[FOM] Relevant logic and paraconsistent logic

[A.P. Hazen]

    I've been slow in responding to Joseph Vidal-Rosset's question, 
and I'm not sure I have much to add  to the previous replies by 
Lance, Heck, Mares and Avron: extra bibliography, perhaps.

     (1)  Relevant logics are systems in which contradictions don't 
**imply** everything (A&~A)->B is not a thesis).  Also, dually,in 
which not everything implies a tautology (A->(Bv~B) -- or, to show 
that this isn't about intuitionist scruples -- A->(B->B) is not a 
theorem.  In giving a "possible worlds"  semantics,  analogous to the 
Kripke  semantics of modal and intuitionistic logics, for such a 
system we must allow there to be "worlds" in which contradictions are 
true (and worlds in which tautologies fail).  Thes peculiar "worlds," 
however, need not include the "actual" world: it is demonstrable that 
the standard relevant logics are consistent (i.e. do not have 
contradictions as theses), and are, in addition, "classically 
complete," in the sense that they prove all the theses of classical 
logic (the -> connective of these logics  is non-classical, not 
definable in  terms of truth functions).

	The formal work on the semantics  of relevant logics does 
not, of course, commit one to any particular stance  on the 
ontological status of possible worlds (and there have been violent 
philosophical disagreements on such topics among the relevant 
logicians!).  Some of the early papers on the semantics of relevant 
logics, by Richard Routley and Robert K. Meyer, avoided the term 
"world," referring to the points in the model neutrally as "set-ups." 
Another  early contributor to the semantic study of these logics, 
Alasdair Urquhart, suggested that the points be thought of as 
THEORIES.  (This was in line with some of the motivational remarks 
made even earlier by Anderson and  Belnap, who suggested relevant 
logic was the appropriate framework for the history of philosophy: 
one might want to say that a certain thinker was "committed" to P but 
not to Q, even if that thinker was also committed  to some 
contradiction on another topic.

        	(BIBLIOGRAPHY: There are now a number of expository works on 
the semantics of relevant logics.  "Entailment, vol. II" by Anderson, 
Belnap and Dunn has a 145-page chapter devoted to semantics.  More 
recent general introductions to relevant  logics usually discuss the 
Routley-Meyer ("Kripke-style") semantics: one possibility would be 
the chapter (pp. 1-136) on "Relevance Logics" by Dunn and Restall in 
volume  6 of the second  edition of Gabbay & Guenthner,eds., 
"Handbook of Philosophical Logic" (2002).  Its ancestor, Dunn's 
chapter (pp. 117-124) in volume 3 of the first edition of said 
hanbook 1986, is intrinsically very good, but has  enough potentially 
confusing typos to make me a little hesitant in recommending it for 
private study.  ... Mark Lance has written on interpreting relevant 
logics in terms of "commitment".)

	(2)  The connection between "dialetheism" -- the view of 
Graham Priest that there are true contradictions -- and the 
Foundations of Mathematics is, I think, in need of further 
clarification.  Very early on in the study of relevant logics it was 
suggested  that one application would be the formulation or a set 
theory based on the "naive" comprehension scheme.  Such a system 
would (unlike the pure logic) have to be negation inconsistent (it 
would imply that the Russell set (exists, and) both is and is not a 
member of itself).  For theories based on classical -- or, for that 
matter, intuitionistic -- logic, negation inconsistency is equivalent 
to TRIVIALITY (a theory being trivial if every sentence of its 
language is a theorem).  Relevant logics, however, admit non-trivial 
inconsistent theories.  Someone accepting such a theory would be a 
dialetheist.  		...		As other's have mentioned, 
relevance logics are not the only possible bases for such theories. 
There is a nice survey of (often 3-valued or 4-valued) systems in 
Feferman's "Toward useful type-free systems, I" (JSL v. 75 (1984), 
pp. 75-111).  The logics  of these systems typically differ from 
relevant logics in NOT having an implication connective supporting 
both modus ponens and a reasonable deduction theorem: Feferman 
comments (p.  95) that "nothing like  sustained  ordinary reasoning 
can be carried on in" them.  (This is a bit out of context: the 
comment was about the systems considerd early in his paper, and 
perhaps shouldn't be generalized.)

	(3) Relevant logic and dialetheism are NOT enough, however, 
to allow for a non-trivial "naive" set theory.  Consider Curry's 
strengthening of Russell's Paradox, about the set (for any chosen 
sentence C) of all things such that, IF they belong to themselves, 
then C.  ({x:x epsilon x-> C})  In a set theory based on naive 
comprehension and any of the most familiar relevant logics, this 
produces triviality.  (BIBLIOGRAPHY: Curry's Paradox is  discussed, 
among other places, in the Feferman paper  just cited.  Its 
application to "naive"  set theories based  on a variety of relevance 
logics is nicely discussed by Robert K.Meyer, Richard Routley, and 
J.M.Dunn, in (the English philosophical journal) "Analysis," vol. 39 
(1979), pp. 124-128.)

	(4) In the other direction, the project of using a 
weaker-than-classical logic to allow a non-trivial "naive" set theory 
does not require dialetheism: the proposition that the Russell set is 
a member of itself might be allowed to lack a truth value (to be  a 
"gap") rather than being asserted to be a dialetheic "glut" (both 
truth values).  	...
		(This was in effect oberved by Frederic Fitch in his 
1952 book "Symbolic Logic": leaving the implication connective,  and 
so Curry's paradox, aside, Russell's paradox can be blocked by using 
only a negation operator like Nelson's "constructible falsity") (= 
"strong negation").)  ...
			Ross Brady was able to show that a 
non-trivial set theory based on the naive comprehension scheme was 
possible in a weak relevance logic (a weaker logic than those 
considered by Meyer, Routley and Dunn in their "Analysis" essay). 
One way of doing this produces a non-trivial but negation 
inconsistent theory: this is favored by dialetheists like Priest. 
Brady also discovered, however, that a negation CONSISTENT naive set 
theory was possible in a slightly weaker logic (and showed this, I 
think, by methods similar to those needed to establish the 
non-triviality of his other system).  My own (non-expert) view is 
that this shows that, EVEN for those who want to save "naive" set 
theory by working in a weak relevant logic, Dialetheism is an 
optional extra.

       	(The BIBLIOGRAPHICAL bit that justifies my whole posting: 
Ross Brady's monograph, "Universal Logic" has JUST been released by 
CSLI press (distributed, I think, by University of Chicago Press), in 
hardcover if you are a library and (we hope affordable) paperback for 
human beings.  It is THE place to go if you want to learn about naive 
set theory in weak relevance logics!!!!!!!!!)

	(5)  Logical notes: The treatment of the implication 
connective in the logics allowing non-trivial "naive"  set theories 
is like that of "lollypop" in  Girard's Linear Logic, at least in 
that the "Law of Contraction"
	[P->(P->Q)]->[P->Q]
is not provable in  them.  (Otherwise they would trivialize, by 
Curry's paradox.)  Relevance logics typically DIFFER from Linear 
Logic in that principles corresponding to all the laws of 
DISTRIBUTIVE lattices hold  for the extensional ("additive" in 
Girard-speak, but relevance logicians were distinguishing 
"extensional" from "intensional" connectives years before Girard 
introduced  his terminology) hold in them.  These distribution laws 
seem to make  the logics more complicated  in some ways: most 
propositional relevant logics are undecidable, unlike propositional 
Linear  Logic without the exponentials.	...	The logic of Brady's 
negation-consistent "naive"  set theory lacks the Law of Excluded 
Middle.

---

Allen Hazen
Philosophy Department
University of Melbourne
(Colleague of Graham Priest, sometime  student of Alan Anderson and 
Nuel Belnap.  All of whom at one time or another have criticized me 
for being to classical or too Quinean in my philosophy of logic.)