Joshua Cuthbert
October 10, 2000
MA 315
Logic vs. Truth
The question keeps coming up, with no apparent answer in sight. Just
what is this propositional calculus anyways? Douglas Hofstadter defines it
as a formal systemÉone which depends on a form of reasoning involving
only the correct usage of the words ÔandÕ, ÔorÕ, ÔifÉthenÕ, and ÔnotÕ
(Hofstadter, 181). But that says little about how this concept relates to truth
in the real world. To me, propositional calculus is just one version of the
truth that can exist in our every day lives.
ÒIn case it is not yet apparent, the symbol Ô^Õ is meant to be acting
isomorphically to the normal everyday word ÔandÕ Ó (Hofstadter, 186).
Similarly, the symbols ¬, ∨, ⇒, ⇔ enjoy such connections as well. But
then how does this relate to absolute truth? For example, can a statement
such as P ∨ ¬P always prove true in the Ôreal worldÕ? This seems to make
sense, for example, we could say ÔThe sun is out or the sun is not out.Õ This
kind of statement is known as a ÔtautologyÕ in propositional calculus, a
statement that will always yield truth. Mathematics at times can provide
these clear answers, but when taken in the real world there always seems to
be room for a little error. In the earlier example, one would be hard-pressed
