This is one of the largest and most active areas of philosophic
logic. Prior's book *Past, Present and Future* (1968) is an
extremely thorough and lucid account of what has been done in the
field. We have already mentioned the four propositional operators
*F*, *G*, *P*, *H* which Prior discusses. He regards these as modal
operators; then the alternativeness relation of the semantic theory
is simply the time-ordering relation. Various axiomatizations are
given, corresponding to deterministic and nondeterministic tenses,
ending and nonending times, etc; and the problems of quantification
turn up again here with renewed intensity. To attempt a summary of
Prior's book is a hopeless task, and we simply urge the reader to
consult it. More recently several papers have appeared (see, for
instance, Bull 1968) which illustrate the technical sophistication
tense-logic has reached, in that full completeness proofs for various
axiom systems are now available.

As indicated above, the situation calculus contains a tense-logic (or rather several tense-logics), in that we can define Prior's four operators in our system and by suitable axioms reconstruct various axiomatizations of these four operators (in particular, all the axioms in Bull (1968) can be translated into the situation calculus).

Only one extra nonlogical predicate is necessary to do this:
it is a binary predicate of situations called *cohistorical*, and is
intuitively meant to assert of its arguments that one is in the
other's future. This is necessary because we want to consider some
pairs of situations as being not temporally related at all. We now
define *F* (for instance) thus:

The other operators are defined analogously.

Of course we have to supply axioms for `cohistorical' and time: this is not difficult. For instance, consider one of Bull's axioms, say , which is better (for us) expressed in the form . Using the definition, this translates into:

which simplifies (using the transitivity of `>') to

that is, the transitivity of `cohistorical'. This axiom is precisely
analogous to the *S*4 axiom
, which corresponded to
transitivity of the alternativeness relation in the modal semantics.
Bull's other axioms translate into conditions on `cohistorical' and
time in a similar way; we shall not bother here with the rather
tedious details.

Rather more interesting would be axioms relating `shrug' to
`cohistorical' and time; unfortunately we have been unable to think
of any intuitively plausible ones. Thus, if two situations are
epistemic alternatives (that is, then they may or
may not have the same time value (since we want to allow that *p* may
not know what the time is), and they may or may not be cohistorical.

Mon Apr 29 19:20:41 PDT 1996