Propositions can be true in different ways. Some propositions are necessarily true, in the sense that they could not possibly be false. (Tautologies are necessarily true, for example.) Other propositions are contingently true, in the sense that they happen to be true although they could have been false. Similarly, some propositions are necessarily false, in the sense that they couldn’t possibly be true, while others are contingently false: they happen to be false, although they could have been true. These different ways of being true or false are sometimes called modes of truth and falsity.
Modal logic is designed to deal with logical structure involving the modes of truth and falsity. More broadly, modal logic deals with the notions of possibility and necessity. However, the words ‘possible’ and ‘necessary’ can mean different things, and it is important to distinguish between these meanings. There are numerous types of modality, corresponding to different meanings of ‘possibility’ and ‘necessity’:
A proposition is logically possible iff it is not a contradiction; it is logically necessary iff it is a tautology. For example, the proposition penguins are purple is logically possible because it is not contradictory (even though it is false). However, a proposition of the form (P • ~P) is logically impossible, since that is a contradiction; and a proposition of the form (P ∨ ~P) is logically necessary, since that is a tautology.
A proposition is conceptually possible iff it is conceivable; it is conceptually necessary iff its negation is inconceivable. Unlike logical modality, conceptual modality depends on what we can conceive or imagine. We cannot conceive something that is logically impossible, but there are many logical possibilities that are beyond our imagination, so conceptual possibility is a narrower concept than logical possibility. Moreover, conceptual possibility is somewhat subjective, since people may differ in their imaginative capabilities.
A proposition is epistemically possible for someone iff it could be true, for all he or she knows; it is epistemically necessary iff it must be true, given what he or she knows. Like conceptual modality, epistemic modality is a subjective notion. If you know something that I don’t know, your knowledge might rule out some possibilities that remain epistemically possible for me. For example, if you know that dolphins are mammals but I mistakenly think they are fish, the proposition dolphins have gills might be epistemically possible for me but epistemically impossible for you (since you know that mammals have lungs, not gills).
A proposition is physically possible iff it could be true, given the laws of physics; it is physically necessary iff it must be true, given the laws of physics. For example, the existence of a perpetual motion machine is physically impossible because it would violate the laws of thermodynamics.
Metaphysical modality is difficult to define precisely, but it depends on facts about identity, essence, and the way in which words refer to things. For present purposes, an example will suffice to illustrate how metaphysical modality differs from the other kinds of modality. In ancient times, astronomers identified what they initially believed to be two separate celestial objects: Hesperus (the evening star) and Phosphorus (the morning star). It was later discovered that Hesperus and Phosphorus are the very same planet, now known as Venus. There is a sense in which the proposition Hesperus is Phosphorus is necessarily true, since both of those names refer to the very same celestial body. However:
Nevertheless, there is still a sense in which the claim that Hesperus is Phosphorus couldn’t possibly be false, because both names refer to the same planet, Venus. This remaining sense of necessity is called metaphysical necessity: it is metaphysically necessary that Hesperus (the evening star) is the same planet as Phosphorus (the morning star).
Any of the above varieties of possibility and necessity can be represented by modal logic. However, we must exercise caution when dealing with arguments that involve more than one type of possibility or necessity. When an argument employs more than one sense of possibility, watch out for fallacies of equivocation! (The fallacy of equivocation is the mistake of using an ambiguous word to mean different things in different parts of an argument.) Consider the following example:
The laws of physics imply that if it’s possible to travel faster than the speed of light, then it’s possible to travel backwards in time. It is possible to travel faster than light, in the sense that there is nothing logically contradictory about moving faster than 300,000 km/s. Therefore, it’s possible to travel backwards in time.
Although the argument resembles the valid form modus ponens, the first premise involves physical possibility, while the second premise refers to logical possibility. So, the second premise does not really mean the same thing as the antecedent of the conditional. The argument is not a genuine instance of modus ponens.