Critical
Thinking
Across the
Curriculum Project
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Critical
Thinking
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If P then Q and If Q then P
Now this statement is badly formed, because we cannot tell which logical operator is modifying each of the statements. The addition of some parentheses, however, will make things much clearer:
(if P then Q) and (if Q then P)
That should help - now we can see that what we have is two conditionals linked b a conjunction. The question now becomes: "which logical connective is the primary one?" To decide this, look at what is contained by the parentheses. In order to find the truth value of the conjunction, you would first have to find the truth value of each of the conjuncts, i.e.- the conditionals. So you would have to work out a truth table for each of the conditionals first - in effect working from the innermost set of parentheses outward. Following these steps may help-- Ask yourself:
Here is an example of the truth table for the above statement:
| (P>Q)&(Q>P) | ||||
|---|---|---|---|---|
| P | Q | (P>Q) | & | (Q>P) |
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | F | F |
| F | F | T | T | T |
An argument has a VALID form if and only if the form is such that:
If the premises were all true, then the conclusion must be
true.
If you notice, the definition of validity is expressed as a conditional
statement. This means that we can an argument in the form of a conditional
statement which has as the antacedent the conjunction of the premises (of
the argument) and which has as the consequent the conclusion of the argument.
Once we have done this, we can construct a truth table where the conditional
will be the main connective and the conjuction of the premises will be
a subordinate statement. If all of the truth values under the main connective
(the conditional) are true (if the statement is a tautology) then the argument
is VALID since it will not be possible for all of the premises to be true
and the conclusion false. If any of the truth values under the main connective
are false, then the argument is INVALID since is is possible for the premises
to all be true and the conclusion false (this is the only situation in
which a conditional is false. See the truth table
for a conditonal to verify). Below are the truth tables for Modus
Ponens and Affirming the Consequent (respectively):
| Modus Ponens | ||||
|---|---|---|---|---|
| P | Q | ((P>Q) | &P) | >Q |
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | F | T |
| F | F | T | F | T |
| Affirming the Consequent | ||||
|---|---|---|---|---|
| P | Q | ((P>Q) | &Q) | >P |
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | F | T |
Notice that in the final row of the truth table for Modus Ponens, the truth value of the conditional is always true (a tautology if it were a statement), while in the final row of the truth table for Affirming the Consequent, there is one row where the conditional is false. This row represents the combination of truth and falsity of the simple statements which makes it possible for the conjunction of the premises to be true, and the conclusion false.
Copyright
© 1996
Critical Thinking Across the Curriculum Project
Longview Community
College , Lee's Summit, Missouri - U.S.A.
One of the Metropolitan Community Colleges
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Permission to reproduce these resource pages is granted for
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Inquiries to: michael.connelly@mcckc.edu
Last modified: 03/02/04