The semantic argument improved even more

The core premise of my semantic argument is an entailment consisting of an antecedent and a consequent. An early version of the core premise states that if two concepts have the same reference set, then they are identical. Over the last few years, I have been able to improve the core premise by weakening its consequent. By improving the core premise I mean rendering it more likely to be true. Introducing a weaker consequent indeed improves the core premise, since it renders it less demanding and thus increases its plausibility.

As I have shown in the linked paper, rather than claiming as a core premise that, for any two concepts, sameness of reference sets entails sameness of meaning, it suffices for the derivation of the semantic argument's conclusion—namely, that there are no contingent universal properties—to claim that sameness of reference sets entails that it is a conceptual truth (and thus necessary) that the references of the two concepts coincide. Moreover, I have shown that it is in fact already sufficient to only claim that sameness of reference sets entails that, necessarily, the references of the two concepts coincide. This is the core premise in its current form. In all cases, the scope of the concepts is restriced to concepts with existential import that correspond to properties that are both positive and attaching.

A further way of improving the core premise is to strengthen its antecedent rather than weaken its consequent. Indeed, a stronger or more demanding antecendent further increases the plausiblity of the core premise. This is precisely the strategy I propose to adopt here in order to improve the core premise even more. Rather than requiring, as the current antecedent does, that two concepts have the same reference set, I introduce a significantly stronger antecedent condition. To this end, I first introduce some terminology

Let S be a set of concepts. Define D(S) as the set obtained by replacing each complex concept in S with its component concepts. For example, D({object, blue car}) = {object, blue, car} and D({sandbeach, blue}) = {sand, beach, landform, blue}. Moreover, define R(S) as the union of the references of the concepts in S. The current core premise of the semantic argument can be stated as follows. Let S1 and S2 be two concepts. If R(D({S1})) = R(D({S2})), then, necessarily, the references of S1 and S2 coincide.

The proposed even more improved version of the core premise is obtained by replacing the current antecendent—namely, R(D(S1)) = R(D(S2))—with the following recursively defined infinite conjunction of propositions:

R({S1}) = R({S2}) and
R(D({S1})) = R(D({S2})) and
R(D(D({S1}))) = R(D(D({S2}))) and
R(D(D(D({S1})))) = R(D(D(D({S2})))) and
R(D(D(D(D({S1}))))) = R(D(D(D(D({S2}))))) and

...ad infinitum.

This new antecedent contains the current core premise's antecedent as its second conjunct and is thus stronger than the current core premise. It can be stated more compactly as

R(D^n({S1})) = R(D^n({S2})) for all natural numbers n.

Here D^0(S)=S and D^(n+1)(S)=D(D^n(S)). The resulting further improved core premise thus states that if the above more demanding antecedent condition obtains, then the references of S1 and S2 necessarily coincide. As can be shown by verifying the derivation of the semantic argument's conclusion in the linked paper above, this even more improved core premise still entails the conclusion of the semantic argument. Therefore, the semantic argument itself is improved even more.

Moreover, I take it that the intuition underlying the even more improved core premise is more compelling than the intuition underlying its predecessor. Consider two concepts that are not only compared at the level of their component concepts, but are recursively decomposed ad infinitum into their constituent component concepts at every level of conceptual decomposition. Suppose that, at each such level, the unions of the references associated with the corresponding decomposed component concepts of both concepts coincide exactly. Under these conditions, it is plausible that the two concepts in question are identical, or at least necessarily have the same reference.

This is precisely the claim expressed by the further improved core premise. Shifting from a single-level comparison of reference sets to a fully recursive decomposition of component concepts at each level of decomposition, substantially reduces the space for potential counterexamples. Indeed, exhaustive referential coincidence at every level of conceptual analysis appears to leave no room for genuine conceptual divergence. Therefore, the intuitive reason to accept the newly proposed core premise is significantly stronger than the intuitive reason to accept its predecessor. The same thus holds for accepting the semantic argument's conclusion.