possibly true, but it is necessarily possibly true. If we only have the limited possibility of p’s truth, that is, if we don’t have a proof of p’s truth, then that very possibility of p’s truth must also entail its necessarily possible truth. That is, if we have not yet established p’s truth, and only know its actual truth in terms of proof, but only have its truth as a possibility, then this very limitation to our knowing p’s truth, in terms of its being possible that p were true, then this limitation would also make it the case that if it is also possibly necessarily true, then that limitation can’t rule out the possibility that it is possible that p’s truth is also necessary. Thus:
i) ◊p⊃◊p [= is true]
ii) ◊p⊃◊p [= is true]
Whereas p’s possible truth in i) above entails that its possible truth does not also rule out its necessary truth, then it is possibly a necessary truth as well. With ii) above, it is p’s status as a possible truth that it is itself possibly necessarily true. Instead of p’s possible truth entailing also its possible necessary truth, ii) applies the necessity operator ‘’ to p’s being possibly true, makes it necessary that it is also possible that p is also a necessary truth. So necessity is predicated of p’s possibly necessary status. With ii), on the other hand, the necessity operator is applied to p’s status as a possibility. That is, p’s being possibly a truth is itself a necessity. Rather than it being possible that p’s possible truth may also be a necessary truth, ii) states that p’s status as a possible truth is itself necessary. It is not saying that ◊ could be ◊p. That is, p could be necessary. With ii), it is predicated with the property of necessity not of the possibility of p’s possible necessity, but p’s possible truth or status is itself necessary. It is not saying that p could be true of necessity. It is saying p’s possible truth is itself a necessity; not that this possible p could be necessarily p. But there is a problem. Our
◊p ⊃◊p
is not one of the axioms so far discussed. Therefore it may be false. However,
◊p ⊃◊p
is Axiom S5 of the set of four. Our version either doesn’t exist, or is simply not a member of this set of axioms.
In addition, in none of the axioms discussed do we find the possibility operator directly before the necessity operator. Thus:
◊p
We do find it the other way around:
◊p
But that on its own makes little sense. p’s possible status of its truth also being necessary requires the whole axiom:
◊p ⊃◊p
That is, ◊p only makes sense if that very necessity of p’s being possible is itself an entailment from the antecedent which simply says that p is possible - ◊p. If p is possibly true, then p’s very possibility of being true is itself necessarily the case.
