## Negation of Existentials

Recall from Chapter 4 that the expression "nothing" may be replaced by "not something". In PL, we are treating,

(1) Nothing is a mouse.

as a paraphrase of,

(2) It is not the case that something is a mouse.

In PPL, "Something is mouse" is symbolized as,

 (3)∃ M1 M1: is a mouse

So the original statement, ‘Nothing is a mouse’, is symbolized in PPL as,

(4) ~ M1

If we follow the rules for replacement from section 4.2, negations of existentials are easy to symbolize.

How should we treat a statement like ‘There are no mice’? We have no rules for replacement to implement here, but it seems that it has the same meaning as ‘Nothing is a mouse’. If this is so, then it too should be symbolized as,

(5) ~ M1

This appears to be a successful technique for writing statements of the form of ‘There are no x's ’. In general, such statements should be treated as negations of existentials.

Exercises

Statements

• Symbolize in PPL using the dictionary provided
1. Nothing is a flea. (F1: is a flea)
2. There are no chairs. (C1: is a chair)
3. No dogs allowed. (D1: is a dog; A1: is allowed)
4. No-one can solve this problem. (P1: is a person; S1: can solve this problem)
5. Nothing caused the event. (C1: caused the event)
6. There's no-one who can stop me now. (P1: is a person; S1: can help me now)

Solutions

1. ~ (F1)
2. ~ (C1)
3. ~ (D1 & A1)
4. ~ (P1 & S1)
5. ~(C1)
6. ~(P1 & S1)