Quantifier Negation Rules
We discussed the matter of the negation of quantified propositions back in Chapter 7. In the section, "Quantified Negations" we noted that
| ~(x)Fx | is logically equivalent to | (∃x)~Fx |
| ~(∃x)Fx | is logically equivalent to | (x)~Fx |
We will refer to these as our basic quantifier negation rules.
In Chapter 8, we showed that an A categorical proposition is logically equivalent to the negation of an O categorical proposition, and that an E categorical proposition is logically equivalent to the negation of an I categorical proposition. Thus we have the following categorical quantifier negation rules:
| (x)(Sx ⊃ Px) | is logically equivalent to | ~(∃x)(Sx & ~Px) |
| (x)(Sx ⊃ ~Px) | is logically equivalent to | ~(∃x)(Sx & Px) |
and it also goes the other way:
| ~(x)(Sx ⊃ Px) | is logically equivalent to | (∃x)(Sx & ~Px) |
| ~(x)(Sx ⊃ ~Px) | is logically equivalent to | (∃x)(Sx & Px) |
These equivalences form the Quantifier Negation Rules (Q.N.R.) for QL. They function just like the Equivalence Rules. A wff can be replaced with a logically equivalent wff. You can think of teh Q.N.R. as an extention of the Equivalence Rules, but you must cite Q.N.R. specifically when you use a quantifier negation equivalence.
The last four equivalences, the categorical quantifier negation rules, are actually superfluous. We can derive them from a combination of the basic quantifier negation rules and our equivalence rules. Here's a proof of:
| ~(∃x)(Sx & ~Px) |
| (x)(Sx ⊃ Px) |
| 1. | ~(∃x)(Sx & ~Px) | Premise |
| 2. | (x)~(Sx & ~Px) | Q.N.R., 1 |
| 3. | (x)(~Sx v ~~Px) | DeM. 2 |
| 4. | (x)(~Sx v Px) | D.N. 3 |
| 5. | (x)(Sx ⊃ Px) | C.E. 4 |
Of course, to show equivalence, we need to show that the biconditional of the two wffs is a theorem. That's left as an exercise for you! And here's a summary of the Quantifier Negation Rules.
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