What we don't need to save ordinary conditionals from

It’s been a while since I posted anything. Mostly this is because life has been pretty hectic lately. In September we spent a few weeks in Gdańsk, but now we’re moving every few weeks between various places in the UK, visiting different universities and trying to get some research done meanwhile. This semester I’m mostly based in Bristol as a British Academy Visiting Fellow, working with Oystein Linnebo on the dynamic approach to abstraction principles, and doing some directed reading (on groundedness with Hannes Leitgeb and on axiomatic theories of truth with Leon Horsten). These days, however, we’re hanging out in Scotland, currently visiting Arche Research Centre in St. Andrews, taking off for a few other places tomorrow.

I gave a talk here on Tuesday about nominalistic approaches to neologicism, and decided to stick around for the Arche/CSMN graduate conference. Today, I managed to catch a talk by Ernest Lepore followed by an interesting talk about counterfactuals by Daniel Berntson (with a commentary by Guðmundur Andri Hjálmarsson). Daniel’s talk was titled Saving Ordinary Counterfactuals and was devoted to the problems that quantum indeterminacy (or related phenomena) are supposed to raise for our intuitions about ordinary counterfactuals. The whole thing was very clear and quite interesting. I do have one minor worry, though - I don’t think the problem that Daniel is trying to address exists... Counterfactually: if there were such a problem, Daniel’s approach would be a neat way to approach it. But let’s start from the beginning...

The problem

Intuitively we accept the counterfactual:

(1) If I were to throw a champagne glass off the top of Empire State Building, it would break.

Supposedly, quantum mechanics informs us also that:

(2) There is some chance that a glass thrown off the top of the Empire State Building will quantum tunnel to the moon without breaking.

If indeterminism is true and (2) expresses objective probability, (2) seems to entail (3):

(3) If I were to throw a champagne glass of the top of the Empire State Building, it might safely quantum tunnel to the moon.

This entails:

(4) If I were to throw a champagne glass off the Empire State Building, it might not break.

Now, Daniel suggests that "(4) puts pressure on us to give up (1)" and that there is an "inescapable clash" in the infelicitous assertion (5):

(5) If I were to throw a champagne glass off the Empire State building, it would break; and furthermore, it might not break.

The strategy

In order to save the truth of (1) within the Lewis-Stalnaker approach, Daniel suggests replacing:

A>B iff all of the closest A-worlds are B-worlds.

with

A>B iff the vast majority of the closest A-worlds are B-worlds.

The underlying idea now would be that (1) is made true because most of the closest worlds where the glass is dropped are worlds where it is broken, whereas (4) emphasizes that not all closest possible worlds are worlds where the glass is broken.

Say we put aside the issue of how we are to count possible worlds and they ratios if there are infinitely many of those. There still are some problems that come along with this solution. Most prominently, agglomeration (A>B, A>C hence A>B&C) and transitivity (A>B, A^B>C hence A>C) fail. To fix these issues, Daniel introduces the notion of being almost true, and says that certain claims, even though they aren’t strictly speaking true on this semantics, are still almost true, like when we have a counterfactual which doesn’t preserve probability ratio, but whose consequent is only slightly less probable than the antecedent. There are some bells and whistles to play around with here, but this should be enough for the set-up.

The worry

First, observe that a necessary (but not obviously sufficient) condition for thinking that (5) is a problem is the acceptance of (1) and (4). Strictly speaking, so far Daniel has shown how to preserve the truth of (1), but didn’t say explicitly how to make sense of (4).

In fact, Daniel introduces might-conditionals by saying:

A >m> B iff ~(A>~B)

That is, a might conditional A >m> B is supposed to come out true iff it is not the case that ~B is true in the vast majority of the closest A-worlds.

Alas, this reading of might-conditionals doesn’t support the truth of (4), because given that all the relevant worlds where the glass is broken are worlds where it is not the case that it tunnels safely to the moon, (4) still comes out false if it is to be read as a might-conditional.

Second, it’s not clear where the clash really is. I would certainly be worried if I had intuitive reasons to believe:

(5’) If I were to throw a champagne glass off the Empire State building, it would break; and furthermore, it wouldn’t break.

But so far, I don’t. (5) certainly doesn’t entail (5’). Given that (4) shouldn’t be constructed as a might-counterfactual if its truth is to be preserved, what job exactly is "might" doing in (4) and (5)?

Well, I’m inclined to say that that are at least two ways to accept (1), (4) and (5) even if we play along with non-probabilistic Lewis-Stalnaker semantics.

Story 1. (1) says that in every closest possible world, where I drop the glass, it’s broken, whereas (4) says that there are still some accessible but less similar worlds, where even though the glass is dropped, it’s not broken.

Story 2 (1) says that in every closest possible world, where I drop the glass, it’s broken, and (4) says that it is possible that the glass is dropped and not broken.

Personally, I prefer story 2, because it assigns less content to "might", and because (5) as it is, when read along the reading suggested in story 2, entails that the world where the glass is broken is not the closest one anyway.

I don’t even have the intuitions that (5) displays any sort of clash to start with. Imagine I say:

If I were to join you for conference drinks tonight, I would be hungover the next day. Well, in fact, it’s highly unlikely that I would decide to drink only water, in which case I might feel good the next day even if I go. So if I were to join you for conference drinks tonight, I might feel good the next day, but I think I wouldn’t.

This doesn't seem to contain any contradiction whatsoever.

Interestingly, we did a poll and around half of the audience thought that (5) was problematic, and around half that it wasn’t.

The "Modal argument" paper is forthcoming

A paper we wrote with Agnieszka Rostalska (I very roughly outlined an early version here) is forthcoming in Philo. A rather final draft of the paper is available online here. The paper is devoted to the clarification and criticism of Swinburne's modal argument for the existence of the soul. Before I paste the abstract and acknowledgments below, one more remark.

When giving this paper at various places, one sort of reactions encountered came from people with good background in logic, but no previous experience with philosophy whatsoever. The reaction boils down to a rather blind stare and comments like "who cares about arguments for the existence of the soul?" or "Why is anyone doing this stuff?". The answers are simple. "Philosophers." to the first question. "Because it's more interesting than using complex mathematical tools to solve problems that only two to three people in the world care about." to the second one.

I prefer to use slightly less elaborate mathematical machinery to deal with philosophically motivated issues than to get into very complex and hermetic issues in, say, inaccessible set theory or computer science. This doesn't mean they aren't interesting. I just find philosophical problems more entertaining and important. And I think it is, in a sense, the responsibility of a philosopher and a logician to spend some time looking at what philosophical arguments are around about claims people care about and what can be said about their correctness, instead of locking themselves in the ivory tower of elaborate and detached purely mathematical problems. But again, it's a matter of choice.

Abstract

Richard Swinburne (Swinburne and Shoemaker 1984; Swinburne 1986) argues that human beings currently alive have non-bodily immaterial parts called souls. In his main argument in support of this conclusion (modal argument), roughly speaking, from the assumption that it is logically possible that a human being survives the destruction of their body and a few additional premises, he infers the actual existence of souls. After a brief presentation of the argument we describe the main known objection to it, called the substitution objection (SO for short), which is raised by Alston and Smythe (1994), Zimmerman (1991) and Stump and Kretzmann (1996). We then explain Swinburne's response to it (1996). This constitutes a background for the discussion that follows. First, we formalize Swinburne's argument in a quantified propositional modal language so that it is logically valid and contains no tacit assumptions, clearing up some notational issues as we go. Having done that, we explain why we find Swinburne's response unsatisfactory. Next, we indicate that even though SO is quite compelling (albeit for a slightly different reason than the one given previously in the literature), a weakening of one of the premises yields a valid argument for the same conclusion and yet immune to SO. Even this version of the argument, we argue, is epistemically circular.

Acknowledgments

We would like to express our gratitude to all the people who discussed these issues with us and commented on earlier versions of this paper. We are grateful to participants of the events where the paper has been presented: Workshop & Young Researcher's Day in Logic, Philosophy and History of Science in Brussels, 2008, Jeffrey Ketland's Omega-seminar in Edinburgh, 2008, and Formal Methods in the Epistemology of Religion in Leuven, 2009. The main ideas of this paper originated after a number of discussions about philosophy of religion and mind with Professor Jack MacIntosh (Calgary). Comments provided by Professor Richard Swinburne (Oxford), who was in the audience when this paper was presented in Leuven in June 2009, were also very helpful, and it was interesting to learn that Professor Swinburne agrees with all our main points, apart from our final assessment of the modified argument. It was Lara Buchak (Berkeley) who observed that our version of the argument developed in response to SO results from a weakening of one of the premises. We also owe gratitude to Paul Draper for his invaluable editorial comments.

Trends in Philosophy of Mathematics (day 3, talk 1)

On the third day the schedule was a bit more complicated, we had to choose between one of two parallel sessions. The choice was difficult, so I will be unable to comment on some of really interesting talks that I was unable to attend. If I don’t talk about a certain talk it’s because I was unable to make it to it because the alternative talk was more related to what I’m working on. For now, the first talk of the day.

Assadian: Crispin Wright and his Hero

Wright, defending the epistemic accessibility of prima facie impredicative Hume’s Principle tells a story of a fictional character (named Hero) who initially knows second-order logic and possesses a bunch of sortal concepts referring to concrete objects, but doesn’t understand the concept of number. Wright then argues that the Hero can process in stages in order to gain the understanding of the concept of natural numbers.

Stage 1 - The Hero introduces HP for the initial domain that he possesses a grasp of.

Stage 2 - The Hero now understands without circularity the truth conditions of Nx:Fx = Nx:Gx, where neither F nor G contain further occurrences of further numerical terms. The Hero also knows that Nx:Fx=s is false for any term s referring to an object present at Stage 1. In this sense, he seems able to solve the Caesar Problem for he comes to accept:

(NE) no object whose identity is grounded in anything else than HP can be identical to a number.

Stage 3 - The Hero moves on to understanding truth conditions of identity statements of terms containing embedded occurrences of numerical operators.

Assadian takes issue with Wright’s account of how the Hero learns NE, which is of key importance in solving CP. NE seems to hinge on the possession of a complete characterization of natural numbers (=identity conditions for them), and this is not something that the Hero can have at stage 2. For it seems that to understand the claim that the characterization of numbers doesn’t go beyond its identity conditions dictated by HP, one has to be able to grasp those identity conditions already.On the other hand, understanding of complex terms containing embedded numerical operators seems to require that NE be already known, if CP is to be solved. Moreover, if NE is not available at stage 2, the Hero is unable to solve CP at that stage.

A slightly different neologicist approach to the problem is to assume that different sorts of objects in general have different criteria of identity - once one postulates the existence of maximal categories of that sort and ads some fairly convincing assumptions about them, two theorems can be proven. The first says that for any category C for any two objects that fall under that category, they are identical iff they satisfy identity conditions corresponding to that category. The other one says that no object belongs to two different categories. It is needed if we are to exclude the possibility that both numbers and persons constitute a single larger category with its specific identity conditions.

Assadian argues that even though if those theorems are present, CP is solved, it’s quite implausible that those theorems are available to the Hero at stage 2, because theorem 2 already says something about the whole category of all numbers and their identity conditions. On the other hand, if those theorems are to be introduced only at stage 3, it is unclear why the Hero would be able to solve CP already in stange 2 to start with.

Although I generally agree that non-iterative approaches to abstraction principles have been so far unable to solve CP (among the iterative ones there’s Linnebo’s and mine, and I think mine can handle CP and it is quite unclear whether Linnebo’s does - but this is a whole different story), I really would have to see the proofs in detail - what would have to be checked is whether (i) the assumptions used to prove theorems 1 and 2 are convincing, (ii) theorems 1 and 2 follow from those assumptions, (iii) theorems 1 and 2 really allow to solve CP (i.e. to prove negations of mixed identity statements, or something to that effect), and (iv) whether no undesired consequences follow from the same assumptions. But the stuff seems interesting.

Qualms about the (non-)circularity of NE aside, what I’m rather worried about is the justification of the framework in which NE even makes sense. I mean, I have pretty hard time understanding the idea of objects such that there is nothing else to learn about them apart from their rather coarse-grained identity conditions. I am perfectly fine with coarse-grained or relative identity claims, or true fake identity between fake singular terms, but the idea that there really are objects such that the only way we can learn anything about them is through abstraction principles seems suspicious. Of course, it is an attempt to deal with epistemic challenges to Platonism about mathematics, but I don’t think a blunt answer of the sort "How can we know something about numbers? Well, we learn something about them through abstraction principles and there’s nothing else to learn" is satisfactory. I would need a more elaborate and convincing metaphysical story which would convince me to accept the existence of such things, and which would explain why those objects should enjoy this particular status.

The adaptive logics book has moved

Due to server issues, the book I mentioned before has moved. Here. I also fixed the original link.

Live from Trends in Logic VII (day 2)

Today we had four quite exciting talks. The first one, given by Oystein Linnebo (Bristol) was devoted to A Partial Defense of Frege's Basic Law V. Oystein started off with the intuitions that there is some pressure to accept Frege's BLV (which says that extensions of two concepts are identical iff exactly the same objects fall under that concept). After criticizing the limitation-of-size approach to restricted versions of the comprehension principle, he went modal-and-iterative about BLV. That is, BLV was used to capture how new sets are formed at new stages using the objects already existing in previous stages, and modal operators were thrown in to express the intuition that we're talking about the possible ways our set-formation process can go. This gives a fairly intuitive criterion for a plurality determining a set: it has to have the same elements across possible worlds. Proof-theoretically, once you take S4.2 as the underlying modal logic, throw in some trans-world extensionality principles for pluralities and sets and introduce the potential plural collapse ("it is necessary that for any xx it is possible that there is a y such that y is the set of xx's"), you can get (a reinterpretation of) Zermelo set theory minus infinity and foundation.

The second talk, by Leon Horsten, was devoted to the relation between numbers and counting systems. Leon defended and described the view dubbed computational structuralism: it's kinda like structuralism, but you take arithmetic to be about the structure of arithmetical notational systems. The basic idea is that if one has a recursively introduced notational systems (so that the symbol denoting the successor is computable), and the addition function is also computable, the system is isomorphic to the intended omega-sequences.

Michael Resnik and Stewart Shapiro both talked about qualms that arise around identity conditions for structures and their elements (or positions in them). Resnik, roughly, was arguing that in certain contexts identity claims (and questions about identity) of certain structures doesn't make sense, whereas Shapiro was rather inclined to say that it's not as much the identity questions that are misled, but rather that certain terms may seem and behave like singular terms, despite referring indeterminately to many objects.

Live from Trends in Logic VII

It's the first day of Trends in Logic VII, aka Trends in the Philosophy of Mathematics. So far, we're past an opening, an opening leture by Ryszard Wójcicki, and a splendid conference dinner.

Ryszard Wójcicki, an excellent "hardcore" logician known for his work on consequence operations and Polish-style meta-theory of propositional calculi, has recently decided to think about more philosophical issues. He was talking about Two sources of mathematical truth. The main gist was that the key "source" of mathematical truth was "conceptual realities" (the other source being empirical domains). Alas, I didn't quite get what being a source of truth is, how conceptual realities are supposed to be different from mathematical structures, what their ontological status is, and why they're supposed to exist. Having said that, it was interesting to hear a real "hardcore" researcher say what he thinks about the philosophical status of his own field.

My general impression is that if a "hardcore" scientist of any specific sort suddenly starts to philosophize, it's bound to be slightly weird stuff from the philosopher's perspective (it's not as bad as a philosopher trying to do science, though). What slightly surprised me was that this also holds for logicians. On the other hand, I do think that one of the problems that analytic philosophy in Poland is facing is that there are many excellent logicians doing highly technical stuff but having no philosophical interests or well developed intuitions, and there are many philosophers with highly developed intuitions, but with almost no grasp of logic or attention to arguments and details whatsoever.

A rant about "deductive"

or
Don't diss the logician

I’m on my way back from The Second Conference on Concept Types and Frames in Language, Cognition and Science in Dusseldorf. It was a nice conference that gathered linguists, cognitivists, philosophers of science and logicians interested in the functional approach to concepts.

One of the things that surprised me was that both experienced cognitivists (like Paul Thagard) and younger researchers still stick to the distinction between inductive and deductive types of reasoning and attach that much importance to it. Interestingly, “deductive” in their use has a pejorative content and the term is sometimes used condescendingly to emphasize that whatever it is that logicians do is boring and useless and that pretty much the only source of insight and real knowledge are “inductive inferences” taking place in “the real brain”. So, here’s a short rant about this sort of attitude (Frederik is reading over my shoulder and tossing in his remarks).

To start with, I don’t think I know a logician alive who still uses the word “deductive” in any serious ahistorical context. This is because the notion is so worn out that different people associate it with many different things. Instead, more specific terms are used that separately capture different things that you might mean when you say “deductive”.

Roughly, a consequence operation is, for instance, often simply thought of as a set of pairs of sets of sentences. It is called structural if it’s closed under substitution. That’s one thing that you might have in mind: deductive means defined in terms of rules (and maybe axioms) which essentially make no distinction between formulas of the same syntactic form. Another way you can think about these things is to require that a deductive consequence should be simply truth-preserving (vaguely: it’s impossible that the consequence is false when the premises are true). This interpretation is not syntactic, but rather model-theoretic. A truth-preserving consequence doesn’t have to be structural and a structural consequence doesn’t have to be truth-preserving. Another sense you might associate with being deductive is being both structural and truth-preserving (in which case, you still get a multitude of consequence operations, depending on what language and model theory you pick, and what you take to belong to your logical vocabulary). Yet another interpretation you can take is to say that something is a deductive consequence of a given set of premises if it follows from them by classical logic – this notion is sometimes used by those cognitivists who think that logic is classical logic. Although this consequence is structural, whether it’s truth-preserving when it comes to natural language is a matter of what you think about the correctness of certain natural language inferences. For instance, you might be a relevantist – in which case you’re inclined to say that the classical logic allows you to infer too much. Yet another notion simply requires a deductive consequence to satisfy Tarski’s conditions, or some of them, or some of them and some other conditions of a similar type. Yet another idea is to make no reference to a formal system whatsoever and assume that a sentence A is a deductive consequence of a sentence B iff “If B, then A” is analytic (standard qualms about analyticity aside). So in general, the logician’s conceptual framework is full of notions more precise than “deductive”, and the word “deductive” seems unclear and a tad outdated.

But let us even suppose we fix on the notion of being deductive as being validated by classical logic (this seems to be the best you can do if you want to make it easy for the cognitivits to argue that deductive inferences are uninformative). Why on earth would you think that deductive reasoning can only give you boring and useless consequences that you already were aware of, unless you say so because what you take to be the most prominent example of a deduction is one of the slightly obvious syllogisms, most likely employing Socrates and his mortality?

The thing is, human beings are not logically omniscient (I myself, for instance, often feel dumb when I stare at a deductive proof I can’t grasp after half an hour). In fact, the history of mathematics is a good source of examples where prima facie well-understood premise sets led to surprising consequences. Just because the truth of a conclusion is guaranteed by the truth of the premises doesn’t mean that once we believe the premises we actually are aware that they lead to this conclusion. Take the Russell’s paradox. A rather bright dude named Frege spent years without noticing a fairly simple reasoning whose conclusion was to him somewhat surprising. Take Godel’s incompleteness theorem(s). A rather known set of mathematical truths together with a bit of slightly complicated deductive reasoning led to one of the most important discoveries in the 20th century logic, which stunned a bunch of other not-too-dumb mathematicians. If you still think that deductive inferences give nothing but boring and obvious conclusions, think again!

Two points about the opposition between the deductive and the inductive. First of all, unless you define inductive as non-deductive, the distinction is not exhaustive. For instance, if inductive inferences are supposed to be those that lead to a general conclusion, we’re missing non-deductive inferences with particular conclusions (like in History, one uses certain general assumptions and knowledge about present facts to surmise something particular about the past). In this sense, the deductive-reductive distinction introduced by the Lvov-Warsaw school sounds a bit neater (look it up).

Another thing is that people often speak of inductive inferences as if they didn’t have anything to do with deduction (the following point was made by Frederik). Quite to the contrary, certain facts about what is deducible and what isn’t lie always in the background when you’re assessing the plausibility of an inductive inference. For instance, you want the generalization you introduce to explain certain particular data you’re generalizing from, and one of the most obvious analysis of explanation uses the notion of deducibility. Also, you don’t want your new generalization to contradict your other data and other generalizations you have introduced before: but hey, isn’t the notion of consistency highly dependent of your notion of derivability?

Having said that, I also have to emphasize that this doesn't mean that I take non-deductive inferences (whatever they are) to be uninteresting; indeed, the question of how we come to accept certain beliefs other than by deducing them (whatever this consists in) from other beliefs is a very hard and interesting problem. What I oppose to, rather, is drawing cut and dry lines between these types of reasoning and saying that only one of them is interesting.

A book on adaptive logics in progress...

Diderik Batens is working on a book about adaptive logics. He made drafts of first few chapters available online and invites comments. Here.

Frames, Frames and Frames

1. The paper on dynamic frames has been accepted and is forthcoming in the Logic Journal of the IGPL. As I understand their self-archiving policy, it can't be publicly accessible for 12 months after it's published by OUP. Hence, I'm making the final version available now, it'll be available till the official publication. If you feel like grabbing it before it disappears, it's here.

2. In the same vein, in Ghent this Friday (August 21) we're having a mini-workshop on frame theory. If you're around at that time, feel free to swing by. There's gonna be an outing afterwards.

Title: Frames, Frames and Frames

Time: Friday, August 21. 17:00-19:00 (There will be three talks, 30 minutes each + discussion)

Place: Room 2.19, Centre for Logic and Philosophy of Science, Universiteit Gent, Blandijnberg 2

Talks:

1. Capturing dynamic frames. It's based on the paper I just mentioned: I explain what frames are, how certain frames can be expressed by sets of first-order, formulas, and how an adaptive strategy can be applied to a reasoning with a conceptual framework when faced with an anomaly.
2. Induction from a single instance and dynamic frames. It reports the content of a joint paper with Frederik Van De Putte; basically, we discuss how the background knowledge needed for a distinction between plausible and implausible cases of induction from a single instance can be formulated within frame theory, and how the theory provides a nice framework for talking about this sort of reasoning as relying on certain second-order inferences.
3. Similarity and dynamic frames. I'm talking about Bugajski's algebraic semantics for similarity relation, indicate its weaknesses, and provide a relational semantics that's simpler and which satisfies more of Williamson's requirements for 4-place similarity relation. Then, I discuss Bugajski's argument to the effect that interesting similarity structures can be generated by a set of properties only if those properties aren't sharp. To criticize it, I describe how non-trivial similarity structures can be generated by sets of sharp properties, if these are viewed within the framework of dynamic frame theory.

NCM 09 (part 2)

... and the postponed report on Non-Classical Mathematics 2009 continues...

The second talk was given by Giovanni Sambin. He talked about his minimalist foundation and about a way constructive topology can be developed over a minimalist foundation. It's quite interesting to see how much stuff can be done constructively. Also, Giovanni is a devoted and areally charming constructivist. I was chatting with him at a pub one night and only by finding myself almost converted to constructivism I knew it was time to go home.

By the way, among many inaccurate things that are being said about Godel's theorem (like these) you can find a remark that Godel's incompleteness and undefinability proofs/theorems don't work in intuitionistic mathematics. Actually, they do. And the person to talk to is Giovanni, who worked out all the details making sure everything is constructive.

Arnon Avron talked about a new approach to Predicative Set Theory. Roughly,the underlying principles of the predicative mathematics are:

I. Higher-order constructs are acceptable only when introduced through non-circular definitions referring only to constructs introduced by previous definitions.

II. The natural number sequence is a well understood concept and as a totality it constitutes a set.

It is well known that Feferman has pursued the project and has shown how a large part of classical analysis can be developed within it. The system, however, is not too popular, partially because it uses a rather complex hierarchy of types, which makes the theory more complicated than, say, ZFC.

Arnon Avron discussed an attempt to simplify predicative mathematics by getting rid of the type hierarchy and developing a type-free predicative set theory. The idea is that the comprehension schema is restricted to those formulas that satisfy a syntactically defined safety relation between formulas and variables. The relation resembles a syntactic approximation to the notion of domain-independence used in database theory, and the intuition is that acceptable formulas define a concept in an acceptable way independent of any extension of the universe.

A replacement for Journal Wiki

Here is a new database (by Andrew Cullison) gathering data pertaining to philosophy journal experience. I mentioned its being in preparation before. Now it seems to be up and running (although, the journal wiki data import is yet about to happen). It is certainly more user-friendly than its predecessor.

Tahko's paper on modal epistemology online

I see Tuomas Tahko, besides posting a bunch of pictures from his recent trip, posted also his paper on modal epistemology. It's quite interesting. Title and details below.

Two-Dimensional Modal Semantics, Conceivability, and Modal Epistemology

ABSTRACT The combination of two-dimensional modal semantics and conceivability purports to be very powerful: it upholds modal rationalism, explains a posteriori necessity, and even accounts for metaphysical impossibilities—all this while committing to only one modal space, conceptual modality. In this paper I will examine whether two-dimensional modal semantics and conceivability can produce a complete account of modal epistemology and argue that they cannot. We will see that the framework fails to account for metaphysical modality or to deal with metaphysically substantial, essentialist statements because it is unable to distinguish between trivial and substantial modal truths.

Leszek Kołakowski has passed away

Leszek Kołakowski, an important figure in political philosophy and an interesting Polish thinker has passed away.

Brian Leiter, despite his severe criticism of Kołakowski actually cared to post a short note about this, too. [pointed out by Ziel of the Polish blog Jakies Przepisywania z Prasy Wszelakiej fame in personal communication]

Marriage and Domestic Partnership

Okay, I know, this isn't logic-related. Anyway, I see Elizabeth Brake's (University of Calgary) Stanford Encyclopedia entry on Marriage and Domestic Partnership is now available.

Leitgeb, "about", Yablo (again)

A paper I already mentioned has been accepted and is coming out in Logique et Analyse soon. I'm keeping the copyright and I like open-access stuff, so the most recent version is available here. Title and (updated) abstract below:

Leitgeb, "about", Yablo

Leitgeb (2002) objects against the clarity of the debate about the alleged (non-)circularity of Yablo's paradox, arguing that there are actually two notions of self-reference and circularity at play. One, on which Yablo's paradox is not circular, is defined via the reference of the constituents of a sentence, and another, on which the paradox is circular, is defined via syntactic mappings and fixed points. More importantly, Leitgeb argues that both definitions aren't satisfactory and that before we can undertake a serious debate about the circularity of Yablo's paradox we first need to clarify the notions involved. I will focus on Leitgeb's criticism of the first definition and will argue that the problems arise not as much on the level of our definition of circularity as on the level of our definition of reference of sentences (aboutness). Leitgeb's main worry is the failure of a requirement called `Equivalence Condition', which says that if a formula is self-referential, any formula logically equivalent to it should also be self-referential. I will argue that preservation under logical equivalence is unreasonable with respect to self-reference, but is indeed needed with respect to aboutness. Since Leitgeb's own tentative notion of aboutness doesn't satisfy the requirement, I will suggest another approach which fixes this problem. I also explain why the intuitions that circularity should satisfy the equivalence condition are misled. Next, I argue that the new notion of aboutness is not susceptible to slingshot arguments. Finally, I compare it with Goodman's notion of absolute aboutness, emphasizing those features of Goodman's approach that make his notion inapplicable in the present discussion.

I would like to express my gratitude to all the people who discussed earlier versions of this paper with me: Hannes Leitgeb, Jeffrey Ketland, Karl Georg Niebergall, Diderik Batens, Joke Meheus, Maarten Van Dyck, Stefan Wintein, Martin Bentzen, Christian Strasser, Ghent Centre for Logic and Philosophy of Science members, and the participants of PhDs in Logic workshop (Gent 2009).