Classes, Why and How

Thomas Schindler, (2019) Philosophical Studies 176:4-7-435. 

Thomas (who I know from when he was a JRF at Clare College) advances a type-free theory of classes. This is not my field, and I’m hardly in position to evaluate the techical success of his attempt—though it seems plausible enough to me! My interest was piqued because (a) Thomas, and (b) his approach is to limit the “range of significance” of predicates or propositional functions, such that attempting to evaluate them at the excluded points (“singularities”) is impermissible, and yields no outcome. If one treats the problem cases—the set of all sets that aren’t members of themselves, for example—as such singularities, then one thereby avoids the paradoxical outcomes. (From one angle, this is Russell’s approach too, but abandons Russell's constraint that ranges of significance must be clumped into types.)  Thomas is inspired, he says, by some remarks of Gödel, but this approach also resembles my own favorite response to the semantic paradoxes, cassatio

Cassatio treats the paradoxical sentences as semantically improper.  The Liar, for example, has no truth value, because it fails to express a proposition and thus has no truth conditions. Crucially, cassatio does not ascribe some third, non-classical truth value (such as µ)—an approach which only invites so-called revenge paradoxes. (µ is not T, so “this sentence is not true” generates a paradox even if you try to assign it µ.) Rather, on cassatio, the Liar has no truth value at all, because it doesn’t assert anything. Calling it true and calling it false are both wrong, but non-paradoxical, in much the same way calling the planet Mars “true” or “false" would be.

As I say, I think Thomas’s approach is plausible; and I can’t evaluate its technial success. But from the point of view of a cassatio fan, I have two questions/suggestions: 

— Why are the singular points singular? A successful treatment of the paradoxes, it seems to me, needs to explain why things go wrong there, and not merely exclude them arbitrarily for convenience. (Though if we are arbitrarily excluding things, excluding fewer is better—on which count Thomas’s approach, if consistent, is better than Russell’s, for excluding only the paradoxical cases.)

Cassatio excludes the Liar on the basis that it viciously delegates its truth conditions to itself. If McX says “P” and I say “what McX says is wrong”, then there is no problem: I’ve delegated my truth conditions to McX’s utterance. But with the Liar, we follow the reflexive arrow of delegation around in a circle forever, never getting to a genuine assertion. 

I suspect that a similar approach can explain why the Russell set is problematic: the intension S = {x|x∉x}, when applied to S itself, circularly delegates S’s inclusion conditions to itself. So perhaps here cassatio can supplement Thomas’s approach.

—What about the Truthteller? An intuitive way to motivate cassatio is to think about “This sentence is true”. There is no paradox there: it is true if it is true, and false if it is false. But which is it? There is no answer to that question, because the Truthteller doesn’t make any real claim. Cassatio treats the Truthteller as improper for the same reason as the Liar: it doesn’t, despite appearances, ultimately say anything.

The Truthteller’s set analogue is S = {x|x∈x}, applied to S. Is it a member of itself? It is if it is, and it isn’t if it isn’t, so no paradox; but which is it? Again you follow a reflexive arrow of delegation forever without getting to an actual inclusion condition. 

What does Thomas’s approach have to say about that case? I would assume it is ripe for the same treatment, so that both problem cases are excluded. This is slightly more restrictive than excluding only the paradoxical cases, but still much less so than Russell’s.

Anyway, fun!