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Notes
“The Logic of Existence,”Philosophical Studies, 7:49–64 (1956).
I use the symbolism ‘Λa · F(a)’ to denominate that characteristic exhibited by a if and only if a satisfies the conditionF.
Op. cit., p. 57.
Leonard appears to hold (p. 58, ibid.) that every object has a contingent property, but his discussion here overlooks abstract objects. His proof that any individual x “exists” in the sense of (4) is in error on precisely this point.
By substitution of ‘∼ϕ’ for ‘ϕ’ (8) could be reformulated as Λx · (ε ϕ) (ϕx & ◊ (ε3 y) ∼σy). In comparison with (4) we see that the only point of difference is that the individual which possibly does not exhibit σ need not be x itself.
It is assumed that “t” is so defined that (y) (ψy ⊃ ϕy) ⊃ ϕ(ex)ψx. This assumption is false in Principia Mathematica in view of the particular definition of “t” used in that work (see *14). But other, alternative definitions of “Y” are available for which this assumption is satisfied. One such is the contextual definition: F(tx)ϕx =Df(y) (yε the x's such that ϕx ⊃ Fy). With this definition, the assumption is readily seen to be satisfied. Here it is easily shown that (tx)ϕx is specified as follows: (1) if there is exactly one individual for which ϕ holds, say a, then (tx)ϕx = a (this can be seen by letting F{X} be X = z in this case), (Z) if there is no individual for which ϕ holds, then (tx)ϕx = Λ (let F{X} be X ⊃ z), so that in this case (tx)ϕξ is the null class, and (3) if there are several individuals for which ϕ holds, then (tx)ϕx is some random or representative element of the x's such that ϕx. The definition under consideration is thus very close to that given by Frege inGrundgesetze der Arithmetik (vol. 1, Jena, 1873, p. 19), who defines (tx) ϕx to bea, Λ, or (all of) the x's such that ϕx, respectively in these three cases.
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Rescher, N. Definitions of “existence”. Philos Stud 8, 65–69 (1957). https://doi.org/10.1007/BF02304902
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DOI: https://doi.org/10.1007/BF02304902