I agree with the points that Chris and Pat made about contexts,
but I'd like to add a few comments. (01)
JB>> I had thought that time was yet another type of context. Is it
>> different somehow? (02)
PH> You CAN treat time as a context, if you like contexts. But that is
> not the only way to tackle a logic of time: the modal temporal logics
> pre-date context logics by about 60 years. (03)
I agree. A context approach to time can be reduced to a noncontextual
approach by adding an extra argument for time to every relation.
For example, consider the sentence (04)
At time t, the cat Yojo was on a mat. (05)
In a context approach, you could translate that to a conceptual graph
of the following form: (06)
[Time: *t]<-(PTim)<-[Situation: [Cat: Yojo]->(On)->[Mat] ] (07)
But the same sentence could be represented with the following CGIF
statement, in which the dyadic relation On is replaced by a triadic
relation that has an extra argument for time: (08)
[Time: *t] [Cat: Yojo] [Mat *x] (On Yojo ?x ?t) (09)
Following is the equivalent CLIF: (010)
(exists ((t Time) (x Mat)) (and (Cat Yojo) (On Yojo x t))) (011)
This method might look simple for such a short sentence. But with
many different time points with many relations, the context boxes
group related information in a more readable notation. Whether that
grouping is simpler for reasoning depends on the type of problem. (012)
I'd also like to recommend some references. The first is to the work
of Arthur Prior, who developed the modal approach in detail. Following
is a brief article about Prior and his work: (013)
http://plato.stanford.edu/entries/prior/
Arthur Prior (Stanford Encyclopedia of Philosophy) (014)
Another article: (015)
http://www.hum.au.dk/ckulturf/pages/publications/ph/life.htm
Per Hasle: Life and Work of Arthur N. Prior (016)
Interesting quotation from above: (017)
"I think I'm right in saying that Arthur initially accepted
Russell and Whitehead as establishing the parameters of modern
logic; developed doubts as he read more ancient and medieval
logicians, and then sought to formalize a tensed logic which
would deal with tensed statements." (018)
One source that Prior probably read was Ockham's _Summa Logicae_,
which was finished around 1323. Part II presents Ockham's theory
of propositions, which states truth conditions for Latin sentences,
very much along the lines of Tarski's model theory. Ockham isn't
as formal as Tarski, but his book is a very readable introduction
to the approach. See (019)
Ockham, William of, _Ockham's Theory of Propositions_,
translation of Part II of Ockham's _Summa Logicae_ by
A. J. Freddoso & H. Schuurman, University of Notre Dame Press,
Notre Dame, IN, 1980. (020)
Unlike Tarski, who covered only first-order logic, Ockham stated
truth conditions for modal, temporal, and causal statements.
Ockham's approach to those subjects was far too brief to cover
all the issues and problems, but it was undoubtedly one of the
inspirations for Prior's work. (021)
I'd also like to mention a very good analysis and comparison of
several different formalizations of time: (022)
Gergely, Tamás, & László Úry (1991) First-Order Programming Theories,
Springer-Verlag, Berlin. (023)
Unfortunately, the list price is $125, but Powell's bookstore, which
I recommend, has a copy for $11. (024)
Gergely and Ury present temporal logics and dynamic logics. But they
argue that the simplest and most general approach is a first-order
method that quantifies over time. I agree with them, although I admit
that the other approaches are useful for some purposes. For example,
certain proofs may be very short in a modal approach to temporal logic,
but more complex in FOL. (025)
John Sowa (026)
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