In the case of counting LEM as an in-rule for disjunction, what happens — I suggest — is that we get this sort of lapse of intelligibility unless we can back it up with the kind of semantic/metaphysical considerations characteristic of realism. Regular Or-in and Or-out underwrite reasoning by cases. But if we reason by cases from LEM counted simply as a further primitive Or-in rule, we can wind up— modulo certain collateral hypotheses which may impress as locally plausible — making disjunction-free claims that impress as unjustified (for instance, that there is a sharp cut-off in a soritical series for “bald”.)

Whether the N-rules, conceived as determinant of the inferential roles both of the cardinality operator and =, fall out on the right side of this distinction is another matter. I don’t know of any strong prima facie reason to doubt it. The symmetry of the two rules seems enough to ensure their harmony wrt “N”. So the issue is plausibly taken as whether the identity contexts that N-in lets us prove, justify the claims that they ought to justify in the light of identity-out. That’s a question of detail (essentially it is the question whether every open sentence “A…” defined for terms of the form, “N(F)”, corresponds to a condition, X…, on concepts for which 1-1 correspondence is a congruence.)

I guess we won’t really understand these matters until, if ever, we have a better handle on the notion of harmony and its connections with ‘making sense’.

For that, something like Fa, a=b |- Fb is what is usually asked for (with real questions concerning what’s appropriate to substitute in the F slot). Then it depends on whether one’s logic is going to give you Fa and ~Fb, but if it does (if, say, it can tell us that 0 is even and that 1 is not, for example), we’re done: we get a≠b.

Independently of your thoughts on autonomy as a principle, it’s an interesting question to ask what conceptual resources suffice to give a proof of ∃x ∃y (x ≠ y), and what different minimal bases there might be.

We can define a narrow notion of ‘logical’ where all logical truths are autonomous. But it’s no part of logicism that mathematical truths are logical in this narrow sense. In fact perhaps there are no existential claims that are narrow in this sense; or perhaps just the fact that something exists is logical.

What is distinctive of logicism is that we have a special reason to believe in the truths of mathematics. And this reason is analogous in certain respects to the reasons we have to believe in logical truths in the narrower sense. In particular, it might be that mathematical truths can be properly derived from the inferential rules constitutive of mathematical concepts.

This does leave open a kind of possibility that a classical logician who is also a logicist might want to exploit. Perhaps it is partially constitutive of → that from (p → ⊥) → ⊥ we can infer p. Or it is partially constitutive of ∨ that in from zero premises we can infer p ∨ ¬p. Perhaps that’s a second intro rule for ∨ that we can only understand once we comprehend ¬.

The classical logician who says this won’t say that Pierce’s Law is logical in the narrow sense, any more than ∃x ∃y (x ≠ y) is. But Pierce’s Law does have, they say, the same kind of justification as mathematics does, because it can be derived from inferential roles.

My guess is that you’d say that such a position is coherent, but as a matter of fact, we have reason to think that these rules (either double negation elimination or law of excluded middle) are not part of the inferential role of → or ∨. Is that right?

Is it legitimate for putative inferential role characterisations to work on more than one concept at a time? Well, why not – provided they are at least consistent with prior partial characterisations?

What does any of this have to do with logicism? Not much, I think, unless we insist that all logical truths, properly so regarded, are autonomous truths. But logicism is an epistemological theses and I am not sure that much of epistemological interest hangs on the distinction between those truths that can be deduced using just the proof theory for the constants featuring explicitly in them and those that rest in addition on truths established via the proof theory for additional notions that ‘disappear’ under e.g. an application of the in-rule for their main operator. Presumably no existential statement can count as a logical truth except under the more relaxed rubric.

Another way to go, in line with Crispin’s general approach, is to take the N rules to somehow play a role in a larger family of rules for singular terms, and hence, for the existential quantifier, since those rules depend on the range of singular terms for their own field of applicability.

Then the N rules tell us something of how numerical terms work, and since they’re singular terms, they feed into the existential quantifier rules.

Then since identity rules are difficult — I think the best way to understand rules for identity use predicate ‘variables’ (following Stephen Read’s nice paper “Identity and Harmony” Analysis 2004) — I don’t find it surprising that you get the failure of the subformula property and a proof of something beyond what is given in first order logic with identity alone.

What this means for logicism is another matter.

I was thinking that the N-in and N-out rules were rules for N, not =. So the rules for = would just be reflexivity of identity, and indiscernibility of identicals. It’s true that if N-in and N-out are rules for =, then we have a proof of ∃x ∃y (x ≠ y). But I’m unsure why we should think Hume’s principle provides rules for identity, rather than rules for numbers.