Definition
A possible world is a complete, alternative way things could have been. The actual world is one such way; alongside it we can imagine countless others — worlds in which the coin landed tails, in which you took a different job, in which the dinosaurs never died out. Each is total: it settles every question, leaving nothing undecided.
Possible worlds are the central semantic device of modal logic. They give precise content to the otherwise slippery talk of necessity and possibility by turning modal claims into claims about quantification over a space of worlds.
Why it matters
How it works
With a space of worlds in hand, modal operators become quantifiers. 'It is necessary that P' is true at a world when P holds at every world accessible from it; 'it is possible that P' is true when P holds at at least one such world. The mysterious modal vocabulary is thereby reduced to the familiar logic of 'all' and 'some'.
The accessibility relation does the fine-tuning. It specifies, for each world, which worlds count as its genuine alternatives. If accessibility is reflexive, every necessary truth is actually true; if it is transitive, necessity iterates. Adjusting these properties generates the whole family of modal systems. Philosophers still debate whether possible worlds are real entities or useful fictions — but as a tool, the device is indispensable.