Indirect Truth Tables
Indirect truth tables provide us with a shorter and faster method of testing
validity, especially when used to test arguments that have a large number of
different propositions. For example, an argument containing seven terms
would require 128 lines to construct a truth table! With indirect truth tables,
we can do any argument in no more than three lines.
Begin by writing the argument out using the same structure we used for
ordinary truth tables. Separate the premises with a slash and put a double
slash by the conclusion:
P  (Q  R) / R  S // P  S
We begin by assuming that the argument is invalid. That is, we assign truth
values assuming that the premises are true and the conclusion false. Thus:
P  (Q  R) / R  S // P  S
Then we test our assumption by filling in the truth values, working from the
information we get from the conclusion:
P  (Q  R) / R  S // P  S
Notice that we are forced into a contradiction. That is, if the conclusion is
false, then one of the premises must be false also. That is the definition of
validity--it is impossible to have true premises and a false conclusion.
Check this one:
A  (B v C) / C  (D  E) / ~B / A  ~ E
So far, we have needed only one line to test the argument. In some
circumstances, we will need two or three, depending. The following
argument requires two lines. Why?
(M v N)  O / O  (N v P) / M  (~ Q  N) / (Q  M)  ~P // N  O
Now: (#11, page 324)
(A v B)  (C  D) / (~ A v ~ B)  E // (~ C v ~ D)  E
We can also use indirect truth tables to test sets of statements for consistency
just as we did with regular truth tables. Indirect tables, however, allow us to
test a greater number of statements without having the truth tables become
unwieldy. To begin testing, write the statement out on a line, separating
them with a single slash mark. We then assume that they are consistent,
indicating such by placing a "T" under each main operator. Then fill in the
values. If they can be filled in consistently, then the statements are
consistent. If a contradiction is found, then they are not consistent.
K  (R v M) / K  ~ R / M  ~ K
The previous requires only one line. If, however, all of the statements can be
true in more than one way, then more than one line is required. For example,
all of the following can be true in three ways, thus we may need three lines.
Remember, if one is filled in consistently, then stop! They are consistent.
A  (B  C) / C  ~ A / B v A / B  C
Try this dandy argument:
N v ~O / P v O / P  Q / (N v Q)  (R  S) / S  (R  T) /O  (T  U) // U
Here are some groups of statements:
A v B / B  (C v A) / C  ~ B / ~ A
(G V ~ Q )  ( F v B) / ~ ( F v Q) / B  N / ( F v N )  Q