% Part: applied-modal-logics
% Chapter: temporal-logic
% Section: introduction

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\olsection{Introduction}

Temporal logics deal with claims about things that will or have been
the case. Arthur Prior is credited as the originator of temporal
logic, which he called \emph{tense logic}. Our treatment of temporal
logic here will largely follow Prior's original modal treatment of
introducing temporal operators into the basic framework of
propositional logic, which treats claims as generally lacking in
tense.

For example, in propositional logic, I might talk about a dog, Beezie,
who sometimes sits and sometimes doesn't sit, as dogs are wont to do.
It would be contradictory in classical logic to claim that Beezie is
sitting and also that Beezie is not sitting. But obviously both can be
true, just not at the same time; adding temporal operators to the
language can allow us to express that claim relatively easily. The
addition of temporal operators also allows us to account for the
validity of inferences like the one from ``Beezie will get a treat or
a ball" to ``Beezie will get a treat or Beezie will get a ball." 

However, a lot of philosophical issues arise with temporal logic that
might lead us to adopt one framework of temporal logic over another.
For example, a future contingent is a statement about the future that
is neither necessary nor impossible. If we say ``Richard will go to the
grocery store tomorrow," we are expressing a claim about something that
has not yet happened, and whose truth value is contestable. In fact,
it is contestable whether that claim can even be \emph{assigned} a
truth value in the first place. If we are strict determinists, then
perhaps we can be comfortable with the idea that this sentence is in
fact true or false, even before the event in question is supposed to
take place---it just may be that we do not know its truth value yet.
In contrast, we might believe in a genuinely open future, in which the
truth values of future contingents are undetermined. 

As it turns out, a lot of these commitments about the structure and
nature of time are built in to our choices of models and frameworks of
temporal logics. For example, we might ask ourselves whether we should
construct models in which time is linear, branching or even circular.
We might have to make decisions about whether our temporal models will
have beginning and end points, and whether time is to be represented
using discrete instants or as a continuum. 

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