  We describe a graph-theoretic syntax for self-referential formulas as well as
a four-valued logic to include contradictory and independent formulas. We then
explore the degree to which generalized truth tables can be realized in our
theory, and go on to describe a model theory for sentential calculus, wherein
models are allowed to include contradictions (such as the ``Liar'') and
formulas that result from them as an integral part of their structure. This
sets the groundwork for a sequel in which we construct models of set theory
that include contradictions.
