Editing Modal logic (section)

==== Basic notions<!--'Necessary proposition' and 'Necessary propositions' redirect here--> ====

The standard semantics for modal logic is called the ''relational semantics''. In this approach, the truth of a formula is determined relative to a point which is often called a ''[[possible world]]''. For a formula that contains a modal operator, its truth value can depend on what is true at other [[accessibility relation|accessible]] worlds. Thus, the relational semantics interprets formulas of modal logic using [[model (logic)|models]] defined as follows.<ref>Fitting and Mendelsohn. ''[https://books.google.com/books?id=5IxqCQAAQBAJ First-Order Modal Logic]''. Kluwer Academic Publishers, 1998. Section 1.6</ref>

* A ''relational model'' is a tuple <math> \mathfrak{M} = \langle W, R, V \rangle </math> where:
# <math> W </math> is a set of possible worlds
# <math> R </math> is a binary relation on <math> W</math>
# <math>V </math> is a valuation function which assigns a truth value to each pair of an atomic formula and a world, (i.e. <math> V: W \times F \to \{ 0,1 \}</math> where <math>F</math> is the set of atomic formulae)

The set <math> W </math> is often called the ''universe''. The binary relation <math>R</math> is called an [[accessibility relation]], and it controls which worlds can "see" each other for the sake of determining what is true. For example, <math>w R u</math> means that the world <math>u</math> is accessible from world <math>w</math>. That is to say, the [[State of affairs (philosophy)|state of affairs]] known as <math>u</math> is a live possibility for <math>w</math>. Finally, the function <math>V</math> is known as a [[valuation function]]. It determines which [[atomic formula]]s are true at which worlds.

Then we recursively define the truth of a formula at a world <math>w</math> in a model <math>\mathfrak{M}</math>:

* <math>\mathfrak{M}, w \models P</math> iff <math>V(w, P)=1</math>
* <math>\mathfrak{M}, w \models \neg P</math> iff <math>w \not \models P</math>
* <math>\mathfrak{M}, w \models (P \wedge Q) </math> iff <math>w \models P</math> and <math>w \models Q</math>
* <math>\mathfrak{M}, w \models \Box P</math> iff for every element <math>u</math> of <math>W</math>, if <math> w R u</math> then <math>u \models P</math>
* <math>\mathfrak{M}, w \models \Diamond P</math> iff for some element <math>u</math> of <math>W</math>, it holds that <math>w R u</math> and <math>u \models P</math>

According to this semantics, a formula is ''necessary'' with respect to a world <math>w</math> if it holds at every world that is accessible from <math>w</math>. It is ''possible'' if it holds at some world that is accessible from <math>w</math>. Possibility thereby depends upon the accessibility relation <math>R</math>, which allows us to express the relative nature of possibility. For example, we might say that given our laws of physics it is not possible for humans to travel faster than the speed of light, but that given other circumstances it could have been possible to do so. Using the accessibility relation we can [[Logic translation|translate]] this scenario as follows: At all of the worlds accessible to our own world, it is not the case that humans can travel faster than the speed of light, but at one of these accessible worlds there is ''another'' world accessible from ''those'' worlds but not accessible from our own at which humans can travel faster than the speed of light.