# [FOM] Relevance Logic and Tennant

Edwin Mares Edwin.Mares at vuw.ac.nz
Thu Sep 5 18:25:46 EDT 2013

Re: Charlie Silver's post on relatedness logic:

Epstein's relatedness logic is not a form of relevance (or relevant) logic. It is an interesting system (or family of systems), though. The idea is that an implication A->B is true iff (i) the corresponding material implication is true and (ii) A and B bear the relation of relatedness to one another. This relation is a primitive notion but can be understood as an intensional semantic connection of some sort.

In the formal semantics, there is a relatedness relation r on propositional variables. r is extended to a relation R on formulae such that we can prove that R(A,B) iff there are p in A and q in B such that r(p,q). Thus, if r is reflexive, A -> (B -> A) is a theorem of the logic.

Relevance logic, in the Anderson-Belnap tradition, is constructed to avoid the paradoxes of material and strict implication (that's the point of that form of relevant logic!). Thus, we don't think of Epstein's systems as relevance logics.

This is not to say that Epstein hasn't constructed good systems. They just capture different intuitions about implication than our systems do.

Edwin Mares
Philosophy
Victoria University of Wellington
P.O. Box 600
Wellington, New Zealand
+64 (0)4 4635234
________________________________________
From: fom-bounces at cs.nyu.edu [fom-bounces at cs.nyu.edu] on behalf of Charlie [silver_1 at mindspring.com]
Sent: 02 September 2013 05:05
To: Foundations of Mathematics
Cc: Harvey Friedman; rle at AdvancedReasoningForum.org
Subject: Re: [FOM] Relevance Logic and Tennant

Not knowing as much as I should about Dick Epstein's "Relatedness Logic" doesn't stop me from wondering why it's never mentioned alongside of Relevant (Relevance) Logic(s).  Is it considered "fatally flawed"?  If you think so, can you, Harvey, or anyone else, provide any details or reference?

Charlie Silver

On Aug 31, 2013, at 12:03 AM, Harvey Friedman <hmflogic at gmail.com<mailto:hmflogic at gmail.com>> wrote:

In communication offline with Neil Tennant, he has pointed out that in his posting of http://www.cs.nyu.edu/pipermail/fom/2013-August/017528.html , he has not indicated a rejection of the validity of thinning on the right, thickening on the left, or related valid inference rules of classical logic.

Various forms of "rejection" on the part of those investigating or proposing adoption of relevance logic, is reasonably standard. E.g.,

http://plato.stanford.edu/entries/logic-relevance   "They claim that these formulae fail to be valid if we interpret → as representing the concept of implication that we have before we learn classical logic. Relevance logicians claim that what is unsettling about these so-called paradoxes is that in each of them the antecedent seems irrelevant to the consequent. ... In addition, relevance logicians have had qualms about certain inferences that classical logic makes valid. For example, consider the classically valid inference ..."

http://rationalwiki.org/wiki/Relevance_logic  "Relevance logic attempts to capture formally this intuitive idea, that the premises must be relevant to the conclusion for the implication to be true."

http://consequently.org/papers/rle.pdf   page 7. "Thus, most notoriously the disjunctive syllogism (of Section 2) is counted as invalid."

http://johnmacfarlane.net/142/relevance-handout.pdf    2 Options
No one wants to reject ^ elimination. These options have all been tried:
(a) Reject _ Intro (a.k.a. “disjunctive weakening”)
(b) Reject the transitivity of entailment
(c) Reject Disjunctive Syllogism

http://www.st-andrews.ac.uk/~slr/Relevant_Logic.pdf   Chapter 6   "Much of Anderson and Belnap's argument for rejecting ex falso quodlibet
and setting up a logical system in which implication is non-truth-functional
depends on such claims as that `A and not-A' is not relevant to B, and
that relevance is a necessary condition for validity."  ... "The idea that validity requires a relevant connection between premises and
conclusion has a long history."

ALSO, in the case of intuitionistic logic and constructive mathematics, many of the key figures, such as Brouwer and Bishop, have explicitly rejected certain classical logic inferences as either invalid or incoherent.

>From offline, Tennant is clearly putting forth a more nuanced position.

The interesting and relevant(!!) part of my posting is most of it, where I do not (apparently incorrectly) ascribe views to Tennant. See http://www.cs.nyu.edu/pipermail/fom/2013-August/017528.html and http://www.cs.nyu.edu/pipermail/fom/2013-August/017535.html.

Harvey Friedman

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