Positive and negative reference
Where some people might write
($x)(Px & Qx)
i.e. something is both P and Q, or some P are Q, I might write:
E (p, q)
where E means ‘there is’ and the comma represents ‘concept intersection’. This is an important operator in the field of Description Logic (see Baader et al (2003) below), an area of logic concerned with knowledge representation. It may often be glossed as ‘p which is q’ or ‘q which is p’ and indeed (p,q)=(q,p). It also corresponds to set intersection – see Venn diagram opposite – but works directly with the concepts not the sets.
It is very easy to bring in relationships, thus:
E(p, Rq)
i.e. Some p is/are an R of (a) q, eg. p is the father of (a) q.
Now here we may express an inverse:
E(R-1p, q) or indeed E(q, R-1p)
which means that some q has p as father. The inverse of father-of is not son-of (any more say than dog is the inverse of cat, whatever Violet Elizabeth Bott of Just William might think.)
Now we may here introduce an operator NON, to give
E(p, NON q)
i.e. Some p are not q, which we may rephrase as ‘not all p are q’.
That’s easy: and indeed we can now do something where Descriptive Logicians usually fear to tread - invert NON just as we have done with R (with the meaning suggested perhaps ‘father-of’.)
E(NON-1p, q) or again E(q, NON-1p)
Again the same as above, so as well as saying some p are not q, we can say not all p are q, OR q are not the only p. In certain contexts NON-1 means ‘not all’ and in others, ‘not the only’. Furthermore, as we can see from the above Venn diagram, we have here a concept which is only meaningful in the context of what surrounds it.
Now from the above discussion I can pick out that I would use e to describe ‘something’, so that (p, e) means ‘p which is something’, or ‘p which there is’ or simply p, so that in effect (p, e)=p. Thus e can be treated as that concept which when combined with p in concept intersection gives p, in effect an ‘identity co-ordinate wrt concept intersection’. (As every philosopher will know, wrt=with respect to, or sometimes meaning without respect to... wrt the editor.)
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Note that instead of ‘exists’ we might say ‘is a member of the universe of discourse’: if I say there’s a dragon I read about, it doesn’t mean there exists a dragon I read about, but the dragon is a member of the universe of discourse.
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And likewise ‘nothing’ is simply NON e, where (p, NON e) = NON e, an ‘annihilating co-ordinate wrt concept intersection’. But does ‘nothing’ - NON e – have any meaning except in the context of what surrounds it?
To my mind, 'nothing' only has meaning either in a sentence of the form 'it is not the case that ... something...' or the form 'everything... NON....' For example, 'nothing is perfect' can be interpreted as 'It is not the case that something is perfect' or 'Everything is NON perfect.'
But it seems to me that anything starting with NON-1 is an anti-concept, and next issue I shall argue that magic is an anti-concept.
Reference
F. Baader, D. Calvanese, D. L. McGuinness, D. Nardi, P. F. Patel-Schneider (2003) The Description Logic Handbook: Theory, Implementation, Applications. Cambridge: Cambridge University Press.
ISBN 0-521-78176-0
By Martin Prior
The Philosophy Takeaway 'Something/Nothing' Issue 24
