July 28, 2007

Sieg's Survey

As Gregory Wheeler notes at Certain Doubts, Wilfried Sieg, Christian Schunn and Melissa Nelson are conducting a survey for those teaching introductory logic courses. It's part of project investigating the effectiveness of different ways of teaching such a course. I just took the survey and it consumed only a few minutes of my time.

July 21, 2007

Oswald was Indeed CIA!

I just stumbled on a couple of interesting declassified documents floating around the internet, and I couldn't resist sharing them with my readers. The first was written by the director of the CIA in 1964 and asserts that Lee (Harvey) Oswald did recon for the agency in the late 50s and reveals in the last paragraph that Oswald's famous 1959 "defection" to Russia was an agency assignment. Oh, that's nice. So when exactly was Oswald NOT on assignment?

The second document includes an attached 1947 memo that links Oswald's assassin Jack Ruby (formerly, Jack Rubenstein) to Congressman Richard Nixon as an informant of some sort. WTF?

Don't know when these docs were declassified or even if they are authentic, but here they are.

UPDATE: Here's a blog devoted to questions about the authenticity of the first document, also known as the McCone-Rowley Document.

Reasoner 1(4)

The fourth issue of the Reasoner is now available online. Here are the contents:

  • Interview with Brendan Larvor
    David Corfield

  • Conceivability, Possibility and Counterexamples
    Anand Jayprakash Vaidya

  • A Counterfactual Account of Essence
    Berit Brogaard and Joe Salerno

  • Knowledge, Truth and Justification in Legal Fact Finding
    Déirdre M. Dwyer

  • The Principle of Agreement
    John L. Pollock

  • A Note on Kripke's Puzzle about Belief
    Christian Constantinescu

July 12, 2007


Today Knowability turns one. We've had just under 20,000 visits, 14,000 uniques (first timers or those returning after an hour), and 8,000 first time visitors (based purely on a cookie). Contributions have been made by guest authors, Bryan Frances and John Greco, and much support was received from Brit over at Lemmings and from other weblog administrators. Thanks guys.

Epistemology and Methodology of Jaakko Hintikka

The Danish Research School in Philosophy History of Ideas and History of Science is sponsoring a Hintikka Symposium
November 16-17, 2007

Roskilde University, Denmark

Invited Speakers

* Adam Didrichsen
* Vincent F. Hendricks
* Jaakko Hintikka
* Stig Andur Pedersen
* Ahti-Veikko Pietarinen
* Robert Stalnaker
* Frederik Stjernfelt
* Tim Williamson

Program and Organizing Committee

* Vincent F. Hendricks
* Frederik Stjernfelt
* Stig Andur Pedersen

July 08, 2007

Remembering Justin D'Arcy Isom

I met Justin when we were both graduate students at the Ohio State University. He received his B.A. in philosophy at the University of Texas. He traveled much of the world alone and his ordinary descriptions of the world around him were often worth more than a thousand words. This was sort of remarkable since Justin had lost his sight at the age of one to a retinal nerve cancer. What I didn't realize at the time was that Justin's overall condition would degenerate. He died on April 23, 2007.

Justin and Neil Tennant devised a braille-like proof system so that Justin could more fully participate in discussions about (intuitionistic) proofs. Neil shares his memory of Justine in the department newsletter, Logos.

July 06, 2007

Second-order logic and Gödel’s first incompleteness theorem (Frances)

That’s a very impressive title for a blog entry, but I promise that things will not get very complicated in what follows. For I am no logician. I’m not even a philosopher of logic. So, I don’t know much about logic. But I know just enough to be confused about something having to do with second-order logic and Gödel’s first incompleteness theorem.

Let L be the formal first-order language with a name for zero and function symbols for the successor function, addition, and multiplication (and no other non-logical symbols). So L is a pretty simple formal language. Let N be the interpretation of L that has as its domain the set of natural numbers {0, 1, 2, …} and assigns zero to the name ‘0’, addition to the function symbol ‘+’, etc. So N is the natural way to interpret L. Arithmetic is the set of sentences of L that are true under N. So, arithmetic is a set of sentences of a formal language, where each sentence is most naturally interpreted as an arithmetic truth. This will be an infinite set.

Arithmetic counts as a theory of L because it is “closed under logical consequence”, which means that any sentence of L that is entailed by the sentences in arithmetic is in arithmetic. Arithmetic contains all its logical consequences; that’s what makes it a “theory”.

Now a theory T in language L is decidable when there is an algorithm for deciding whether any given sentence of L is a theorem (member) of the theory. So a theory T in L is decidable when there is some algorithm that when fed a sentence S of L will tell you in a finite number of steps whether or not S is in T.

Roughly put, a theory T is axiomatizable just in case everything in T is a logical consequence of some nice subset of T. The nice subset of T generates, so to speak, everything in T (just like how the central Euclidean claims generate all the truths of plane geometry). The member sentences of the nice subset are axioms of T.

The subset could be infinite! But even so it has to be nice. That means: there has to be an algorithm that when fed a sentence of L will tell you in a finite number of steps whether or not the sentence is one of the axioms. That is, the set of axioms has to be decidable.

The famous consequence of Gödel’s first incompleteness theorem is that arithmetic isn’t axiomatizable. This is surprising! You’d think it wouldn’t be that hard to write down a nice set of axioms (say, the Peano axioms) for generating all the truths of arithmetic. But it can’t be done.

Now suppose we add second-order quantifiers and variables to L. Now let arithmetic* be the set of first-order and second-order sentences of L that are true under N. Arithmetic* contains everything in arithmetic plus some more arithmetic truths (the second-order ones).

The cool thing, in my opinion, is that arithmetic* is axiomatizable. You can write down one long sentence that generates, through logical consequence, everything in arithmetic*. So when you hear someone say ‘Gödel famously showed that arithmetic isn’t axiomatizable’, you should say to them ‘Wait a damn minute buddy. First-order arithmetic isn’t axiomatizable, but second-order arithmetic is’. So you can write down just one relatively simple sentence that generates all the (first-order) truths of arithmetic—-but it will generate a whole bunch of other arithmetic truths as well, the second-order ones.

I take it that not that many non-logicians and non-philosophers of logic are aware of that result. Hopefully I’m right about it and as a consequence this blog entry is worthwhile.

But now we come to my question: why is it thought to be such a big honking deal that arithmetic isn’t axiomatizable? I take it that people who know about these things think the second-order axiomatization is not terribly significant. But why?

I guess some people are allergic to second-order logic, but I don’t know anything about that issue. What if you think second-order logic is a-okay?

Is the problem connected to this: whereas there is an algorithm that when fed a first-order sentence S of L that’s logically true, will tell you in a finite number of steps that S is indeed logically true, but there is no algorithm that when fed a second-order sentence S of L that’s logically true, will tell you in a finite number of steps that S is indeed logically true? If so, why does that matter so much?