(q 
r)
rValid and Invalid Argument Forms
1. The validity of the following argument is confirmed by the critical rows of the truth table as shown below.
| p
(q
r) | |
| ~ p | |
| q
r |
| p | q | r | p
(q
r) | ~ p | q
r |
| T | T | T | T | F | T |
| T | T | F | T | F | F |
| T | F | T | T | F | F |
| T | F | F | T | F | F |
| F | T | T | T | T | T |
| F | T | F | F | T | F |
| F | F | T | F | T | F |
| F | F | F | F | T | F |
2. An invalid argument form can likewise be demonstrated by truth tables.
| p
(q r) | |
| ~(p
q) | |
| r |
| p | q | r | p
(q
r) |
~(p q) | r |
| T | T | T | T | F | T |
| T | T | F | T | F | F |
| T | F | T | T | T | T |
| T | F | F | T | T | F |
| F | T | T | T | T | T |
| F | T | F | F | T | F |
| F | F | T | F | T | T |
| F | F | F | F | T | F |
While rows 3, 4 and 5 indicate valid (true) premises, the 4th row reveals a false conclusion (indicated by dark blue); therefore, the above argument form is invalid. Notice that it is possible to have multiple critical rows, and remember that for an argument to be valid, all critical rows must have true conclusions!
A valid argument for a propositional
well formed formula (wff) say P1
P2
P2
...
Pn 
Q is a valid argument when it is a tautology (where the P's are propositions).
In this context, when we consider truth tables and the conclusion is connected
with the premise(s) using an implies (i.e., ),
the following statements can be made:
Valid implies that the argument must be true for all instances (i.e., all rows end in true)
Invalid implies that the argument is not true for all instances
An argument is satisfiable if at keast one instance is true, and is not satisfiable if all instances end in false.
Note, that a valid argument is satisfiable, invalid arguments may be satisfiable unless they are not satisfiable (just think about it).
