Sunday, 30 December 2012

Mathematical Explanation in Science, Conservativeness and Necessity

Mathematical explanation in science has become a hotly debated topic over the last ten years or so. There were, of course, discussions of this topic in various places, going back many decades and no doubt centuries! Physicists frequently commented on the issue (the most famous example being Eugene Wigner's "The Unreasonable Effectiveness of Mathematics": one can also find assorted comments in many places - for example in Feynman, Weinberg and Penrose). First book length discussion by a philosopher, at least that I am familiar with, is Mark Steiner's The Applicability of Mathematics as a Philosophical Problem. In that work, some very intriguing suggestions are made; in particular, the suggestion of a kind of pre-established, anthropocentric, harmony between mind and the mathematical nature of reality.

Here I'm more interested in how these debates have developed over the last decade or so. The question is:
do mathematical statements ever play an explanatory role in scientific theory?
Since I am some kind of metaphysical Pythagorean, I think reality is mathematical, my answer is: well, yes, obviously.

However, the usual setup for such discussions involves a presupposition that the world is non-mathematical, and the question arises whether mathematics plays a role in explaining the non-mathematical. Since I reject this nominalistic assumption, I don't feel any great urgency to answer that demand. For the physical quantities (mass, length, etc.; and field quantities with values in abstract spaces, etc.) that play such a crucial role in scientific theories are already mathematical entities - they are usually functions of some sort.

Still, suppose we simply grant the nominalist's basic assumption. That there is a way the world is that doesn't involve quantities, field quantities, symmetry groups, etc. There is a "nominalistic way the world is". (But, to repeat: I do not believe this is true.) The most perspicuous way of formalizing this, following the work of John Burgess, is to think of an interpreted 2-sorted theory language, $L$, with variables ranging over concreta and abstracta. The primitive predicates then fall into three kinds: those that express properties of, and relations amongst, concreta ("primary"); those that express properties of, and relations amongst, abstracta ("secondary"); and those that express relations between concreta and abstracta ("mixed").

The purely nominalistic statements, as it were, are simply those that involve no variables (and, a fortiori, no secondary or mixed predicates) for abstracta. Mathematical statements may be pure or mixed.

Then the question becomes: do either pure or mixed mathematical statements ever play a role in explaining nominalistic statements? Now one needs to be careful here, because the mixed mathematical statements come in two kinds. The usual laws of physics are mixed statements. (This is a quick formulation of one of the premises used in the Indispensability Arguments against nominalism.) For example, Maxwell's Law,
(1) $\nabla \cdot B = 0$
is, under analysis, a mixed statement. The same is true of Newton's Laws, the Laws of Quantum Theory, General Relativity, and so on. It is true of more mundane scientific laws, like the Gas Laws, and laws of optics and so on, of course.

However, there are mixed statements which are nothing like laws of physics. Two examples are:
(2) There are $n$ Fs if and only if the number of Fs = $n$.
(3) There is a set $X$ such that, for all $c$, $c \in X$ iff $\phi(c)$.
The primary difference, as it seems to me, between a mixed law of physics such as (1), and the mixed statements (2) and (3), is that the first is contingent, while the latter two are necessary. Of course, a nominalist will be inclined to claim that (2) and (3) are not necessary: rather, they are false! But the nominalist has a way of handling this, due to Field, that I come on to in a moment.

Statements such as (2) and (3) I shall call axioms of applicable mathematics. Other examples would include Hume's Principle. In fact, as far as I can tell, the primary such axioms are either Hume's Principle (from which (2) is derivable in the setting of second-order logic) or Comprehension Axioms (such as (3)).

In the setup favoured by nominalists, one imagines that mixed laws of successful physics, such as (1), have either been nominalized away by some kind of paraphrase or elimination, or can be downgraded to the status of "nominalistically adequate", as opposed to being "true" (or "approximately true"), as a realist requires. So, we can, at least temporarily, forget about (1) and other such laws of physics: contingent claims that some physical quantity has certain properties.

On the other hand, the status of applicable axioms---that is, mixed statements such as (2) and (3)---is that they are required to be conservative. It is this conservativeness property that gives them a deflated role. They are useful in some sense, but untrue. They are convenient scaffolding for formulating science, but, being conservative, they are insubstantial.

In fact, one can show that (2) and (3) are conservative in the required sense. In particular,
if the comprehension principle (3) implies a nominalistic statement $S$, then $S$ is a logical truth.
Let me slim this down a bit,
If $\exists X \forall c(c \in X \leftrightarrow \phi(c)) \vdash S$, then $\vdash S$.
Such results correspond to Hartry Field's Principle C, in his Science Without Numbers.

How does this relate to the contemporary debate about mathematical explanation in science? I think it points to a very important feature:
The applicable mathematical axioms are necessities. If they imply a purely nominalistic statement, then that statement is a necessity too. Consequently, contingent nominalistic statements cannot be explained by applicable mathematical axioms.
The examples that have been given in the literature have tended to obscure this point. This is not to deny that there examples of mathematical explanation! On the contrary, I think there are. It is simply that if applicable mathematical axioms (such as comprehension axioms for the existence of sets of concreta) imply a purely nominalistic statement, then that statement must be a necessity (a logical truth, in fact).

Even so, such necessities may themselves be quite interesting, particularly if they are conditionals. Suppose we abbreviate the applicable mathematical axioms that we are assuming, in the background as it were, as $AM$. We may have,
$AM \vdash \phi \to \theta$
where $\phi$ and $\theta$ are purely nominalistic claims. So, with the mathematical axioms, we can deduce that: if the concreta do this, then they do that. This conditional $\phi \to \theta$ is, in the end, a complicated logical truth: that is,
$\vdash \phi \to \theta$
So, there is a derivation $\Gamma$ of $\phi \to \theta$ using some sound and complete deductive system for FOL. It may also happen that the shortest such $\Gamma$ is huge (e.g., contains at least $10^{10^{10}}$ symbols, say), and thus one cannot deduce this conclusion in practice without using mathematics, for speed-up reasons.

If this is right, then the only output we can get from applicable mathematical axioms at the purely concrete level must be complicated logical truths. Does it make sense to talk of explaining a logical truth? I'm not sure, but actually, I think it might; but that's a topic for another M-Phi post.

(Postscript: I've included no references or links, so if I have time to update this, I'll try to add some references and links.)

Thursday, 27 December 2012

There's Glory for You! - update

A few months ago I put a short dialogue, "There's Glory for You!", on M-Phi about the nature of language and languages (interpreted languages, conventionalism, metasemantics, the modal status of semantic and other linguistic facts, cognizing a language, semantic indeterminacy, and idiolects) based on a well-known snippet from Lewis Carroll's "Through the Looking Glass", where Humpty Dumpty declares that when he uses a word, it means "just what I choose it to mean -- neither more nor less".

I gave a rewritten version of this as a talk in Oxford a month ago at the Philosophical Society, with my colleague Anandi Hattiangadi as Alice. I now have put it on in case anyone is interested. (It's probably not suitable as a "proper" philosophical article.)


Philosophers have sometimes debated whether there are, or aren't, holes. I say there are holes, and I once spent a while thinking about how one would try to define "hole" adequately. The conclusion I came to was based on the notion of a hole in a manifold---roughly, a region has a hole in it if that region cannot be smoothly contracted to a point. I hadn't thought about it for ages, but pleasingly, this is the definition given by Eric Weisstein at Wolfram Mathworld:
A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point.

Thursday, 20 December 2012

Nordic Logic Spring 2013


Nordic Spring School in Logic, 
May 27-31, 2013,
Sophus Lie Conference Center,
Nordfjordeid, Norway

The first Nordic Spring School in Logic is organized under the auspices of the Scandinavian Logic Society and is supported by the Department of Mathematics of the University of Oslo.

The Sophus Lie Conference Center ( is located with a view to one of the famous fjords of Norway, in an area particularly attractive with its nordic exotic nature and bright nights at that time of the year. Nordfjordeid can be reached by plane from Oslo, Bergen, or Trondheim to Sandane airport, or by coach from each of these cities.

The school programme will comprise the following short courses on a variety of important topics in mathematical, computational, applied and philosophical logic, given by leading experts in their fields:

- Alexandru Baltag and Sonja Smets (University of Amsterdam): Dynamic logics for interactive belief revision
- Dag Normann (University of Oslo): The semantics of higher-order algorithms
- Erich Graedel (RWTH Aachen): Logics for dependence and independence
- Joan Bagaria (ICREA, University of Barcelona): Set theory
- Katrin Tent (University of Muenster): Model theory
- Kim Larsen (Aalborg University): Logic and formal verification
 - Michael Rathjen (University of Leeds): Proof theory: From arithmetic to set theory
- Nina Gierasimczuk (University of Amsterdam) and Jakub Szymanik (University of Groningen): Logic, computability and cognition
- Thierry Coquand (University of Gothenburg): Type theory and univalent foundations
- Wolfgang Thomas (RWTH Aachen): Logic, automata and games
- Øystein Linnebo (University of Oslo and Birkbeck, University of London): Philosophy of logic: The problem of absolute generality

The program will be divided into two parallel streams, one mainly on mathematical logic and the other mainly on computational, applied and philosophical logic. The courses will target mainly PhD students, but will also be of interest for young (and not so young) researchers in logic and its applications. Some of the courses will be accessible to advanced master students, too. Besides the series of courses, the school program will also include a half-day excursion to the famous glacier Briksdalsbreen, on Wednesday, May 29.

The registration for the school will commence on January 15, 2013. The number of participants will be limited, and requests for participations will be accepted in the order of registration until the limit is reached. However, 50 of the available places will be reserved for master and PhD students until the early registration deadline.

Early registration: MARCH 15, 2013. Late registration: MAY 1, 2013

The students accommodation will be provided in the conference center, mostly in double rooms. These are located in 9 pavilions with 4 double + 2 single rooms each.  Each pavilion has common kitchen and bathroom facilities. The cost of accommodation in a double room with full board in the center, from (arrival) Sunday, March 26 until (latest departure on) Saturday, June 1 is NOK 3490. There are also alternative possibilities for accommodation in hotels or guest houses nearby. The registration fees, covering the scientific programme, conference facilities, refreshments, and the excursion, are as follows:

Early Registration:  Student fees: NOK 1500. Regular fees:  NOK 2000 Late Registration: Student fees: NOK 2000. Regular fees:  NOK 2500

Instructions on the registration procedure will be posted on the school website:

Organizing Committee: Dag Normann (University of Oslo, Norway): Chair Elisabeth H. Seland (University of Oslo, Norway): Administrative support Fredrik Engström (University of Gothenburg, Sweden): Website Valentin Goranko (Technical University of Denmark) Scientific Committee: Benedikt Loewe (Universities of Amsterdam and Hamburg) Erik Palmgren (University of Stockholm, Sweden) Juliette Kennedy  (University of Helsinki, Finland) Marc Bezem (University of Bergen, Norway) Patrick Blackburn (University of Roskilde, Denmark) Valentin Goranko (Technical University of Denmark), Chair

Enquiries on registration and logistics: Enquiries on the scientific programme:

Monday, 17 December 2012

Two ERC-funded PhD studentships in Philosophy (Leeds)

The School of Philosophy, Religion and History of Science at Leeds is offering two studentships for students commencing a full-time PhD in Philosophy in October 2013. The students will form part of the ERC-funded research project The nature of representation.

Your PhD project will be in the areas of Language or Mind, broadly construed. A variety of projects in these areas may be suitable for funding in connection with this project. Please feel free to contact the project director, Prof. Robert Williams, for more information on this opportunity (email: 

You should be willing to participate in project activities, including regular meetings and workshops with project members. You will also be expected to assist other members of the project team who will be organizing these events.

Project details

Mental representation – perception and cognition – unites humans and other animals. Linguistic representation differentiates humans from other animals. But representation in either form is a phenomenon that cries out for explanation. How does one thing-a volley of sensation, a pattern of neurons firing in the head, or a sequence of sounds or written marks-"stand for" or "represent" another? However we answer this question, what are the means by which we find out about it? And for what purposes do we need to appeal to representation in the first place?

The Nature of Representation is a five-year project, funded by the European Research Council, on the metaphysics and epistemology of representation. The project is located at the University of Leeds under the direction of Robert Williams (School of Philosophy, Religion and History of Science). The project team will include two postdoctoral researchers (2013-17) and two PhD studentships (2013-16). Regular workshops will be held on project themes.

Major project themes include:

The explanatory role of representation. For what explanatory purposes do we need to appeal to (mental or linguistic) representation? 

Grounding representation. Can we say in nonrepresentational terms what the world must be like, in order for there to be representation? 

Representational media. How are the vehicles of representation (e.g. words, sentences) to be individuated? Does mental content have such vehicles?   

Knowing what's represented. How do we find out about what represents what?

Studentship information

The studentship is tenable for up to 3 years (full-time). Renewal of the studentship each year is subject to satisfactory academic progress. Applicants should normally have, or expect soon to be awarded, a Masters degree in a relevant discipline. International students are eligible to apply for this studentship.  The studentships will comprise of a maintenance allowance of £18,189 p.a. In addition to these amounts, the project funding allows significant further funding for approved research and travel expenses. Please note that you will be liable to pay academic fees for the duration of your course (in session 2012/13 the rate for UK/EU students was £3,828 p.a. and the rate for international students in session 2013/14 will be £13,100).

The closing date for applications is 1st  March 2013. You should also arrange for three academic references to be sent to us by this date.

Applications for the studentships should be made in accordance with the standard procedure for applying to do a postgraduate research degree in Philosophy at Leeds. See here for complete instructions and requirements.

Please specify in your application that you wish to be considered for the award. 

Correspondence regarding the application procedure may be directed to

Saturday, 15 December 2012

MA Program in Logic and Philosophy of Science at LMU Munich
Deadline for application for the 2013-2014 academic year: July 15th, 2013.

Studying Logic and Philosophy of Science at the LMU Munich

Since October 2012 the Faculty of Philosophy, Philosophy of Science and Study of Religion at Ludwig-Maximilians-Universität Munich is running an international MA program in Logic and Philosophy of Science.

LMU has a long-standing tradition in logic and philosophy of science. In recent years this tradition has been revived through the foundation of the Munich Center for Mathematical Philosophy, and by Hannes Leitgeb and Stephan Hartmann moving to the LMU to take, respectively, the chair in logic and philosophy of language and the chair in philosophy of science. Both Leitgeb and Hartmann have been awarded an Alexander von Humboldt Professorship by the German Alexander von Humboldt Foundation. They are now working together to co-direct the MCMP.

The MCMP hosts a lively community of university faculty, MCMP fellows, students, and visitors. The MA students will be able to take part in a great number of academic activities at LMU. The MCMP itself is hosting two weekly research seminars on the topics of logic, philosophy of science, and mathematical philosophy from visiting speakers – while also we get to hear what our own members are working on during our internal work-in-progress seminars.  In addition, the MCMP regularly organizes reading groups, workshops, and conferences. See the MCMP website for details.

We have strong ties to similar institutions internationally, in Germany and in Munich. We regularly host visitors who do their research at our Center or who teach here. Additionally, the journal Erkenntnis is based at the MCMP.

Overall, the Faculty of Philosophy, Philosophy of Science and Study of Religion at LMU is one of the largest in the German-speaking world. It covers all areas of philosophy and has a strong interdisciplinary approach towards subjects like physics, neuroscience, mathematics, statistics, linguistics, computer science, and more.

The MA Program details

– MA website:

– Questions and contact : Olivier Roy (Program Coordinator),

– Language of program and courses: English.

– Next starting date: October 14th, 2013 (Beginning of the Winter Semester). Unless in case of transfer from other MA programs, there is no admission to the MA in Logic and Philosophy of Science in the summer semester.

– Duration: The MA is a two-year (four semesters, 120 ECTS) program. We are also working on introducing a one-year option, but this doesn't apply for 2013.

– The MA is primarily aimed at preparing doctoral research in philosophy. It is, however, a self-standing degree which can be taken as such, or in preparation for doctoral research in other disciplines (computer sciences, linguistics, etc.)

– Structure: In each semester, all students in the program attend a compulsory MA seminar (the "Master's Colloquium").  Beyond that, students can choose from a broad range of additional courses in logic, philosophy of science, and related areas until their final semester. The final semester is devoted to writing a MA thesis.

– Entry requirements: Academically, an average mark in one's undergraduate studies of at least 2,0 in the German system is required (which corresponds to at least an upper second-class honours degree in the British system or the international equivalent). It is possible to apply for a place in the program even if one has not yet finished an undergraduate degree. In such a case we would make a conditional offer. The program is not just open to philosophy and logic undergraduates, but also to undergraduates from relevant scientific disciplines such as: mathematics, physics, biology, chemistry, neuroscience, computer science, engineering, economics, linguistics, psychology, cognitive science, the social sciences or related disciplines.
If your first language is not English and you do not have a degree from an English-speaking university, you will need to supply us with evidence of your language skills as far as speaking and writing in English is concerned. We demand at least a 6.5 in all IELTS bands (or equivalent).

– Fees and financial support: Every student at LMU has to pay EUR 500 per semester for their studies. We will be able to award, on the basis of track record, a limited number of entry grants consisting of a tax-free, monthly stipend of EUR 1000 for one year. The possibility of extension for another year is subject to positive review after the first year.  If you are interested in applying for an entry grant please say so in your cover letter.

– Application: We will accept applications until July 15th, 2013, but we strongly recommend to apply before that date. Please submit your application electronically to the program coordinator ( with "Application MA in Logic and Philosophy of Science" as the subject.

Your application package should include:

(1) A cover letter in which you explain why you would like to join the program. If you wish to apply for an entry grant please mention it explicitly in the letter.
(2) A CV (including the list of courses that you have taken as an undergraduate).
(3) A copy of your undergraduate final grade certificate or transcript.
(4) A seminar paper or published article that you have written, the subject of which can vary. It should give evidence of clear thinking, systematicity, logical argumentation, and an awareness of methodological questions, which are all particularly relevant to our MA program.
(5) Two letters of reference, which should also address your special potential in the very areas that are covered by our MA program.
(6) Proof of English language proficiency (IELTS or equivalent), in case your first language is not English and you do not have a degree from an English-speaking university.

If necessary, applicants may also be interviewed. If you have questions about that part of the application process, please contact the program coordinator (

Secondly, and in addition, please apply simultaneously to the International Affairs Office at LMU which will check for general requirements that concern Master studies at LMU in general. At you can download the set of application documents for Master degrees at LMU. There you will also find information about which documents to submit to the International Affairs Office, how to submit them, and whom you can contact about this other part of the application process in case of questions.


Friday, 14 December 2012

Post-doc position in Groningen - Infinite Regresses

The Faculty of Philosophy of the University of Groningen, The Netherlands,

has an opening for a

Postdoc Position in Philosophy (three years fulltime)

Vacancy number 212285

Job description

The postdoc will conduct independent research on historical and/or systematic aspects of infinite regresses in philosophy. The position is situated within the research project on infinite regresses in epistemology, funded by the Dutch National Science Foundation NWO. The project is carried out in the Department of Theoretical Philosophy. The project leader is Prof. Jeanne Peijnenburg; informal inquiries may be directed to Prof. Peijnenburg.


Candidates for this postdoc position:
·  have a Ph.D. in philosophy or in another field that is demonstrably relevant;
·  have a background in, or familiarity with, the regress problem in epistemology or in the history of philosophy;
·  have an excellent command of English and are prepared to present their research results in English;
·  are willing and able to teach a small number of courses.

Conditions of employment

The University of Groningen offers a salary that is determined in accordance with the current scales as set out in the collective labor agreement for the Dutch universities (CAO) and has a range between € 3,195 and € 4,374 gross per month dependent on qualifications and experience. The position is temporary, with a maximum of three years (initial appointment of two years, with the possibility of renewal for a third year). A part-time appointment at 0.8 fte for four years is possible.


Since its foundation in 1614, the University of Groningen has enjoyed an international reputation as a dynamic and innovative centre of higher education. The Faculty of Philosophy is a rich and lively community of excellent lecturers and researchers. For more information on the faculty:

How to apply

To apply for the postdoc position, please send an e-mail to Mrs Anita Willems-Veenstra ( and attach:
·  in one file: a letter of application with reference to the vacancy number (also including an explanation of why you are particularly qualified for this project) and a curriculum vitae (including a list of publications, a short summary of your dissertation, and the names and contact information of two academic references);
·  a writing sample (such as a research paper or part of your dissertation).

The closing date for applications is April 8, 2013.
Interviews are planned for the the first half of May.

Monday, 10 December 2012

FEW 2013 Announcement

Believe it or not, this will be our 10th FEW!

We are very pleased to announce that FEW 2013 will be co-located with The Rutgers Epistemology Conference (REC), which is also celebrating its 10th meeting.

Both meetings will take place at the Hyatt Regency in New Brunswick.

FEW will take place on May 8 and 9 and REC will take place on May 10 and 11. There will be a "bridge" session on the morning of the 10th. The "bridge speaker" will be Brian Weatherson.

The FEW schedule will be much shorter (and more relaxed) than usual. We will have the following nine (9) invited speakers, including our "bridge" speaker (there will be no contributed papers this year).
Rachael Briggs
Jim Joyce
Matt Kotzen
Hannes Leitgeb
Christian List
Sherri Roush
Robert Stalnaker
Katya Tentori
Brian Weatherson
If you'd like to serve as a "chair" or a "discussant" for one of the FEW talks (so that your name can go on the online schedule, which would probably allow you to use research money to cover your trip), please email me and we'll work something out.

We hope that FEW participants will stick around for REC. The invited speakers at REC (aside from the "bridge" speaker) will be:
Tamar Szabo Gendler
Sanford Goldberg
Matthew McGrath
Susanna Siegel
David Sosa
And, the discussants for REC will be:
Adam Elga
Pascal Engel
Imogen Dickie
Baron Reed
We hope you'll be able to join us this year --- at the intersection of formal and traditional philosophy.

The Modus Ponens-Modus Tollens asymmetry and non-monotonic logics

(Cross-posted at NewAPPS)

A well-known phenomenon in the empirical study of human reasoning is the so-called Modus Ponens-Modus Tollens asymmetry. In reasoning experiments, participants almost invariably ‘do well’ with MP (or at least something that looks like MP – see below), but the rate for MT success drops considerably (from almost 100% for MP to around 70% for MT – Schroyens and Schaeken 2003). As a result, any theory purporting to describe human reasoning accurately must account for this asymmetry. Now, given that for classical logic (and other non-classical systems) MP and MT are equally valid, plain vanilla classical logic fails rather miserably in this respect.

As noted by Oaksford and Chater (‘Probability logic and the Modus Ponens-Modus Tollens asymmetry in conditional inference’, in this 2008 book), some theories of human reasoning (mental rules, mental models) explain the asymmetry at what is known as the algorithmic level (a terminology proposed by Marr (1982)) – that is, in terms of the mental process that (purportedly) implement deductive reasoning in a human mind. So according to these theories, performing MT is harder than performing MP (for a variety of reasons), which is why reasoners, while still trying to reason deductively, have difficulties with MT. Other theorists defend that participants are not in fact trying to reason deductively at all, so the asymmetry is not related to some presumed competence-performance gap. (Marr’s term to refer to the general goal of the processes, rather than the processes themselves, is ‘computational level’ – the terminology is somewhat unnatural, but it has now become standard.) Oaksford and Chater are among those favoring an analysis at the computational level, in their case proposing a Bayesian, probabilistic account of human reasoning as a normative theory not only explaining but also justifying the asymmetry.

In the reasoning literature, most of those who have rejected the so-called ‘deduction paradigm’ have gone probabilistic. One exception is the work of Stenning and van Lambalgen (2008), who take a (qualitative) non-monotonic logic known as closed-world reasoning as their starting point to investigate human reasoning both at the algorithmic and the computational levels. For reasons that are too convoluted to go into here, I am not entirely satisfied with either the probabilistic approach or the closed-world reasoning approach. I like the idea of non-monotonic logics and the qualitative perspective they offer, but Stenning and van Lambalgen’s approach is too ‘syntactic’ to my taste. Instead, I’ve been working with the semantic approach to non-monotonic logics originally introduced by Shoham (1987) and further developed in the classic (Kraus, Lehmann and Magidor 1990) to account for a number of psychological phenomena pertaining to reasoning, such as so-called reasoning biases (see slides of a talk here) and now the MP-MT asymmetry. This group of theories is often referred to as ‘preferential logics’. (It is worth noting that on many but not all non-monotonic logics, MP holds but MT does not.) I take the semantic approach of preferential logics to be not only technically useful to study reasoning phenomena, but also to be descriptively plausible. (And here is a little plug: Hanti Lin is doing amazing theoretical work with a similar framework.)

Shoham’s semantic approach to non-monotonic logics is beautifully simple: take a standard monotonic logic L and define a strict partial order for the models M of L, which is viewed as a defining a preference relation: M1 < M2 means that M2 is preferred over M1. The non-monotonic relation of consequence then becomes:

A => B iff all preferred models of A are models of B.

This relation is non-monotonic because A & C may have preferred models that are not preferred models of A alone. So with the addition of C, it may no longer be the case that B holds in all preferred models of A & C, even if it is the case that B holds in all preferred models of A alone.

What about MP and MT? If the conditional is given a defeasible interpretation corresponding to the defeasible consequence relation as defined above, then what we have is not the classical version of MP, which allows for no exceptions (no cases where the antecedent is the case but not the consequent), but rather something that could be described as defeasible MP (a terminology used for example by D. Walton and collaborators). Do we obtain a defeasible MT as well?

Now, the first thing to notice is that the preferential consequence relation has a built-in asymmetry: it refers to the preferred models of A, but to all models of B. So this consequence relation does not contrapose: assuming that, for all models M and all propositions P, either P or not-P holds in M, for the relation to be contrapositive it would be required that all preferred models of not-B are also models (preferred or otherwise) of not-A. But there may well be non-preferred models of A which are also (preferred) models of not-B. Thus, the definition is not satisfied for not-B => not-A. By the same token, MT does not hold, and the MP-MT asymmetry is both explained and justified.

In fact, adopting this framework also suggests that the sky-high rate of success with MP in experiments may not be an indication that participants are in fact reasoning deductively-indefeasibly. This is because in the case of MP, the defeasible and the indefeasible responses coincide; in the case of MT, however, the responses come apart, suggesting that at least some of the participants (those who do MP but not MT) may be reasoning defeasibly all along.