 Instead of talking about if-then-ism in this connection, one can describe Putnam’s position also as a kind of “universalist structuralism” (see again Reck &amp; Price 2000), since it involves universal quantification over relevant systems and since our two structuralist slogans above seem satisfied.  Often an objection to this position is the “non-vacuity problem”. It is based on the observation that if-then statements of the given form are vacuously true if there is nothing that satisfies the antecedent, e.g., if there is no model of the Dedekind-Peano axioms. (As this would also be the case for, say, “\(2+3=6\)”, the overall result is clearly not desirable.) In response, one can invoke axiomatic set theory as providing the needed models.  But from Putnam’s point of view this has two drawbacks: First, it relies on a realist and foundationalist view about set theory, it would seem, thus undermining the thrust of if-then-ism. Second and more basically, it forces us to treat set theory differently from other mathematical theories, on pain of circularity. As a way out of these dilemmas, Putnam suggests to employ modal logic. However, the details are again left open and largely unexplored, especially for the case of set theory.