There are actually many different kinds of modal logics, depending on what axioms you choose, and what kind of extra concepts are incorporated. However, I will choose a fairly basic set of axioms, and explain them as I go.
What we are trying to do here is model the philosophical concepts of necessity and possibility. If we say a proposition is "necessary", that means that it is true in all possible worlds. The world which we happen to live in is one of those possible worlds. Therefore, if a proposition is necessarily true, then it is true. We will take this as an axiom, which I will state symbolically.
Axiom T: p pThe letter "p" represents a proposition, a statement which is either true or false. "p" could be the proposition "God exists", or it could be the proposition "The moon is made of cheese." When we put a "" in front of the proposition, we are modifying the proposition with the concept of necessity. For example, if "p" is the statement "Snow is white", then "p" is the statement "Snow is necessarily white", meaning that snow is white in all possible worlds. Stated in English, Axiom T says that if a proposition is necessarily true, then it is true.
However, it is not the case that if a proposition is true that it is necessarily true. For instance, the statement "I have black hair" happens to be true in this world, however, I may have dyed my hair blue in other possible worlds. Therefore, we cannot say without qualification that pp. However, it is a general truth in logic that the statement "pq" is equivalent to "not-qnot-p". (From this point forward, I will replace the word "not" with the symbol "¬".) "¬q¬p" is called the contrapositive of statement "pq". If any statement is true, then its contrapositive must be true as well.
The contrapositive of Axiom T is ¬p ¬p. Remember, p can be absolutely any proposition, so we're going to replace it with the proposition "¬q". Therefore, by Axiom T, q ¬¬q. The sequence of symbols "¬¬" may seem like meaningless jargon, but it in fact represents the concept of possibility. Saying ¬¬q is the same as saying that there exists a possible world where q is true. It may or may not be the world which we live in, but it's possible. Possibility is such an important concept that we're going to give it its own symbol, "".
Definition: p means ¬¬pAs we proved in the last paragraph, p p. This makes sense, since if a proposition is true in our world, then it must be true in at least one possible world, namely, our own.
We would like our concepts of necessity and possibility to be properties which pertain to the entire set of possible worlds, not just our own. For instance, if it's possible that dogs exist, then that just means that there exists at least one possible world where dogs exist. Therefore, no matter which of the possible worlds we lived in, we could still correctly state that it's possible that dogs exist. While the statement "dogs exist" may be true in some worlds but not others, the statement "dogs possibly exist" is either true or false in all possible worlds. That is the reasoning behind the following two axioms, known as Axiom 4 and Axiom 5.
Axiom 4: p pAnother thing we would like is for our axioms to be true in all possible worlds. It would be quite absurd if, for instance, Axiom T was true in our world, but not in others. And so we take this as a new axiom.
Axiom 5: p p
Axiom N: All axioms, as well as theorems proven from them, are necessarily trueApplying Axiom N to Axiom T, we could say (p p). But what useful things could we say from a statement of the form (p q)? The answer to that is our last axiom.
Axiom K: (p q) (p q)This means that if p q is true in all possible worlds, and if p is true in all possible worlds, then q is true in all possible worlds.
- Definition: p denotes the statement that p is "necessary", or true in all possible worlds.
- Definition: p, or ¬¬p denotes the statement that p is "possible", or true in at least one world.
- Axiom T: p p "If a statement is true in all possible worlds, then it is true in our world."
- Axiom 4: p p "Necessity is a property of all possible worlds."
- Axiom 5: p p "Possibility is a property of all possible worlds."
- Axiom K: (p q) (p q) "If p implies q in all worlds, and p is true in all worlds, then q is true in all worlds"
- Axiom N: All axioms, as well as theorems proven from them, are necessarily true