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This work log charts our progress in our research into *logical
pluralism*, the view that there is no *One True Logic* but
rather, there are *Many* True Logics.

We don't update this log *daily* but we do endeavour to log in
advances on the project, pointers to net resources related to our work, and
anything else we find interesting and related to our research project.

This log is for our personal use, though the general public may well find it amusing. Of course, we would appreciate any pointers to other work you think we would enjoy, or any comments on our work.

If you're interested in our work, the place to start is in our joint paper Logical Pluralism, which you can download as a pdf file.

*Greg Restall* <Greg.Restall@mq.edu.au>

*JC Beall* <j.c.beall@utas.edu.au>

**Reminder** (mostly to JC): Review Lemmon's 1957 'Is There only One Correct System of Modal Logic?', *Proceedings of the Aristotelian Society (Supplementary Volume)* 33:23--40.

JC

Every (philosophical) position starts somewhere, providing little to no argument for that particular starting place. Arguments enter the scene as the position is developed. The situation is no different with our pluralism. In response to various comments and questions on our position (many emails, some "live"), I figured I'd better make clear our starting point. We've made this clear elsewhere; in fact, we've made it clear all along. Still, if questions persist, then we need to either repeat or clarify. Here, I repeat; in the book, we clarify.

Our starting point is what we have previously called `(V)', pronounced `Vee' (and not to be confused with Frege's relevant *five*). Perhaps we'll drop this name in the book, but for now I use it. (V) is simply an account of validity or consequence, or (as we see it) a schema for accounts of validity or consequence. Taking validity or consequence to be the *follows from* relation, (V) can be expressed thus:

(V) A conclusion,Our starting point, then, is (V). What to make of (V) is at the heart of our pluralism. One question to ask concerns (V)'s invocation ofA, follows from premises,P, if and only if every situation in which (each element of)Pis true is one in whichAis true.

In assessing the validity of an entailment every variety of situation must be considered.With (V), combined with Dunn's doctrine, pluralism

For what it's worth, we do take both (V) and Dunn's doctrine to be fairly common, fairly uncontroversial. Where controversy arises concerns the variety of situations sanctioned. We think that there are various sorts of situation (case, world, point, etc.), including some that may be represented by (negation-) complete and consistent sets of sentences. On this score, many philosophers and philosophical logicians agree. But we see no reason not to recognize natural variations on this (uncontroversial) sort of situation. For example, once we have complete situations, why not recognize a natural twist, namely *in*complete situations? We see no good reason against recognizing incomplete situations as a natural sort of situation. More controversially, however, we likewise see no good reason against recognizing *in*consistent situations as a natural sort of situation. And, indeed, we recognize all these sorts of situation, each being variations on a commonly recognized sort of situation (again, the sort represented by complete and consistent sets of sentences). Defending the admission of these varieties requires argument, and this is (in part) what our book attempts to do. The point of this little entry, however, is not to do that. (Again, the point of this entry is simply to answer a fair few queries I've received lately on our position and where its arguments rest, where its starting point lies, etc.)

There is, perhaps, one other "starting point" of our position, or at any rate a thesis we maintain without argument. The thesis in question is that `all' is a quantifier the range of which varies with context. (NB: Exactly how to cash this out is something we are working on. An earlier entry below, briefing noting comments from Daniel Nolan on a recent UQ talk by Greg, reflects some of the issues that need to be settled. For now, though, I just put the point in a simple, though currently tentative, fashion.) Combining this view about `all' with Dunn's doctrine and (V) gives the heart of Pluralism: There are many different consequence relations governing English. Depending on the range of `all' in `all situations', one specifies different consequence relations underwriting the English language. (Interestingly, some philosophers have apparently taken us to be saying that there are different English languages, each corresponding to a different consequence relation so understood. This has never been our position, as far as I can remember, but apparently some have thought as much. In response to this sort of misinterpretation Greg has written a very nice comparison of *our* version of pluralism with that of Carnap's, where Carnap discusses a position closer to the misinterpretation mentioned above.)

Hope this helps. Thanks to all who've asked questions. The book will fill this stuff out a lot more, and also answer numerous pressing questions that I haven't even touched here.

JC

This is a nice little paper by the inimitable (literally!) Jay Garfield. He talks more about the epistemic situation of the *dog* (who appears in many places in the literature on relevant/relevance logic). He also endorses our pluralism. (Actually, I'm not sure if Jay means to emphasise our pluralism, or some other kind of contextualism, but it's not his fault if we're not completely clear about it all.) Anyway, it's a great read.

G

Last week I visited the University of Queensland, to give a version of the Carnap paper. The seminar went well, and I think I finally got an important distinction straight in my thought: The relevant distinction involves what I take to be the meaning of the word "valid" in English, as a pluralist. As pluralists, we have three options, I think:

- The word "valid" is ambiguous.
- The word "valid" means something like "truth preserved in all situations" and the scope of the "all" here is contextually determined in some way.
- The word "valid" means something like "truth preserved in all situations" and the extension of "situations" is indeterminate by practice and can be made more precise in different ways.

Thanks to Daniel Nolan for helping me get just a little clearer on this.

G

Come and join us at the Logical Pluralism 2001 Conference in Hobart on July 4!

G and JC

Another piece of the pluralism puzzle will be out there. My paper Constructive Logic, Truth and Warranted Assertibility has been accepted in Philosophical Quarterly.

G

OK, before I get to sleep, I may as well point to my shiny new encyclopedia entry on Curry's Paradox. Go, read and be enlightened.

Comments, as ever, are welcome.

JC

His influence on philosophy of logic remains large. He will be missed.

*The New York Times* offers further remarks:
W V O Quine.

JC

Stephen Vickers has a nice introductory talk on non-classical logic. Most fun for us is these slides, where he basically endorses logical pluralism. As he says, the more structures you have, the fewer validities you get. And that's OK.

G

Well, *our* answer to this is yes and no. Mine (currently) is *no* and JC's -- well, I'll let JC say what he thinks, but I think that JC's answer should be, like Graham's, *yes and no*. But anyway, read the paper by Lorenzo Peña. It's fun!

G

There's some nice work of Bob Coecke who is doing some interesting things with Quantum logic. It's the kind of thing I ought to have a pluralistic opinion on. (Though my friends/colleagues will know that I'm not necessarily committed to finding a place for quantum logic in the Grand Scheme of Things.)

I found these papers originally in the logic section of the arxiv preprint archive.

G

Both of us have seen this but at least one of us (JC) keeps forgetting to officially record the record. I'm just recording that we need to address some of the rather interesting and relevant (to pluralism) pieces in volume 3 of the Nordic Journal of Philosophical Logic. That's all.

JC

We need to think more about *contexts*, their nature and structure. Perhaps one useful source of information is the work of Bouquet and Serafini, whose ESSLLI course focused on both philosophical and formal aspects of contexts. Their course notes, etc., may be found here, from which one can also access the official course page.

JC

JC pointed me to this interesting paper by Rumfitt. It's a good comprehensive discussion of the proof theoretical analysis of the meaning of logical constants. It surprisingly doesn't mention Stephen Read's "Harmony and Autonomy" paper, which goes over similar ground (though is more formal, and less generally philosophical.)

Tennant's Anti-realist Aporias paper in the same issue of Mind also has a nice collection of material to mine for our project.

G

(P.S. I'm back, after a more tedious semester of teaching and administration than I had hoped. Hopefully, there'll be more pluralism from now on!)

Quick Note: We need to look carefully at Richard Epstein's textbook, *Propositional Logics: The Semantic Foundations of Logic*, published by Wadsworth. I recall an earlier version of the book published (I think) by Kluwer. At any rate, the book, qua textbook, looks great, but I think that it may have something for our project too. For example, one of the blurbs on the back reads thus:

In this first and only book to unify many different logics within a common spectrum of semantic analysis, world-renowned logician Richard L. Epstein presents modern logic so readably that both beginning and advanced students will benefit. Epstein helps readers to see logic as the formalization of reasoning that needsand deservesa semantic foundation.

Epstein attempts to answer questions such as "What is logic?", "What is a proposition?", and "What is a connective?", all questions that are germane to our project. His big goal is to *unify* the plethora of so-called logics available today.

His chapters go as follows: (I) The Basic Assumptions of Propositional Logic; (II) Classical Propositions Logic; (III) Relatedness Logic: The Subject Matter of a Proposition; (IV) A General Framework for Semantics for Propositional Logics; (V) Dependence Logics; (VI) Modal Logics; (VII) Intuitionism; (VIII) Many-Valued Logics; (IX) Paraconsistent Logic [concentrates mostly on J_3]; (X) Translations Between Logics; (XI) The Semantic Foundations of Logic.

I haven't had time to read the book yet, but it seems to me that Epstein has remarks that will be useful for our (pluralism) book. I can't yet tell for sure but Epstein seems to want to see different logics as reflecting either different notions of truth or different notions of proposition --- neither approach sitting perfectly well with our version of pluralism. His basic view, I think (without reading much at all), may perhaps be summed up thus: "Each logic, other than classical logic, is based on some aspect of propositions in addition to form and truth-value; different aspects give rise to different structural conditions on the semantics, yielding a spectrum of semantics" (127). Now, what we have to figure out is just this: How does Epstein's talk of "different aspects of propositions" fit with our talk of different cases, truth-conditions, etc.? This is a must-learn question.

Just FYI: Epstein also has a second volume dedicated to predicate logic, and it looks good too; however, for our purposes we can probably just go through his first volume on propositional logic.

JC

Last week I gave a talk at Pittsburgh's Center for the Philosophy of Science [no link provided -- use Google yourself to get to it! -- I'm paying for net access at Kinko's and every second counts, or costs]. It was really fun, with a great audience, and a good feel. Lots of people thought that my take on Carnap was good, and that pluralism sounded interesting and fruitful.

Best was lots of time to talk about pluralism with Nuel Belnap. He has an interesting line on pluralism with respect to *grammar*. I'll tell you more about how this intersects with my stuff sometime.

Tomorrow it's off to Waterloo, and on Monday, I talk at the University of Toronto. More later, when I get cheaper net access.

G

Stewart Shapiro has a nice, crisp summary of classical logic, a contribution to the Stanford Encyclopedia of Philosophy. Shapiro begins his essay thus:

Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics. The language is, or corresponds to, a part of a natural language like English or Greek.The deductive system is to capture, codify, or simply record which inferences are correct for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions, or possible truth conditions, for at least part of the language. [Italics mine.]

We agree with all this, of course. I should like only to point out that, though I (perhaps unlike Greg?) don't like the terminology, speaking of "correct inferences" would seem to lead easily to some sort of pluralism, as does the terminology of "truth conditions, or possible truth conditions" (terminology with which I'm more comfortable). Of course, as Greg pointed out a few entries earlier, Shapiro himself is a pluralist, though perhaps not exactly in the same sense as we (Greg and I) are.

Shapiro's essay is not at all concerned with the philosophy of logic; and I do not intend to suggest that it is. I simply wanted to post applause for crispness and clarity of the piece, and also note the little point above about pluralism.

JC

This is a useful paper which provides some interesting Medieval perspectives on semantics, ontology, and the relation between the two. The paper is by Dr. Klima.

Jc

Just a short note to say that upon reading (at last!) Stewart Shapiro's *Foundations Without Fondationalism*, I've realised that he's explicitly a pluralist. His case *for* Second Order Logic is not a case *against* first order logic. The book contains a lot of helpful scene setting clarifying distinctions such as the distinction between idealised justification and logical consequence. (For Shapiro, an argument from *A* to *B* might be valid without *A* giving you a justification for *B*, for example.) Anyway, I'm in broad agreement with the distinctions he draws, and it's good to know that we don't need to cover all of that introductory ground by ourselves.

G

This is Stephen Read's contribution to the forthcoming *festschrift* for Bob Meyer (edited by Ed Mares). This paper is useful to us in many respects and it demands a careful reply. There are points on which we (Greg and I) will agree with Read, but also fundamental points of disagreement. Here is the abstract:

It is often suggested that truth-preservation is insufficient for logical consequence, and that consequence needs to satisfy a further condition of relevance. Premises and conclusion in a valid consequence must be relevant to one another, and truth-prservation is too coarse-grained a notion to guarantee that. Thus logical consequence is the intersection of truth-prservation and relevance.This situation has the absurd consequence that one might concede that the conclusion of an argument was true (since the argument had true premises and was truth-preserving); yet should refuse to infer the conclusion from the premises, in the absence of demonstration of the relevance of the premises to the conclusion.

The error lies in giving insufficient attention to the notion of truth-preservation. Relevance is no separable ingredient in the analysis of logical consequence, but a necessary condition of it. If an argument really is truth-preserving, then that in itself is enough to show that the premises are (logically) relevant to the conclusion.

There are many things we should say in response, though I do not now have time to say all of them. Two big points are these. First, we can and do agree that there has been "insufficient attention to the notion of truth-preservation". *Our* question, however, is: Truth-preservation over *what sort of case*? If one answers by pointing to inconsistent and incomplete cases, then the notion of truth-preservation is relevant indeed, just as Read wants. In that case, as Read says, "[i]f an argument really is truth-preserving, then that in itself is enough to show that the premises are (logically) relevant to the conclusion." On this we agree firmly with Read. By our lights, however, other answers to the question of truth-preservation are equally warranted, answers that do not involve relevance --- e.g., taking cases to be complete or consistent or both. *This* is where our main disagreement with Read arises. Where he holds that relevance "is no separable ingredient in the analysis of logical consequence, but a necessary condition of it" we do not disagree (per se); however, we hold that his claim is accurate only on some notions of consequence, notions that are no more ``real" consequence than others for which relevance is not a necessary condition.

The second big point responds to paragraph two of Read's abstract. (I *think* that Greg and I agree here. If not, then this is simply my own response, though one based on pluralism.) Why is the situation described (in said paragraph) *absurd*? I am not sure. The important point, though, is just this. There is no dilemma here for a pluralist. The pluralist may claim that the given argument is not relevantly valid; however, if she thinks that the premises are true and that the argument is truth-preserving over some range of cases (that includes the actual case), then she may infer the conclusion *on the basis of those premises*, all the while maintaining that the argument is indeed not a relevantly valid one.

There is so much more to be said about Read's paper, but I do not have time here. (Indeed, I'm now late for an appointment!) I should also note that, aside from the interest that this paper provokes in pluralists, another wonderful feature is its bibliography, and in particular the handful of very useful quotations that the paper contains.

(A side note. Based on his previous work from darker, unilluminated days, Greg is portrayed as something less than a pluralist in Read's paper.)

JC

Last night at Gleebooks I picked up a copy of Richard Mason's Before Logic, a small tract on why doing logic isn't philosophically more foundational than every other area of philosophy. That is, there's some work that you need to do *before* doing logic. I don't want to argue about that thesis at the moment. I do want to direct you one nice quote I found in the introduction.

the symbolism must come later and in response to a need, from which it follows, of course, that whoever wants to create or develop a symbolism must first study those needs.Isn't that nice! I must dig that up.

Hilbert, in a letter to Frege, 4 October 1895. In G. FregePhilosophical and Mathematical Correspondence, edited by G. Gabriel et al., Translated by H. Kaal. (Blackwell, 1980), page 34.

G

Eli Dresner's paper, entitled Boolean Algebras and Natural Language: A Measurement Theoretic Approach seems reminiscent, to me, of what Bob Meyer has always said about propositions. They're abstractions away from what we don't want to worry about in utterances. Bringing the analogy to *measurement* to light seems like the right sort of thing to do, and Dresner's paper seems to do that clearly. What's fun for a pluralist, of course, is that we say that we can measure our claims in different ways, with different measuring scales. More on this later, as I have time to think about how to put it.

G

A new paper has left the building. It's rough, but it's currently my favourite. Here's the abstract:

In this paper, I distinguish different kinds of pluralism about logical consequence. In particular, I distinguish the pluralism about logic arising from Carnap'sThere. Comments, as ever, are welcome.Principle of Tolerancefrom a pluralism which maintains that there are different, equally “good” logical consequence relations on the one language. I will argue that this second form of pluralism does more justice to the contemporary state of logical theory and practice than does Carnap's more moderate pluralism.

G

Teaching logic is lots of fun. This time around I've gained lots more respect for different ways of accounting for logical consequence. In particular, in my intro class, I teach propositional logic by way of truth tables and tableaux. Lots of people think that truth tables and tableaux are the same technique, but I don't think that's right. The traditional account of truth tables is *inside-out*. The account determines the semantic significance of a sentence as a function of the semantic significance of its constituents. The traditional account of tableaux is in reverse. You take a complex proposition, and determine the semantic significance of its constituents in terms of the significance of the complex sentence.

Note that none of this is to say that one way of determining the significance of an expression (or the way that an expression gains its significance) is primary. Different accounts say different things here, and it's hard to see that a commitment to pluralism means we have to take a particular branchpoint here. Even if we say that logical consequence is primarily a matter of truth preservation in all circumstances (whatever you say circumstances are) we're not committed to one branchpoint. This analysis of validity is commonly taken to be a "realist" account which determines the significance of a sentence in an inside-out fashion, but that's just not the case. Nothing tells us that our account of "A is true in x" is to be analysed inside-out as opposed to outside-in.

None of this is particularly sharp or crisip yet. There's much more to be done in clarifying this. I thought I'd get it down before I forget.

G

There's a nice discussion between Dirk van Dalen, Bill Tait and Fred Richman which went on at the FOM email list. The archives are here.

G

Here are some incohate thoughts about a route into pluralism which doesn't go through our definition (V), which analyses validity in terms of truth preservation in all cases.

The desire for such a route into pluralism comes about as follows: for an inferentialist like Brandom, the content of an expression is constituted by its inferential role. To say *A* is to commit yourself to *A* and its consequences. To be a sapient is to be involved in the game of giving and asking for reasons. Clearly any philosophy of language or content like this is one that we have to respond to. It seems like the core notions of this account have something to do with *logic*, and certainly, at first blush, seem to pick out a single notion of logical consequence. This is a problem for us. It's not that we necessarily want to hitch ourselves to this philosophical bandwagon (though bits are attractive to me) -- it's that if we want to rule *out* a philosophical analysis of content and meaning we really ought to be able to clearly say how and why.

Now, we seem to have lots of options here.

- First, note that the notion of inference Brandom uses isn't a form of deductive validity. It's a kind of non-monotonic conditionality. [He mentioned this in discussion in one of his papers I went to. I haven't dug up the reference from
*Making it Explicit*yet/] If this kind of conditionality is basic in forming content and in concept acquisition, it's no skin of our noses, because it's not any kind of deductive logical consequence, so it can't be the*One True Consequence Relation.* - Second, note that we can cook up various kinds of consequence relations out of his basic primitives. It looks to me like a notion of relevant entailment embodied in commitment preservation. If commitment to
*A*involves commitment to*B*, then it seems that*A*relevantly entails*B*, and vice versa. (There's some original work of Mark Lance and Phil Kremer on this stuff.) A kind of classical consequence can be cooked up out of compatibility notions, if you're careful enough. An argument from*A*to*B*is classically valid if and only if*A*&*~B*is not consistent, and consistency can be dfeined in terms of compatibility. (This depends on how compatibility is cashed out.) - Third, note that this is not going to get us to the rich kind of pluralism we have discussed. We don't have a whole raft of different consequence relations coming out of a whole raft of different classes of cases until we define those cases. Here, we're defining consequence relations differently, in terms of the inferential semantic primitives Brandom helps himself to. For Brandom, formal semantics a matter of regimenting our understanding of those primitives by showing how they can be semantically modelled using notions like possible worlds or states or whatever.

G

OK, before I get to sleep, I may as well point to my shiny new encyclopedia entry on substructural logics. Go, read and be enlightened.

Comments, as ever, are welcome.

G

This paper is not completely pluralist, but it's not anti-pluralist either. Here's what I think a fan of paraconsistent logics, like me, should think about the law of non-contradiction. And the law of the excluded middle, for good measure. (Duality is your friend.)

Comments, as always, are welcome.

And yes, the conference was great fun. Yes, there's too much to think about. No, I don't think that pluralism is anodyne. Yes, I'll write up responses to all of the good questions I got at the conference. No, it isn't done yet.

Now, it's time for *sleep* having that paper out of the way.

G

Here's a few quick points after Greg's AAP talk. (I'm in Koji's office, and have very little time, so this is very sketchy...) (1) Jerry Seligman suggests that we'd show the metaphysical neutrality of pluralism more clearly by showing that a constructivist should be a classicalist, rather than vice versa (as in Greg's talk). 2) Graham Priest complains that he can't distinguish pluralism from relativism. By 'relativism', he tells me (in conversation), he means that all the given logics are equally good. "Good for what?" I ask. Answer: "Good for whatever". Well, we think that the logics are all equally good at the canonical application (of modelling genuine consequence relations over our language); however, we don't think they're equally good for any given aim (obviously). (3) Hartry Field thinks our pluralism is interesting only in as much as it keeps closely to necessary truth preservation, where in the end there's no determinate answer as to what "real possibility" or "absoluate possibility" amounts to. (That is, the various restrictions of the quantifier over the various "circumstances", *which* of these is the "right restriction" is always indeterminate.... or something like this. Need to speak with Hartry some more. He wants copies of the first two papers, too, and we must send him these.)

JC

We're off to two conferences: First, the Australian Association for Logic conference up in sunny Noosa, and then the Australasian Association for Philosophy conference in Brisbane. We're both giving some pluralist papers, and we'll report back here when we get reliable net access.

G

An exercise. What is suggestive in this quote? What do we agree with, and what do we disagree with?

The first attempts to cast the ship of logic off from theNice.terra firmaof the classical forms were certainly bold ones, considered from the historical point of view. But they were hampered by the striving after 'correctness'. Now, however, that impediment has been overcome, and before us lies the boundless ocean of unlimited possibilities.

I will be writing more on this. I think I can see how our position differs from Carnapian tolerance, and this places us in an interesting historical context. However, I must think more about this, before I write.

G

Note to self: Examine Artificial Intelligence: A Modern Approach for its use of logic in modelling reasoning. (Link found at the wonderful discussion at Wes Felter's Hack The Planet Prime.)

G

In the quest to minimise the number of errors in my paper on constructive logic, I offer up Draft 3.

It's being given (in two bits) at the Australasian Association for Logic and Australasian Association for Philosophy Conferences in July. So no doubt there'll be a Draft 4 after the audience has had their go at it.

G

In this paper Peirce tells us what he thinks is the source of logical validity.

G

Sol Feferman asked proof theorists some questions about the state of their discipline, late last year. Here are their answers.

They're interesting for pluralists like us, as I think that the general outlook of the replies (to wit: proof theory gives us fine control of the content of theories, relative consistency strengths, constructive content of proofs, etc.) coheres very well with pluralism. Proof theorists, by and large, don't see their discipline as providing a single unitary ground for theorising or theories.

G

Colin McGinn has a new book coming out with OUP, called *Logical Properties*. The OUP website tells us a little about the book.

Identity, existence, predication, necessity, and truth are fundamental philosophical concerns. Colin McGinn treats them both philosophically and logically, aiming for maximum clarity and minimum pointless formalism. He contends that there are real logical properties that challenge naturalistic metaphysical outlooks. These concepts are not definable, though we can say a good deal about how they work. The aim ofThis sounds interesting, apart from the last sentence, which just sounds silly, considering how much philosophy is already in philosophical logic! Still, there are things there for us to think about, as usual.Logical Propertiesis to bring philosophy back to philosophical logic.

The book is due in December 2000, so we won't get to do the detailed thinking for a while.

G

This page is interesting; however, I have nothing more to say about it at the moment. (I list the page as one of those proverbial *things that make you say `hmmm...'*.) I *will* say that there is a need for clearly articulating the neutrality of logic with respect to metaphysics. Alas, such is our job. (See Greg's post on Dipert's view of logic for more evidence.)

JC

This entry is purely light relief. It is good to know that pluralism beats monism by 25 to 8.

G

Nice folks at Durham have some lecture notes on Carnap available for a viewing.

And Randall Dipert has his rather odd metaphysical paper (from the *Journal of Philosophy* 1997) online for viewing too. This paper, on "The Mathematical Structure of the World" is rather less pluralist, but worth a scan.

Also worth responding to from Dipert are his views on logic, which are nicely expressed, and completely wrong.

G

Just endorsing JC's claim that Huw's stuff is relevant to our inquiry. Indeed it is, because plenty of people will think that we must be metaphysical pluralists of exactly this type.

I think that we *can* be, and this kind of metaphysical pluralism is consistent with our logical pluralism (well, that's assuming that it's consistent with *anything*) but it's by no means mandatory.

I think, of course, that Huw is the late 20th Century (early 21st Century?) Carnap *Redivivus.* For more metaphysical pluralism, see Carnap's *Aufbau*.

G

There is a somewhat recent paper by Huw Price that may be of interest to us, namely his *Journal of Philosophy* (89:1992) paper, Metaphysical Pluralism, the footnotes to which may be found here.

JC

I don't know about you, but I get kind-of excited when people talk of

Christopher Gauker has a paper [pdf] on the same, which I have just printed out, and will read. He's by no means a pluralist, but at least he says more about what acontext-relative logical validity

G

We need to get a hold of Stalnaker's paper

"Impossibilities"in order to understand more fully the kinds of prejudices (informed or otherwise) people have against impossibilities.Philosophical Topics, 24 (1996), 193-204.

This is not to say that Stalnaker's view is *prejudiced*. I really like his stuff. He will provide a good guide to the orthodox views on impossibility which plenty of people (especially coming out of the US, it seems) keep saying whenever we present work on pluralism.

G

JC and I are talking about epistemology and logic today. We're worrying about what to say as pluralists about the epistemic status of logical theory and logical commitments. I find this discussion hard, because I know so little epistemology, and what I *do* find out, I don't really like.

One bit of epistemology I am sort-of attracted to is the work of John Post. He has a rather radical line that justification is non-transitive, and as a result rather more can be justified than you might think. Logical principles can be justified by stuff, without falling into unhelpful circularity, because justification needn't flow all the way down the chain, which *might* loop. Hmmm. Sounds loopy, but it doesn't sound out of the question.

G

Last night I completed a few revisions on my constructive logic paper. It's been uploaded to the servers today. I think the paper is grammatically correct now.

It's perhaps too much to ask that it be philosophically correct too.

G

When driving down to Verona Sands and other points south from Hobart, we got to thinking about pluralism. We've found a distinction between our two views. Greg has been a *restricted* pluralist, for whom if premisses are true, and conclusion false, the argument is invalid. That is, the actual world is a member of any class of cases used to define validity.

JC, on the other hand, is a liberal pluralist, for whom *any* class of cases defines a relation of logical consequence. What gives?

Well, Greg asked JC. Do you mean to say that you can have an argument you take to be *valid* with premises you take to be *true* and yet have no reason to think that the conclusion is true too? (As you must, if you have no reason to think that the actual world is one of the cases quantified over in the definition of validity used.)

Well, JC said to Greg. If you are also committed to the world being one of the cases in question, you do.

Aha. Let's say that we're *committed* to a logic if you're committed to the world being one of the cases quantified over. Then if you're committed to a logic, and you're committed to the premises of an argument valid in that logic, you indeed do have some reason to believe the conclusion.

We need to think about this some more, but this might be a way to get the best of Restricted and Liberal pluralisms.

G

The titles of these papers look promising for our work. I'm posting this as a reminder that when time is available the given papers should be examined.

JC

We gave a talk today at the University of Tasmania philosophy department; the talk was called `Logic, Impossibility, and Contradiction: Arresting the Slide from Paraconsistency to Dialetheism'. The focus of the talk was Graham Priest's recent argument (from the Ghent Paraconsistency Conference) according to which any paraconsistent logician ought *rationally* to be a dialetheist. We reject this argument, and the talk said why we so reject it.

We received a few good comments, but in any event we seemed to clarify our thoughts on paraconsistency and pluralism. Now we have to write up the paper, and then present it at the AAL 2000 conference. When it's written, it'll be posted up here.

JC

Our Pluralism Paper (that's a pdf file) has at last been accepted for publication in the Year 2000 Special Logic Issue of the Australasian Journal of Philosophy.

If you hear a large sigh of relief, that would be ours.

G and JC

Yesterday I uploaded a rough draft of Constructive Logic for All. Read and enjoy!

For the next week or so I'll be in Tasmania, with JC, so expect more pluralism action, which we shall hopefully post here!

G

Jamie Tappenden has a few downloadable papers worth reading, one on the pragmatics of negation, denial, etc., and another discussing Frege's view of logic.

JC

This is a reminder that we need to spell out our position on various traditional topics, including especially the *a priori* nature of logic, the *topic neutrality* nature of logic, etc.. Related to this is an interesting question, namely: What do we say about the *revisability* of logic? Usually, the revisability debate is carried on under the presupposition of logical monism. What would revisability amount to in the context of pluralism? ....Things to work on.

JC

This is an interesting project (in progress) by John Halleck. It will be interesting to see how it develops. (I'm throwing it up here just for future reference.)

JC

Our pluralism book, like any book of which Greg is an author, will have leading quotations at the start of each chapter. A potentially useful tool in this regard is Mathematical Quotations Server, which affords keyword searches, etc.. Try `logic or validity' for an amusing (and sometimes provocative) start.

JC

I've not been *completely* slack over the last week. I've been getting some thoughts together. First, some of my thoughts on Mike Resnik's noncognitivist monism can be found in ``Data for Explanation'' [PDF ~ PS] and my thoughts on Priest's nice new argument from paraconsistency to dialethesim are in ``Pluralism and Paraconsistency'' [PDF ~ PS]

Both are short two to three page notes, very rough and in progress. *Caveat Lector*.

G

The great group in logic and metaphysics at St. Andrews is putting together a program on which we should keep a close eye: viz., the Arche program. Note that two professorial fellows include Graham Priest and Stewart Shapiro, each of whose work is relevant to pluralism. They are also planning to construct online bibliographies for the philosophy of maths, philosophy of logic, and related areas relevant to the pluralism project.

JC

This is not pluralist, but fun to note. There are two nice papers online by Robin Hirsch on Marxism and logic. You can find them at this page. It's good to know that there are Marxist logicians out there.

G

We have significantly revised and updated our original paper. Let us know if you have any comments on it!

G

This is merely a "to do" post; I'm posting something to be done as soon as the teaching-grading-etc schedule cools down.

Harvey Friedman has an online paper with an interesting-sounding title, a title that suggests the paper may be of potential interest to us. The paper in question is `A Complete Theory of Everything: Validity in the Universal Domain'. This is something to check out when time permits.

JC

In his *Elements of Intuitionism* (1977) Michael Dummett writes that 'surprisingly, negation is definable in intuitionistic arithmetic' (1977:35). He gives (in effect) the definition:

(*) ~A =df (A-->0=1)Neil Tennant, in his contribution to the Gabbay-Wansing (eds)

Interestingly, Roy Cook (cook.266@osu.edu) and Jon Cogburn (jcogbu1@lsu.edu) have argued that the problem is serious on any view. In their recent `What negation is not: intuitionism and "0=1"' (Analysis 60.1, Jan 2000, pp5-12) they argue that Dummett's view has to be incorrect. Why? Answer:

Defining negation in terms of `0=1' does not isolate a unique interpreation of the operator, and in addition fails to guarantee the logical or metaphysical unacceptability of assertions of form A&~A. This results from circularity...in the definition; `0=1' only attains its absurdity in virtue of arithmetic asioms inwhich negation occurs ineliminably. (6)To establish their point they give a model (where zero is the only element and identity is the successor function, plus and times defined as per usual) that satisfies each of Dummett's seven given (1977:33-34) axioms for intuitionistic arithmetic except for the first -- viz., (x)~(x=0). Then all negation-free sentences of the language are (classically) true in the model. When (*) is considered here, (*) implies that ~A (a conditional with true consequent) is true in the model for any A.

Cook and Cogburn proceed to discuss natural objections, etc. At any rate, the paper is well-written and worth a look. The question is whether pluralism has anything illuminating to say on this interesting topic. This is worth considering.

JC

David Israel is a researcher at SRI with a longstanding interest in logic, AI and semantics. His paper on the role of logic in AI is very clear, and contains an excellent discussion of the role of logic as a determiner of valdi inference, as opposed to logic as a tool for constructing or analysing *derivations*. I think that this is an important distinction for us, and it will help us to keep it in mind in our discussion of the proof-theory/model-theory distinction. (I hesitate calling this the *syntax*/*semantics* distinction, because some parties to the debate consider that the intro- and elim- rules for connectives *do* give the meaning of the connectives, and hence, that the proof theory is a semantics.)

More of Israel's papers can be found here.

G

I spent a few hours reworking my Research Links page. Now people are (mostly) categorised, and it should be easier to find researchers in fields you're interested in. An I can remember who-is-who a lot better, now, too!

Now to read some of the new papers I've downloaded, to see what might be relevant for us.

G

CSLI have had an excellent publication arm for years, but now they're excelling themselves by reprinting cheaply important but out-of-print texts. Two which we like are: Etchemendy's The Concept of Logical Consequence and Barwise and Perry's Situations and Attitudes.

On a sadder note, I heard about a week ago that Jon Barwise had lost his battle with cancer on Sunday March 5. The logic community is seriously the poorer.

G

John Etchemendy's `Reflections on Consequence' is now online. There are also a few of Solomon Feferman's papers worth studying, including some on logicism, some on intuitionism, and related topics.

JC

Jack Copeland has his old critical papers on the semantics of relevant logics online. Of special interest to us will be On When a Semantics is not a Semantics, Pure Semantics and Applied Semantics, and What is a Semantics for Classical Negation?

G

James Robert Brown has a nice little focussed discussion between Douglas Bridges and Geoffrey Hellman, on the usefulness of constructive reasoning in applied maths. We pluralists like this, because we think we can diagnose the discussion a little better than either side. We can understand what Bridges and other constructivists are doing, accepting 99% of their rhetoric. However, it's no skin off our (pluralist) noses if you can't constructively prove all that you might want to *prove*. It's grist for our pluralist mill that certain things might only be provable classically (or in another logic stronger than intuitionistic logic).

G

Just a quick note on some potentially useful readings: There are two *Monist* volumes that contain useful papers for the project. One is entitled Uses and Abuses of Logic in Philosophy, which contains papers presupposing various views of logic that we might consider. Another volume is entitled *Logic as a Field of Knowledge* (Jan. 1989, Vol.72 No.1), which contains some potentially useful stuff on the epistemology of logic. (Unfortunately, the latter volume is evidently too old to make the webpage! Still, it has a few interesting papers.)

JC

Now the maths students at Vanderbilt are getting a pluralist logic course. I like the look of the unit, and there are some nice resources on this page for us to mine. The instructor, Eric Schechter seems like an enlightened mathematician.

G

There's an old view proposed by (an early) Putnam according to which alternative logics (non-classical logics) are ultimately competing theories of *truth*. The exciting project of the Italian Pluralists hints at this sort of view.

The problem of truth in non-classical logics is a fundamental issue in the philosophy of logic. Today we have a variety of systems more or less different from classical logic. What does it mean, however, 'different' in this context? To answer this question we must take into account the reasons which prompt some authors to build non-calssical logical systems. Sometimes they are purely formal. But often the construction of a nonstandard system is due to the conviction that classical logic is wrongand contemplates a too narrow notion of 'truth'. (My emphasis)

Of course, in some sense different logics *are* different accounts of truth; at least, in some sense, they are different accounts of 'truth-in-a-case' (as it were). Still, I think that we should emphatically get away from this sort of picture, the picture of different logics being different, and competing, accounts of truth. Instead, we should see *truth itself*, or at least *truth-in-a-case*, as being multi-faceted; to understand the *whole* notion is to understand how truth behaves *in each case*.

The suggestion, in short, is as follows. Let the *Putnamian picture* of different logics be the one according to which different logics are merely different and *competing* accounts of truth. If we go with this picture at all, we should simply reinterpret the main line. What *we* will see in the picture is this: Different logics are different accounts of truth in so far as they are different *parts* of the one big account of truth; they are each accounts of how truth behaves in such-and-so a case.

This needs to be carefully clarified and, then, scrutinized. Many philosophers, I think, are inclined toward the Putnamian picture, including (as above) the Italian pluralists. We need to make clear how *our* pluralism sees the Putnamian picture (as it were).

JC

It looks like there are *lots of* Italian Logical Pluralists out there. Note the logic/geometry illustration, the reference to Carnap and the way they distinguish instrumentalism from pluralism. However, they do seem to be what I'd call contextualists: they think that different subject matters or contexts bring forth the requiremnent for different logics. So, I don't think that they're as thouroughgoing in their pluralism as we are. It looks to me like they don't agree with us that the *very same argument* might be appropriately called valid (in logic X) and invalid (in logic Y), for there's no difference in subject matter or context there at all.

Still, there's lots we can agree on. Maybe we should take a research trip to Italy. It's been a while for me!

G

Ivor Grattan-Guinness' paper
Are Other Logics Possible?: MacColl's logic and some English reactions, 1905-1912 is a nice little introduction to the context of Hugh MacColl's work on intensional or modal logic. Not only that, but he has a section entitled *The Fight for Logical Pluralism* and he explicitly construes MacColl as a pluralist. Might he be the first logical pluralist we can find? Can we go earlier? Was Peirce a pluralist? What about the Port-Royal School? How early can we go?

The rest of that issue of the Nordic Journal of Philosophical Logic is on MacColl, and there are other articles worth reading in it too.

G

Achille Varzi provides us with
more conventionalism to consider. JC pointed me to that paper, which argues for a conventionalism about logic. He argues that a logic is a theory in a very strict model-theoretic sense of the term *theory*. The idea is that any logic is simply given by a class syntactic units which receive a fixed interpretation on a class of models. Different logics are given by choosing different syntactic units (is identity one of them? is necessity?) or by choosing different fixed interpretations of the syntactic units so chosen. Intuitionistic and classical logic interpret negation differently, as one example.

I think I like this. It is a nice discussion of how *broad* one can take one's class of appropriate models. He says that there's no principled distinction between logical connectives and anything else. We find this line congenial. However, there are issues. For example, he writes:

... the difference between, say, classical and intuitionistic propositional logic reflects a different way of explaining the meaning of negation; the difference between classical and relevant logic reflects a different way of explaining the meaning of entailment; and so on. I suppose the same applies to most cases of logical ``unorthodoxy'': the way some logical terms or operations areThis is good, but deceptive. One might think that the notion of ``admissibility'' in play here is an all-or-nothing affair. Our pluralism regards admissibility as requiring more than one extension. Different models (or `cases' or whatever) are used for different purposes, or in styles of reasoning. In fact, such a pluralism is required, given Varzi's denial of an all-or-nothing matter concerning connectives.understoodchanges as we move from one system to another. So much I take to be obvious. The point that I intend to address -- and that I regard as not obvious -- is that this change of understanding can to a great extent be treated in plain model-theoretic terms. To endorse a certain logic is to select a certain class of meaning structures as the only admissible models (relative to a given language). To endorse a different logic is to select a different class of admissible models. And so on.

There's even *more* conventionalism about of course. Carnap comes up again and again these days, and we must come down on some kind of opinion on the *Aufbau* and Carnap's own conventionalism too.

G

I have changed the look of the page. Let me know if it renders worse in your browser.

G

There are a lot of non-pluralism-related chores to do this semester, including teaching, preparing the move to UConn, and more. But one thing that is becoming increasingly urgent (especially in response to queries by philosophers) is the following task: *Write a paper that explains the similarities and differences between (our) logical pluralism and Rescher's Conventionalism*. This has been on my mind for months, but it has to be done soon. Too many philosophers have asked this question in response to hearing of our pluralism.

JC

We like Steve Yablo's work. His webpage contains a few online papers. His *A Paradox of Existence* has some good metaphysics in it which we'll need. He explicitly addresses the status of claims like our (V) of validity, which seem to connect uncontroversially accepted ideas such as validity (or proportion, or possibility) with controversial entities such as models (or numbers or possible worlds). We mightn't *agree* with his broadly fictionalist approach, but we must certainly point to it.

G

I'm back in town after the trip. Some talks had to be cancelled because of a personal tragedy (a death in the family meant I needed to cut the trip short) but the Edinburgh, Glasgow, St. Andrews and Leeds pluralism talks went really well. More logging here soon.

G

Well, it's not a *world* tour. Just a tour of England, the Netherlands and Scotland. If you want to catch up with what's happening in logical pluralism, you'll be able to catch me giving pluralism talks at the following spacetime locations.

- Utrecht, January 19
- Leeds, January 27
- Keele, February 2
- Edinburgh, February 4
- Glasgow, February 8
- St. Andrews, February 9

It should be fun! Now I have to run along and finish organising my life.

G

Well, it seems like I can even add enries from home from my Newton (running lynx over telnet). My Europe schedule is firming up with a confirmed date in St. Andrews and likely dates in Edinburgh and Glasgow. I'll be busy!

G

This is a hectic time of year, and it isn't improved by all of the busyness of trying to get my trip organised. Still, I've managed to do a few things.

- I've submitted the Constructive Logic paper to the SEP conference.
- I've lined up talks in Amsterdam, King's, Leeds. Utrecht, Keele and St. Andrews are still in the pipeline.
- I've marked a lot of philosophy essays ...

G

It's been a long time between updates. I've not only been doing rather a lot of marking, but I've managed to get a short paper written. It's on constructive logic, truth and warranted assertibility. (In fact, that's its title.) The idea is a simple one. Shapiro and Taschek (in a 1996 *Journal of Philosophy* paper have argued that simply using
intuitionistic logic and its semantics, one can show that
there are no gaps in warranted assertibility. That is, if a statement *S* is not warrantedly
assertible, then its negation *~S* is. Tennant (any *Analysis* paper in 1995) has also argued for this conclusion, on similar grounds. In this paper, I show that
these arguments fail, albeit in interesting ways.

We *need* to do this, as I'm not about to endorse the anti-realist thesis that if *S* is true, it's warrantedly assertible -- even though I *do* think that the semantics for intutionistic logic is worth endorsing. How can I do this? Well, read the paper.

Feedback, of course, is always welcome. I'm thinking of submitting this to the 2000 SEP conference, as it seems an appropriate topic.

This paper should form a part of the chapter on constructive logic in the pluralism book. I've uncovered some issues (in terms of expressibility and incompleteness) which I think deserve fuller attention.

G

This is a new book by Tomasz Placek on different motivations for intuitionism and constructive logic. I must read this. Alas, it's rather *expensive* and although our library at Macquarie has it, it's not yet on the shelves. Waiting, waiting. I'm filling my train journeys with Dummett's *Logical Basis of Metaphysics*. That is required preparation for my trip to the UK.

G

One of us must attend this conference. We should also provide another joint paper. (Deadline: 3 Jan 2000.) One idea is to present the revised Contents paper, which is to be revised by the end of January. Another idea is a paper that discusses the relation between pluralism and truth. Hilary Putnam famously argued that many-valued logics are merely competing theories of truth. (This is crudely put, of course, but the point is clear enough.) A paper examining and criticizing Putnam's view from a pluralist perspective would be very useful; it would be interesting in itself, but also it would further clarify our position against a famous perspective on 'deviant logics'.

JC (from US)

The FOM (foundations of mathematics) discussion list has a number of interesting posts. It's too regular for me to subscribe (though I have been a subscriber in the past) -- but the search engine on the main page seems to serve up some decent results. Searches for relevant logic, intuitionism and even *polylogism,* give results worth examining further.

G

We need to look into the philosophical views of Polish logicians. Some of them,
including Bochenski, said some interesting things about (what they called) *logical relativism*. The Polish Philosophy Page is a great source for us, and we need to check it regularly for new entries, especially the
Papers page.

JC

Phil Bricker and Daniel Nolan think alike; each of them independently suggested that we (G and I) need to discuss the connection between Lewis's pluralism about counterparts and pluralism. As for Rescher's views, JC is already thinking about this, and will continue to do so!

One quick thought about issues of conventionalism and the like.... Why shouldn't our pluralism be neutral on this point, just as it's neutral on points of metaphysics generally? For example, our pluralism is neutral with respect to the 'nature' of possible worlds; they may be concrete (a la Lewis) or otherwise. Pluralism about logic is independent of this issue. My suggestion is that pluralism may be independent of conventionalism (and related issues) too.

JC

Here's another page which Google brought to light. This Czech student doesn't *endorse* pluralism but at least he mentions it.

G

Daniel mentioned in conversation today that we pluralists should consider things like the pluralism (at least implicit) in David Lewis' account of counterpart relations, and in Rescher's conventionalism or conceptualism about logic (and about just about everything). I don't think I'm a *conventionalist* about logic, or about counterparts, though I need to be able to explain what makes our pluralism like and unlike these pluralisms already there in related literatures.

G

Our pluralism, as expressed in 'Logical Pluralism' and 'Defending Logical Pluralism' (available from our publication pages), *demands* that consequence be reflexive and transitive. Consequently, we rule out various familiar systems, including, e.g., the Martin and Meyer S-for-Syllogism system, which rejects A|–A on grounds of circularity. (Likewise rejected, of course, is Tennant-style 'relevant logic' in which transitivity fails.) The trouble, at least with the M-and-M system, is that aside from foot-stomping or fist-shaking we really have no good explanation as to why there is not a genuine non-reflexive consequence relation. In support of saying that other logics capture genuine consequence relations, we have elsewhere ('Defending') invoked early words of J.M. Dunn and said merely that we recognise a sense of 'follows from' in which (say) arbitrary A does not follow from arbitrary B&~B. Well, why isn't there a sense in which A doesn't follow from A? In our 'Logical Pluralism' we left the issue open, and our opinions on the matter remain unsettled. But in the end, this is a big problem. (Well, at least one of us thinks so!)

Two things to do along this line. First, look very closely at any of the work Dana Scott has done wrt the formal side of the problem. Second, look *anywhere* in an effort to find philosophical considerations that may be useful. Granted, as Greg has said (in print and person), one must put one's foot down somewhere. He is correct. Still, it would be nice to have some sort of explanation as to why *here* rather than *there* is our stomping ground (as it were).

JC

David Pym at Computer Science at Queen Mary and Westfield College in London has done some interesting work on the logic of *bunched implications*, which recasts in the terminology of computing and category theory what relevant logicians like Belnap, Dunn, Read and Slaney have been doing for years -- paying attention to the ways that premises are combined in arguments. This is what *substructural logic* is all about.

Still, he's getting interesting results for quantifiers. We must pay some more attention to this.

G

In this entry I'm talking off the top of my head. It's rough, but I want to put it down on `paper' and this is as good a place to do it as any.

Last night I spent a little time thinking about pluralist accounts of negation. The idea is pretty simple (and some of it has featured in our paper *Defending Logical Pluralism* currently available from our publications pages). Instead of treating boolean negation, intuitionistic negation and a relevant negation as different connectives, a pluralist can think of them as different analyses of consequence of the *one* negation. This is all well and good as far as it goes. The question engaging me was simple: how far *does* it go?
We know that there is a single semantics encompassing boolean, intuitionistic and relevant negation (that was given in *Defending*). Why stop there? What about *dual intuitionistic* negation? This is a paraconsistent negation defined by setting

where `T` is the true constant satisfying `C |- T` for all propositions `C`.

This kind of negation differs from boolean, intuitionistic, and relevant negations in rejecting contraposition and other de Morgan laws. (The Dunn style semantics is helpful here in understanding the different behaviour. Intuitionistic logic has a *universal* flavour, and dual intuitionistic logic is *existential*).

It *sounds* like we have two very different ways of formalising negation. What *I* want is some kind of semantics in which at *some* points negation acts with the universal clause, and at *others* it acts with the existential clause --- *and* it works in a uniform way all over the model.

Why? Well, it's by analogy with what we've done so far. In an intuitionistic model, we say that the *classical* points are those which are endpoints of the ordering. I want to do the same thing here, but on a much more general scale.

I think I can do it, too.

G

Our paper "Contents", which was presented at the 1999 AAL conference in Melbourne, is being revised. The revision will be completed no later than January 2000. A very early (pre-AAL) draft is available from each of our respective publications pages.

JC

One of us at least should go to WCP #2 in Brazil. It looks like a lot of fun. However, the deadline for paper submission is less than a month, and I don't know if we have anything in particular to submit. I must rummage through my files when teaching finally finishes.

G

The *final final* copy was delivered to Routledge today, and it's off to the printers this week. The book may well hit the shelves in December.

G

RBJ seems to mention logical pluralism in his web resources in logic. The relevant pages are Are Foundations Necessary? and Logical Foundations and Formal Verification. There is no doubt that his interests are related to ours, though his focus is more computational than ours.

G

We found out today that the ARC Large grant was awarded. The project has funding. It's not as much as we had hoped, but it should be enough to do some interesting research in the next three years.

G

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