Modal logic extends classical logic by adding one or more modes. If there’s only one mode, it’s usually denoted □. Curiously, □ can have a wide variety of interpretations, and different interpretations motivate different axioms for how □ behaves. Modal logic is not one system but an infinite number of systems, depending on your choice of axioms, though a small number of axiom systems come up in application far more than others.

For a proposition *p*, □*p* is often interpreted as “**necessarily** *p*” but it could also be read, for example, as “it is provable that *p*“. Thanks to Gödel, we know some theorems are true but not provable, so *p* might be true while □*p* is false.

**Kripke semantics** interprets □*p* to be true at a “**world**” *w* if *p* is true in all worlds accessible from *w*. The rules of a logic system transfer to and from the set of models for that system, where a model is a directed graph of worlds, and an oracle (a “valuation function”) that tells you what’s true on each world. Axioms for a logic system correspond to requirements regarding the connectivity of all graph models for the system.

All this talk of what worlds are accessible from other worlds sounds a lot like science fiction. For example, if the planet Vulcan is accessible from Earth, and *p* is the statement “The blood of sentient beings is red,” then *p* is *true* on Earth, but not *necessarily true* on Earth since it’s not true on Vulcan, a world accessible from Earth.

Modal logic defines ◇by

◇*p* = ¬ □ ¬ *p.*

For a proposition *p*, ◇*p* can be read as “**possibly** *p*.” A proposition ◇*p* is true on a world *w* if there is some world accessible from *w* where *p*is true.

So in the Star Trek universe, if *p* is the statement “Blood is green” then *p* is false on Earth, and so is □*p*, but ◇*p* is true because there is a world accessible from earth, namely Vulcan, where *p* is true.

You could have all kinds of fun making up rules about which worlds are accessible from each other. If someone from planet *x* can reach planet *y*, and someone from planet *y* can reach planet *z*, can someone from *x* reach *z*? Sounds reasonable, and if all worlds have this property then your Kripke frame is said to be **transitive**. But you could create a fictional universe in which, for whatever reason, this doesn’t hold.

Is a world accessible from itself? Depends on how you define accessibility. You might decide that a non-space faring world is not accessible from itself. But if every world is accessible from itself, your Kripke model is **reflexive**. If a Kripke frame is reflexive and transitive, the corresponding logic satisfies the *S*4 axioms. (More on this in the next post.)

Johan van Benthem gives an example in his book Modal Logic for Open Minds that’s scientific but not fictional. If you define a “world” to be a point in **Minkowski space-time**, then the worlds accessible from a given world are in that world’s future light cone. Propositions in this logic satisfy

◇□ *p* →□◇ *p*

and in fact the logic satisfies a system of axioms known as *S*4.2.