Examples 6.5
Indirect Truth Tables
These things are useful, but not somewhat tricky to catch on to.
The basic idea is as
follows: instead of doing an entire truth table for an argument with lots of components
and lots of premises, we’ll just do a short cut by assuming the argument’s invalid
(premises are true while the conclusion is false), then working backward we’ll see
whether it’s
possible
to fill in the truth values to make such a row (in a truth table).
Intuitively, if it’s possible to make such a row (where the premises are true and the
conclusion is false), then we know that argument is invalid.
If it’s not possible, then we
know the argument is valid.
So, instead of explaining, let’s just look at this concept in action, and maybe it will make
more sense.
Start with this argument:
If ~A then (B v C)
~B
If C then A
First, lay out the argument like a truth table:
If ~A then (B v C)

~B

If C then A
Then lay out the truth values
under the operators
to make the argument invalid.
If ~A then (B v C)

~B

If C then A
T
T
F
Now, since the conclusion can only be falsified by ONE arrangement of truth values (i.e.,
where the antecedent is true and the conclusion is false—according to the truth function
for conditionals), we can fill out the truth values for the components.
(I’m gonna bold the
TVs under the main operators so things don’t get (more) confusing).
If ~A then (B v C)

~B

If C then A
T
T
T
F
F
Now, all we have to do is go back and fill in the rest of the table.
We know that all of the
‘C’s have to be T, and all of the ‘A’s have to be false—so go do it. Also, we’re can flip
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Barrett
 Logic, Conclusion, Logical connective

Click to edit the document details