Saturday, December 3, 2011

Godelizing the Yablo sequence

Ever couldn't sleep or eat thinking what happens when truth is replaced with provability in the omega liar? Now you can put your qualms to rest, this issue (needless to say, of ultimate relevance to Great Philosophical Questions) has been handled! Cezary Cieslinski and I have finished drafting a paper about this. (Of course, the entailment of corollaries about The Meaning of Life is so obvious that we didn't bother stating them.)
The paper is available here.

Abstract. We investigate  what happens when `truth'  is replaced with  `provability' in Yablo's paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by  the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Godel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of the existential Yablo paradox. We also look at a formulation which employs Rosser's provability predicate.  

Sunday, November 27, 2011

Lesniewski book draft

Finally, I have finished drafting and proofreading a book about Lesniewski's systems. To some extent it is based on my PhD thesis, but it has gone through major revisions and around 1/3 of it is new material. Anyway, if you feel like accessing the draft in exchange for a promise to let me know what you think by June 2012, please get in touch.
Synopsis. Stanisław  Leśniewski (1886--1939), a Polish logician, a representative of the Lvov-Warsaw school and Alfred Tarski's PhD supervisor, developed his philosophically (and nominalistically) motivated foundations of mathematics as an alternative to the system of Principia Mathematica. He constructed three systems: a generalized propositional calculus called Protothetic, his own (higher-order) logic of predication dubbed Ontology,  and a theory of parthood known as Mereology. This books presents, explicates and critically discusses Leśniewski's work and some more recent developments stemming from it. In contrast to the technical literature of the subject, this book is accessible to philosophy students with basic logic training.

Friday, November 18, 2011

The TE paper now available in Synthese

The draft about logic and platonic thought experiments I talked about before has now been superseded by the final version now available (open access) in Synthese. Here

(By the way, Springer normally charges around 2000 Euro for publishing open access. But if at least one of your affiliations is with a Polish academic institution, it's free. I'm not sure why.)

Saturday, October 15, 2011

A conference on Mereology, spring 2012

Janusz Czelakowski, Tomasz Połacik and Marcin Selinger are organizing the 17th conference on Applications of Logic in Philosophy and Foundations of Mathematics. This edition is about Mereology. It's planned for May 7-11, 2012 and it will take place in a very pleasant mountain town in Poland. More details here.

Friday, August 19, 2011

Yablo's paradox: Bueno & Colyvan vs. Ketland

I am looking at the debate between Jeff Ketland on one hand and Bueno & Colyvan on the other, pertaining to the consistency of Yablo sentences. (See HERE for a wider list of references compiled by Jeff. While I think B&C have a few brilliant remarks, I also think their arguments are sometimes unsatisfactory, and I want to comment on these points. This is all rather sketchy, and presumably half-baked.

1. The provability of the existence of the Yablo sequence. 

B&C complain:
"it's not we know that the Yablo list exists. Priest's argument seems to presuppose the existence of the list, in order to establish that to derive a contradiction from the latter, a fixed-point construction is required." (Analysis 63, 156).
For me, it seems like a very unusal approach to the diagonal lemma (DL). DL is an existence claim: it says that given a certain formula, another formula satisfying a certain equivalence exists. My impression is that DL is used by Priest to prove the existence of Yablo sentences, and the fact that certain equivalences hold for the formulas (part of the application of DL under consideration) whose existence has been thus proved is used in the derivation of a contradiction. But let's move on.

Ketland (Analysis 64,166-167) says:
"...if one extends the language L of arithmetic by adding a 1-place prediate symbol T to form a language LT, the uniform diagonalization theorem shows that there is an LT-formula Yx() such that the Uniform Fixed-Point Yablo Principle is provable in PA."
Some notational details aside, he means that PA proves 
(x)[Y(x) <-> (y>x)~T(Y(y))]  
He continues:
"To stress, it is a theorem of mathematical logic that the Yablo list exists."
B&C (in an unpublished (?) ms "Yablo's paradox rides again", available HERE)  pick on this sentence by saying:
"...what is Ketland taking mathematical logic to be? Given the context, and his argument that the existence...follows from PA supplemented with a truth predicate, presumably he considers that combination to amount to mathematical logic. But, clearly, this combination is quite a bit more  than mathematical logic - unless one assumes, without argument, the truth of logicism!"
Okay, sure, Ketland was probably a bit hasty when he said the existence of Yablo sequence is a theorem of logic: for it is conditional upon PA extended with a truth predicate. (Although, this of course hinges on what you mean by mathematical logic). Anyway, his main point holds: the existence of the Yablo sequence is not assumed without argument, it is proven within a relatively uncontroversial theory which no side of the debate wants to reject (and indeed if you look at the quote I gave, Ketland's statement is conditional).

B&C continue: "Still, if the list is a theorem of Peano arithmetic suppelemented with a truth predicate, isn't that enough? Not really." The problem is that they didn't say what it is not enough FOR. I think it is definitely enough to respond to their qualm about the existence of the sequence. 

To be fair, we find out what it is not supposed to be sufficient for soon enough. B&C, on the assumption that the use of DL is a mark of self-reference, point out that using DL for proving the existence of the sequence doesn't show the sequence to be self-referential, at least unless you also show that any way of specifying the sequence has to use a fixed point construction. (They move on to sketching a way of specifying the sequence without a fixed point; evaluate it on your own). 

2. The consistency of the Yablo sequence.

One of Ketland's main claims is that the sequence isn't really inconsistent: rather, it's omega-inconsistent (for a proof see his Sythese paper from 2005, vol. 145, 295-302). B&C attack this claim. They focus on an argument given by Ketland earlier (Analysis 64, 165-172), where he relied on the compactness theorem and used the claim that every finite subset of Yablo sentences is consistent. They oppose:
"There is a very interesting mistake in this argument, though. While it is true that each finite subset of Yablo sentences is not paradoxical, it is not true that each subset is satisfiable. Consider, for example, the subset consisting of just the first two sentences ... Since there is no s_k for k>2 in this set, s_2 is vacuously true. But this means that s_1 is straightforwardly false. That is, [the set] is not satisfiable. [but it] is not paradoxical ...: there is a consistent valuation function." (B&C, "Yablo's paradox rides again")
I'm not convinced. The fact that no S_k with k>2 is in the set doesn't make s_2 vacuously true: such s_{k}s still exist (well, we can prove that on uncontroversial assumptions, see point 1), only NOT IN THIS SET. Just like the fact that you're temporarily considering the set of even numbers doesn't make the odd numbers come out of existence.  

3. The inconsistency of Yablo sequence.

B&C still claim that the sequence is inconsistent. They start with assuming each instance of a Yablo biconditional:
Y_i <-> (m>i) ~T(Y_m)
and each instance of the local disquotation (for i in omega):
T(Y_i) <-> Y_i
This gives them each instance of:
T(Y_i)<-> (m>i)~T(Y_m)
Then, the crucial move occurs: they say that "since each instance was arbitrary", the uniform homogeneus Yablo Principle is derivable:
(n)[T(Y_n)<-> (m>n)~T(Y_m)]
And this one is clearly inconsistent (see Ketland 2005 for a proof).

But here is a problem: the uniform homogeneus Yablo principle doesn't follow directly from the local instances. It follows on the assumption that the model of arithmetic you're looking at is standard (and indeed, the instances weren't completely arbitrary: they were restricted to standard numerals). But this is exactly what Ketland's 2005 consistency proof hinges on: the Yablo sequence can be satisfied in a non-standard model!

4. Non-standard numbers

In their 2003 (alleged) derivation of contradiction B&C move from the untruth of s_1 to the truth of some sentence s_k with k>1. In his 2004 response Ketland claims from the untruth of s_1 that it doesn't deductively follow that "there is at least one true sentence in the Yablo list" (2004:171). B&C insist that the negation of s_1 simply is equivalent to the claim that for some k>1 s_k is true (and they're right), from which they infer that there is a true sentence on the Yablo list. B&C rejoin (ms): 
"if Ketland is right, and k might stand for a non-standard natural number, so what? The nonstandard models of the natural numbers are also well-ordered, and that's all that matters here. There will still be a first true sentence in the list independent of whether the index ranges over standard or non-standard numbers. And our argument goes through either way."
Well, Ketland's response is slightly misleading, for it doesn't emphasize the (possible) difference between being on the Yablo list, and being a Yablo sentence. On one interpretation (interpretation A) the Yablo list is the list of Yablo sentences obtained for natural numbers as indices. In another (B) the Yablo list is just the set of all Yablo sentences, ordered by the less-then relation, no matter whether the numbers involved are standard or not. 

Now, on interpretation (A) Ketland is right: just because for some k>1 s_k is true, it doesn't follow that some sentence on Yablo list is true, because k, the witness, might be non-standard. 

But on interpetation (B), Ketland's response is inadequate, and B&C's point partially holds. It does follow that for some k>1 s_k is true.

This doesn't mean they can derive a contradiction, though! In their reasoning they relied on the local Yablo disquotation principle which for n in omega says:
T(Y_n) <-> Y_n
The thing is, if k is non-standard, the local disquotation principle doesn't apply to it, and this blocks the inference.

Of course, in response you might start with the uniform Yablo disquotation principle:
(x)[T(Y(x)) <-> Y(x)]
and this will allow you to deduce a contradiction (Ketland 2005 admits that much).

Anyway, Ketland's main point still holds: the sequence BY ITSELF is consistent, as long as you don't use uniform Yablo fixed-point principle, or uniform Yablo disquotation principle. Whether we should go for the uniform or only for the local is a separate question I don't intend to tackle here.

Tuesday, April 26, 2011

The myth-busting paper on definitions, final version

Busting a myth about Lesniewski and definitions, a paper I wrote with Severi Hamari (I talked about it a while ago) is now forthcoming in History and Philosophy of Logic. I also posted an updated version of the paper on my academia profile

  • One important change is that the discussion of  Nemesszeghys's views in section 9 contained a serious error. I missed it proofreading the first 20 times, but managed to catch this right before submition.
  • Minor modifications are here and there.
  • Another change is that we streamlined the references and provided more bibliographical details.
Abstract A theory of definitions which places the eliminability and conservativeness requirements on definitions is usually called the standard theory. We examine a persistent myth which credits this theory to S. Lesniewski, a Polish logician. After a brief survey of its origins, we show that the myth is highly dubious. First, no place in Lesniewski’s published or unpublished work is known where the standard conditions are discussed. Second, Lesniewski’s own logical theories allow for creative definitions. Third, Lesniewski’s celebrated ‘rules of definition’ lay merely syntactical restrictions on the form of definitions: they do not provide definitions with such meta-theoretical requirements as eliminability or conservativeness. On the positive side, we point out that among the Polish logicians, in the 1920s and 30s, a study of these meta-theoretical conditions is more readily found in the works of J. Lukasiewicz and K. Ajdukiewicz.

Tuesday, April 12, 2011

An Episode in the History of Polish Art and Logic

Stanisław Ignacy Witkiewicz (aka Witkacy) (1885 – 1939) was a Polish painter, playwright, novelist and philosopher. He did a variety of things which not too many of us would describe as the usual day in the office: he wrote a manual (well, sort of) on using drugs (aka Unwashed Souls) and actually used drugs while painting. Some of my favorite pieces are spread over this post (HT to, where you can find more of them)).

Interestingly, Witkacy interacted with some of the Polish logicians, including Leon Chwistek (known for his work on type theory) and Alfred Tarski. He even painted portraits of both of them. Chwistek:

Tarski's portrait seems to be in Tarski's home in Berkeley (correct me if I'm wrong). But Victor W. Marek has posted a picture of it (here, 11th picture from the top).

(Nota bene, the above is not a portrait of Tarski! You actually have to follow the link to see it! ;))
Now, to the point. Witkacy  read Tarski's 1933 and scribbled his comments on the margins (sorry, both things in Polish). The Warsaw University Library has recently made a copy available online. The comments are pretty cute, some of them quite funny, but my guess is Witkacy at some point got lost in the text. It's not quite clear whether he was on drugs while reading. My favorite page contains Witkacy's illustration of the concept of metalanguage (see here for a better version):

(Some other marginalia are also available, see the list here)

Monday, April 11, 2011

Reading "Language, sense and nonsense"

For a while now, I've been forcing myself to read Language, Sense and Nonsense. A Critical Investigation into Modern Theories of Language by G. P. Baker and P.M.S. Hacker. I started reading it, because the main theses sounded interesting and controversial: the authors clam that modern philosophy of language and linguistics are based on false identification of main problems and severe misconceptions.

The book is meant to show that "the most of what goes by the name of `theories of meaning' or `scientific study of language' needs not remedial readjustment, but wholesale abandonment." [x]  The authors criticize  modern linguistics and philosophy of language  on account of assuming that (i) any natural language has a deep structure of a "(correct) formal, function-theoretic, logical calculus" [2], (ii) the task of the philosophy of language is to construct a theory of its meaning which would elicit "the underlying principles of construction of any language in virtue of which we can construct and understand the infinite array of meaningful sentences" [3], (iii) that our ability to understand an infinite assembly of sentences shows the existence of compositional meaning principles, (iv) that there is a sharp distinction between syntax, semantics and pragmatism [6]. Why did I have to force myself? Well, for one, they are pointlessly and rhetorically rude:
If the true Philosopher's Stone is at last almost within our reach, if a theory of meaning, once properly constructed, holds within it the key to the great problems of philosophy, if grammar holds the key to the structure of the human mind, then indeed this wonderful insight and advance must be hailed with fanfares. And philosophers, together with theoretical linguists, must bend their wills to a united effort to grasp this treasure. then they may go on to explain the deep mysteries of our ability to understand new sentences, to discover what really exists (e.g. whether events are essential for our `ontology'), to reveal what is innately known to the human mind, to uncover the true logical form of our thoughts and the essential nature of our understanding. But the Last Trumpet has been blown with tiresome regularity in the history of philosophy, and false prophets have been legion. If the promises held out by the possibility of constructing a theory of meaning are false promises, and if the very idea of such a theory of meaning as is currently envisaged is incoherent, then this too must be proclaimed, the incoherences made clear and the hopes dashed. For then, far from being at last upon the true path of science, theorists are merely pursuing yet another monstrous chimera.
[W]e shall focus upon just those topics which are introduced in most theories of meaning with the barest of explanation, taken to be altogether perspicuous and treated with nonchalance. We shall probe the seemingly clear notion of the truth-conditions of a sentence, which is commonly taken to be the key to any cogent semantic theory. We shall place pressure upon the apparently obvious distinction, within every sentence, between its descriptive content [...] and its force. We shall test the soundness of the supposition that a language is a system, a calculus consisting of a network of hidden rules tacitly employed whenever we speak or understand what is spoken. And we shal examine whether the question of how it is possible to understand setences never heard before really is as deep as it is commonly taken to be. In general we shall resist by argument the theorists' habit of frog-marching the neophyte straight to a ceremony of initiation ito the full mysteries of the modern science of language. We shall unmask their conceptual conjuring tricks and break the mesmerizing force of their incantations by critical questioning. Our method will be the clarification of concepts. [11-12]
A few other samples of their style:
The issues we examine are important ... The misconceptions we identify ramify widely, contributing greatly to the barren mythology of late twentieth century culture. Hence this book is written with more polemical passion than is common in the typical reserved and detached forms of academic philosophy. For this we make no apology. [x]
 It was rather unclear to me how the popularity of the criticized view justifies the unusually polemical style. In this case, people being criticized are more likely to be convinced by cold, calculated arguments rather than by rhetorical and emotional ramblings.

Another sample:
[W]e [...] demonstrate a readiness to demolish large parts of what pass for significant modern intellectual achievements. But our ultimate purpose is not to persuade linguists or philosophers that their theories are false... It is rather to suggest that their endeavours are futile because pointless and misconceived. [13].
Sometimes, you can forgive the style, if the arguments are good. But these really aren't. Most of them are superficial and hasty straw-man strategies which are unlikely to convince anyone who doesn't buy into treating insults as real arguments. 

For a while, I was thinking about writing a short paper pointing out what went wrong in the book. But when I gave it some thought, I decided that it really wouldn't be short, and that since the book dates back to 1984, I wouldn't be addressing a really new and living concern either. Then, I also discovered a review in Mind (New Series, Vol. 94, No. 374 (Apr., 1985), pp. 307-310) by Jane Heal, and it turns out that it pretty much sums up my views about the book (below, the key fragment):
The  first thing  to be said about these books is that they are extremely aggressive  in  tone.  The  violence  and  frequency  of  pejorative  terms  is  striking.  Phrases like 'grotesque  conceptual  confusions'  (LSN  p.  I2),  'wastelands of the intellect' (LSN p.  13),  or 'frantic attempts to justify the bogus demands of a misguided theory' (LSN p. 94) are to be found peppered throughout the books. Where they do not condemn outright Baker and Hacker proceed by sneer, by loaded rhetorical question and by self-congratulation ('The several criticisms add up to a devastating indictment' LSN p. 24I).
This  might be  no more than robust intellectual knockabout. If  one  thinks something is nonsense one should be entitled to say so. But Baker and Hacker go  further and cast  aspersions not  only  on  the  intelligence but  also on  the intellectual integrity of  their  opponents. They  are  accused of  'intellectual  opportunism' (LSN  p.  9),  'conceptual conjuring tricks' (LSN  p.  12),  'mystery mongering' (LSN p.  I9),  and so on. One might say of  such abuse that it  is  totally out of  place in  a work with serious pretensions to philosophical scholarship. One could remark further that the repugnance it excites will lose Baker and Hacker what small chance they had of gaining converts. But from the point of view of  the philosophical enterprise neither of these is the gravest charge. The real danger is that hostility will prevent them approaching their subject in  such a spirit that they can feel the genuine attractions of the opposing views and hence put themselves in a position to see the real weaknesses there, if any; some level of sympathetic understanding is a prerequisite for truly efficacious demolition. Have Baker and Hacker avoided this danger? Have they, despite all the faults, something important to say?
The general problem area they have identified in both books is important; the correct interpretation of Wittgenstein's views on rules and meaning and their relevance to  modern theoretical approaches to  language are topics that many philosophers rightly want to see debated. But the detailed handling of the issues by Baker and Hacker is constantly disappointing. In  effect they discharge an enormous blunderbuss of arguments in the direction of their opponents. Most of the missiles spray out wildly into space and even those that are on target do not wound fatally.

Wednesday, March 30, 2011

Munich group and blog

Probably you all know about the new research group at Ludwig-Maximilians Universitat in Munich (Munich Center for Mathematical Philosophy). More info about the group can be found in the most recent volume of The Reasoner (vol 5 no 4, April 2011). 

What you might not know yet is that the group started a new blog, M-Phi. (The administrators kindly invited me to contribute - since I might have hard time matching the high level of contributions it might take a while before I actually post anything there).

Friday, March 18, 2011

Postdoc in logic and phil of sci, Calgary

The Department of Philosophy at the University of Calgary invites applications for a one-year postdoctoral fellowship starting on September 1, 2011. The area of specialization is logic or the philosophy of science. The fellow will be expected to have a well-defined research project, teach one course in the area of specialization, and participate in the research activities of the Department. All requirements for the PhD must have been completed by the starting date and no earlier than September 2007. The stipend is $50,000 Canadian per year. Applications will be accepted until April 15, 2011 or until the position is filled. Details.

Monday, March 7, 2011

"Platonic" thought experiments: how on earth?

I have posted a draft of the TE paper online now (the title is the same as the title of this post). Here.
Abstract. Brown (1991a,b, 2004, 2008) and Bishop (1999) argue that thought experiments (TE) in science  cannot be arguments and cannot  even be represented by arguments. They rest their case on examples of TEs which either proceed through a contradiction to each a positive resolution (Brown calls such  TEs "platonic") or are used by different people with opposite results. This, supposedly, makes it impossible to represent them as arguments for logical reasons: there is no logic that can adequately model such phenomena. (Brown further argues that this being the case, "platonic" TEs provide us with irreducible insight into the abstract realm of laws of nature). I argue against  this approach by describing how "platonic" TEs can be modeled within the logical framework of adaptive proofs for prioritized consequence operations. To show how this mundane apparatus works, I use it to reconstruct one of the key examples used by Brown, Galileo's TE involving falling bodies. I also address Bishop's qualms about the clock-in-the-box TE which Einstein and Bohr employed when they disagreed about the uncertainty principle.

Many thanks to Christian Strasser,  Frederik Van De Putte, Erik Weber, Rawad Skaff and Joke Meheus for reading and discussing with me earlier versions of this manuscript and to all the people who discussed this topic with me: Graham Priest, Diderik Batens, Anouk Barberousse, Peter Simons, Margherita Arcangeli, Gillman Payette, and the audiences in Geneva, Paris and Ghent, where I gave talks based on this material.

Monday, February 28, 2011

Friday, February 25, 2011

Teddy bears, guns and modal logic

It's the time of the year when a new semester starts in Poland and I'm in Gdansk for a while (it's annoyingly and unusually cold, it feels like Calgary for some reason -  seems I haven't escaped after all. Damn you, global warming!). Anyway, one of the courses I'm teaching is non-classical logic and I'm using Graham Priest's awesome book. If you've ever taught modal logics, you probably observed that it's sometimes difficult to get the students to remember which normal modal logic is related to which properties of the accessibility relation. Here's a trick I invented last year, feel free to use it (just give credit where it's due).

First off, Priest uses Greek letters to denote the main properties of the accessibility relation: 

  •  \rho stands for reflexivity
  •  \sigma stands for symmetricity
  •  \tau stands ofr transitivity
  •  \eta stands for extendability

The main logics worth remembering in a basic course are T, D, B, S4 and S5:

- T is determined by the class of \rho-models
- D is determined by the class of \eta-models
- B is determined by the class of \rho\sigma-models
- S4 is determined by the class of \rho\tau-models
- S5 is determined by the class of \rho\sigma\tau-models

Here's a mnemotechnic to help people remember this.

First, you want people to remember the ordering of the logics:
T, D, B, S4, S5
instead, (make them) memorize:
TeDdy Bear with 45S
The coding here is quite obvious.

Next, you want people to remember the ordering:

\rho, \eta, (\rho \sigma), (\rho \tau), \(rho \sigma \tau)

instead,  (make them) memorize:
R stands for \rho, E stands for \eta, S stands for \sigma, T stands for \tau.

Thus, the matching of modal logics with properties of the accessibility relations is encoded by:

a Teddy bear with  45s rests.
Two problems:
  • You still have to remember that the first two logics are just \rho and \eta, and that the remaning one involve \rho
  • You have to remember to repeat "st", because it stands for "first \sigma, then \tau, then \sigma and \tau together".
Now, get people to imagine a teddy bear with a gun (preferably with a 45).

One option is to use this (source):

Another is to use this:

Or you can use this:

(I was also thinking of taking a picture of a teddy bear with a 45, I have everything I need apart from a teddy bear and a 45.)

Of course, the key sentence is not the best English phrase (I think there are at least some fragments of Yeats' poetry which trump its genius; not so sure about J. Conrad though), and there are some weak points (it doesn't extend easily to non-normal modal logics, you have to remember at least two extra assumptions I mentioned, and so on). So, if you have a better mnemotechnic, please share.

Wednesday, February 16, 2011

Reichenbach on the philosophical insignificance of G's second incompleteness theorem

Nowadays, one standard answer to various exaggerated claims about Gödel's second incompleteness theorem (the unprovability of consistency) is that even if an interesting mathematical theory could prove its own consistency, this wouldn't help us much because inconsistent theories also prove their own consistency, (so we would still have no idea whether the theory is consistent). (For instance, I think this sort of remarks, without further references, can be found in Franzen's and Smith's books, but I don't have them handy now).
In a somewhat uninspired moment of mine, I wondered why this rather straightforward observation didn't get through to the wider philosophical audience earlier and when it was formulated. The earliest mention I run into so far (although, it's not like I spent days browsing stuff systematically) is in Reichenbach's 1948 unpublished lecture notes (which, by now, have been published in 1978). The fragment comes from the first volume:  
This would mean that the proof of consistency of the language L could be given within L. A simple analysis shows that this would not improve the situation, since in this case our proof of consistency of L would be of value only if we were sure that L is consistent. In case L were not consistent, we could also deduce the statement of the consistency of L, with the qualification that then the negation of the statement were deducible too. Thus if the consistency of L were deduced within L, this fact would not prove the consistency of L.
So, one reason might be that the remark occurred in lecture notes which were unpublished for a while (by the way, if you know of Reichenbach's point being published earlier, I'd appreciate a reference), and even when they were published, it was in a two-volume set of Reichenbach's papers, which very few non-specialist would buy (or grab from a library shelf, for that matter).
But I don't think this is the whole story. The point is simple enough and certainly came to minds of many bright people who gave the issue proper consideration. The reason why this is often omitted by (hack?) philosophers is not that they are too dumb to get this point (well, at least some of them aren't). Rather it is that wishful thinking sometimes prevents people from considering objections to their view. The perspective of reaching strong philosophical conclusions by relying on some fancy mathematical apparatus, which (they think) would lend splendor to the philosophical claims themselves, is quite tempting.
Another reason is that Reichenbach's point applies to cases where one wants to claim that mathematics is somehow epistemologically flawed and that there is no mathematical certainty because mathematics cannot prove its own consistency. (... and if real mathematical knowledge is unattainable, then no knowledge is, science is useless, blah blah, blah...). It doesn't help much when one wants to suggest that since we, humans, can establish consistency claims, and mathematical "devices" cannot, our minds are not computing machines. In this argument, it seems that no deep epistemological considerations of the former sort are needed.
Incidentally, it seems that despite the fact that I've seen some philosophers using both strategies in one passage (or breath) both stances are incompatible. Either you say that mathematics is unreliable and then you cannot rely on it in an argument against computational theories of mind, or you rely on mathematics, give an argument in philosophy of mind, but cannot simultaneously deny the reliability of mathematics. (By the way, (1) even if you do the latter, you admit that mathematics can be done reliably by humans and the fact that a mathematical theory cannot prove its consistency has no bearing on whether humans can use it sensibly, (2) the sketched argument against computational theories of mind also doesn't work, but that's a whole different story). 

Monday, January 31, 2011

CLMPS registration - what's up with that?

Like some of you I had to spend some time submitting my stuff to one of the biggest logic events these years, CLMPS. What I found slightly surprising is how complicated the submission procedure is.

1. Before you submit your materials you have to actually register as a participant. This seems slightly impractical, especially since some of people probably decide whether they will participate only when they find out whether their proposal has been accepted.

2. When you register, you have to fill a pretty detailed form which contains all sorts of unusual questions:

  • you're supposed to mark your function at a conference (for instance you may choose "participant without contributed paper" or "participant with contributed paper). While some people will know their status, some people will have no idea before the submission and acceptance notification.
  • They also ask you whether you plan to attend the opening ceremony and the welcome banquet on July 19,
  • which concert you prefer to attend on July 23,
  • which trip you will be in a mood for on July 24,
  • how many people will accompany you at the conference,
  • what your exact arrival and departure dates are, and
  • which kind of hotel and what kind of room you want to stay in.
I mean, seriously, what happened to spontaneity? I have no idea what the answer to all those questions is in my case. Ah, but there is a note in case I change my mind:

Please note that demands for modification of your registration data after payment must be submitted to the LOC in writing by mail, fax or e-mail. No modification will be possible through the registration form. Also note that no modification will be possible after May 31st, 2011.
I'm inclined to think that at some point the organizers will be swamped receiving all kinds of messages from people who will decide to change their plans and, say, go to a jazz concert rather than a classical concert, or to come with a partner, or to come one day later, or stay in a different room, and so on. I know the conference will have a lot of participants and the organizers need some head start on getting things ready, but isn't this a bit too much? And won't it, eventually, devour more of their time, given the need to deal with later changes case-by-case? Why not move the form modification deadline to May 31? Has anyone had any experience with earlier events in this series and the submission procedures? Has it always been like that?

Tuesday, January 25, 2011

Monday, January 17, 2011

Ontological Proofs Today, Bydgoszcz

Mirosław Szatkowski, Anthony Anderson, Jonathan Lowe, Dariusz Łukasiewicz, Richard Swinburne and Daniel von Wachter are putting together a conference in Bydgoszcz (Poland), titled Ontological Proofs Today (6-8 Sept 2011).

Alas, parts of the details are available only in Polish and the information available isn't too specific. But I would expect them to update the info some time soon. In case you have more urgent queries, I guess you can always drop the organizers a line.

Thursday, January 13, 2011

Vickers and the criterion of arithmetical truth

I'm reading The problem of induction by John Vickers. The entry is quite comprehensive and enjoyable. There is a thing which seems a bit hasty, though. Vickers, at some point, makes a distinction between the problem of finding a method for distinguishing reliable inductive habits and the problem of saying what the difference between reliable and unreliable inductive habits is. While the distinction is quite sensible, I'm not sure what to think about the example Vickers uses to clarify it (section 2):
The distinction can be illustrated in the parallel case of arithmetic. The by now classic incompleteness results of the last century show that the epistemological problem for first-order arithmetic is insoluble; that there can be no method, in a quite clear sense of that term, for distinguishing the truths from the falsehoods of first-order arithmetic. But the metaphysical problem for arithmetic has a clear and correct solution: the truths of first-order arithmetic are precisely the sentences that are true in all arithmetic models.
A few remarks come to mind:
  • Prima facie, the incompleteness of first order arithmetic (let's fix our attention on first-order true arithmetic TA1) would show that the epistemological problem of finding a method of deciding whether an arithmetical claim is true is unsolvable only if the only way to find out whether an arithmetical sentence is true would have to employ a complete axiomatization of true arithmetical sentences.
  • Another way to put his is that I'm not sure if incompleteness itself entails the lack of epistemological criterion; after all there are negation-incomplete and decidable theories (think of the set of theorems of classical propositional logic). Yes, negation-completeness entails decidability, but the converse doesn't hold.
  • For instance, if second-order model-theoretic consequence relation were decidable, we would have a decision method for first-order arithmetical truth despite the incompleteness theorem. It would suffice to check if a first-order arithmetical claim is a model-theoretic consequence of the axioms of second-order Peano arithmetic (which, modulo second-order semantic consequence, is complete).
  • Perhaps one could circumvent the above concern by saying that if there were any epistemological criterion for arithmetical truth, then the true arithmetic would be recursively axiomatizable. This, alas, isn't immediately obvious and is quite sensitive to what your background epistemology is. In a scenario where one can have a direct contact with God (assume there is one) and God correctly responds to any queries we have about arithmetical truth, we do have some epistemological criterion for arithmetical truth, but this fact alone tells us nothing about the axiomatizability of arithmetical truth.
  • Now, the set of first-order arithmetical truths in fact is undecidable. But this follows rather from the Church-Turing thesis, the undefinability of truth and the arithmetical definability of all recursive sets. If the set of FO arithmetical truths were recursive, it would be definable. But Tarski's theorem already states that it isn't. So the set of FO arithmetical truths isn't recursive, so by the Church-Turing thesis it is not decidable.
  • Still, there is a gap between the claim that the set of arithmetical truths is undecidable and the claim that the epistemological problem for first-order arithmetics is unsolvable. This follows only if every epistemologically respectable criterion of arithmetical truth has to be a decision method. While this seems quite likely to me, there is still some maneuvering space for all those people/crackpots who disagree.
  • It seems there is more to epistemological concerns about arithmetic than just incompleteness. If the incompleteness theorem failed, this would show that there are complete, sufficiently strong and consistent arithmetical theories. This by itself doesn't give us a full-blown epistemological criterion for arithmetical truth, because we also need some epistemological account of why the axioms of a given complete theory are true (after all, there are complete and false theories).
  • Another worry is that Vickers defines FO arithmetical truths as those claims that hold in "all arithmetic models". Presumably he means all models of first-order arithmetic. The problem with this definition is that there are non-standard FO arithmetical models. For instance, I like to think that the Godel sentence for Peano Arithmetic is true. Yet, there is a model in which it is false. So Vickers's definition would exclude the Godel sentence from arithmetical truths. Also, since there are models in which the negation of the Godel sentence for PA is false, it also wouldn't be included in the set of arithmetical truths on Vickers's definition, thus yielding the set of arithmetical truths incomplete.