A note on proof
"Proof" is going to be an important topic of this course; one
that runs as a thread through everything we do.
Perhaps the most basic form of proof is what we might call
"Proof by calculation". You've been doing this in algebra for
years already. If someone asks you to prove that 15% of 88.30
is greater than 12.50, you would simply show them the
calculation .15*88.30 = 13.2450 > 12.50. The point is that
this kind of caclulation is so basic, that we all consider it to be
self-explanatory. I could do it myself if I didn't want to
take your word for it. It will be the same for propositional
logic. Each of these calculations like
"false⇔false evaluates to true" is a proof of a result. A
boring proof, but there you go. It's "boring" because it is a
one step proof with the justification always being "because
that's what the table in the definition says".
It's important to recognize what's going on here, though,
because the idea that we always bottom out with "because
that's what the definition says" is important. When we get an
proof down to that level, there is no arguing and no room for
disagreement ... unless you can't agree on what the definition
actually is. In mathematics, this shouldn't happen often. In
life, however, this happens very often.