2. Over view
• Why this matters
• Proofs are like games
• Procedure for constructing formal
proofs
• Sample Exercises
3. Why this matters
• We now have two ways of ascertaining validity:
– Constructing counterexamples, a fairly natural way of
ascertaining validity, but only as reliable as your insight
and imagination.
– Truth tables, a very reliable way of ascertaining validity,
but is both unwieldy and not very natural.
• Formal proofs of validity (“natural deductions,”
“formal inferences”) are more rule-governed than
constructing counterexamples, but less unwieldy
than truth-tables.
– A nice balance
4. Proofs are like games
• Football
• Initial field position
• Rules of play (offside, holding, no forward
lateral)
• End zone
5. Proofs are like games
• In football, you move from your initial
field position, in accordance with the
rules of play, to the end zone.
• In formal proofs, you move from your
premises, in accordance with the rules of
inference, to the conclusion
6. Playing the game
• You will be given an argument. It is your task to show that
the conclusion follows validly from the premises. To do this:
1. List the premises as the first lines of the proof. Mark them
with an “A” for Assumption.
2. Apply the basic rules of inference to the premises and then
to the subconclusions that result from those applications.
• Every line in the proof should have a proposition and a
rationale for why you are entitled to assert that
proposition.
3. Follow step 2 until you get the desired conclusion. Then you
win!
7. Proofs are really,really like games
• Many people can learn to play chess
correctly, but it takes some talent and
practice to play chess strategically.
• Many people can use the basic rules of
inference properly, but it takes some
talent and practice to prove things
strategically
8. Some helpful Strategies
• Recognize patterns.
• Think about the big picture, then worry about the
details.
• Reverse engineer.
• Using “clean-up” procedures, i.e., try to establish
common patterns between different premises and
intermediate conclusions in the proof.
• Tease things out of the premises, i.e., use the rules of
inference to draw interesting conclusions.
• Cut the fat, i.e., use the rules of inference to eliminate
statements that occur in the premises but not in the
conclusion.
• Know the rules that cause roadblocks for you.
9. Revers engineer
• Look at the conclusion, and ask yourself how
you might get there from the premises you
have.
• In other words, imagine your 2nd to last step,
3rd to last step, etc. until you get back to the
premises.
10. A siltely
• ~Q→P, (R & S) →~Q, R, ~T↔S, ~T ├ P
• A counterexample here is going to be very
hard to follow.
• A truth table will require 32 rows.
• So, what can a formal proof do with this
puppy?
