4
"God is a person" and "God is an impersonal force," together violate the LONC. A genuine
contradiction between beliefs demonstrates that at least one of those beliefs is false.
Logicians have identified some propositions or arguments that violate or appear to violate
the LONC. The first two concern propositions. Contraries are two statements that cannot both
be true, though both could be false (e.g. "There are only two divine beings" and "There are
twelve divine beings;" for there could be another number, like one). Contradictories are two
statements such that if one is true, the other must be false, and vice versa (e.g. "God is all-good"
and "God is not all-good").
9
All agree that at this level such violations prove that at least one of
the statements is false. However, the other violations are more complex. A paradox is a set of
arguments or statements all of which seem to be valid (or meaningful) but appear to lead to a
contradiction or some clearly false conclusion or sense.
10
One example, a version of the "Liar's
Paradox," consists simply of one sentence: "This statement is false." Logicians work at solving
paradoxes, and some have been shown not to violate the LONC. For example, Quine
distinguished between "veridical paradoxes," apparent contradictions that are composed of
claims all of which can be shown to be true (e.g. a person can be 23 years old on his 6th
birthday--if he was born on Feb. 29) and "falsidical paradoxes," that have been proven to be
invalid in some way (e.g. Zeno's paradoxes).
11
Other paradoxes still remain to be resolved either
way. Paradoxes are valuable to logicians for they can illuminate mistakes in reasoning through
the attempt to solve them.
12
However, labeling something a paradox is to recognize the potential
violation of the LONC without assuming that the problem is necessarily a genuine contradiction.
In contrast, the term antinomy is reserved for a falsidical paradox consisting of two arguments
which individually seem valid but (like contradictories) are mutually exclusive and so