The Logic of Conditionals

1
The Logic of Conditionals
Language, Proof and Logic
Chapter 8
2
Informal methods of proof
8.1.a

• Valid steps
• 1. Modus ponens From P?Q and P, infer Q.
• 2. Biconditional elimination From P and either
  P?Q or Q?P, infer Q.
• 3. Contraposition P?Q ? ?Q? ?P

The method of conditional proof To prove P?Q ,
temporarily assume P and derive Q based on that
assumption.
3
Informal methods of proof
8.1.b
Proving Even(n2)?Even(n) Assume n2 is even. To
prove that n is even, assume, for a
contradiction, that n is odd. Then, for some k,
n2k1. Hence n2(2k1)(2k1) 4k24k12(2k2k)
1. Hence n2 is odd, which is a contradiction.
So, n is even. Proving the same using
contraposition Assume n is odd. Then, for some
k, n2k1. Hence n2(2k1)(2k1) 4k24k12(2k2
k)1. Hence n2 is odd
4
Informal methods of proof
8.1.c

• Proving biconditionals To prove a number of
  biconditionals, try to
• arrange them into a cycle of conditionals, and
  then prove each
• conditional.
• 1. n is even
• 2. n2 is even
• 3. n2 is divisible by 4
• Cycle (3)?(2)?(1)?(3)

5
Formal rules of proof for ? and ?
8.2
? Elim P?Q P Q
? Intro P Q P?Q
? Intro P Q Q
P P?Q
? Elim P?Q (or Q?P) P Q
You try it, pages 208, 211
6
Soundness and completeness
8.3.a
FT --- the portion of F that contains the rules
for ?,?,?,?,?,?. P1,,Pn ?-T Q --- provability of
Q in FT from premises P1,,Pn. CLAIM FT is
sound and complete with respect to tautological
consequence.
Soundness If P1,,Pn ?-T Q, then Q is a
tautological consequence of P1,,Pn.
So, once you see that Q is not a tautological
consequence of P1,,Pn, you can be sure that
there is no way to FT-prove Q from P1,,Pn.
Completeness If Q is a tautological consequence
of P1,,Pn, then P1,,Pn ?-T Q.
So, once you see that Q is a tautological
consequence of P1,,Pn, you can be sure that
there is an FT-proof of Q from P1,,Pn, even if
you have not actually found such a proof.
7
Soundness and completeness
8.3.b
Would the system be sound if it had the following
rule for exclusive OR ?? How about the same
rule with the ordinary ??
P?Q P S Q T S?T
See page 215 for a proof idea for the soundness
of FT.