Fregean semantics for a realist ontology.
Nino B. Cocchiarella · Notre Dame Journal of Formal Logic · 1974
T* is a logistic system 1 designed to represent the original ontological context behind RusselΓs paradox of predication.It encompasses standard second order logic, hereafter referred to as T, but goes beyond it by allowing predicate variables to occupy subject positions in its formulas.Because of a violation of the restrictions imposed for the proper substitution of a formula for a predicate variable, RusselΓs argument fails in T*.Indeed, not only is T* consistent but it is also a conservative extension ofT. 2 Nevertheless, T* is not without its oddities.E.g., although "the Russell property" of being a property which does not possess itself does not exist in the ontology of T*, the modified Russell property of being an individual which is indiscernible (in the sense of having all properties in common) with a property which that individual does not possess does exist in this ontology.Instead of leading to a contradiction, Russell's argument applied to the modified Russell property shows that the principle that properties which are indiscernible are co-extensive is disprovable in T*, i.e., according to the ontology of T*, there are properties which are indiscernible (in the sense indicated above) but which nevertheless are not co-extensive.3 It has been suggested that one way of understanding this result is to construe occurrences of predicates in subject positions as referring, not to the properties which occurrences of the same predicates in predicate positions designate, but instead, to individual objects associated with these 1.The author was partially supported by NSF grant GS-28605. Cf. [2], §6.3. Ibid., §5.We should avoid using 'identical' in place of indiscernible' here.In[3], Meyer has shown that according to T* there exists no relation which satisfies full substitutivity, and, accordingly, insofar as full substitutivity is taken to be a necessary feature of identity, there is and can be no identity relation in the ontology of T*.