Definition 3.2.2. Proposition Generated by a Set.
Let
be any set of propositions. A proposition generated by
is any valid combination of propositions in
with conjunction, disjunction, and negation. Or, to be more precise,
If
then
is a proposition generated by
and
If
and
are propositions generated by
then so are
, and
Note: We have not included the conditional and biconditional in the definition because they can both be generated from conjunction, disjunction, and negation, as we will see later.
If
is a finite set, then we may use slightly different terminology. For example, if
we might say that a proposition is generated by
and
instead of from
Within any level of the hierarchy, work from left to right. Using these rules,
is taken to mean
These precedence rules are universal, and are exactly those used by computer languages to interpret logical expressions.
A few shortened expressions and their fully parenthesized versions:
A proposition generated by a set
need not include each element of
in its expression. For example,
is a proposition generated by
and