Conditional Proof
Section 7.5
Conditional proof is a method for obtaining a line in a proof sequence (either
the conclusion or some intermediate line) that frequently offers the
advantage of being both simpler and shorter to use than the direct method.
Further, there are some proofs that are impossible to derive by the direct
method so that some form of the conditional method must be used. The
method consists of assuming the antecedent of the required conditional,
deriving the consequent, then discharging this sequence.
1. A  (B  C)
2. (B v D)  E
/A  E
Important: the first decision is to figure out what should be assumed in order
to obtain the desired result. While anything whatsoever may be assumed,
only the correct assumption will obtain the desired result. The best place to
start is with the antecedent of the conditional to be obtained. For example, if
the statement to be obtained is (KL)  M, then KL should be assumed. As
above, indent it and tag it with ACP. It can then be used to obtain the
consequent of the conditional. Once obtained, the assumption is discharged,
and the discharged line is tagged CP and is sealed off from any subsequent
lines. Note that in the discharge line, the assumption is listed as the
antecedent and the derivative as the consequent. Thus, if you assume
G v J and derive H v J, then the discharged line reads: (G v J)  (H v J).
Conditional proof may also be used to derive a line within the scope of a
proof. Consider:
1. G  (H  I)
2. J  (K  L)
3. G v J
/H v K
Note that the lines that are closed off from the proof are unavailable for later
use. Once sealed, they are gone. They are applicable only during the
assumption. Once discharged, they can no longer be used. Also note that any
assumption must be discharged.
1. L  [M  (N v O)]
2. M  ~N
/L  (~M v O)
From the homework section:
#5
1. A  ~ (A v E) / A  F
#8
1. P  (Q v R)
2. (P  R)  (S  T)
3. Q  R
/T
#12
1. F  (G  H)
/ (A  F)  (A  H)
#13
1. R  B
2. R  (B  F)
3. B  (F  H)
/RH
#15
1. C  (D v ~E)
2. E  (D  F)
/ C  (E  F)
#20
1. A  [B  (C  ~D)]
2. (B v E)  (D v E)
/ (A  B)  (C  E)