Wednesday 21st October, 1914: In his diary, LW records that they have been told that they will be going back to Krakow, and that ‘that would not be disagreeable to me’. All day he has been in Sandomierz. He records having worked (on philosophy) with confidence, but being too tired in the evening, and prone to depression. ‘But take heart!’ he urges himself (GT1, S.46). LW asks himself whether Russell’s definition of zero isn’t nonsensical. Can we speak of a class ẍ(x ≠ x) at all? Or even of a class ẍ(x = x)? Is either x ≠ x or x = x a function of x? Mustn’t 0 then be defined by means of the *hypothesis* (Ǝφ):(x) ∼φx? Something analogous would also hold of all the other numbers. This throws light on the whole question about the existence of numbers of things. He then defines: 0 = a {(Ǝφ):(x) ∼φx.a = û(φu)} Def. 1 = â {(Ǝφ)::(Ǝx).φx.φy.φz ⊃y = z:a = û(φu)} Def. The proposition must *contain* (and thus show) the *possibility* of its truth, but not more than the *possibility*. He remarks that he *had* thought that the possibility of the truth of the proposition φa was tied up with the fact (Ǝx,φ).φx, but this can’t be right since it’s impossible to see why φa should only be possible if there is another proposition of the same form. ‘Φa surely does not need any precedent’, since if there existed only the two elementary propositions ‘φa’ and ‘ψa’, but ‘φa’ was false, it should not make sense only if ‘ψa’ was true (NB, pp.16-17. See Potter, pp.110-111).
Posted on: Tue, 21 Oct 2014 06:22:42 +0000
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