Negation of Existentials

An additional section to be read at the end of 9.4:Quantifiers

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,

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)