Critical Thinking
Across the
Curriculum Project
Truth Tables for Compound Statements:
The most difficult parts of constructing truth tables are the initial set-up
of the table, and remembering the truth tables for the different logical
connectives. Lets deal with the first problem:
Remember - A simple statement can only have two possible truth values
- It can be true, or it can be false. This means that there are only two
possible truth values for any given statement. When dealing with Compound
statements (simple statements whose truth value is altered by one or more
logical connectives) it is essential to know how many simple statements
are involved. If only one simple statement is being modified, then only
two rows are needed (as in negation). If two simple statements are involved,
then four rows will be needed to exhaust all of the possible combinations
of truth and falsity. Here are two truth tables which (hopefully) will
illustrate the difference between all of the possible combinations of truth
and falsity for a simple statement and two simple statments:
Truth Table for One simple statement
| A |
|
T
|
|
F
|
Truth Table for Two simple statements
| P |
Q
|
|
T
|
T
|
|
T
|
F
|
|
F
|
T
|
|
F
|
F
|
. If three simple statements are being combined using logical connectives,
then eight rows will be involved in order to capture all of the possible
combinations of truth and falsity for those simple statements. Here is
an example of such a Table:
Truth Table for Three simple statements
|
P
|
Q
|
R
|
|
T
|
T
|
T
|
|
T
|
T
|
F
|
|
T
|
F
|
T
|
|
T
|
F
|
F
|
|
F
|
T
|
T
|
|
F
|
T
|
F
|
|
F
|
F
|
T
|
|
F
|
F
|
F
|
Notice that the pattern of the combinations of truth and falsity from the
table for two simple statements appears in the columns under Q and R for
the first four rows, and then repeats itself for the second four rows (5-8).
Likewise, in the table for two simple statements, the pattern for the combinations
of truth and falsity for one statement are repeated twice in the column
under Q. Once you know how to set them up, you can then concentrate on
remembering the truth tables for each of the logical connectives. The tables
for each of these follow below:
Truth Table for Conjunctions
|
1st Conjunct
|
2nd Conjunct
|
Statement
|
|
A
|
B
|
A and B
|
| true |
true |
true |
| true |
false |
false |
| false |
true |
false |
| false |
false |
false |
Truth Table for Disjunctions
|
1st Disjunct
|
2nd Disjunct
|
Statement
|
|
A
|
B
|
A or B
|
| true |
true |
true |
| true |
false |
true |
| false |
true |
true |
| false |
false |
false |
Truth Table for Conditionals
|
Antacedent
|
Consequent
|
Statement
|
|
A
|
B
|
A > B
|
| true |
true |
true |
| true |
false |
false |
| false |
true |
true |
| false |
false |
true |
The conditions of truth and falsity for conditionals is the hardest of
all the compound statements to understand. Perhaps it is best to remember
that a conditional is a hypothetical statement - such that it is asserting
that IF...some state of affairs were true, THEN.... some other state of
affairs would be true. The tricky thing about this is that if the antacedent
of the conditonal is false, then just about anything follows from it. So,
for example, the statement, "if today is Wednesday then tomorrow is Tuesday"
would be true today (Monday) but false the day after tomorrow.
This is correct (in a sense) IF you are considering the truth value
of the entire conditional. IF it was true that "today is Wednesday" on
any given Monday, Then time would be so screwed up that tommorow could
be Tuesday. What throws most people off the track is that they are confusing
the truth value of the conditional statement with the truth value of the
simple statements which make it up. In essence, anything can be true if
a contradiction can be true.
Truth Table for Negations
|
Statement
|
Negation
|
|
A
|
not A
|
| true |
false |
| false |
true |
Truth Tables can also be constructed to determine the validity of an
argument form. These are a little more complicated, but if you understand
the basic principles used to construct Truth Tables for Statements, then
you can do it for arguments or more complex statements
(with multiple logical operators) as well.
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Copyright
© 1996
Critical Thinking Across the Curriculum Project
Longview Community
College , Lee's Summit, Missouri - U.S.A.
One of the Metropolitan Community Colleges
An Equal Opportunity/Affirmative Action Employer
Permission to reproduce these resource pages is granted for
non-profit educational use provided the above information
is retained on all copies.
Inquiries to: michael.connelly@mcckc.edu
Last modified: 03/02/04