[FOM] The unbearable ghastliness of EFQ, and Tim Chow's focusing question

[Tennant, Neil]

Tim Chow has focused on the critical issue:

________

Consider:

A. From a contradiction follows anything. [EFQ]
B. The material conditional P => Q is true if P is false.

Does the disagreement boil down to whether there is any distinction
between A and B?  As I understand it, Neil Tennant sees a distinction and
says that Core Logic utilizes B but not A.  Harvey, perhaps, sees no
important distinction between the two, or at least claims that actual
mathematical practice does not observe a distinction?

________


To confirm: the Core logician sees a tremendously important distinction between (A) and (B). (A) is bad; (B) is just fine.

This is because Core Logic relevantizes at the level of the turnstile, not at the level of the object-language conditional. The latter kind of relevantizing is what goes on in the tradition deriving from Anderson and Belnap. In that tradition, Disjunctive Syllogism and similar moves are rejected; and the result is the utter crippling of the deductive reasoning that one needs for mathematics.

That Core Logic (like its classicized extension) actually succeeds in relevantizing deduction while leaving intact the truth-table-characterized meaning of the conditional connective, is evident from the metatheorem I commented on earlier, which is proved in ‘The Relevance of Premises to Conclusions of Core Proofs’ [pdf]<https://u.osu.edu/tennant.9/files/2015/02/core_relevance_revised-11cnfkt.pdf>, Review of Symbolic Logic, 8, no. XXX, 2015, pp. XXX-XXX, DOI: http://dx.doi.org/10.1017/S1755020315000040

Distinguishing (A) from (B) goes hand-in-hand with being prepared to accept (wisely) that the so-called Deduction Theorem can fail in certain cases, in the direction

P1,...,P(n-1) |- Pn->Q  ==> P1,...,P(n-1),Pn |- Q

Counterexample:

~A |- A->B  but not ~A,A |- B.

The Core Logician countenances such failures with equanimity, resorting as usual to the upbeat observation that the failure of yet another cherished but over-valued 'general principle' can be turned into potential epistemic gain. For, suppose you have a proof D to justify the claim that

P1,...,P(n-1) |- Pn->Q

Suppose further that you propose to assume (or prove) Pn, via D', say, so as to be able to 'deduce' Q from the total collection of premises involved in D and D'. Stop! Proceed with caution! ... For you do not know whether this new collection of premises is inconsistent---or at least, whether the 'proof' you thereby obtain will not be trading illicitly on some hidden inconsistency. Remedy: find the reduct [D,D'], in order to discover what you have really proved.

Anticipated objection (from a Harveyian): "But I want to get on proving more things now, from Q! I don't want to waste hyperexponential time calculating [D,D']!"

Sigh in Core-Logician's response: "Carry on then, Harvey, if you are sure that the combined premises of D and D' are consistent. But remember the cautionary tale of the grad student who might find that Q follows from some proper subset thereof, a fact that you might be overlooking because of your lack of curiosity about [D,D']."

As I said before, we all have competing demands on our time. For my part, I would prefer to see Harvey going on to prove yet more theorems, and leave to posterity the mopping-up operation of minimizing the premise-sets needed for those results.

Neil Tennant
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