# What is necessity in modal logic?

I understand that something that is necessary must exist. However, what is the exact definition of necessity?

Thank you.

• Welcome to the forum. Very broadly, p is necessary in world w iff p is true in all worlds that w can access (where the definition of accessibility differs from modal system to system). Consequently, an object exists necessarily (as far as w is concerned) iff it exists in all worlds that w can access. (How existence across worlds is understood exactly, is more a matter of metaphysics and less a matter of modal logic.) For more details, have a look at e.g. this article: plato.stanford.edu/entries/logic-modal – MarkOxford Nov 13 '17 at 12:34
• I understand what you have said; however, what, exactly, is necessity? Perhaps it could be understand by contrasting it with contingency? And thank you for your answer. – CMK Nov 13 '17 at 14:05
• I’m not entirely sure what type of answer you are looking for, apart from the formal definition. Do you mean ‘How is the necessity operator interpreted’, i.e., are you asking about the semantics of modal logic? Apart from that, I suppose one could say that necessity and contingency are different modes of being true. Perhaps it helps to think of the temporal analog: ‘necessarily, p’ is a bit like ‘always, p’. – MarkOxford Nov 13 '17 at 14:37
• Maybe giving an example, and contrasting it with something which is possible or impossible might help CMK? – Daniel Goldman Nov 13 '17 at 15:33
• Mark, you used "necessary" in the definition of "necessity"; I believe that that is where the misunderstanding arises. I was told that if something is necessary, it must exist. If something is contingent, it's existence is dependent upon certain conditions. Is this true? Thank you both. – CMK Nov 13 '17 at 16:20

Necessity means that the truth of a proposition follows in ALL possible worlds consequent to that proposition.

This is contrasted with Possibility which means that the truth of a proposition follows in SOME world consequent to the initial frame of reference.

• Thank you. How would you describe what makes something necessary and what something else merely possible? And is possible another word for contingent? – CMK Nov 22 '17 at 0:55
• I imagine, the state of affairs dictates what is Necessary and what is Possible, but your question certainly shines a light on my ignorance--I do not know what it is about the elements/features of states of affairs that "makes something necessary". Is it possible that a thing simply just IS necessary, IS NOT necessary, or is just not the kind of thing that it makes sense to talk about as being necessary or not? – cafeTechne Nov 25 '17 at 17:13
• I understand. I always thought that a contingent entity was one that required the existence of something else to exist. Without that thing, the contingent entity would not exist. That was just based on one dictionary definition of contingency. A necessary entity could be one that does not depend on the exist of anything else to exist, and so must exist. That doesn't make much sense, though. These are just my ideas, and are not generally accepted it seems. I haven't seen any definition of these things that are generally accepted. Thanks for your insight. – CMK Nov 25 '17 at 18:05
• @CMK: Well, it sort of depends whether you want to use Kripke's definition of possible worlds, which is fairly standard (basically, "a possible world is a set of predicates and truth values"), or something more exotic like Lewis's modal realism and counterpart theory (basically, "a possible world is a parallel universe that really exists"). – Kevin Dec 12 '18 at 19:15

It seems that what you are asking for is a reductive analysis of necessity -- one that doesn't use cognate notions such as possibility or possible worlds in its analysis.

One way to go about seeking such an analysis is by accepting the definition in terms of possible worlds that others have mentioned, but then proceeding to give an analysis of what possible worlds are (which doesn't itself appeal to the notion of necessity).

The best known attempt to do this is David Lewis' in On the Plurality of Worlds. That's also probably a good source for learning about other approaches. Needless to say, it's very controversial whether any of these approaches succeed - many philosophers think they don't, but think that modal logic is useful anyway.

Saul Kripke provides a straightforward mathematical definition of necessity and possibility:

There is a set of (mathematical) objects called "possible worlds." Each world assigns[1] truth values to all statements of first-order logic whose predicates and operands we care about, is internally consistent (does not entail any contradictions), but it may disagree with other worlds. Additionally, there is a binary relation between possible worlds called the "accessibility relation." We may impose requirements on this relation, and in particular we often require it to be reflexive (but not always; see for example deontic logic).

Then the necessity (box) operator means "the enclosed statement is satisfied in every world reachable from the current world." The possibility operator is defined as the dual of the necessity operator (i.e. read diamond as "¬☐¬"), and boxes and diamonds may nest arbitrarily. So, for example, ☐☐A means "A is satisfied in every world which is two steps away from the current world via the accessibility relation." Statements are usually considered "true" if they are satisfied in every world. In Kripke's formulation, no particular world is distinguished as "the actual world," but to do so may be philosophically desirable.

The requirements we impose on the accessibility relation can be translated into axioms about the behavior of the box operator. For example, if the relation is reflexive, then the axiom ☐A → A is produced, which we usually abbreviate as axiom T. More colloquially, "anything that is necessarily true is true."

Every modal logic which adheres to the above mathematical structure must at least admit axiom K, which is ☐(P → Q) → (☐P → ☐Q) (If P necessarily implies Q, then the necessity of P implies the necessity of Q). Every modal logic must also admit axiom N, which states that all tautologies or theorems are necessary (that is, if A follows from the axioms we are using, then ☐A is also a theorem, because A must be true in every world or it would create a contradiction).

In many cases, modal logic is used for purposes other than possibility and necessity. It retains the same mathematical structure, but the box and diamond are reinterpreted, for example as "It is ethically permissible/obligatory that X" in deontic logic. So when discussing modal logic generally, we should be careful not to read ☐A as "necessarily A" unless we're sure of our logical context.

This definition is not universally accepted. In particular, David Lewis offers a wildly different account of possible worlds as "real places" that just happen to be physically inaccessible to us (like parallel universes). He then goes on to develop a theory of "counterparts" as an alternative to transworld identity, which he dislikes (broadly, this is the idea that P(x) may refer to the "same x" in multiple different worlds, which seems problematic if worlds are interpreted as literal places rather than as mathematical objects).

[1]: This is a flat lie. The worlds do not actually assign truth values. Rather, truth values are assigned by a second binary relation between worlds and modal statements. This is a subtle distinction, but in some contexts, we want to consider whether a statement is true for every choice of this second relation, rather than just a particular choice.