Temporal, Deontic, and Epistemic Logic

The relational semantics was presented for necessity and possibility, and nothing in it mentions either. A frame is a set with a relation, and reading the relation as temporal order, as normative preference or as epistemic indistinguishability gives three subjects with their own axioms, their own applications and their own well-known failures.

Each of the three has a characteristic difficulty, and in each case the difficulty is the same kind of thing: a principle that looks obviously right in the intended reading turns out to be too strong.

Temporal logic

Prior's tense logic has four basic operators, two future and two past, and the interaction axioms fix how they relate: what will be the case will always have been going to be the case. Assumptions about time appear as modal axioms in the expected correspondence. Transitivity of the order validates the $\mathsf{4}$ axiom; density validates a further schema; linearity of the past validates the principle distinguishing branching from linear time.

Branching time is where the philosophical interest concentrates. On a branching frame each moment has a unique past and several futures, and this is the setting for Aristotle's sea battle: the statement that there will be a sea battle tomorrow is true on some branches and false on others. Prior's Ockhamist and Peircean readings answer differently, the first evaluating a future-tensed sentence relative to a selected branch and the second requiring truth on all of them, and the choice determines whether excluded middle holds for future contingents. The three-valued treatment of the many-valued article is a third option, and the branching frame makes clear what the three options are choices among.

The applied descendant is larger than the philosophical one. Amir Pnueli proposed in 1977 that temporal logic be used to specify programs, and the resulting systems, linear temporal logic and computation tree logic, are the basis of model checking, which verifies that a finite state system satisfies a specification by exhaustive search. The technique is used on processor designs and protocols, and the branching that Prior introduced for the sea battle is the branching of a program's possible executions.

Deontic logic

Seriality of the relation gives that axiom, and the reading is that some deontically acceptable alternative always exists, which is to say that the obligations are jointly satisfiable. Reflexivity is deliberately absent, since what ought to be the case need not be.

The system's difficulties are old and well documented. Ross's paradox notes that from an obligation to post the letter, the normal logic derives an obligation to post the letter or burn it, which sounds like a permission to burn. The good Samaritan paradox derives, from the obligation to help the robbed man, an obligation that a man be robbed. Both stem from the closure of the box under logical consequence, and both are treated by the neighbourhood semantics of the earlier article, which is one of the main reasons non-normal systems are studied.

The failure is structural. Formalising the second conditional with the obligation outside the conditional and the third with it inside, or the reverse, produces one of the two defects, and no assignment of scopes avoids both. Since contrary-to-duty obligations are the normal case in law and in ethics, where the interesting question is usually what to do after something has gone wrong, this is a serious limitation rather than a puzzle case. The responses have been dyadic deontic logic, taking conditional obligation as primitive rather than defining it, and the preference semantics of Hansson and Lewis, ordering worlds by betterness and reading an obligation as holding at the best antecedent worlds. The parallel with the counterfactual semantics is exact and was noticed at the time.

Epistemic logic

The axioms carry epistemological commitments and the correspondence makes them explicit. Reflexivity gives factivity, that what is known is true, and is uncontroversial. Transitivity gives positive introspection, that knowing implies knowing that one knows, and is disputed. The euclidean condition gives negative introspection, that not knowing implies knowing that one does not, and is rejected by most epistemologists, which is why S5 is used for idealised agents and information-theoretic applications rather than as an account of human knowledge. Williamson's anti-luminosity argument targets the introspective principles directly and is the main contemporary case against them.

The problem is the deepest of the three, because it is not a matter of choosing the right axiom. Necessitation and distribution are what make a system normal, and any relational semantics validates both. Every proposed remedy therefore changes the semantics: impossible worlds, where an agent's epistemic alternatives include logically incoherent ones, as the non-normal article described; neighbourhood models, where the closure conditions are imposed selectively; awareness structures, due to Fagin and Halpern in 1988, which add a syntactic filter so that an agent knows only what they are aware of; and syntactic or algorithmic treatments of knowledge, which model it as derivability within a resource bound.

The multi-agent setting is where the subject has been most productive and where the omniscience problem is least pressing, since the questions concern the relations between agents' knowledge rather than the closure of any one agent's. Common knowledge, defined as the infinite conjunction of everyone knows that everyone knows, is the notion that the muddy children puzzle and the coordinated attack problem turn on, and it has a clean modal characterisation as knowledge along the transitive closure of the agents' relations. Dynamic epistemic logic adds operators for announcements, and delivers the result that some true sentences become false when announced, the Moore sentences being the standard example. The applications in distributed computing and game theory are substantial and are what the field is mostly about now.

Next

This tier has surveyed a great many logics: classical, intuitionistic, modal in several readings, substructural, many-valued, free. The final article asks the question that survey raises, which is what makes any of them logic, and whether the plurality is a disagreement or a division of labour.