Saturday, September 19, 2009

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.

Saturday, September 12, 2009

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.

Saturday, September 5, 2009

Thursday, September 3, 2009

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.

Wednesday, September 2, 2009

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.