The final section of McGee’s paper is called “A rule for “everything”‘. He argues that “the semantic values of the quantifiers are fixed by the rules of inference”. The claim rests on noting that (i) two universal quantifiers governed by e.g. the same UE and UI rules will be interderivable [McGee credits “a remarkable theorem of J.H. Harris”, but it is an easy result, which is surely a familiar observation in the Gentzen/Prawitz/Dummett tradition]. McGee then claims that, assuming the quantifier rules don’t misfire completely [like the tonk rules?], this implies that (ii) they determine a uniquely optimal candidate for their semantic value. And further, (iii) “the Harris theorem … gives us reason to anticipate that, when we develop a semantic theory, it will favor unambiguously unrestricted quantification.”
The step from (i) to (ii) needs some heavy-duty assumptions — after all, the intuitionist, for example, doesn’t differ from the classical logician about the correct quantifier rules, but does have different things to say about semantic values. But McGee seems to be assuming a two-valued classical background; so let that pass. More seriously in the present context, the step on to (iii) is just question-begging, if it is supposed to be a defence of an absolutist reading of unrestricted quantification. Consider a non-absolutist like Glanzberg. He could cheerfully accept that the rules governing the use of unrestricted universal quantifiers fix that they run over the whole background domain, whatever that is (and that a pair of quantifiers governed by the same rules would both run over that same domain): but that leaves it entirely open whether the background domain available to us at any point is itself contextually fixed and can be subject to indefinite expansion.
Of course, says the anti-absolutist, there is no God’s eye viewpoint from which we can squint sideways at our current practice and comment that right now “(absolutely) everything” on our lips doesn’t really run over all the exists. “Everything” always means everything. What else? But that isn’t what the anti-absolutist denies, and so it seems that McGee fails to really engage with the position.