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,
|(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.
- Symbolize in PPL using the dictionary provided
- Nothing is a flea. (F1: is a flea)
- There are no chairs. (C1: is a chair)
- No dogs allowed. (D1: is a dog; A1: is allowed)
- No-one can solve this problem. (P1: is a person; S1: can solve this problem)
- Nothing caused the event. (C1: caused the event)
- There's no-one who can stop me now. (P1: is a person; S1: can help me now)
- ~∃ (F1)
- ~∃ (C1)
- ~∃ (D1 & A1)
- ~∃ (P1 & S1)
- ~∃(P1 & S1)