Kreisel's squeezing argument and Shapiro's categoricity results suggest second-order logical consequence has a determinate extension fixed by the standard semantics, independent of set-theoretic indeterminacy.
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Shapiro(The statement refers to her specific theoretical requirements)
A philosopher (Laurie Shapiro) who studies the philosophy of mind and cognitive science; she has argued specific criteria for when we should consider something a real scientific category.
Standard semantics(as a way of interpreting logical systems)
The most straightforward interpretation of what logical statements mean in terms of the real world—basically, the 'normal' way of understanding truth.
determinate extension(as used in philosophy of language and metaphysics)
Having clear, definable boundaries—knowing exactly what things count as examples of something and what doesn't.
logical consequence(WL II, 391–395; noted as similar to Tarski 1956, 419)
A proposition s is a logical consequence of a set of premises σ if and only if s follows from σ with respect to the sequence of all extra-logical simple ideas contained in σ or s.
squeezing argument(philosophy of logic)
Kreisel's argument that uses proof theory to support the model-theoretic definition of logical consequence, depending on the language having a sound and complete proof system