How to Misapply QL Rules
The generalization and instantiation rules are very straightforward, but it's easy to misapply them. The crucial thing to remember is that you can only instantiate quantified wffs. For example, the following is correct:
| 1. | (x)(Mx ⊃ Px) | premise | |
| 2. | (Ma ⊃ Pa) | U.I, 1 |
but the following is incorrect:
| 1. | (x)Mx ⊃ (x)Px | premise | |
| 2. | (Ma ⊃ Pa) | U.I, 1 |
The reason is simple: In the second example, the premise is a
conditional. It is not a universally quantified wff. For the same reason,
you can't instantiate a part of a wff. So the following is a misapplication of
EI:
| 1. | (Ma ⊃ (∃y)Py | premise | |
| 2. | (Ma ⊃ Pb) | E.I, 1, flag a |
The same thing goes for the generalization rules. Here's another misapplication involving generalization:
| 1. | (Ma ⊃ Pb) | premise | |
| 2. | (Ma ⊃ (∃y)Py) | E.G, 1 |
Here the attempt is made to existentially generalize from Pb. But Pb is the consequent of the conditional, and you can't apply the rule to part of a wff.
If you're trying to a generalization rule, but what you infer is not a generalization, you've got a problem. For example:
| 1. | (Ma ⊃ Pa) | premise | |
| 2. | (x)Ma ⊃ (x)Px | U.G, 1 |
What we've attempted to infer on line 2 is a conditional. It is not a universal generaliation!
Another source of error is the use of negation.
| flag a | SP1 | ||
| ~Ma | (derived from earlier lines | SP1 | |
| ~(x)Ma | UG, |
This is not universal generalization, because the inferred line is not a universally quantified wff! What one can get legitimately is:
| flag a | SP1 | ||
| ~Ma | (derived from earlier lines | SP1 | |
| (x)~Mx | UG, |
which is a universally quantified wff!
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