Banff report by Restall

Greg Restall has an extensive report (and here, here and here) from the Mathematical Methods in Philosophy conference in Banff. There are slides from Restall's own talk, which incidentally was on Modal Models for Bradwardine's Truth. Bradwardine was a 14th century Oxford logician, author of the Insolubilia, a medieval work on paradoxes which is currently being translated by my supervisor Stephen Read.

Is Quine defending classical logic?

Quote of the day:

Classical quantification theory enjoys an extraordinary combination of depth and simplicity, beauty and utility. It is bright within and bold in its boundaries. Deviations from it are likely, in contrast, to look rather arbitrary. (Quine, 'Existence and Quantification', 1969)

Nothing is quite as inspiring as an outrageous claim to start a long day of work. Everyone working with philosophy of logic knows Quine to be an ardent defender of classical logic (CL). I'm not going to go into the details of Quine's work here - it suffices to say that most relevant material can be found in his Philosophy of Logic (1970; 1986 2nd ed.). Rather, my concern is what it means to say that Quine, or anyone for that matter, is defending CL. In particular, what exactly is it that's being defended?

This is one of many questions that was discussed in last semester's Quine seminar in Arché. But it came back to me recently when reading an article by Beziau et al. In 'What is Classical Propositional Logic?', they discuss CL as a special case of how difficult it is to define a logic. Needless to say, there are many structures that characterises a logic, and, as it turns out, with different metalogical results (e.g., on one characterization propositional CL comes out as decidable, on another not (see p. 6)).

So, what is Quine defending? He's not, for example, defending an axiomatic system. CL can be given in a range of different proof-systems. Is he perhaps defending a set of theorems? Perhaps, but then what about the semantics. Surely, he is defending CL with a particular interpretations of logical constants in mind, not just a pure formalism. Ok, so he's defending a set of theorem plus classical semantics. What is classical semantics? Oh, it's just what we normally see in introductory books in logic, i.e., model theory.

But here we run into a little problem. If Quine is defending classical semantics, he will endorse the standard Tarskian interpretations of logical constants, including quantifiers. However, Quine's view on semantics deviate on a crucial point from the so-called classical account. Whereas Tarski, and also contemporary model-theory, has objectual quantifiers, Quine defends substitutional quantification (see Quine 1986 pp. 49-56, 91-94). True, if there had been no significant difference between the logics produced by these different accounts, it wouldn't make much different. But this is not the case: there are important metalogical differences between classical semantics and its substitutional counterpart. The resulting logic is only weakly complete, the upward Löwenheim-Skolem theorem fails, and so does compactness (see Shapiro 1991, pp.243-246 for details).

Maybe Quine is more logically deviant than he is normally given credit for.

Relativism about truth

Sven Rosenkranz, a research fellow of Arché, is visiting St. Andrews this week. On Thursday he gave a talk in the vagueness seminar entitled 'Frege, Relativism and Faultless Disagreement'. Now, recently I've been building up an interest for relativism about truth. Not only is it an Arché project, it's also threatening to become quite relevant to my own work on disagreement about logic (this, however, will be the topic of a later post).

Sven's talk spells out a dilemma for the relativist, but it's not my intention to go into that here. Rather, I'd like to comment on an innocent looking characterization of perspective. Take relativism about truth to be, roughly, the position that truth is relative to a perspective. Attributing it to Max Kölbel, Sven provided the following gloss: a perspective is a function from propositions to truth-values. Since I haven't actually had time to look up Kölbel's article (Faultless Disagreement, 2003), I'm not going to hold this characterization against anyone; instead I simply want to explore how we could spell it out.

So, let the function be f: P → V, where P is the set of propositions and V is the set of truth-values. Without yet knowing precisely which values we want P to contain, it's already clear that no proposition p ∈ P can be mapped to more than one value. In other words, there are no 'glutty' propositions. Take the simple case where V = {T, F}: with a total function we get bivalence, while a partial function allows for 'gappy' propositions. Presumably, perspectives are not restricted to non-gappy/non-glutty propositions, so some changes are appropriate. One way to go is simply to replace V with ℘(V). With f mapping to the powerset, we can let p be gappy by mapping it to ∅ and glutty by mapping it to {T, F}.

But this generalization still leaves much to be desired. For, why stop with ℘(V)? Surely, although it soon gets hard to provide the values with intuitive content, there is nothing preventing perspectives from taking values from ℘℘(V), etc. What is more, perspectives are probably not exhausted by powersets constructed from {T, F}. Genuinely new values could constitute perfectly acceptable perspectives, and combinations of these are as viable as combinations of T, F. In other words, an adequate characterization of perspectives should not exclude, say, continuum-valued logics, nor should it exclude gaps or gluts. Perhaps, then, we would do well to simply skip the idea of perspectives as function from propositions to values, but rather let the assignment be relational. (For an application of relational assignments of truth-values, see Priest (2001), ch. 8.)

Sven's talk went on to offer the following principle (also from Kölbel's paper):

(TR) It is a mistake to believe (assert) a proposition that is not true in one's own perspective (Kölbel (2003): 70).

With the set of values generalized, one would think that the rule of assertion is precisely one of those things relative to a perspective.


(PS. Thanks to Henri over at Theorem(e) for pointing me to this brilliant html tool.)

What is a logical notion?

Yesterday, the Arché Philosophy of logic seminar had a two hour session on Gómez-Torrente's 'The problem of logical constants' ( 2002). Most of the time was spent discussing the well known definition of 'logical notion' in Tarski (1986; published posthumously) and the related dispute between Gómez-Torrente and Gila Sher. There was several interesting remarks made about Tarski's definition, but the one I intend to briefly follow up here has to do with the possible extensions of the Tarskian approach.

Let me first introduce Tarski's idea to readers unfamiliar with the paper. Famously, Tarski ended his 1936 paper on logical consequence (new translation here) with a remark about the division between logical and extra-logical notions: "Underlying our whole construction is the division of all terms of the language discussed into logical and extra-logical". [Tarski, 1936, p. 418, old trans.] At the time Tarski was pessimistic about the prospects for a non-arbitrary division, but later in his career he returned to problem, giving the lectures that were to be edited and published by John Corcoran after Tarski's death. Roughly, Tarski's aim was to give a mathematical characterization of the traditional set of logical constants, e.g., Principia type languages. In particular, the scope of the project was intended to be the same as for his work on truth and logical consequence.

Let us turn to the definition itself. Tarski takes a notion to be an object of any type in a hierarchy of types. Let U be a universe of individuals, and let a permutation p on U be an one-to-one function mapping U onto itself. Any permutation p on U, induces a uniquely determined permutation p' for any derivative universe U'. We say that an object O in U' is invariant under all permutations p' if p'(O) = O.

An object O in some U', then, is a logical object if it is invariant under every permutation p of U. A logical constant in L is a symbol denoting a logical object. (For further details see Tarski & Givant [1985], p. 57)

Gómez-Torrente seems convinced that Tarski's definition does indeed give a necessary condition for logical constanthood. In fact, the results appear immediately quite impressive:

(1) Truth-functional connectives are logical constants. T is defined as the universal set, F as the empty set. Functions over these are of course invariant under any permutation.

(2) Likewise, the first-order existential and universal quantifier are logical constants. The denotation is a function assigning T to the universal set, F to all other subsets; and a function assigning F to the empty subset and T to all others. In fact, all finite-order quantifiers turn out to be logical.

(3) The empty set and the universal set are logical, and so are cardinality sets. There are four binary relations which are logical, identity, diversity, the universal relation and the empty relation.

(4) Individuals and predicates are, as should be expected, extra-logical. More importantly, the membership relation is also extra-logical.

From a more general perspective than Tarski's, however, it is an obvious complaint that his definition is silent about most of modern logic. The problem is simply that these are languages either not in the range considered by Tarski, or logics not prone to the semantic treatment he prefers. Yet, reading through these texts it struck me that there's probably something interesting to be said about an extension to Kripke semantics. This would enable us to consider modal operators in general, and in particular we could apply the definition to intuitionistic logic through it's well-known Kripke semantics.

Raising the point in the seminar, I received a very interesting reply from one of my fellow PhD students: Apparently, some work has already been done on this by extending the idea of permutations to Kripke frames. If we permute W, the set of worlds, then only the S5 modality is invariant. The weaker modalities are all sensitive to permutations because the modalities are functions taking a subset of W to either W (T) or the empty set (F). As such, if the R relation, i.e., some subset of WxW, has non-trivial structure (say, it's transitive and reflexive), then, since changes in W spill over into R, the function is not invariant. Since the S5 modality has a trivial R relation, this turns out to be a logical notion.

Unfortunately, I haven't been able to trace down any references yet, so I would greatly appreciate any hints on where to find remarks on such an extension.

Logica Universalis

The first issue of the new journal, Logica Universalis, is now available online. The list of content is here. There is a preface by Jean-Yves Béziau, the editor of the book with the same title, and one of the organizers of UNILOG.

(Through Theorem(e))

Helping Studia Logica

Just got this by email. Hat tip to Elia.

Dear Colleagues,

the Institute for Scientific Information runs a list of "master journals" http://scientific.thomson.com/free/essays/selectionofmaterial/journalselection/. This ISI list is considered in many countries as a base for evaluating scientific journals and scientific publications. Unfortunately, Studia Logica is still not on the ISI list, while many journals of lower quality are listed there. Springer (Studia Logica's publisher) has assured me that Springer has been taking care of the matter in the last few years. Nevertheless, their efforts so far didn't have any effect. For this reason I believe that an independent action of Studia Logica's editors is needed.

This is why I kindly ask for your help. It will take only a few minutes of your time. On the ISI web page there is a "recommending journal form" http://scientific.thomson.com/forms/isi/journalrec/. I would like to ask all the people cooperating with Studia Logica to complete it and submit it to the ISI.

Could you please follow the instructions provided below. The more recommendations the ISI will receive, the better. Could you please also encourage your cooperators and students to submit the ISI form.


Thank you in advance for your help.

With best regards,

Jacek Malinowski
Studia Logica, Editor-in-Chief

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How to proceed

1. Visit http://scientific.thomson.com/forms/isi/journalrec/

2. Complete the form using the following data:

- Journal title: Studia Logica

- Journal URL: www.StudiaLogica.org

- Publisher name: Springer

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- Unique features distinguishing this journal: please write here your opinion in this matter or write just "Applying Formal Methods to Philosophical Problems"

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4. Please let us know that you filled in the form. We will collect these data. The list of those who recommend Studia Logica might be helpful in our further interaction with the ISI.


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Arché Studentships Announcement again

The Arché Studentships announced for next academic year have now been updated, including a more generous stipend for several of the studentships. Note also the lack of nationality restriction, implying that US and other overseas students are eligible for four of the studentships. Especially for US students, it's worth mentioning that in addition to the stipends, you get paid for tutoring. Below are the details.

Arché is offering up to six PhD studentships for research in contemporary epistemology and/or philosophy of language commencing in September 2007. Successful applicants will work primarily with Arché Professors Jessica Brown, Herman Cappelen and Crispin Wright.

Prospective applicants are encouraged to get in touch about their planned research before applying. Two of the studentships are associated with the AHRC funded Basic Knowledge research project. The two Basic Knowledge studentships, which are open only to UK/EU applicants, provide full coverage of tuition fees and a maintenance grant of up to £12,500 per year for a maximum of three years (subject to satisfactory progress). Two further studentships, open to applicants of all nationalities, also provide a yearly maintenance grant of up to £12 500 and full coverage of fees. A further two studentships are likewise open to all applicants and provide full fees but a maintenance award at a lower level.

Applicants who wish to be considered for these studentships must apply for admission to the St Andrews Philosophy Graduate Programme through the Postgraduate Admissions Office by the 7th of March. Please indicate in your application that you wish to be considered for these studentships.

Further Particulars (PDF)

Two books

Reading Logic Matters, Cambridge logician Peter Smith's blog, I've kept myself updated on how things are going with his new book, An Introduction to Gödel's Theorems. In fact, I've been thinking about setting up a reading group early in the summer (when the semester is over) to go through the book. The book should be out in July, so hopefully we'll get it fresh from the print. As a supplement I would like to read through some of the philosophical literature on the Gödel's theorems (yes, we are all aware that there are some awful philosophical remarks out there, but there are also sensible ones). If anyone has proposals for sound philosophical papers on the topic, please let me know. I'm browsing around myself, but I'm sure there's a lot of interesting material published in the last few years.

In fact, Peter Smith recently toyed with the idea of starting a new project - this time on Gerhard Gentzen's consistency papers. I can do nothing but encourage the idea; there's obviously not enough accessible material on Gentzen's work.

Another book that stirred my interest recently was an anthology edited by Thomas Bolander, Vincent F. Hendricks, and Stig Andur Pedersen called Self-Reference. The book is out 13th of February, and should be interesting reading for our philosophy of logic seminar. There are contributions by Vann McGee, Graham Priest, Raymond M. Smullyan, and many more.

An aside: After subscribing to the SEP's RSS feed I just noticed that there's a brand new entry on dynamic logic. Check it out here.

Philosophy framed

Jason Stanley just provided an excellent link to photographer Steve Pyke's online pictures of famous philosophers. These pictures were originally published in the book Philosophers from 1993. Now, Pyke has continued his work, capturing a new generation of philosophers. Same quality? Anyone? Oh, sorry, I meant the photographs. You're welcome to comment on both.

I urge everyone to take a close look at the captions. These are brief words from the respective philosophers on their own philosophy and, more generally, the nature of philosophy. Note Kripke's laconic appended remark "I have also done some work of a technical or mathematical character". Some? That's a rather modest claim. Hartry Field opts for the humorous approach with this carefully argued passage: "A nice thing about philosophy of the sort I do is that it can never be used to justify wars or oppress the disadvantaged or anything like that. This follows from a more general principle." Yet, he can't challenge H. L. A. Hart's meta-remark: "To be frank I think the idea of a 50-100 word summary is an absurd idea... I advise you to drop it."

On a more serious note: Timothy Williamson has supplemented his own picture with words of a certain finesse. In a time when logicians too frequently underestimate the range of their discipline, thoughts such as these ought to be widely distributed.