∀ and ω

Abstract

I first briefly rehearse the two substructural solutions that I’ve elsewhere proposed to the semantic and vagueness paradoxes. I then ask what the correct principle of universal generalisation is. The traditional answer to this question is represented by the familiar principle to the effect that, provided that τ does not occur free in either Γ, Δ or φ, if Γ⊢ Δ, φτ∕ξ holds, Γ⊢ Δ, ∀ ξφ holds. I argue for interpreting such principle as in effect licencing the inference from ‘anything’ to ‘everything’. I then proceed to offer five arguments against that inference. The first three arguments rely on considerations concerning the preface paradox, the failure of agglomeration for counterfactual implication and free-choice permission respectively. The last two arguments connect back with the semantic and vagueness paradoxes. I show how the inference from ‘anything’ to ‘everything’ would wreak havoc for the workings both of my non-contractive solution to the semantic paradoxes and of my non-transitive solution to the vagueness paradoxes. I then inquire into what a more adequate generalisation principle should be, and argue in favour of a suitably generalised version of the ω-rule, defending it from several prominent objections. I then trace back the quantificational phenomena studied in the paper, in particular those most directly related to the semantic and vagueness paradoxes, to their sentential root concerning the behaviour of conjunction. I sketch a metaphysical view making sense of the failure of the conjunctive analogue of the traditional generalisation principle, and close by bringing out some positive implications such view has for our logical freedom.