I have mentioned Neil Tennant’s system(s) of what he calls Core Logic once or twice before on this blog, in friendly terms. For the very shortest of introductions to the core idea of his brand of relevant logic, see my post here on the occasion of the publication of his book on the topic. (And there is a bit more info here in a short note in which I respond to some criticisms on Neil’s behalf — unnecessarily, it turned out, as he published his own rejoinder.)

I notice that Neil has now written a piece outlining his developed ideas on Core Logic for *Philosophia Mathematica. *If you want to know more, this might be a good place to start. You can download this paper here.

David MakinsonI have noticed that Joseph Vidal-Rosset has a blog (https://www.vidal-rosset.net/) containing a recent proof purporting to show that if Tennant’s classical core logic is paraconsistent, then it satisfies EFQ, contrary to its rationale. A technical argument, passing by refutation systems and antisequents. Let’s see whether/how Tennant or other core-religionists respond to it.

Neil TennantPeter, I’m happy for David Makinson and all your readers to know that this comment comes from Neil Tennant, who is also happy to confess his core-religionism. He found Core Logic in Plato’s heaven and has been studying it seriously for a great many years.

Jo Vidal-Rosset has once again failed to grasp that unrestricted Cut is NOT a rule of Core Logic. Nor does Jo employ the carefully defined notion of admissibility that I employ when discussing the properly restricted form of Cut that I call ‘cut with potential epistemic gain’.

Here I address the byzantine argument that Jo furnishes on his blog, to which you have conveniently supplied a link.

Look at the final step (an application of unrestricted Cut) in his purported proof of the invertibility of the rule he calls DNS1. (It’s right before his ‘Remark 1’.) The single turnstiles purport to be the single turnstile of Core Logic.

Now jump ahead to Jo’s supposed coup de grace labeled (3) at the end of his post. He is there instantiating Delta with ~A, phi with A, and psi with B.

So the final step of Cut just referred to has a conclusion-sequent containing both ~A and A among its premises. Cut is not a rule of Core Logic. The Core logician will simply drily remark that Core Logic can prove a subsequent of the would-be ‘target sequent’ for that would-be, but prohibited, application of cut. The subsequent in question is “~A, A:#”. Big deal! The leftmost, topmost step in Jo’s final proof-display is crashingly fallacious.

Jo wishes to criticize Core Logic as somehow inconsistent because (1) it furnishes no proof of the First Lewis Paradox ~A,A:B; and (2) it guarantees what I call ‘restricted cut with epistemic gain’. The latter principle is that from a core proof of Delta:phi and a core proof of Gamma,phi:psi one can effectively determine a core proof of some subsequent of Delta,Gamma:psi. Jo makes the fundamental mistake of thinking he can pin on the Core logician a commitment to using the irrelevance-inducing rule of unrestricted cut. He needs to understand that if Core Logic is to be framed as a sequent system (which of course it can be), then Reflexivity is the only structural rule available. One cannot use Thinning; and one cannot use Cut. Jo’s attempted critiques of Core Logic (both in his blog, and on the email list fom) all make the fatal mistake of invoking uses of Thinning and/or of Cut that he assumes can be visited upon the Core logician when deriving results within the Core system (and therefore also meta-results about the deducibility relation in Core Logic).

I must simply invite Jo to read my papers on Core Logic in the 2012 and 2015 volumes of the Review of Symbolic Logic; and to read my book Core Logic (OUP, 2017). Carefully. Please.

Best regards,

Neil