Monday, February 18, 2013

This sentence is refutable (dualizing Goedel)

While the standard informal interpretation of Goedel's sentence:
[G]   I am (/This sentence is) not provable.
is quite well-known, it's dual sentence:
[DG] I am (/This sentence is) refutable.
studied, for instance, by Smullyan, isn't. Yet, pretty much like you can run an argument for incompleteness using the former, you can also run a parallel argument using the latter. Just because it's fun to see how this works (if you're geeky enough), here's how it goes (it's quite easy). 

For simplicity let's assume the background theory is sound (it proves only truths) and sufficiently expressive.

One of the easiest arguments for incompleteness using [G] goes like this.

  1. Suppose [G] is false. Then (because of what it says) it is provable, which contradicts soundness. So [G] is true.
  2. If [G] is true, it is not provable, so we have the first half of incompleteness.
  3. Given that [G] is true, its negation is false.
  4. If ~[G] is false, it cannot be provable, given soundness. This is the second half of incompleteness.
An analogous argument for [DG] is:

  1. Suppose [DG] is true. Then, by what it says, its negation is provable.
  2. If ~[DG] is provable, it is true (by soundness), so [DG] is false. 
  3. Assuming [DG] is true we inferred that it is false. So, unconditionally, [DG] is false.
  4. If [DG] is false, then (by soundness) it is not provable.
  5. If [DG] is false, ~[DG] is true, which means [DG] is not refutable. That is, ~[DG] is not provable either. 

Friday, February 15, 2013

CFP: Entia et Nomina III


The third conference within a series of logico-philosophical workshops I've been organizing is coming up. It will take place in Gdańsk, Poland (July 15-17, 2013). Accordingly, a call for papers is due. (The fourth one will take place in 2014 undercover as a Trends in Logic conference.) So here it is. Please distribute this information among your potentially interested colleagues.
(PDF version here.) 

Gdańsk University (Poland) and Centre for Logic and Philosophy of Science at Ghent University (Belgium) invite submissions of papers related to the application of formal methods in philosophy, especially outside the narrow field of philosophy of logic and language. 

DETAILS

  • We plan around 12 presentation slots.
  • Each speaker will be given 30-60 minutes to present, depending on the length of the paper.
  • Each paper will be sent ahead of time to a participant who isn't its author with a request for a commentary.
  • Each presentation will be followed by 10-15 minutes of a commentary by another participant.
  • Each commentary will be followed by 15 minutes of discussion.
  • Each paper will be blind-reviewed by two referees, comments will be sent ahead of time to the author with (possible) request to revise the paper before forwarding it to the commentator.
  • Depending on the number of submissions, we might be unable to provide comments on rejected papers.
  • The language used is English.

As we think it is better to submit a paper  to a good journal than to a proceedings volume, there will be no proceedings volume.

SUBMISSION

Full papers, prepared for blind-review (accompanied by an email providing author details) should be sent in PDF format to entiaetnomina2013@gmail.com by April 1, 2013.


Rafal Urbaniak
Agnieszka Rostalska
Aleksandra Szulc