## Wednesday, 4 December 2013

### CfA: Summer School on Mathematical Philosophy for Female Students

LMU Munich, July 27 - August 2, 2014
http://www.mcmp.philosophie.uni-muenchen.de/news/mathsummer2014/

The Munich Center for Mathematical Philosophy (MCMP) is organizing the first Summer School on Mathematical Philosophy for Female Students, which will be held from July 27 to August 2, 2014 in Munich, Germany. The summer school is open to excellent female students who want to specialize in mathematical philosophy.

Since women are significantly underrepresented in philosophy generally and in formal philosophy in particular, this summer school is aimed at encouraging women to engage with mathematical methods and apply them to philosophical problems. The summer school will provide an infrastructure for developing expertise in some of the main formal approaches used in mathematical philosophy, including theories of individual and collective decision-making, agent-based modeling, and epistemic logic. Furthermore, it offers study in an informal setting, lively debate, and a chance to strengthen mathematical self-confidence and independence for female students. Finally, being located at the MCMP, the summer school will also provide a stimulating and interdisciplinary environment for meeting like-minded philosophers.

LECTURERS: Rachael Briggs (ANU), Catrin Campbell-Moore (MCMP), Sebastian Lutz (MCMP), Conor Mayo-Wilson (MCMP), Gil Sagi (MCMP), Sonja Smets (Amsterdam), Florian Steinberger  (MCMP)

ORGANIZERS: Stephan Hartmann, Catherine Herfeld, Hannes Leitgeb, Kristina Liefke

APPLICATION: For details about the application procedure, have a look at the above-mentioned webpage. The deadline for application is 15 February 2015.

MCMP EVENT POLICY: Have a look at

http://www.mcmp.philosophie.uni-muenchen.de/events/event-policy/index.html

MCMP MASTER PROGRAM: Interested students might also consider to apply for the MCMP’s master program in Logic and Philosophy of Science (which addresses male and female students). For details, click

http://www.mcmp.philosophie.uni-muenchen.de/students/ma/index.html

MCMP MEDIA PAGE: You might also want to have a look at our media page with more than 400 recorded lectures which you can watch on iTunes U.:

http://www.mcmp.philosophie.uni-muenchen.de/media/index.html

## Wednesday, 27 November 2013

### CFP: Deductive and Mathematical Cognition Philosophy Conference -- Bristol

The Deductive and Mathematical Cognition Philosophy Conference will be held at the University of Bristol on 7th and 8th April 2014. The conference aims to investigate the implications of recent empirical developments in the study of deductive and mathematical cognition for established questions in the philosophy of mathematics and logic. We hope to provide an environment for interdisciplinary discourse between philosophers and those working within the relevant empirical disciplines. The conference will spend one day focussing on each field, the first day (April 7th) on Mathematical Cognition and Philosophy of Mathematics and the second (April 8th) on Deductive Cognition and Philosophy of Logic.

Invited Speakers:
Catarina Dutilh Novaes (Groningen)
Helen De Cruz (Oxford)
Bart van Kerkhove (Brussels)

Details:
The call for papers is open to any from a diverse range of fields, including but not limited to philosophy, logic, mathematics, psychology, cognitive science, history and anthropology. At least two spaces are reserved for early career academics and graduate student submissions.
For the first day we welcome submissions that focus on the implications of recent findings in the study of mathematical cognition for traditional issues in the philosophy of mathematics. Suggested topics include:
• Presentations of experimental work that is of interest to philosophers of mathematics.
• Do recent findings about the nature of mathematical cognition support certain positions in the ontology of mathematics?
• Do these findings support Structuralism?
• Do these findings support Fictionalism?
• To what extent are mathematical entities mind-independent?
• What can recent findings in the study of mathematical cognition tell us about the nature of mathematical knowledge?
• Is mathematical knowledge a priori / a posteriori?
• How do we acquire arithmetical knowledge?
•  How do we acquire geometrical knowledge?
• What role does the historical development of mathematical notation play in determining the nature of mathematical knowledge?

For the second day of the conference, we are looking for papers on a wide range of topics introducing empirical sources of information and insight to philosophical questions concerning logic. Such questions may be metaphysical, epistemological or methodological.Topics include but are by no means limited to:
• Presentations of empirical work into the nature of deductive processes.
• Implications of empirical work for issues in the epistemology of logic
• Is logic innate?
• Can we acquire knowledge of logical principles through introspection?
• Implications of empirical work for the foundations of logic
• What is the subject of logic?
• Are we deductively rational? If not what are the implications for the prescriptive role of logic as a guide to correct reasoning?
• Should we construct and assess our logics using data from the study of deductive reasoning processes?

Papers should be submitted via Easychair by 15th February 2014 in the following format.
1) A cover letter including the author’s name, university affiliation, contact information, title of paper, topic area, word count, and an abstract of no more than 250 words.
2) A paper prepared for blind review. Submissions should not exceed 4,000 words and should be suitable for a 40-minute presentation.

The registration for delegates that are not presenting a paper is £20, with a reduced fee of £10 for students.

The conference is supported by The British Society for the Philosophy of Science

## Saturday, 16 November 2013

### Hilbert's Infinite Hotel in 60 seconds

There are 5 other installments in the wonderful series '60-second Adventures in Thought' by the Open University: Zeno's paradox, the grandfather paradox, the Chinese room, the twin paradox, and Schrödinger's cat. Go check them all out!

## Friday, 15 November 2013

### 300,000 pageviews!

Today M-Phi passed the mark of 300,000 pageviews in all-time history. Quite a milestone! I think I speak on behalf of all M-Phi bloggers when thanking readers for their support, and for the many fruitful debates at different posts. Personally, I can say that many discussions led me to revise my views, and that my research and thinking as a whole have greatly benefited from these interactions.

We'll do our best to keep it coming!

## Thursday, 14 November 2013

### A dialogical analysis of structural rules - Part II

(Cross-posted at NewAPPS)

Two weeks ago, I wrote a post proposing a dialogical perspective on structural rules. In fact, at that point I offered an analysis of only one structural rule, namely left-weakening, and promised that I would come back for more. In this post, I will discuss contraction and exchange (for both, I again restrict myself to the left cases). (I will assume that readers are familiar with the basic principles of my dialogical approach to deductive proofs, as recapped in my previous post on structural rules.)

Contraction, in particular, is very significant, given the recent popularity of restriction on contraction as a way to block the derivation of paradoxes such as the Liar and Curry. What does contraction mean, generally speaking? Contraction is the rule according to which two or more copies of a given formula in a sequent can be collapsed into each other (contracted); in other words, the idea is that the number of copies should not matter for the derivation of the conclusion:

A, A => C
-------------
A => C

(If two copies of A give you C, then one copy of A will give you C just the same.) Logics that reject contraction are based on the idea that the number of copies does matter; in the case of linear logic, the most notable non-contractive logic, formulas are seen as resources, which are no longer available once they are used (not that particular copy, in any case), and are sensitive to the number of copies available. (The standard example for resource sensitivity in linear logic is money required to buy a pack of cigarettes, which I think reveals something about the time and place in which the system was developed… Two 5-franc notes do not buy you the same amount of cigarettes as one 5-franc note!) In other words, linear logic has a plausible story to tell on why, given the purpose of the logic (to keep track of resources), the number of copies does matter.

Prima facie, in a dialogical setting, once a proposition is stated and granted, it becomes part of the public domain, as it were, and may be used as many times as necessary – at least by those who explicitly committed to it. So contraction may appear to be unproblematic in this setting. What a proposition stated or granted does is to produce a discursive commitment for the speaker in question, but it also licenses her to refer back to this commitment whenever necessary – in other words, it also creates an entitlement that can be ‘used’ as many time as one wishes. (I’m deliberately using Brandomian terminology here.)

However, one may well conceive of particular kinds of dialogical interaction where, every time a premise is required so as to license a conclusion, a fresh copy of it is required. We would need a story on why, once having granted a particular premise to proponent, opponent might then refuse to grant the same premise when it is asked again by proponent; if opponent will always have to grant premises he has granted before, in practice there is no need to go through the procedure of actually generating the new copies (new commitments).

One reason why discursive commitments may have to be modified during the interaction is if the reasons one had to commit to a statement at a given point no longer hold (say, due to changes in the world, or incoming new information); in that case, the possibility of retracting a commitment may seem plausible after all, in particular if discursive commitment is time-relative. But notice that this is a very different phenomenon from the idea of formulas being used as resources in linear logic; a given statement may no longer be ‘available’ even if it hasn’t been used yet, but simply because there are good reasons to revise one’s prior discursive commitments.

In a similar vein, my friend and former colleague Dora Achourioti has been developing a (thus far unpublished, I think) account of the truth operator where its function is precisely to turn a given statement into something that can be used as many times as one wishes; it becomes a limitless resource (she explicitly adopts a multi-agent perspective, and uses notions from linear logic to formalize this insight). So the presupposition is that this does not hold for other, ‘regular’ occurrences of statements not affected by a truth operator, and thus that contraction does not hold unrestrictedly.

So I conclude that, while contraction is prima facie a very plausible principle in a dialogical setting, there may be purely dialogical reasons to restrict contraction, but which are different from linear logic’s rationale for contraction restriction.

What about exchange? It is not a structural rule that is much discussed in the paradoxes literature, but it is interesting in its own right for different reasons. While contraction entails that the number of copies does not matter, exchange entails that the order in which formulas are presented does not matter.

A, B => C
------------
B, A => C

In a purely model-theoretic conception of (logical) consequence, it is indeed the case that order does not matter, as the sequence (A, B) has the same models as the sequence (B, A). In a dialogical setting, however, it is not at all obvious that order should not matter. This is because every new discursive commitment – every new premise granted by opponent – creates an update in his commitments; naturally, dialogues are intrinsically dynamic processes (and here you see that I am a real Amsterdam child!). Indeed, depending on the specific rules for different kinds of dialogical interaction, a premise A may be granted if it is proposed at a given stage of the debate, but rejected if it is proposed at a different stage (not in the very same interaction, but in an alternative interaction involving the same statements).

For example, the regimented kind of disputations known as obligationes (very popular among Latin medieval logicians) is inherently dynamic. If the starting point of the disputation is the proposition ‘Every human is running’ (which should be accepted if it is possible, even if it is not true, given the rules of the game), and then ‘You are a human’ is proposed, the player (in this case called ‘respondent’) must grant it as irrelevant for the starting point (it is not entailed by it or incompatible with it) and true. Then, if ‘You are running’ is proposed, the player must now accept it, as it follows from her two previous commitments, even though it is false (presumably, she is not running while disputing!). If however, given the same starting point ‘Every human is running’, ‘You are running’ is proposed first, respondent should deny it as irrelevant and false. Then, if ‘You are a human’ is proposed, it should be denied, even though it is true, because this follows from accepting ‘Every human is running’ and denying ‘You are running’. So different responses are required to the same statements depending on their order of presentation.

(However, in other dialogical situations, the order of presentation of premises may not matter; in Aristotle's Topics, for example, the recommendation is that questioner gets answerer to commit to all the premises he will need before he starts drawing conclusions (Book VIII, chapter 1).)

So I conclude that exchange is not a plausible principle from the point of view of the dialogical conception of proofs if we take into account the inherently dynamic nature of dialogues. In dialogues, the order of presentation of premises may well matter.

(I had intended to talk about cut too, but this post has again reached the reasonable length for the genre. Cut is complicated because it is related to the fundamental property of transitivity, so it cannot be discussed in haste. Maybe in another post?)

## Wednesday, 13 November 2013

### PhD Position in Logic, Rational Choice or Meta-Ethics at the University of Bayreuth

The Department of Philosophy at the University of Bayreuth invites applications for one PhD position (3 years, E13 TV Z, 0.5 FTE) starting from 01 April 2014 or soon after. The Department is a rapidly growing research and teaching environment at the intersection of Philosophy and Economics and offers excellent career development opportunities. For further details please visit www.pe.uni-bayreuth.de.

The successful applicant will conduct his or her doctoral research under the supervision of Prof. Olivier Roy. The applicant will be required to undertake a small amount of teaching in accord with the Bavarian Higher Education Law (BayHSchG). We encourage applications in the following areas:

-    Philosophical Logic (especially epistemic logic or deontic logic)
-    Rational Choice Theory (especially foundations of decision and game theory)
-    Philosophy of Action and Meta-Ethics (especially theories of rationality and normativity)

The list is not exhaustive. Excellent projects in related areas will also be considered.

Applicants should have advanced training in philosophy (typically a Master degree) or in a discipline closely related to philosophy. Projects involving computer science or economics are very welcome.

Applicants should submit a cover letter, a CV, a two-page outline of their proposed PhD research project, BA and MA degree certificates and transcripts, and two academic letters of reference. The complete application package, excluding the letters of reference, should be submitted as a single PDF file. Applications should be submitted electronically to the Secretarial Administrator for the Department of Philosophy, Miss. Sonja Weber (Sonja.weber@uni-bayreuth.de) by 15 January 2014. The letters of references should be sent separately by their author, also to Miss. Sonja Weber.

For further information about the post, the P&E programmes, professional opportunities, and the Department of Philosophy, please contact Prof. Olivier Roy by e-mail: olivier.roy@uni-bayreuth.de.

The University of Bayreuth is an equal opportunity employer and aims to increase the number of female faculty members. Applications from female candidates are, therefore, explicitly encouraged. The University of Bayreuth was accredited as a Family Friendly University by the Hertie Foundation in 2010. Persons with disabilities will be given priority if equally qualified.

## Wednesday, 30 October 2013

### A dialogical analysis of structural rules - Part I

(Cross-posted at NewAPPS)

As some of you may have seen, we will be hosting the workshop ‘Proof theory and philosophy’ in Groningen at the beginning of December. The idea is to focus on the philosophical significance and import of proof theory, rather than exclusively on technical aspects. An impressive team of philosophically inclined proof theorists will be joining us, so it promises to be a very exciting event (titles of talks will be made available shortly).

For my own talk, I’m planning to discuss the main structural rules as defined in sequent calculus – weakening, contraction, exchange, cut – from the point of view of the dialogical conception of deduction that I’ve been developing, inspired in particular (but not exclusively) by Aristotle’s logical texts. In this post, I'll do a bit of preparatory brainstorming, and I look forward to any comments readers may have!

In a nutshell (as previously spelled out e.g. here), the dialogical conception is based on the idea that a deductive proof is best understood as corresponding to a semi-adversarial dialogue between two fictitious characters, proponent and opponent, where proponent seeks to establish a final conclusion from given premises, and opponent seeks to block the establishment of the conclusion. Proponent puts forward statements stepwise, which she claims follow necessarily from what opponent has already granted in the course of the dialogue. Opponent can grant these statements, or else he can object that a given statement does not follow necessarily from what he has granted so far, by providing a counterexample (a situation where premises hold but conclusion does not). Another move available to opponent is: "why does it follow?" This would correspond to an inferential step by proponent that is not sufficiently perspicuous and compelling for opponent; proponent must then break it down into smaller, individually perspicuous inferential steps. The game ends when proponent manages to compel opponent to grant her final conclusion.

The motivation behind this dialogical conceptualization is what could be described as a functionalist approach to deductive proofs: what are they good for? What is the goal or function of a deductive proof? On this setting, the main function of a deductive proof is that of persuasion: a good proof is one that convinces a fair but ‘tough’ opponent of the truth of a given statement, given the (presumed) truth of other statements (the premises).

I believe that this dialogical/functionalist approach allows for a philosophically motivated discussion of the different structural rules. This becomes important in the context of the recent surge in popularity of substructural approaches to paradoxes (just yesterday I came across the announcement for a very interesting workshop taking place in a few weeks in Barcelona precisely on this topic). A number of people have been arguing that in many of the paradoxes (the Liar, Curry), it is the availability of contraction that allows for the derivation of the paradoxical conclusion (as I discussed here and here). I have expressed my dissatisfaction with many of these approaches given the lack of an independent motivation for restricting contraction: to say that contraction must be restricted solely because it gives rise to paradox is nothing but a ‘fix-up’ which does not take us to the core of the phenomenon. What is needed is a reflection on why contraction was thought to be a legitimate principle in the first place, and arguments against this presumed original rationale for contraction (or any other rule/principle).

Naturally, restrictions on structural rules are not a new idea: indeed, relevant logicians have offered arguments against the plausibility of weakening, and linear logicians pose restrictions on contraction. But in both cases, what motivates restriction of these structural rules are not paradox-related considerations; rather, it stems from independent reflection on what the logical systems in question are good for – in other words, something resembling the functionalist approach I am defending here. Linear logic is usually described as a logic of resources, and as such it matters greatly how many copies of a given formula are used – hence the restriction on contraction. Relevant logics require a relation of relevance between premises and conclusion, which can be disrupted by the addition of an arbitrary formula – hence the restriction on weakening.

Now, as it turns out, the dialogical conception of deductive proofs has its own story to tell about each of these structural principles. Let us start with weakening, which is the following structural rule in its sequent calculus formulation (I will restrict myself to left-weakening, as right-weakening poses a range of other problems related to the concept of multiple conclusions):

A => C
--------------
A, B => C

In first instance, if necessary truth-preservation is the only requirement for the legitimacy of proponent’s inferential steps, then weakening may seem as an entirely plausible principle: if opponent has granted A and is then compelled to grant C because C follows of necessity from A, then whatever additional B that comes up in the dialogue and is granted by opponent will not invalidate the move to C. That's simply the property of monotonicity, one of the core components of a deductive proof.

However, there is much more to be said on weakening from a dialogical perspective, and now it becomes useful to distinguish different kinds of such dialogical interactions. In the spirit of the purely adversarial interactions described for example in Book VIII of Aristotle’s Topics, it is in fact in the interest of proponent to confuse opponent by putting forward a large number of statements, some of which will be irrelevant to her final conclusion. In this way, opponent will not ‘see it coming’ and therefore may be unable to guard himself against being forced to grant the final conclusion. So in a purely adversarial setting, weakening is in fact strategically advantageous for proponent, as it may have a confusing effect for opponent.

By contrast, in a context where the goal is not only to beat the opponent by whichever means, but also to produce an explanatory proof – one that shows not only that the conclusion follows, but also why it follows – weakening becomes a much less plausible principle. Indeed, for didactic purposes for example, it makes much more sense to put on the table only the information that will in fact be relevant for the derivation of the conclusion, precisely because now confusing the interlocutor is the opposite of what proponent is trying to accomplish. And indeed, as it turns out, Aristotle’s syllogistic – which, according to some scholars, was developed to a great extent so as to provide the general framework for the theory of scientific explanation of the Posterior Analytics – is a relevant system, one where weakening is restricted. That is, for explanatory purposes, weakening is a pretty bad idea. So in other words, depending on the goal of a particular dialogical interaction of this kind, weakening will or will not be a legitimate principle.

(As this post has already become quite long, I will leave contraction, exchange and cut for a second installment later this week. So stay tuned!)

UPDATE: Part II here.

## Tuesday, 22 October 2013

### CFP: Trends in Logic XIV (Off-stream applications of formal methods)

Trends in Logic XIV
Entia et Nomina workshop
Ghent University, Belgium, July 8-11, 2014

Off-stream applications of formal methods

Theme

Logicians have devoted considerable effort to applying formal methods to what are now considered core disciplines of analytic philosophy: philosophy of mathematics, philosophy of language and metaphysics. Researchers in these ﬁelds have been accused of sharpening their knives without actually cutting anything of interest to those outside of philosophy. The focus of formal methods is changing and our intent for this conference is to further counter the impression of idleness with respect to philosophy at large. The focus of the workshop is to be on those applications of formal methods in philosophy which might be of interest to people working on philosophical questions of more direct relevance to human life.

We plan three sessions with the following invited speakers:

Session 1 Applications of formal methods in philosophy
Diderik Batens, Centre for Logic and Philosophy of Science, Ghent University (Belgium)
Krister Segerberg, Uppsala University (Sweden)
Katie Steele, London School of Economics and Political Science (UK)

Session 2 Applications of formal methods in social philosophy
Gabriella Pigozzi, Universite Paris-Dauphine (France)
Martin van Hees, University of Amsterdam (Netherlands)
John F. Horty, University of Maryland (USA)

Session 3 Applications of Bayesian methods in philosophy
Luc Bovens, London School of Economics and Political Science (UK)
Lara Buchak, University of California, Berkeley (USA)
Richard Pettigrew, Bristol University, (UK)

Format

Authors of contributed papers are asked to submit extended abstracts and full papers, prepared for blind-review by January 6, 2014. Extended abstracts should be no more than 2000 words. Authors of accepted papers will have 30-60 minutes to present their work, depending on the length of their papers. Each paper will be followed by two commentaries from other participants. Accepted participants might be asked to comment on at least one talk. 5-10 minute commentaries will be followed by 10-15 minutes of discussion. All accepted papers will be made available to the participants ahead of the conference.

More details

http://entiaetnomina.blogspot.be/p/trends-in-logic-xiv.html

trendsinlogic2014@gmail.com

## Friday, 4 October 2013

### Epistemic Utility Theory project at Bristol: new webpages

There is now a website, a Facebook page, and a Google Calendar for my 'Epistemic Utility Theory: Foundations and Applications' research project at University of Bristol.  These give lots of information about events that we'll be holding as part of the project.

## Wednesday, 2 October 2013

### Operationalization 2013: An interdisciplinary workshop at the edge of experimental psychology and analytical philosophy

The workshop will take place at the FRIAS in Freiburg im Breisgau (Germany) the 15th and 16th October 2013.

The workshop brings together psychologists and philosophers in order to investigate questions of concept operationalization that are common to both disciplines. The questions we tackle combine theoretical and empirical features. Some examples: How should we devise elicitation procedures for complex and potentially ambiguous conceptsHow should we work out predictions that are precise enough to be put to empirical testHow can we connect the philosophical literature to the long-lasting empirical tradition in psychology?

The workshop is organized into three types of sessions: keynotes by researchers who have already worked successfully about operationalizing epistemic concepts, methodological tutorials which will present contemporary statistical procedures, and discussion sessions where planned experimental projects will be discussed.

Keynote speakers

Ulrike Hahn (Birkbeck, University of London, UK)
Edouard Machery (University of Pittsburg, USA)
David Over (University of Durham, UK)

Methodological tutorials

David Kellen and Henrik Singmann (University of Freiburg, Germany)

Discussion sessions

(The discussion session focus on ongoing projects and not past work.)

Matteo Colombo (Tilburg University, Netherlands); Nicole Cruz (PH Ludwigsburg, Germany); Marco Ragni and Barbara Kuhnert (University of Freiburg, Germany); Mark Siebel, Jakob Koscholke, and Michael Schippers (University of Oldenburg, Germany), and Marc Jekel (Max Planck Institute for Research on Collective Goods, Bonn, Germany); Jan Sprenger (Tilburg University, Netherlands); Tatsuji Takahashi (Tokyo Denki University, Japan); Matthias Unterhuber (University of Düsseldorf, Germany).

Organization

The workshop is organized by Henrik Singmann (University of Freiburg), Marco Ragni (University of Freiburg), Vincenzo Crupi (Università di Torino and LMU Muenchen), and Jan Sprenger (Tilburg University) and is a a follow-up event to the workshop Operationalizing Epistemic Concepts that took place last year in Aachen. The workshop is co-sponsored by the DFG Priority Program New Frameworks of Rationality (SPP 1516) and the Freiburg Institute for Advanced Studies (FRIAS).

## Sunday, 29 September 2013

### Assistant Professor Position at LMU Munich (MCMP)

The  Chair of Philosophy of Science at the Faculty of Philosophy, Philosophy of Science and the Study of Religion and the Munich Center for Mathematical Philosophy (MCMP, http://www.lmu.de/mcmp) at LMU Munich seek applications for an Assistant Professorship with a specialization in (at least) one of the following areas: Philosophy of Psychology, Philosophy of Social Science, Philosophy of Economics, and Philosophy of Neuroscience. The position is for three years with the possibility of extension for another three years. Note that there is no tenure-track option. The appointment will be made within the German A13 salary scheme (under the assumption that the civil service requirements are met), which means that one has the rights and perks of a civil servant. The starting date is October 1, 2014. A later starting date is also possible.

The appointee will be expected (i) to do philosophical research and to lead a research group in her or his field, (ii) to teach five hours a week in at least one of the above-mentioned fields and/or a related field, and (iii) to take on some management tasks. The successful candidate will have a PhD in philosophy and some teaching experience.

Applications (including a cover letter that addresses, amongst others, one's academic background and research interests, a CV, a list of publications, a list of taught courses, a sample of written work of no more than 5000 words, and a description of a planned research project of 1000-1500 words) should be sent by email (ideally everything requested in one PDF document) to office.hartmann@lrz.uni-muenchen.de by November 20, 2013. Hard copy applications are not possible. Additionally, two confidential letters of reference addressing the applicant's qualifications for academic research should be sent to the same address from the referees directly.

## Thursday, 26 September 2013

### The Mathematics of Dutch Book Arguments

Dutch Book arguments purport to establish norms that govern credences (that is, numerically precise degrees of belief).  For instance, the original Dutch Book argument due to Ramsey and de Finetti aims to establish Probabilism, the norm that says that an agent's credences ought to obey the axioms of mathematical probability.  And David Lewis' diachronic Dutch Book argument aims to establish Conditionalization, the norm that says that an agent ought to plan to update in the light of new evidence by conditioning on it.  As we will see in this post, there is also a Dutch Book argument for the Principal Principle as well, the norm that says that an agent ought to defer to the chances when she sets her credences.  We'll look at each of these arguments below.

Each argument consists of three premises.  The second is always a mathematical theorem (sometimes known as the conjunction of the Dutch Book Theorem and the Converse Dutch Book Theorem).  My aim in this post is to present a particularly powerful way of thinking about the mathematics of these theorems.  It is due to de Finetti.  It is appealing for a number of reasons:  it is geometrical, so we can illustrate the theorems visually; it is uniform across the three different Dutch Book arguments we will consider here; and it establishes both Dutch Book Theorem and Converse Dutch Book Theorem on the basis of the same piece of mathematics.

I won't assume much mathematics in this post.  A passing acquaintance with vectors in Euclidean space might help, but it certainly isn't a prerequisite.

### The form of a Dutch Book argument

The three premises of a Dutch Book argument for a particular norm $N$ are as follows:

(1) An account of the sorts of decisions a given set of credences will (or should) lead an agent to make.

(2) A mathematical theorem showing two things:  (i) relative to (1), credences that violate norm $N$ will lead an agent to make decisions with property $C$; (ii) relative to (1), credences that satisfy norm $N$ in question will not lead an agent to make decisions with this property $C$.

(3) A norm of practical rationality that says that, if an agent can avoid making decisions with property $C$, she is irrational if she does make such a decision.

In this post, I'll present Dutch Book arguments of this form for Probabilism, Conditionalization, and the Principal Principle.  But I'll be focussing on premise (2) in each case.  There's plenty to say about premises (1) and (3), of course.  But that's for another time.

### The Dutch Book argument for Probabilism

The first premise in each Dutch Book argument is the same.  It has two parts:  the first tells us, for any proposition in which the agent has a credence, the fair price she ought to pay for a bet on that proposition; the second tells us the price she ought to pay for a book of bets on a number of different propositions given the price she's prepared to pay for each individual bet.  Thus, we have

(1a)  If an agent has credence $p$ in proposition $X$, she ought to pay $pS$ for a bet that pays out $S$ if $X$ is true and $0$ if $X$ is false.  (In such a bet, $S$ is called the stake.)

(1b) If an agent ought to pay $X$ for Bet 1 and $Y$ for Bet 2, she ought to pay $X+Y$ for a book consisting of Bet 1 and Bet 2.  (This is sometimes called the Package Principle.)

Putting these together, we get the following:  Suppose $\mathcal{F} = \{X_1, \ldots, X_n\}$ is a set of propositions.  And suppose we represent our agent's credences in these $n$ propositions by a vector $c = (c_1, \ldots, c_n)$ where $c_i$ is her credence in $X_i$.  And suppose we consider a book of bets $S$ in which the stake on $X_i$ is $S_i$.  Then we can represent this book by the vector $S = (S_1, \ldots, S_n)$ Then the price that the agent ought to pay for this book of bets is $\sum^n_{i=1} S_ic_i := (S_1, \ldots, S_n) \cdot (c_1, \ldots, c_n) = S\cdot c$ where $S\cdot c$ is the dot product of $c$ and $S$ considered as vectors.

Happily, there is also a nice way to represent the payoff of a book of bets $S$ at a given possible world $w$.  Represent that possible world $w$ by the following vector: $w = (w_1, \ldots, w_n)$ where $w_i = 1$ if $X_i$ is true at $w$ and $w_i = 0$ if $X_i$ is false at $w$.  Then the payoff of $S$ at $w$ is $\sum^n_{i=1} S_iw_i := (S_1, \ldots, S_n) \cdot (w_1, \ldots, w_n) : S\cdot w$  As we will see, these vector representations will prove very useful below.

In this section, we're looking at the Dutch Book argument for Probabilism.

Probabilism  It ought to be that a set of credences $c$ obeys the axioms of mathematical probability.

Let us turn to premise (3) of this argument.  It says that it is irrational for an agent to have credences that lead her to make decisions that will lose her money in every world that she considers possible.  Now, a book of bets loses an agent money if $\mbox{Payoff} < \mbox{Price}$ But recall from above:  the payoff of a book of bets $S$ at a world $w$ is $S \cdot w$; and the price of that book is $S \cdot c$.  Thus, the agent is irrational if there is a book $S$ such that $S \cdot w < S \cdot c$ for all worlds $w$.  Equivalently, $S \cdot (w-c) < 0$ for all $w$.

So the Dutch Book Theorem (that is, premise (2)) can be stated as follows:

Theorem 1
(i) If $c$ violates Probabilism, then there is a book $S$ such that $S \cdot w < S \cdot c$ for all worlds $w$ (equivalently, $S \cdot (w-c) < 0$ for all $w$).
(ii) If $c$ satisfies Probabilism, then there is no book $S$ such that $S \cdot w \leq S \cdot c$ (equivalently, $S\cdot (w-c) \leq 0$) for all worlds $w$ and $S \cdot w < S \cdot c$ (equivalently, $S \cdot (w-c) < 0$) for some world $w$.

We now turn to the proof of this theorem.  It is based on two pieces of mathematics:  the first involves some basic geometrical facts about the dot product; the second involves a neat geometric characterization of the credences that satisfy Probabilism.

First, a well known fact about the dot product.  If $u$ and $v$ are vectors in $\mathbb{R}^n$, we have $u \cdot v = ||u||\, ||v|| cos \theta$ where $\theta$ is the angle between $u$ and $v$.  Since $||u||\, ||v|| \geq 0$, we have $u\cdot v < 0 \Leftrightarrow cos \theta < 0$  And, by basic trigonometry, we have $u \cdot v < 0 \Leftrightarrow \frac{\pi}{2} < \theta < \frac{3\pi}{2}$  Thus:
• To prove Theorem 1(i), it suffices to show that, if $c$ violates Probabilism, we can find a vector $S$ such that the angle between $S$ and $w-c$ is oblique for all worlds $w$.
• To prove Theorem 1(ii), it suffices to show that, if $c$ satisfies Probabilism, there is no vector $S$ such that the angle between $S$ and $w-c$ is oblique or right for all $w$ and oblique for some $w$.
To do this, we need a geometric characterization of the credences that satisfy Probabilism.  Fortunately, we have that in the following lemma due to de Finetti:

Lemma 1 $c$ satisfies Probabilism iff $c \in \{w : w \mbox{ is a possible world}\}^+$.

where, if $\mathcal{X}$ is a set of vectors in $\mathbb{R}^n$, $\mathcal{X}^+$ is the convex hull of $\mathcal{X}$:  that is, $\mathcal{X}^+$ is the smallest convex set that includes $\mathcal{X}$; if $\mathcal{X}$ is finite, then $\mathcal{X}^+$ is the set of linear combinations of elements of $\mathcal{X}$.

Thus, Lemma 1 says that the vectors that represent the probabilistic sets of credences are precisely those that belong to the convex hull of the vectors that represent the possible worlds.

How does this help? Let's take the case in which $c$ violates Probabilism.  That is, $c$ lies outside the convex hull of the vectors representing the different possible worlds.  Then it is easy to see from Figure 1 below that there is a vector $c^*$ that lies inside that convex hull such that, for a given world $w$, the angle $\theta$ between the vector $c-c^*$ and the vector $w-c$ is oblique.  Thus, if we let $S = c - c^*$, we have Theorem 1(i).

 Figure 1: The oval represents the convex hull of the set of vectors that represent the different possible worlds.  If $c$ violates Probabilism, then it lies outside this.  But, by a Hyperplane Separating Theorem, there is a point $c^*$ in the convex hull such that the angle between $c-c^*$ and $x-c$ is oblique for any $x$ inside the convex hull.  Thus, in particular, it is oblique when $x$ is a vector representing a possible world, as required.

Now let's take the case in which $c$ satisfies Probabilism.  That is, $c$ lies inside the convex hull of the vectors representing the different possible worlds.  Then it is easy to see from Figure 2 below that, if $S$ is a vector, then while there may be some worlds $w$ such that the angle $\theta$ between $S$ and $w-c$ is oblique, there must also be some worlds $w'$ such that the angle $\theta'$ between $S$ and $w'-c$ is acute.  Alternatively, it is possible that the angles $\theta$ between $S$ and $w-c$ for all worlds $w$ are all right.

 Figure 2: Again, the oval represents the convex hull of the possible worlds.  If $c$ satisfies Probabilism, then it lies inside.
This completes the geometrical proof of Theorem 1, which combines the Dutch Book Theorem and the Converse Dutch Book Theorem.

### The Dutch Book Argument for the Principal Principle

The Principal Principle says, roughly, that an agent ought to defer to the chances when she sets her credences.  One natural formulation of this (explicitly proposed by Jenann Ismael and entailed by a slightly stronger formulation proposed by David Lewis) is this:

Principal Principle  It ought to be the case that $c$ is in $\{ch : ch \mbox{ is a possible chance function}\}^+$.

That is, the Principal Principle says that one's credence function ought to be a linear combination of the possible chance functions.

Now, adapting the proof of Theorem 1 above, replacing the possible worlds $w$ by possible chance functions $ch$ (represented as vectors in the natural way), we easily prove the following:

Theorem 2
(i) If $c$ violates the Principal Principle, then there is a book $S$ such that $S \cdot ch < S \cdot c$ for all possible chance functions $ch$.
(ii) If $c$ satisfies Probabilism, then there is no book $S$ such that $S \cdot ch \leq S \cdot c$ for all possible chance functions $ch$ and $S \cdot ch < S \cdot c$ for some possible chance function $ch$.

But what does this tell us?  Well, as before, $S \cdot c$ is the price our agent would pay for the book $S$.  But this time, the other side of the inequality is $S\cdot ch$.  And this, it turns out, is the objective expected payout of $S$, rather than the actual payout of $S$.  Thus, violating the Principal Principle does not necessarily make an agent vulnerable to a true Dutch Book.  But it does lead them to pay a price for a book of bets that is higher than the objective expected value of that book, according to all of the possible chance functions.  And this, we might think, is irrational.  For one thing, such an agent will, with objective chance 1, lose money in the long run.  Thus, in the Dutch Book argument for the Principal Principle, premise (1) is as before, premise (2) is Theorem 2, but premise (3) becomes the following:  It is irrational for an agent to have credences that lead her to pay more than the objective expected value for a book of bets.

### The Dutch Book Argument for Conditionalization

Conditionalization is the following norm:

Conditionalization  Suppose our agent has credence $c$ at $t$; and suppose she knows that, by $t'$, she will have received evidence from the partition $E_1, \ldots, E_m$.  And suppose she plans to update as follows:  If $E_i$, then $c_i$.  Then it ought to be that $c_i(-) = c(-|E_i)$ for $i = 1, \ldots, m$.

In fact, the Dutch Book argument for Conditionalization that we will present is primarily a Dutch Book argument for van Fraassen's Reflection Principle, which is equivalent to Conditionalization.  The Reflection Principle says the following:

Reflection Principle  Suppose our agent has credence $c$ at $t$; and suppose she knows that, by $t'$, she will have received evidence from the partition $E_1, \ldots, E_m$.  And suppose she plans to update as follows:  If $E_i$, then $c_i$.  Then it ought to be that:
(i) $c_i(E_i) = 1$ for $i = 1, \ldots, m$;
(ii) $c$ is in $\{c_i : i = 1, \ldots, m\}^+$.

That is, Reflection says that an agent's current credences ought to be a mixture of her planned future credences.  Since Reflection and Conditionalization are equivalent, it suffices to establish Reflection.

Here is the theorem that provides the second premise of the Dutch Book argument for Reflection:

Theorem 3
(i) Suppose $c, c_1, \ldots, c_n$ violate Reflection.  Then there are books $S, S_1, \ldots, S_m$ such that (a) for all $i = 1, \ldots, m$, $S \cdot (w - c) + S_i(w - c_i) \leq 0$ for all worlds $w$ in $E_i$; and (b) for some $i = 1, \ldots, m$, $S \cdot (w - c) + S_i(w - c_i) < 0$ for some world $w$ in $E_i$.
(ii) Suppose $c, c_1, \ldots, c_n$ satisfy Reflection.  Then there are no books $S, S_1, \ldots, S_m$ such that (a) for all $i = 1, \ldots, m$, $S \cdot (w- c) + S_i\cdot (w-c_i) \leq 0$ for all worlds $w$ in $E_i$; and (b) there is $i = 1, \ldots, m$ such that $S \cdot (w- c) + S_i\cdot (w-c_i) < 0$ for some $w$ in $E_i$.

What does this say?  It says that, if you plan to update in some way other than conditioning on your evidence, and thereby violate Reflection, there is a book $S$ that you will accept at $t$ as well as, for each $E_i$, a book $S_i$ that you will accept at $t'$ if you learn $E_i$ such that, together, they will guarantee you a loss.  And this will not happen if you plan to update by conditioning.

How do we prove this?  Theorem 3(i) is the easier to prove.  Suppose $c, c_1, \ldots, c_n$ violate Reflection.  First, suppose that this is because $c_i(E_i) < 1$.  Then let $S = 0$ and $S_j = 0$ for all $j \neq i$.  And let $S_i$ be the book consisting only of a bet on $E_i$ with stake $-1$.  Then $S \cdot (w-c) + S_i(w-c_i) = (-1)(1 - c_i(E_i)) < 0$ for all worlds $w$ in $E_i$.  And $S \cdot (w-c) + S_i(w-c_i) = 0$ for all worlds $w$ in $E_j \neq E_i$.

Second, suppose that $c_i(E_i) = 1$ for all $i = 1, \ldots, m$.  But suppose $c$ is not inside the convex hull of the $c_i$s.  So $c, c_1, \ldots, c_n$ violate Reflection.  Then, adapting the proof of Theorem 1 by replacing the worlds $w$ with the planned posterior credences $c_i$, we get that there is a book $S$ such that $S \cdot (c_i - c) < 0$ for all $i = 1, \ldots, m$.  So if we let $S_i = -S$ for all $i = 1, \ldots, m$, we get $0 > S \cdot (c_i - c) = S \cdot (w-c) + (-S)\cdot (w-c_i) = S \cdot (w-c) + S_i \cdot (w-c_i)$ for all worlds $w$.  This completes the proof of Theorem 3(i).

Now we turn to Theorem 3(ii).  Suppose $c, c_1, \ldots, c_n$ satisfy Reflection. Suppose, for a contradiction, that we have (a) for all $i = 1, \ldots, m$, $S \cdot(w-c) + S_i \cdot(w-c_i) \leq 0$ for all $w$ in $E_i$; and (b) for some $i = 1, \ldots, m$, $S \cdot(w-c) + S_i \cdot(w-c_i) < 0$ for some $w$ in $E_i$.  Our plan is to use this to construct $S'$ such that (a) for all $i = 1, \ldots, m$, $S' \cdot(w-c) \leq 0$ for all $w$ in $E_i$; and (b) for some $i = 1, \ldots, m$, $S' \cdot(w-c) < 0$ for some $w$ in $E_i$.  And we know that this is impossible from Theorem 1(ii).

We construct $S'$ as follows: First, suppose that $X_1, \ldots, X_k$ are the atoms of the algebra $\mathcal{F} = \{X_1, \ldots, X_k, \ldots, X_n\}$.  Then notice that for each book of bets $S = (S_1, \ldots, S_n)$ on the propositions $X_1, \ldots, X_n$, there is a book $S^A = (S^A_1, \ldots, S^A_k, 0, \ldots, 0)$ on the atoms $X_1, \ldots, X_k$ of $\mathcal{F}$ such that $S^A$ is equivalent to $S$:  that is, the payout of $S$ is the same as the payout of $S^A$ at every world; and the price that a probabilistic agent should pay for $S^A$ is exactly the price she should pay for $S$.  Thus, if we have $S \cdot(w-c) + S_i \cdot(w-c_i) \leq 0$, then we have $S^A \cdot(w-c) + S^A_i \cdot(w-c_i) \leq 0$, and so on.  Thus, in what follows, we can assume without loss of generality that $S$ is a book of bets only on the atoms of $\mathcal{F}$.  Then we define $S'$ as follows, where for any atom $X_j$, we write $E_{i_j}$ for the cell of the partition in which $X_j$ lies:
$S'(X_j) := S(X_j) + S_{i_j}(X_j) - \sum_{X_k \in E_{i_j}} c(X_k | E_{i_j}) S_{i_j}(X_k)$
Then we can show that,
$S'\cdot(w - c) = S\cdot(w - c) + S_i\cdot (w - c_i)$
for all $E_i$ and $w \in E_i$.  Suppose $w$ is a world; suppose $X_j$ is the atom that is true at that world; suppose, as above, that $X_j$ lies in cell $E_{i_j}$.  Then we have
\begin{eqnarray*}
S'\cdot(w - c) & = & S(X_j) + S_{i_j}(X_j) - \sum_{X_k \in E_{i_j}} c(X_k | E_{i_j}) S_{i_j}(X_k) \\
& & \ \ \ - \sum_{X_l} c(X_l) [S(X_l) + S_{i_l}(A_l) + \sum_{X_k \in E_{i_l}} c(X_k | E_{i_l}) S_{i_l}(X_k)]\\
& = & S \cdot w + S_{i_j}\cdot w - S_{i_j} \cdot c_{i_j} - S \cdot c \\
& & \ \ \ - \sum_{X_l} c(X_l) [S_{i_l}(A_l) + \sum_{X_k \in E_{i_l}} c(X_k | E_{i_l}) S_{i_l}(X_k)]\\
& = & S \cdot (w - c) + S_{i_j}\cdot (w - c_{i_j}) \\
& & \ \ \ - \sum_{X_l} c(X_l) [S(X_l) + S_{i_l}(A_l) + \sum_{X_k \in E_{i_l}} c(X_k | E_{i_l})\\
& = & S \cdot (w - c) + S_{i_j}\cdot (w - c_{i_j}) < 0
\end{eqnarray*}
by assumption.  This completes the proof of Theorem 3(ii) and thus Theorem 3.