Modal logic


Modal logic is a kind of logic used to represent statements about necessity and possibility. In philosophy and related fields
it is used as a tool for understanding concepts such as knowledge, obligation, and causation. For instance, in epistemic modal logic, the formula can be used to represent the statement that is known. In deontic modal logic, that same formula can represent that is a moral obligation. Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula as a tautology, representing the principle that only true statements can count as knowledge. However, this formula is not a tautology in deontic modal logic, since what ought to be true can be false.
Modal logics are formal systems that include unary operators such as and, representing possibility and necessity respectively. For instance the modal formula can be read as "possibly " while can be read as "necessarily ". In the standard relational semantics for modal logic, formulas are assigned truth values relative to a possible world. A formula's truth value at one possible world can depend on the truth values of other formulas at other accessible possible worlds. In particular, is true at a world if is true at some accessible possible world, while is true at a world if is true at every accessible possible world. A variety of proof systems exist which are sound and complete with respect to the semantics one gets by restricting the accessibility relation. For instance, the deontic modal logic D is sound and complete if one requires the accessibility relation to be serial.
While the intuition behind modal logic dates back to antiquity, the first modal axiomatic systems were developed by C. I. Lewis in 1912. The now-standard relational semantics emerged in the mid twentieth century from work by Arthur Prior, Jaakko Hintikka, and Saul Kripke. Recent developments include alternative topological semantics such as neighborhood semantics as well as applications of the relational semantics beyond its original philosophical motivation. Such applications include game theory, moral and legal theory, web design, multiverse-based set theory, and social epistemology.

Syntax of modal operators

Modal logic differs from other kinds of logic in that it uses modal operators such as and. The former is conventionally read aloud as "necessarily", and can be used to represent notions such as moral or legal obligation, knowledge, historical inevitability, among others. The latter is typically read as "possibly" and can be used to represent notions including permission, ability, compatibility with evidence. While well-formed formulas of modal logic include non-modal formulas such as, it also contains modal ones such as,,, and so on.
Thus, the language of basic propositional logic can be defined recursively as follows.
  1. If is an atomic formula, then is a formula of.
  2. If is a formula of, then is too.
  3. If and are formulas of, then is too.
  4. If is a formula of, then is too.
  5. If is a formula of, then is too.
Modal operators can be added to other kinds of logic by introducing rules analogous to #4 and #5 above. Modal predicate logic is one widely used variant which includes formulas such as. In systems of modal logic where and are duals, can be taken as an abbreviation for, thus eliminating the need for a separate syntactic rule to introduce it. However, separate syntactic rules are necessary in systems where the two operators are not interdefinable.
Common notational variants include symbols such as and in systems of modal logic used to represent knowledge and and in those used to represent belief. These notations are particularly common in systems which use multiple modal operators simultaneously. For instance, a combined epistemic-deontic logic could use the formula read as "I know P is permitted". Systems of modal logic can include infinitely many modal operators distinguished by indices, i.e.,,, and so on.

Semantics

Relational semantics

Basic notions

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 accessible worlds. Thus, the relational semantics interprets formulas of modal logic using models defined as follows.
  • A relational model is a tuple where:
  1. is a set of possible worlds
  2. is a binary relation on
  3. is a valuation function which assigns a truth value to each pair of an atomic formula and a world,
The set is often called the universe. The binary relation is called an accessibility relation, and it controls which worlds can "see" each other for the sake of determining what is true. For example, means that the world is accessible from world. That is to say, the state of affairs known as is a live possibility for. Finally, the function is known as a valuation function. It determines which atomic formulas are true at which worlds.
Then we recursively define the truth of a formula at a world in a model :
  • iff
  • iff
  • iff and
  • iff for every element of, if then
  • iff for some element of, it holds that and
According to this semantics, a formula is necessary with respect to a world if it holds at every world that is accessible from. It is possible if it holds at some world that is accessible from. Possibility thereby depends upon the accessibility relation, 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 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.

Frames and completeness

The choice of accessibility relation alone can sometimes be sufficient to guarantee the truth or falsity of a formula. For instance, consider a model whose accessibility relation is reflexive. Because the relation is reflexive, we will have that for any regardless of which valuation function is used. For this reason, modal logicians sometimes talk about frames, which are the portion of a relational model excluding the valuation function.
  • A relational frame is a pair where is a set of possible worlds, is a binary relation on.
The different systems of modal logic are defined using frame conditions. A frame is called:
  • reflexive if w R w, for every w in G
  • symmetric if w R u implies u R w, for all w and u in G
  • transitive if w R u and u R q together imply w R q, for all w, u, q in G.
  • serial if, for every w in G there is some u in G such that w R u.
  • Euclidean if, for every u, t, and w, w R u and w R t implies u R t
The logics that stem from these frame conditions are:
  • K := no conditions
  • D := serial
  • T := reflexive
  • B := reflexive and symmetric
  • S4 := reflexive and transitive
  • S5 := reflexive and Euclidean
The Euclidean property along with reflexivity yields symmetry and transitivity. Hence if the accessibility relation R is reflexive and Euclidean, R is provably symmetric and transitive as well. Hence for models of S5, R is an equivalence relation, because R is reflexive, symmetric and transitive.
We can prove that these frames produce the same set of valid sentences as do the frames where all worlds can see all other worlds of W. This gives the corresponding modal graph which is total complete. For example, in any modal logic based on frame conditions:
If we consider frames based on the total relation we can just say that
We can drop the accessibility clause from the latter stipulation because in such total frames it is trivially true of all w and u that w R u. But this does not have to be the case in all S5 frames, which can still consist of multiple parts that are fully connected among themselves but still disconnected from each other.
All of these logical systems can also be defined axiomatically, as is shown in the next section. For example, in S5, the axioms, and hold, whereas at least one of these axioms does not hold in each of the other, weaker logics.

Topological semantics

Modal logic has also been interpreted using topological structures. For instance, the Interior Semantics interprets formulas of modal logic as follows.
A topological model is a tuple where is a topological space and is a valuation function which maps each atomic formula to some subset of. The basic interior semantics interprets formulas of modal logic as follows:
  • iff
  • iff
  • iff and
  • iff for some we have both that and also that for all
Topological approaches subsume relational ones, allowing non-normal modal logics. The extra structure they provide also allows a transparent way of modeling certain concepts such as the evidence or justification one has for one's beliefs. Topological semantics is widely used in recent work in formal epistemology and has antecedents in earlier work such as David Lewis and Angelika Kratzer's logics for counterfactuals.