Friday, 23 December 2011

Validity as a Primitive

Earlier this year, there was a bit of discussion on M-Phi about a nice forthcoming (in Journal of Philosophy) paper by Jc Beall and my MCMP colleague Julien Murzi, "Two Flavors of Curry's Paradox".

I've now written a shortish paper (11 pages) summarizing the main idea I had about it - i.e., to take validity as a primitive notion and add it to Peano arithmetic governed by a couple of reasonable principles:
if $\phi$ is valid, then $\phi$
if $\phi \rightarrow \theta$ is valid, then if $\phi$ is valid, then $\theta$ is valid
and a restricted necessitation rule, saying
if $\phi$ is logically derivable, then infer "$\phi$ is valid"
showing that one can get a consistent (indeed conservative) extension. I also add a truth predicate, along with "if $\phi$ is true, then $\phi$", "if $\phi \rightarrow \theta$ is true, then if $\phi$ is true, then $\theta$ is true", along with the principle that validities are true. This is also conservative over PA.

The basic idea of the conservativeness proof is quite simple: just replace "$\phi$ is valid" (and "$\phi$ is true") by "$\phi$ is a theorem of pure logic (in the relevant language)". Then everything comes out as a theorem of PA. Everything is classical.

The paper, "Validity as a Primitive", is here (at academia.edu) and will appear in Analysis.

Thursday, 22 December 2011

A Revolution in Mathematics?

Via the FOM list, I link to a forthcoming (Jan 2012) article by Frank Quinn in Notices of the AMS, entitled "A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today". Rather interesting.

My main comment would be that, although Quinn is defending mathematical logic, the rigour it has brought to understanding mathematical reasoning, and in particular the work of Hilbert and Gödel, Quinn does not mention the semantic definition of truth used in mathematical logic, and which Gödel recognized and which played a role in his discovery of the incompleteness results. (Solomon Feferman's 1984 article "Kurt Gödel: Conviction and Caution", which is also Ch. 7 of Feferman's In The Light of Logic, has a discussion of this.)

I'm not sure of this, but I think Quinn's own view is some kind of formalism, as, more specifically, Quinn writes,
Ironically, it had the same practical consequences because it established “impossible to contradict” as the precise mathematical meaning of “true”.
The suggestion that "$\phi$ is true" be analysed as "$\phi$ is consistent" is not workable, because "$\phi$ is consistent" is not a truth predicate in the technical sense: it does not commute with the connectives and quantifiers. For example, "$\phi$ is consistent" and "$\theta$ is consistent" do not jointly imply "$\phi \wedge \theta$ is consistent". Consistency of a formula $\phi$ or a (recursively axiomatizable) theory $T$ is expressible using a $\Pi_1$-formula ("there is no derivation from $\phi$ (or $T$) ending with $\bot$"), whereas truth of a formula (or soundness of a theory) usually requires a higher level (in arithmetic, for $n>0$, the truth predicate for $\Pi_n$ statements is itself a $\Pi_n$-formula). Consistency and truth are not the same thing. A mathematical anti-realist is better off just saying straight out that the relevant theories are not true: i.e., they're useful-and-consistent-but-not-true. Quinn also doesn't mention the status of mixed sentences, such as "the mass-in-kg of $x$ is $3.21$", "the divergence of the magnetic field $\mathbf{B}$ is zero everywhere", or "the gauge group of the electromagnetic field is $U(1)$", etc., which are at the centre of science. Are these merely consistent? How is truth defined for them?

Suppose I define truth for sentences in the language of arithmetic in the usual way, as follows:
(i) An equation $t = u$ is true iff the values of $t$ and $u$ are identical.
(ii) A negation $\neg \phi$ is true iff $\phi$ is not true.
(iii) A conjunction $\phi \wedge \theta$ is true iff both $\phi$ and $\theta$ are true.
(iv) A universal quantification $\forall x \phi$ is true iff, for all $n$, $\phi(x/\underline{n})$ is true.
As Paul McCartney put it, what's wrong with that? (Ok, aside from the conceivably gratuitous mention of Paul McCartney ...)

Tuesday, 20 December 2011

Festschrift for Martin Stokhof

(Cross-posted at New APPS)

In my years working at the ILLC in Amsterdam (2007-2011), I was fortunate to have Martin Stokhof as my mentor (I always referred to him as my ‘boss’, which he found very strange). Martin is one of the best philosophers and nicest persons I know, and as he turned 60 last year, together with Jaap van der Does, we decided to put together a Festschrift to honor his accomplishments. Yesterday, the official launch of the Festschrift took place, during the Amsterdam Colloquium; it is a web-based-only, open source volume, with a unique printed version (for Martin himself, naturally).

Martin is perhaps best known for his work on the semantics of questions and on dynamic predicate logic, both in collaboration with Jeroen Groenendijk. But Martin is deep down a philosopher much more than a formal semanticist, and in recent years his focus has been predominantly on philosophical topics, in particular Wittgenstein. He has a wonderful book on the early Wittgenstein, World and Life as One: Ethics and Ontology in Wittgenstein's Early Thought (Stanford University Press, 2002). Moreover, he has written insightful and rather critical articles on the philosophical foundations of formal semantics; more generally, he has focused on the methodology of using formal tools for investigating language as a whole, and on the connections between formal and natural languages (I like very much in particular 'Hand or hammer? On formal and natural languages in semantics'). As well put by Barbara Partee in her contribution,

Martin has been addressing [...] foundational problems increasingly in recent years - I’m not always happy to hear his conclusions, but his work is important and valuable, and since he is a semantics insider, he can write critically about formal semantics in a way that semanticists can and must take seriously.

The Festschrift reflects Martin’s wide interests, and contains a range of papers worth reading on their own (i.e. even by those who do not have special connections with his work). A few of them are in Dutch, but the wide majority is in English. By the way, Johan van Benthem’s article was my main source of inspiration for a blog post of a few months back, and now I can finally disclose the source of the quote there!

Here is the table of contents, and here is the link to the Festschrift again.

Albert Visser: Context Modification in Action

Barbara H. Partee: For Martin - Much to Celebrate!

Catarina Dutilh Novaes: Notations in Logic

Chantal Bax: Guiding or Drilling?

Edgar Andrade-Lotero: From Formalization to Philosophical Reflection

Frank Veltman: An Appendix to DPL

Frans Jacobs: Domheid

Fred Landman: Boolean Pragmatics

Göran Sundholm: Some Coherentist Strands in Wittgenstein's Tractatus

Jaap van der Does: Philosophical Interactions

Jan van Eijck: A Conversation with Wittgenstein

Jeroen Groenendijk: Erotetic Languages and the Inquistive Hierarchy

Johan van Benthem: The Dynamic World of Martin Stokhof

Michiel van Lambalgen: Tractatus on Time

Paul Dekker: De Waarheid over de Waarheid

Rob van der Sandt: De Asceet en de Kerkvader

Robert van Rooij: The Paradoxical Wittgensteinian

Tine Wilde: This is not a Festschrift

Thursday, 8 December 2011

Why confirmation ≠ explanation

Well, more precisely, this is about why confirmation and explanatory power cannot be explicated by the same probabilistic measure within a Bayesian framework. The argument below occurred to me while working on a paper, "A second look at the logic of explanatory power" (with Katya Tentori, now forthcoming in Philosophy of Science), wherein a debate is carried out involving Jonah Schupbach and Jan Sprenger's recent work on "The logic of explanatory power" (Philosophy of science, 78, 2011: 105-127). (By the way, I take the following note to provide some motivation for both Jonah's and Jan's project and the alternative line that Katya and I pursue.)


Why it makes sense to ask

There exists a rather effective rule of thumb to generate a decent measure E(e,h) of the explanatory power of candidate explanans h with regards to explanandum e; that is, take a plausible probabilistic measure of (incremental) confirmation C(h,e), and invert the positions of e and h. (To illustrate, Schupbach and Sprenger, 2011, readily notice that their favorite measure of explanatory power entertains just this relationship with a very interesting measure of confirmation originally defined by Kemeny and Oppenheim, 1952, and then revived by Fitelson, 2005.)
In view of these structural analogies, investigating the connection between a probabilistic measure of explanatory power E(e,h) and of confirmation C(h,e) appears appropriate, if not pressing, as a source of theoretical clarification. An instructive possibility to explore is the statement of outright identity between the two notions, of course with the caveat that the hypothesis h at issue be in some explanatory relation (to be separately defined) with evidence e. Always keeping this proviso in mind, the "reductionist" claim to identity would then be as follows:

Reduction (R).
For any e,h, C(h,e) = E(e,h).

(I'll be assuming throughout that statements are contingent and probabilities are regular.) Notice that, while admittedly daring, R seems to closely fit so-called inference to the best explanation (IBE). After all, for advocates of the IBE view, “observations support the hypothesis precisely because it would explain them” (Lipton, 2000, p. 185, emphasis added). But R is of concern even beyond that, if only because it would arguably trivialize the division of labor between two branches of formal epistemology and philosophy of science that are usually seen as distinct. For short, R is not to be dismissed too quickly, i.e., unless a relevant argument is provided to undermine it. Such an argument is sketched in what follows.

Against the reductionist claim

A compelling principle of a model of explanatory power seems to be that the better hypothesis h would succeed in explaining the occurrence of a state of affairs e the worse it would fail in explaining the occurrence of its complementary ¬e. Formally, such an inverse ordinal correlation between explanatory success and explanatory failure with regards to a pair of complementary statements e and ¬e is spelled out as follows:

Symmetry (S).
For any e,h,h*, E(e,h) >/=/< E(e,h*) iff Ee,h*) >/=/< Ee,h).

On the other hand, consider the following condition concerning confirmation:

Final probability incrementality (F).
For any h,e,e*, C(h,e) >/=/< C(h,e*) iff P(h|e) >/=/< P(h|e*).

Condition F states that, for any given hypothesis h, confirmation is an increasing function of the posterior probability conditional on the evidence at issue – a virtually unchallenged assumption in contemporary probabilistic analyses of confirmation.
Notably, the following can be proven:

Theorem. {S,F} is consistent, but {R,S,F} is not.

Relying on both S and F as sound, the theorem above discredits the reductionist claim to identity R. Apparently, probabilistic confirmation and explanatory power cannot be identified, for the two notions are constrained by genuinely distinct principles on a quite basic level. For all its tempting simplicity, thus, the reductionist thesis R turns out to be a naïve view of the connection between confirmation and explanatory power. This is not to say, of course, that there cannot be other meaningfull and systematic relationships. This does mean, however, that one natural candidate formal rendition of IBE is flawed.

References
Fitelson, B. (2005), "Inductive logic", in S. Sarkar and J. Pfeifer (eds.), Philosophy of Science. An Encyclopedia, Routledge, New York, 2005, 384-393.
Kemeny, J. and Oppenheim, P. (1952), "Degrees of factual support", Philosophy of Science, 19, 307-324.
Lipton, P. (2000), "Inference to the best explanation", in W.H. Newton-Smith (ed.), A Companion to the Philosophy of Science, Blackwell, Malden (MA), 184-193.