Next, we move to two basic themes that have risen to prominence since the late twentieth century—not just in modal logic, but also for logical systems generally.
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Computation The basic modal language is a decidable miniature of first-order logic. There are many decision methods for validity or satisfiability exploiting special features of modal formulas — each with their virtues. Well-known methods are selection, filtration, and reduction, for which we refer to the literature Marx But there is a deeper issue here, going beyond the traditional understanding of logical systems. What is the precise computational complexity of various key tasks for a logic, allowing us to gauge its difficulty as a device to be used seriously? These key tasks include testing for satisfiability, but also model checking for truth, as well as comparing models.
Here are the facts for the basic modal logic. This is better than for first-order logic, where this task takes polynomial space. For first-order logic, this is undecidable. These benchmark complexities for logics differ as languages are varied. Complexity awareness may be a new feature to many logicians and philosophers, but computational behavior seems a feature of basic importance in understanding formal frameworks.
Modals and Conditionals
Disjunction is a choice for , conjunction for , negation is a role switch, makes pick a point reachable from the current point, does the same for. A game is won by if the atom holds at the current point, otherwise by. A player also wins if the opponent has no move for a modality.
Fact iff Verifier has a winning strategy for the -game in starting at. Here is an example.
Our first model picture when we introduced the basic semantics induces the following tree for an evaluation game for the formula starting from point 1, with boldface indicating the winning positions for Verifier:. These are indeed the two possible successful ways of verifying in the given model at point 1. This style of analysis is widespread in the current literature. There are also model comparison games between players Duplicator maintaining an analogy and Spoiler claiming a difference , playing over pairs of points in two given models.
This may be seen as a fine-structured way of checking for existence of a bisimulation, where successor states chosen in one model by Spoiler must be matched by successors in the other model, chosen by Duplicator, while atomic harmony always remains. Without providing details, we note that in games like this, a Spoiler's winning strategies in a -round game between and match the modal formulas of operator depth on which the points disagree, b Duplicator's winning strategies over an infinite round game between match the bisimulations linking to.
For instance, proof games find deductions or counter-examples through a dialogue between two players about some initial claim. And always, the logical core notion turns out to match a strategy for interactive play. This completes our sketch of basic modal logic as a meeting place for a wide range of logical notions, techniques, and results.
In Section 4, we look at some concrete modern applications in more detail, and then in Section 5, we identify further general issues to which these give rise, reinforcing the role of modal logic as a conceptual lab. We have given some information on the attractions of the basic system of abstract modal logic. At the same time, it is also important to see that many different concrete interpretations can be attached to this system, and how diverse these are. Knowledge and belief One of the major interpretations of modal logic in use today reads modalities as operators of knowledge or belief Hintikka , Stalnaker , though this reading is itself a subject of ongoing debate.
Languages like this express many further basic epistemic patterns that occur in natural discourse, such as. A major new theme in the epistemic setting is a social one. Not lonely thinkers are essential to cognition, but interaction between what different agents in a group, involving what they know about each other — in patterns such as or. What I know about your knowledge or ignorance is crucial, both to my understanding and to my actions.
For instance, I might empty your safe tonight if I believe that you do not know that I know the combination. Some forms of group knowledge transcend simple iterations of individual knowledge assertions. A key example is common knowledge : if everyone knows that your partner is unfaithful, you have private embarrassment — if it is common knowledge, you have public shame. Technically, this works as follows in our models.
A new common knowledge modality says that holds at every world reachable via a finite chain of uncertainty relations for agents in. For instance, in the following picture, where epistemic accessibility is an equivalence relation, the atomic fact holds in the current world, marked by the black dot:.
In the current world, our semantics yields the following further facts: a agent does not know whether : , b agent does know that , while c it is common knowledge in the group that knows whether :. Incidentally, this is a good situation for to ask the question whether is true: but more on epistemic actions below. Common knowledge treated in a modal style is a widely used notion by now in philosophy Lewis , but also in computer science Fagin et al.
Similar models can represent belief. This is often done a bit crudely by adding one more accessibility relation that is no longer reflexive to allow for false beliefs. Thinking of equivalence classes of the epistemic relation as the total range of what an agent knows, we endow these with binary plausibility orderings that encode what the agent considers less or more plausible.
Then a belief modality is interpreted as saying that is true in all most plausible epistemically accessible worlds. And plausibility models also support a richer notion, namely a binary modality of conditional belief saying that is true in all most plausible epistemically accessible worlds that satisfy.
Unlike the situation with conditional knowledge, conditional belief cannot be defined in terms of absolute belief. Indeed, the logic of conditional belief is much like modal logics for conditional assertions in models with similarity relations Lewis , Burgess , Veltman For a current modal study of how to model genuine notions of knowledge using more sophisticated philosophical intuitions, see Holliday Modalities now get labeled with explicit action expressions to show what they range over.
In dynamic logic — originally designed to describe execution of computer programs, but now used as a general logic of action,. Concrete models of this sort are process graphs describing the possible workings of some computer or abstract machine.
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Another concrete model for dynamic logic are games , where actions are moves available to several players. For instance, in the following game tree, player has a strategy for achieving an outcome satisfying against any play by player :. This strategic assertion is captured by the modal formula p. Again we get a minimal modal logic, this time a two-level system treating propositions and actions denoting transition relations on a par. This joint setup allows for an analysis of important action constructions, encoded in valid principles of dynamic logic:.
A major new feature here is unbounded finite repetition of actions:. This shows in two more axioms. Dynamic logics resemble infinitary fixed-point extensions of classical logic, but with a modal stamp: like the basic modal logic, they are bisimulation-invariant and decidable, forming a core calculus for reasoning about the essentials of recursion and induction. Fixed-point definitions are ubiquitous in computer science, mathematics and linguistics, as many natural scientific notions involve recursion. An elegant powerful system of this kind generalizes dynamic logic by adding a facility for arbitrary fixed-point definitions: the so-called —calculus that we will consider briefly below.
Information update Different kinds of modal logic can also form new combinations. For example, the logics of information change by combining knowledge and action. Our earlier epistemic formulas tell us what information agents have right now, but they do not say how this information changes, through acts of observation, communication, or learning in general.
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To model such cognitive actions, we need to combine epistemic and dynamic logic. One powerful idea here is an information update changes the current epistemic model. In the simplest case, reflecting a ubiquitous common sense intuition, this update mechanism works as follows, decreasing the current epistemic range:. But more subtle cases are possible, even with simple models. For example, a question itself may convey crucial information.
By asking, conveys the information that she does not know whether. Even if did not know the answer at the start, this may tell him enough to settle , and now answer the question. Here is a case where this happens:. But the modeling power of epistemic dynamics is still higher. Suppose that neither nor knew whether , but asks expert , who answers only to.
Then learns whether , is no wiser about , but it has become common knowledge that knows if. The modal logic of update has some delicate features. For instance, a public announcement that some formula is the case need not always result in our learning that holds in the updated model. The reason is that truth value switches may happen when announcing formulas that contain a statement of ignorance. Similar logics can deal with acts of belief change , triggered by the above events! Intuitionistic logic and provability logic Let us now move from information and action to the grand themes of mathematics.
Our earlier models can now be viewed as universes of information stages, and accessibility is upward extension.
Intuitionistic logic is then about persistent assertions that, once established, remain true upward in the information order. The full modal language also contains non-persistent assertions beyond the translated intuitionistic language that fit with some earlier-mentioned epistemic statements such as Moore sentences that may become false after updating with new information.
Another proof-oriented interpretation of the modal language occurs in provability logic Boolos , Artemov Here the box modality gets interpreted as existence of a proof in some formal system of arithmetic. Note that this interpretation contains an existential, rather than a universal quantifier, as noted in our introduction. This view validates the laws of the minimal modal logic , as well as the transitivity axiom, that can now be read as saying that given proofs can be proof-checked for correctness. This expresses a deep fact about arithmetical provability — and in fact, provability logic and its many extensions are decidable modal core theories of high-level features of mathematical provability in theories that have the coding power to discuss their own metatheory.
Temporal and spatial logic Still close to mathematics, another lively application area of modal logic concerns physical rather than human nature.