Nothing of Consequence

I've started a new blog

2011-03-04T10:25:00.000Z

After moving to Munich and LMU I've started a new blog. There will be philosophy and logic, yes, but there will also be other things I'm interested in.

I don't promise higher frequency posting. Coming to a new city, it's easy to find time for blogging. If I get a life, however, productivity might drop. Sounds like a good deal to me.

Mini-course and Workshop on 'Logic or Logics?'

2010-08-07T11:14:00.001+01:00

A little ad for our upcoming FLC event in Arché. Lots of great people will be around to talk about logical pluralism, revision of logic and combining logics.

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The members of the Arché Foundations of Logical Consequence project will be holding a Mini-course and Workshop on the theme 'Logic or Logics?' from 27 September - 1 October, 2010. The aim of the Mini-course is to provide intensive graduate-level instruction on the latest thinking about pluralism and revision in logic, followed by a Workshop dedicated to new research on the same theme.

If you are interested please register now at the following link: http://onlineshop.st-andrews.ac.uk/browse/extra_info.asp?compid=1&modid=2&prodid=150&deptid=29&catid=7

The speakers for the Mini-course are:

• JC Beall (University of Connecticut)
• Carlos Caleiro (Instituto Superior Técnico, Portugal)
• João Marcos (DIMAp / UFRN, Brazil)
• Graham Priest (University of Melbourne/CUNY)
• Greg Restall (University of Melbourne)
• Gillian Russell (Washington University, St Louis)
• Johan van Benthem (University of Amsterdam)

The speakers for the Workshop are:

• JC Beall (University of Connecticut)
• Colin Caret (Arché, University of St Andrews)
• Roy Cook (University of Minnesota)
• Ole Hjortland (Arché, University of St Andrews)
• Greg Restall (University of Melbourne)
• Penelope Rush (University of Tasmania)
• Gillian Russell (Washington University, St Louis)
• Johan van Benthem (University of Amsterdam)

Further inquiries should be directed to arche@st-andrews.ac.uk

FLC Conference

2010-06-10T14:28:00.000+01:00

This weekend is our 1st FLC conference. We have an amazing group of invited and contributing speakers, plus a number of other visitors from all over. The full programme with commentators is here.

Translation of Thomas Bradwardine's 'Insolubilia'

2010-06-07T09:25:00.000+01:00

I just found out that Stephen Read's edition of Thomas Bradwardine's Insolubilia is available. This is the first full translation of Bradwardine's work on paradoxes. (For more on insolubles, see here.) The edition has a long introduction by Stephen Read, comparing Bradwardine's theory both to medieval and contemporary approaches to semantic paradoxes. The text, which contains both the Latin and the English, is based on all thirteen known manuscripts. Highly recommended reading, not only for historians of logic, but for anyone interested in paradoxes.

FLC Audit

2010-06-02T13:55:00.000+01:00

June 9-11th, 2010

Once a year the AHRC-funded projects in Arché have an academic audit. It involves an external auditor who comes to St Andrews to interview project members, and writes a progress report on their work. For the Foundations of Logical Consequence the auditor is Hartry Field. As part of Audit programme the auditor will give a paper next week, and so will the two postdoctoral research fellows in the project, i.e. Colin Caret and yours truly.

Here is the programme:

Wednesday 4-6pm: Hartry Field, "Is There a Problem about Revising Logic?"

Thursday 9-11am: Colin Caret, "Against Model Theory"

Friday 9-11am: Ole Hjortland, "Thrice Denied: Speech acts, categoricity, and the meaning of logical connectives"

More information on the upcoming FLC conference is soon to follow.

Meeting in Honor of Jouko Väänänen's Sixtieth Birthday

2010-06-01T17:51:00.000+01:00

16-18th September, University of Helsinki

The Helsinki Logic Group is organising a conference in honor of Väänänen's 60th birthday. The event is entitled "Set Theory, Model Theory, Generalized Quantifiers and Foundations of Mathematics". There is a great list of invited speakers, and also two tutorials on the 13-15th prior to conference.

Find more information here.

Workshop

2010-05-18T11:07:00.000+01:00

December 7-8th, 2010

A little plug for an upcoming workshop in Amsterdam entitled: "From cognitive science and psychology to empirically-informed philosophy of logic". Quite a mouthful, but an interesting topic. The list of speakers is below, and the workshop also has a call for papers. More information here. I've already seen Pelletier talk about reasoning and generics, a topic that proved really interesting. Much recommended.

Confirmed speakers:

Johan van Benthem: Opening

David Over: "New paradigm phychology of conditionals"

Michiel van Lambalgen: "Logical form in cognitive processes"

Helen de Cruz: "Animal logic, an evolutionary perspective on deductive reasoning"

Rafael Nuñez: "Towards a cognitive science of proof"

Francis Jeffrey Pelletier: "Reasoning with generic information"

Catarina Dutilh Novaes: "Formal languages and the extended mind"

Travel plans disrupted

2010-04-22T14:41:00.000+01:00

Unfortunately, the air traffic ban in Europe prevented me from attending UNILOG'10 in Estoril, Portugal this week. Colin and I were hoping until the end that we could rebook, but it's been impossible to find another flight. A real shame - I've been to both the former UNILOGs, and I was really looking forward to seeing friends again. The good news is that Catarina, my co-teacher of the logical pluralism tutorial, finally made it to Estoril. She's put together a compressed version of the tutorial, so other attendees will still have the chance to see most of the material.

Also, thanks to the organisers for their tireless effort, despite the unhappy circumstances.

With a bit of volcanic cooperation, however, I * will * be on my way to Helsinki on Monday. As announced some weeks back, I'll be there on staff exchange giving lectures for both undergraduate and postgraduate students.

Helsinki visit

2010-04-03T10:32:00.000+01:00

As I've said earlier, I'll be in Portugal between the 17th and 26th, for the 3rd UNILOG Congress. I thought I'd mention that I'll be moving straight from Lisbon to Helsinki the following week. I'll be there for six days on a Socrates staff exchange (an EU programme) between the University of St Andrews and the University of Helsinki. In other words, it's not a research visit, but a number of guest lectures for the students in Helsinki. I'll get back to you about the details and times when it's settled.

I'm much looking forward to meeting both faculty and students in Helsinki. Any suggestions about what else to do in Helsinki are welcomed.

Paraconsistent Foundations of Mathematics Blog

2010-03-05T10:57:00.000Z

Just to note that Zach Weber has set up a blog for the Melbourne based, ARC-funded project: Paraconsistent Foundations of Mathematics. Zach Weber is a post doc on the project, working together with Chief Investigators Graham Priest and Greg Restall. In addition, Francesco Berto, from the University of Aberdeen, NIP, is a research associate.

You can read about the aim and background on the blog. If you want the real thing, go look at Zach's PhD thesis Paradox and Foundation here.

3rd FLC Workshop: Propositions, Context, and Consequence

2010-02-21T08:00:00.001Z

March 20-21st, 2010

It's about time to advertise our upcoming 3rd FLC Workshop, entitled 'Propositions, Context, and Consequence'. This time around the Foundations of Logical Consequence project moves into an area closely connected to the Contextualism & Relativism project. We're having a two day workshop focusing on the relata of consequence, context-sensitivity, indexicality, and more.

The speakers:

Francesco Berto

Catarina Dutilh Novaes

Walter Pedriali

Martin Pleitz

Stewart Shapiro

Hartley Slater

Isidora Stojanovic

Elia Zardini

Programme and more information here.

Categoricity and Absoluteness

2010-02-18T08:00:00.001Z

There are lots of ways of being a logical inferentialist. I've harped on a number of times about proof-theoretic harmony and other proof-theoretic constraints that inferentialists favour. In short, these approaches, which mostly concern themselves with finding, in some sense, appropriate pairs of rule-sets (in natural deduction or sequent calculus), are following in the tradition of Nuel Belnap. His was probably the most influential reply to Prior's venomous tonk-paper. Yet, there were a series of other replies, some of which receive little attention in the contemporary debate about proof-theoretic semantics.

People like J.T. Stevenson ("Roundabout the runabout inference-ticket") and Steven Wagner ("Tonk") had---perhaps like Prior himself---little sympathy for proof-theoretic semantics (in its early versions by Popper and Kneale). Both suggested that the real trouble with tonk is its blatant disregard for the underlying truth-conditional semantics (in fact, Stevenson is even looking for truth-functional semantics). These replies were mostly ignored in the harmony industry, for two different reasons, I suspect: First, a recourse to Tarski style semantics was precisely what the proof-theoretic semanticists were trying to avoid. Second, even if you thought there was something to the idea that when a set of rules determine the meaning of a logical connective, it determines a truth-condition, the above papers offered little or no insight into how to formalise this connection. In fact, both appear to be mostly concerned with classical logic (and boolean values).

Yet, their suggestion comes with some merits. First, inferentialists, although they provide an interesting idea of how concept-acquisition works for logical concepts, had more to say about justification of deduction than about the nature of semantic content---other than that it's not truth-conditional à la Tarski. Wagner suggests a worthwhile Fregean distinction (I here adopt some terminology from Hodes "On the sense and reference of a logical constant"): Although inference rules are constitutive of the sense of a logical connective, the reference is not to be identified with the set of inference rules. Rather, the reference is truth-conditional and it is determined by the sense. There is a hope here of having the best of two worlds. On one hand we get a recognised theory of semantic content; on the other hand we can still adopt the inferentialist's thinking about entitlement to inference.

But despite any initial advantages, the problems described above still remain. How precisely does the relation between inference-rules and truth-conditions work? How do the inference rules carve out the semanic content? And, importantly, is this simply a way of spelling out a relationship between classical inferences and classical semantics?

In an attempt to answer these questions I've worked on some technical notions that can help give us the appropriate go-between for the truth-conditions and the inference rules. Standardly, proof-theory and model-theory is kept together by soundness and completeness, but these results tell us little in way of how the rules of the proof-theory determine what the models look like. We can highlight this in a straightforward way. Assume that we have an antecedent idea of some basic semantic concepts, in particular, the difference between our designated and undesignated truth values. However, assume that we have no antecedent notion of the semantic conditions for the logical connectives in question (over and above that they operate on said values). Consider then the example of classical logic, in a standard (single-conclusion) natural deduction system. Carnap showed that such systems are non-categorical in the sense that the inference rules are sound and complete with respect to two distinct classes of boolean valuations. (In fact, to an infinite number of different ones, see Hardegree's "Completeness and Super-Valuations".) Of course, only one of these classes is the admissible class of classical valuations, in the sense that it can be induced from the atomic assignments via the truth-conditional clauses for connectives. However, the point in the inferentialist context is precisely that these clauses cannot be assumed; they must be determined by the proof-theory. In our example, the non-categoricity amounts to a sort of non-uniqueness result for the proof-system at hand (with respect to boolean valuations).

It is natural, then, to think of categoricity as constraint one would like to impose on a proof-system for the rules to determine the truth-conditions. The idea is as follows: If the rules are sound and complete with respect to only one class of valuations, then we have learned something about the semantic content of the connectives; namely that their truth-conditions must respect whatever conditions the class of valuations yields. If we move to a multiple-conclusion system, Shoesmith and Smiley (Multiple-Conclusion Logic) showed us that we get precisely this. A classical multiple conclusion system is only sound and complete with respect to the class of classically admissible valuations. We have, in other words, a way of carving up the truth tables for classical connectives looking exclusively at the inference rules. Of course, we do apply some background assumption about what the matrices contain (e.g., the values), but we make no assumption about the semantic content of, say, negation and disjunction.

What about non-boolean matrices? Without getting into details, it is crucial that the notion of categoricity at play in Shoesmith and Smiley's work is defined with respect to partitions of formulae into those that a valuation takes to 1 and those it takes to 0 (for the boolean values). When we go (finitely) many-valued, on the other hand, the partitions glosses over distinctions that don't cut across the designated/non-designated divide. Put differently, the definitions aren't fine-grained enough. We can learn something about the consequence relation from the inference rules, since consequence (at least normally) is only concerned with preservation of designated values. But, the truth-functions contain more detailed information, that can not be coded by proof-systems that can be categorical in the above sense.

What is called for is the more fine-grained notion of absoluteness. There is interesting work on this in Dunn and Hardegree's Algebraic Methods in Philosophical Logic. Again, without being too precise, absoluteness holds of a class of valuations just in case the logic induced from the class of valuation, itself induces a class of valuation which is identical to the original. I.e., V = V(L(V)), where V and L are functions such that (a) L(V) is the class of arguments not refuted by any valuation v in V; and (b) V(L) is the class of valuations which refutes no argument in L. (A logic is meant in the minimum sense of a class of arguments; of course the shape of them, e.g., single- vs multiple-conclusion (premise), will matter.)

Absoluteness is a stronger notion than categoricity. In fact, whereas Shoesmith and Smiley proves that every finitely many-valued logic is categorical with respect to a multiple-conclusion system, the result does not hold for absoluteness. Multiple-conclusion, although sufficient in the boolean case, doesn't hold in general. What is needed, it turns out, are generalisations using n-sided sequent systems, where n is the number of values in the logic in question. Such systems have been explored in much detail in Richard Zach's Proof theory of finite-valued logics.

Humberstone to visit University of St Andrews as Carnegie Centennial Professor

2010-02-17T15:44:00.000Z

We've known for a while, but now it's finally public: Lloyd Humberstone (Monash) is visiting St Andrews from March to June, 2012. The University has won a Carnegie Centennial Professorship that will fund the visit and a number of lectures at different Scottish universities. Humberstone will spend the time attached to the FLC project in Arché.

The FLC members are of curse very pleased about this, and we're now gearing up to prepare the visit by reading (selected parts of) The Connectives. If you don't know the unpublished manuscript by Humberstone, you need to have a look.

St Andrews announcement is here.

Future Contingents in Late Medieval Philosophy

2010-02-15T22:20:00.001Z

While Arché's regular schedule is still on break until late March, we've started up our Medieval Logic reading group (MLG) again. So far we've worked mostly with theories of insolubilia and views about logical consequence (e.g. Bradwardine and Buridan), but now we're moving a bit further out in the terrain (at least for me). This semester we're taking on Ockham on future contingents, in particular his Predestination, God's Foreknowledge, and Future Contingents. I haven't really done any work on future contingents since I took a graduate class with Patrick Greenough some years back, so I might try to refresh my knowledge of the contemporary debate as well. Meanwhile, I enjoy reading some historical background material in Calvin Normore's article on future contingents in the Cambridge History of Medieval Philosophy. Here is an interesting paragraph from the text (371):

I wonder if there is a connection to truth-is-only-an-expressive-device deflationism here. Any thoughts? I hope to return with more on Ockham as we start working our way through the text.

FLC Conference Deadline Today

2010-02-15T09:09:00.000Z

Just a reminder that the deadline for the 1st Conference on the Foundations of Logical Consequence is today, Feb 15, 2010. Find more information at the conference website.

Logical Pluralism Mini-Course, UNILOG'10, Estoril, Portugal

2010-02-01T08:00:00.002Z

April 18-25

UNILOG'10, the 3rd of its kind, will be held in Estoril, Portugal. As before, the event will have two parts: First, a school consisting of several mini-courses on different subjects in logic; second, a conference with a number of topics in both open sessions and dedicated sessions. In addition, there is a competition, this time with the title/problem: How to combine logics?

I'm happy to announce that Catarina Dutilh Novaes and I will be giving a tutorial on logical pluralism in the school. In three sessions we'll run through the emergence of logical pluralism, the contemporary debate, in particular Beall and Restall's book Logical Pluralism, and, finally, different reactions to logical pluralism, e.g. Hartry Field, Graham Priest, Stephen Read. Hopefully, we'll also be able to shed some light on how the Universal Logic programme itself connects to contemporary debate of logical pluralism. We hope many of you will be interested in attending.

A full list of contributed talks, tutorials in the school, and special sessions can be found on the UNILOG website. My talk is about proof-theoretic semantics and many-valued logics. I'll post a note here when abstracts are available online.

PS An aside: We're currently in process of planning an event entitled 'Logic or Logics?' in Arché for next autumn. The event will involve both a minicourse and a workshop. I'll post more as soon as speakers and details are settled.

Logic and Experimental Data

2010-01-27T10:00:00.001Z

In the presence of lots of philosophers of methods, it's hard not to spend some time thinking about formal methods in philosophy. For a long time I've wanted to write something about what formal methods in philosophy are, and why we should keep on using them. But, since I've yet to formulate my thoughts on this, I'll compromise and give you some related thoughts about logic and experimental data. More precisely, experimental data and the model- vs proof-theory divide.

Although frequently labelled as one, I'm not at all the sort of hardline proof-theorist who finds no room in his philosophy of logic for model-theoretic techniques. (Does the hardliner exist? Yes - check out this paper by Tennant.) Quite the contrary, I believe the two approaches complement each other, both being equally valuable in formal methods broadly construed. Nevertheless, for particular purposes in one's philosophy of logic, preferring one over the other is sometimes right. Lots of philosophers share the suspicion that inference rules are somehow integral to acquiring and possessing logical concepts. And, accordingly, a lot of energy has been put into exploring the link between this idea and entitlement to infer. (Examples are Christopher Peacocke, Paul Boghossian, and Crispin Wright.)

Yet, in the spirit of experimental philosophy one might wonder whether or not the critical claim about concept-aquisition is empirical (and, further, experiment-susceptible). I started thinking harder about this issue after an introductory talk by Ruth Byrne, a leading expert on the mental model theory in psychology. The mental model theory is a theory of reasoning in psychology which is accompanied by numerous experimental results. Roughly, the theory says that human capacity to imagine mental scenarios or possibilities is what engenders basic inferences. The experimental work involves Wason task studies, studies on counterfactual reasoning, and on how children develop mental models. For some information about the programme, see the Mental Models Website. Actually, I noticed that the psychological theory is not entirely unconnected to its formal counterpart, model theory: In the SEP entry for model theory, Wilfrid Hodgson mentions mental models as cognitive theory support for the importance of model theory in logic.

Correspondingly, proof-theory has its own psychological counterpart. In fact, the mental model theory is a response to the idea that inference is a result of a rule-based mental capacity which relates to formal calculi. Byrne calls it the formal rule theory. As its model-theoretic rival, the formal rule theory offers experimental support for why reasoning as a psychological process is a largely rule-based affair. One source for this view is The Psychology of Proof (1994) by L. J. Rips.

You don't have to subscribe to experimental philosophy in its most rampant forms in order to see the value of connecting the above experimental work with the philosophy of logic. In fact, my hunch is that philosophers - although experimentally crippled (aka "don't try this at home") - can contribute to the experimental work with logical expertise. Actually, philosophers of logic have hypotheses about actual human reasoning that are primed for testing, and which aren't likely to be tested by psychologists who don't know discussions about, say, vagueness and semantic paradoxes. Earlier I've mentioned one admirable adventure into experimental research by Dave Ripley, and another by Jeff Pelletier.

My plan is to start out by checking out The Rational Imagination (2005, MIT Press) by Ruth Byrne.

Studentships, Birkbeck College, University of London

2010-01-26T11:53:00.000Z

PhD studentships in philosophy of logic and mathematics available at Birkbeck College, University of London. More information here. The studentships are associated with Øystein Linnebo's project 'Plurals, Predicates, and Paradox'.

HT: Logic and Rational Interaction

CFP: Conference on the Foundations of Logical Consequence

2010-01-25T08:30:00.002Z

June 11-15, 2010

I'm very happy to announce that there is a call for paper out for the 1st Arché Conference on the Foundations of Logical Consequence. This is the first major event on the FLC project's output schedule. Hopefully, some of my readers will want to contribute a paper to make this a great event. There will be both open sessions for submitted papers, and designated sessions for graduate speakers.

The list of invited speakers are as follows:

JC Beall (UConn)
Josh Dever (University of Texas at Austin)
Hartry Field (NYU)
Michael Glanzberg (UC Davis)
Hannes Leitgeb (Bristol)
Vann McGee (MIT)
Agustin Rayo (MIT)
Dag Westerståhl (Gothenburg)
Robbie Williams (Leeds)

Any questions? Get in touch with me or flc.conf@googlemail.com.

More information about our 3rd FLC Workshop will follow soon.

I Fought the Law, and the Law Won

2010-01-24T14:00:00.002Z

Before the holidays I spent a couple of days visiting The Northern Institute of Philosophy (NIP) in Aberdeen. Ian Rumfitt had been invited up to give three talks on his new book project, The Boundary Stones of Thought. Great seeing friends again in Aberdeen, and a good opportunity to discuss the nature of logical laws. Perhaps some photos will appear on the NIP blog in the near future.

Rumfitt's project interests me a great deal. Although I disagree substantially with lots of what he has to say, I'm very happy to see a book-length attempt at getting to grips with the nature of logical laws. Even more so because Rumfitt's starting point - much like my own - is Dummett's The Logical Basis of Metaphysics [LBoM] (although, of course, the title is a reference which takes us even further back, to Frege's Grundgesetze). Here I'm only going to say something about Rumfitt's set-up, that is, the preliminaries before he presents the details of his theory. Maybe later, when the book is out, I'll look closer at the details.

Rumfitt's over-all aim is a defence of classical logic. Of course, this is ambiguous in a variety of ways. First, an easy defence of classical logic is to defuse concrete objections from the non-classicists camp, say, the intuitionist Dummett. Alternatively, however, a defence might be a wholesale justification of classical logic. Needless to say, this is a great deal harder to achieve (and, to understand). Second, what exactly is being defended is a bit unclear. It turns out that in Rumfitt's case the One True Logic is indeed a classical consequence relation, but, importantly, need not be associated with classical semantics. The hallmark of the latter, as Rumfitt understands it, is the principle of bivalence. So, in short, Rumfitt is willing to sacrifice bivalence to defend a classical consequence relation. Interestingly this separates Rumfitt from some other champions of classical logic, e.g., Timothy Williamson.

(An exception is the conditional. Rumfitt chooses to set this complication aside. For the purposes of the book he is only interested in the correct logic for conjunction, disjunction, negation, and quantifiers, admitting that the conditional most likely requires special (non-classical?) treatment. Of course, this already conflates a number of rival positions where the conditional is precisely the demarcating point.)

Rumfitt isn't going for the latter, wholesale justification of classical logic. He only promises to defend classical consequence against some of its most stubborn attacks - largely the attacks launched by Dummett and Crispin Wright. Instead, following Dummett, he sees the central problem of logical dispute as a sort of circularity charge (see for example LBoM, chs. 8-9). It is no good, it seems, to justify logical laws applying inferences that themselves are warranted by these very laws. The Holy Grail, then, is a framework in which such disputes can be settled, without illicit applications of laws under dispute. More, it ought to be a framework in which the semantics of the involved logical constants are stable across the logics under consideration. This is crucial if one wants the dispute to be more than a case of talking at cross-purposes. (Of course, if you think that the inference rules already determine the semantics of logical constants, this problem takes on an even more critical shape.)

Rumfitt wants to offer a framework where the semantic clauses are stable, but where the consequence relation is sensitive to how one defines the space of logical possibilities. A substantial part of his work is dedicated to defend a particular notion of logical possibility, but I won't go into that here. Rather, I'm interested in how this compares to another view: The logical pluralism of Beall and Restall. Formally speaking the two views, although different in aim, share some features. Both give a sort of all-purpose template for logical consequence, which fluctuates with the choice of model-input. Beall and Restall call them cases while Rumfitt calls them logical possibilities, but the idea is similar. Consequence is schematic. (See Beall and Restall's Logical Pluralism, OUP 2006, p. 29.)

Here is Rumfitt's general account of consequence:

Some premisses A_1, ..., A_n R-relate to a conclusion B if and only if, for any possibility x in S, if A_1, ..., A_n are all true at x then B is true at x too.

As Beall and Restall, Rumfitt requires consequence relations to have the Tarski properties (reflexivity, transitivity, monotonicity), and to be truth-preserving, although perhaps designated-value-preserving would be more appropriate. Rumfitt also insists that consequence-relations are single-conclusion (rather than multiple-conclusion). His reasons are more or less those stated in his 2000 paper ''Yes' and 'No'' and the more recent 'Knowledge by Deduction' (2008), so I won't bother to report them, although I don't finding them convincing. As it happens, Beall and Restall also opt for this constraint, but I suspect that they're not wedded to it. (For Restall's view on Rumfitt on multiple conclusion, see here.)

That is not to say that the two approaches to consequence relations are the same. It all boils down to the type of structures you allow in your schematic consequence relation. For example, Beall and Restall considers not only classical and intuitionistic logic (via classical models and constructions), but also relevant logic (via situations). Rumfitt on the other hand, perhaps following the spectrum in Dummett's LBoM, appears to limit his scope to classical, intuitionistic, and quantum logic. The algebraic framework he proposes allows one to suggest different structure for the space of logical possibilities. Which logical consequence relation one gets depends on the choice of structure.

I'll bypass the objection that these frameworks are not sufficiently encompassing. (Why rule out other substructural candidates?) The genuine challenge is to explain why there is sameness of content for the logical connectives across the rivals. Despite the fact that the semantic clauses are the same, the type of structures allowed obviously make a difference to the resulting logic. Take for example:

For every v,w, v_w(¬A) = 1 iff for every w' s.t. wRw', v_w'(A) = 0.

Is there sameness of content for the negation independent of the structure of R? If there is, well then there is reason for thinking that the classical and intuitionistic negation has the same meaning. Similarly, the semantic clauses of many-valued logics are formulated with MIN and MAX functions, with obvious flexibility depending on the set of truth values. Same content? The problem, put bluntly, is that only formal ingenuity stands between us and the conclusion that any two logical connectives have the same meaning.

I'll return with more once I have the details from the book.

Arché Gone Google

2009-12-20T14:59:00.001Z

Google has been running a campaign called Gone Google where businesses, schools and organisations switch over to Google Apps, a free, web-based platform for communication and collaboration. The Arché Research Centre in St Andrews has started its move over to Google this semester, introducing Google Groups for different projects, Google Docs for minuting and research collaboration, and a shared Google Calendar for the centre's activity.

How does it make a difference?

(1) Google Calendar: Arché now has a new schedule online. The schedule is based on Google Calendar, and is synchronised with our own calendars. If I update the Arché calendar stream in my private Google Calendar (on my phone or my computer), the public Arché calendar is automatically updated and everyone subscribing to the Calendar stream has their private calendars updated as well. This makes for a more efficient and accurate schedule. You can subscribe to both Google Calendar and iCal at the bottom of our Calendar page. If you don't need to see the Arché Calendar stream on a daily basis, you can deactivate it while seeing your other events. Note also that there are two different streams, one for weekly seminars, and one for events (e.g., workshops).

(2) Google Docs: Instead of the endless email exchanges with series of updated attachements, collaborative work - either research or admin - can be done in a shared document in Google Docs. It's perfect for writing seminar minutes of Arché's research projects, but also for writing papers together (at the same time) and commenting on papers. It's even got a LaTeX editor that can do a bit of basic logic, fractions, etc. (A more robust LaTeX options might be on its way as well.) Andrew Cullison writes on length about Google Docs for teaching purposes here. You can also share folders, using them for example as repositories for papers that a reading group is working through.

(3) Google Groups: Maybe this isn't universal, but we've had a hard time maintaining email lists on the university servers. Google Groups gives you an easy way to start and manage email lists without going through university bureaucracy. A typical problem with email lists is that one wants to include people without keeping names and addresses that are no longer valid or relevant. With Google Groups the admin is easy and can be shared between the researchers leading the project, seminar or reading group. You can also share documents through the Groups.

Hopefully, this is just the start. Soon we'll use Google Wave for collaborative research, integrating the above features in one interface.

New website

2009-12-16T12:19:00.000Z

I've migrated over to the google site platform, and I've made a new website. I'll soon start uploading some work in progress. Fortunately, there is an Arché recess for two months in February and March, so I'll have the chance to write up some papers that have been overdue.

Workshop Report: Logic of Denial

2009-12-15T12:08:00.002Z

Here is a short report from the 2nd FLC workshop on the Logic of Denial.

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Workshop Report

The Logic of Denial, October 24th - 25th

2nd Foundations of Logical Consequence Workshop

FLC, The Foundations of Logical Consequence, is an AHRC-funded project run by the Arché Research Centre in the University of St Andrews. The four year project is currently in its second phase, The Structure of Logical Consequence. As part of the regular activity, FLC has just hosted its second workshop, entitled The Logic of Denial.

Since Frege, the analysis of denial as assertion of a negation has been the received view in formal logic. However, recent literature has seen a number of attempts at introducing denial into formal systems as a primitive alongside assertion. The workshop invited its speakers to discuss how one best treats denial in a formal framework. In particular, how does denial correspond to different negations, and what is the relationship between logical consequence and rational demands imposed by the norms of assertion and denial?

The workshop's first day started with Dave Ripley (Institut Jean-Nicod, Paris) on 'Embedding Denial'. He discussed the strategy of introducing a primitive denial operator to address problems with expressive power in paracomplete and paraconsistent approaches to semantic paradoxes. Arguing from broader considerations about the nature of denial, he concludes that these theories ought to include a "complete" and "consistent" denial operator. Heinrich Wansing (Dresden) developed a proof-theoretic semantics for bi-intuitionistic logics with a denial-like negation. BHK (Brouwer-Heyting-Kolmogorov) semantics is extended by applying proofs, disproofs and their duals. In a similar vein, Luca Tranchini (Tübingen) offered a dualisation of proof-theoretic semantics in terms of refutation. For this purpose he developed a natural deduction (single-premise, multiple-conclusion) for dual-intuitionistic logic. Finally, Ian Rumfitt (Birkbeck, London) rounded off the first day by revisiting his bilateralist theory (where content is specified by both assertion- and denial-conditions), and offering arguments for another theory of content-determination: Evidentialism.

Peter Schroeder-Heister (Tübingen) started us off on the second day with an extension of his proof-theoretic semantics, using a denial operator in programming clauses, and developing corresponding harmony principles between assertion and denial rules. Next up was Michael De (Arché, St Andrews), with an investigation into how the norms of denial are affected by falsity-preservation consequence. Colin Caret (Arché, St Andrews) returned to the topic of semantic paradoxes discussed by Ripley the day before. He explored how we can get a denial operator that blocks revenge paradoxes while preserving as many intuitions about denial as possible. Greg Restall (Melbourne) had the honour of closing the workshop. He developed his theory that multiple-conclusion derivations can be interpreted as rational constraints on assertion and denial, extending it to discussing issues such as logicality and tonk.

The workshop had about 30 participants. We hope they all share our opinion that the event created significant impetus to future work with the logic of denial, highlighting common ground for researchers from a number of different fields. If there was one single conclusion from the workshop as a whole it was that formal approaches to denial offer interesting enrichments of expressive power both in theories about semantic paradoxes and in proof-theoretic semantics.

More information about Arché and FLC events can be found at:

http://www.st-andrews.ac.uk/~arche/index.php

The Organisers,

Stephen Read

Colin Caret

Ole Thomassen Hjortland

Gentzen and Normalisation

2009-12-13T16:44:00.000Z

Recently, Arché and the St Andrews computer science department (with Roy Dyckhoff) jointly organised a one-day symposium celebrating the centennial of Gerhard Gentzen's birth, November 24th, 1909. The event showcased cross-disciplinary work in computer science, philosophy and logic in short instalments. As keynote speaker, we invited Jan von Plato (Helsinki), proof-theorist and Gentzen scholar. Arguably, he was responsible for a rather significant upset in our little logic community.

I'm a huge Gentzen fan, in fact I think he deserves a place together with Gödel and Tarski in the history of 20th century logic. Even more than the two latter, Gentzen's work contributed to the formation of an entirely new discipline in mathematical logic, a discipline that now completely dominates theoretical computer science, and contributes significantly to both philosophy and linguistics. Yet, biographically, I knew a great deal more about Gödel and Tarski. (If you want a Gentzen biography, this is what you need.)

One thing I didn't know about Gentzen was that there are hundreds of unpublished pages, stenographed in Gabelsberger shorthand. The transcription of Gentzen's notes is a substantial task, but hopefully in the near future we'll be able to see the full scope and significance of his Nachlass. One thing that has already been uncovered, however, but which surprised me quite a bit, is that Gentzen proved normalisation for intuitionistic logic in 1932. In other words, over 30 years before the result was published in Dag Prawitz's dissertation!

Of course we knew all along that Gentzen must have been on to something like normalisation: After all, that appears to be part of the motivation for him looking for, and proving, the cut elimination theorem (the Hauptsatz) for classical logic, and there are some revealing remarks indicating that he had looked at conversions for intro- and elim-rule pairs. Yet, it is astonishing to learn that a result that wasn't known to logicians until 1965, was not only proved by Gentzen while working on his dissertation, but also excluded from the final version of his dissertation. One explanation, offered by von Plato, is that Gentzen thought the direct proof of normalisation unnecessary as he had the corresponding proof of cut elimination in the sequent calculus.

It gets worse. Not only had Gentzen already proved normalisation for intuitionistic logic (and probably failed to do the same for classical natural deduction), but his handwritten manuscript was in his supervisor's (Bernay's) possession after the war. In fact, and here is the real killer, Bernays showed the manuscript to M. E. Szabo, the editor of The Collected Papers of Gerhard Gentzen, and the latter proceeded to use a picture of the manuscript on a cover page of the edition. Szabo used page two, but if he had used page nine instead, we would all have been staring at the start of the normalisation proof. From von Plato's paper Gentzen's Proof of Normalization for Natural Deduction:

Remarkably, most of page 2 of Gentzen's manuscript is reproduced photographically among the frontmatter of Gentzen's Collected Papers of 1969. The editor indicates that the source for the manuscript is Bernays. It is astonishing that he did not mention in this connection, say, the complete set of detour convertibilities that stand very prominently on top of page 9 of the manuscript, or the announcement of the proof of normalization in Gentzen's summary of his further results at the end of the first introductory chapter, and its realization in chapter three. (249)

The paper by von Plato also includes a translation of the handwritten manuscript. An obvious question is to ask to which extent Gentzen's had figured out the details of normalisation. The answer is that his proof is as detailed as what you find in standard textbooks today, complete with conversions and examples. He calls the maximum formula of a non-normal proof a 'Hügel' (translated by von Plato as 'hillock'). I leave the reader to marvel at the details in von Plato's translation.

PhilPapers survey: Logic

2009-12-10T23:18:00.002Z

The PhilPapers survey has generated a lot of discussion lately. I took the survey myself, and thought that answering the question was a nice exercise in thinking about my own views. In some ways, my own answers surprised me more than the general outcome of the survey.

I suspect that my answers fit well with the Karenin effect. Anna Karenina's husband is the type - we are told - that easily and gladly makes judgements about things of which he knows nothing (art and literature, in particular), but who finds it much too hard to be absolute in his own field (Russian law).

The quintessential logic-question was "Logic: classical or non-classical?" An unsurprising 51,5% answered accept or leaning towards classical logic; against a 15.3% accept or leaning towards non-classical. An honest 12% admitted to being insufficiently familiar with the issue. Myself, I'm a fan of the 'Accept both' answer, thus reading the question as an inclusive or. Nothing wrong with that perhaps, but it makes one wonder what the question here really is about. Classical or non-classical logic for what? - we might see the pluralist add. 3.4% found the question too unclear to answer, and although that was a bit tempting, I suspect that if you go down that path, most questions in the survey would need similar treatment.