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I have been learning about the modal operators of modal logic and it's truth evaluations in different contexts. For instance:

$$\Box A \to A$$ Would be false in Deontic logic, but true in Alethic logic. However, I am struggling to understand the following concept:

$$\Diamond T$$

By the definition of the diamond this means that it is possible that true holds, but what is the meaning of this? Does this mean that there are worlds where true might be false? And what would the meaning/truth assignment be in different contexts?

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See Modal Logic : Possible Worlds Semantics:

The truth value of the atomic sentence $p$ at world $w$ given by the valuation $v$ may be written $v(p, w)$.

Thus, if:

$v(\Box A, w)= \text T$ iff for every world $w′ \in W : wRw', v(A, w′)= \text T$ [i.e. $\mathfrak M,w' \vDash A$],

that formalize the intuition that "a necessary truth holds in every (possible) world", and we assume that : $\Diamond A = \lnot \Box \lnot A$, then :

$v(\Diamond A, w)= \text T$ iff for some world $w′ \in W : wRw', v(\lnot A, w′) = \text F$.

But to say that $v(\lnot A, w′) = \text F$ is the same as : $v(A, w′) = \text T$.

Conclusion : "it is possible that $A$" means : "there are worlds where $A$ holds".


Regarding $\top$, we usually define it as $\lnot \bot \equiv \bot \to \bot$, where $\mathfrak M, w \nvDash \bot$ and thus : $\mathfrak M, w \vDash \top$.

And thus, using an universal accessibility relation $R$ (that reflects the intuition about "logical" necessity), we have that : $\mathfrak M, w \vDash \Diamond \top$.

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  • $\begingroup$ In particular, since $\top$ (by definition) holds in every world, $\Diamond\top$ also does - assuming every world sees some world! This is usually true - most modal logics are at least reflexive, so every world sees itself - but in general can fail. So in fact "$\Diamond\top$" is true at a given world in a Kripke structure, iff that world sees at least one world (possibly itself). $\endgroup$ – Noah Schweber Nov 20 '16 at 18:48
  • $\begingroup$ Thanks! So ◊T is false in a model with only 1 world that has no relation with itself? $\endgroup$ – SilverTear Nov 20 '16 at 19:05
  • $\begingroup$ @SilverTear Yes, and that's not the only example: $\Diamond\top$ is false in any model such that there is some world that isn't related to any world. (I'm using "false in a model" to mean "not true in every world in that model"; I believe this is standard, but I want to clarify just in case.) $\endgroup$ – Noah Schweber Nov 20 '16 at 19:31

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