May 042015


The Open Logic Project is live! From the website:

The Open Logic Text is an open textbook on mathematical logic aimed at a non-mathematical audience, intended for advanced logic courses as taught in many philosophy departments. It is open-source: you can download the LaTeX code. It is open: you’re free to change it whichever way you like, and share your changes. It is collaborative: a team of people is working on it, using the GitHub platform, and we welcome contributions and feedback. And it is written with configurability in mind.

The Project was instigated and shepherded by Richard Zach, with contributions by Andy Arana, Jeremy Avigad, Gillian Russell, Nicole Wyatt and Audrey Yap (besides yours truly).  Go and check it out!

 Posted by at 11:30
Jan 112015

After teaching the course last term, I posted a slightly revised version of the metalogic notes, The Completeness of Classical Propositional and Predicate Logic (C2P2L). Mostly typos, not significantly different from previous versions, but it is linked here for reference.

 Posted by at 11:29
Nov 232014

The past few days have seen the passing of two most remarkable people: the French mathematician Alexandre Groethendiek  (1928-2014) and the Americal philosopher ad logician Patrick Suppes (1922-2014). They were as different as they could be: Groethendieck spent the last two decades of his life in seclusion in the small French village of Ariège, in the French Pyrenees; Suppes was active until the very end, giving a talk in Munich as recently as 2012, an urbane, refined gentleman, whose speech had just a tinge of his native Oklahoma.

Groethendiek had an almost obsessive compulsion to generalize, abstracting from the clutter until just the conceptual scaffolding was left in view. Groethendiek’s student, Pierre Deligne (who himself won the Fields Medal in 1978 and the Abel Prize in 2013) is quoted in the obituary by Le Monde:

He was unique in his way of thinking. He had to understand things from the broadest possible point of view, and once things are so conceived, the landscape becomes so clear that proofs appear to be almost trivial.

Mar 222014

Guy, not recognizably affiliated with the department, on the phone:

For a political philosopher to be in love with anyone is a sad thing.

Wonder what the other end of the conversation was like.

Feb 222014

2013 Moto Guzzi V7 Stone.


For those who were wondering why there hadn’t been any new posts in a while: this beauty from the shores of Lake Como, that’s why.

 Posted by at 16:22  Tagged with:
Oct 282013

The paper on the general interpretation of first-order quantifiers has now been published online by the Review of Symbolic Logic (this direct link requires subscriber/institutional access, or email me for a PDF). The philosophical companion to that paper — “Life on the Range” — is not out yet, but a pre-print can be found here.

 Posted by at 10:04
Aug 232013

Alfred Tarski’s great technical innovation in his Wahrheitsbegriff (1933/36) was the introduction of the auxiliary notion of satisfaction of a first-order formula \varphi(x_1,\ldots,x_n) by a function s assigning objects from some model M‘s domain  to the variables \{x_i : i \ge 0\}  of the language. Once the notion of satisfaction is in place, one can then define truth of a sentence \varphi in a model M, by saying that \varphi is true just in case it is satisfied by some (equivalently: every) assignment. It is this latter notion of truth that is of real mathematical and philosophical interest, of course, but it cannot be defined directly by recursion because in a first-order language (contrary to the propositional case), sentences are not built up from other sentences by means of constructors such as connectives. Rather, it is formulas in general that are so built, with sentences providing a special case. The characteristic clause of satisfaction is the one for quantified formulas:

  • A formula \forall y \, \varphi(x_1, \ldots, x_n,y) is satisfied by an assignment s in M if and only if the formula \varphi(y, x_1, \ldots, x_n) is satisfied, in M, by every assignment that that differs from s at most in the value s(y).

The detour through satisfaction, while technically interesting, it is just that — a detour. It can also sometimes get pedagogically problematic, as students usually grasp the importance of truth while not always appreciating the necessity of satisfaction.

There is an alternative approach, which does away with the notion of satisfaction, but uses a variant notion of a model. According to this approach, we only allow models that are “rich” in that every object a \in M is the denotation of a constant c_a in the language. The truth clause for the universal quantifier is then as follows:

  • A sentence \forall y \, \varphi(y) is true in M if and only if \varphi(c_a) is true in M for each constant c_a.

(I believe this definition is fairly standard in some textbooks, but I can’t track down a definition now, any references appreciated). While this definition sidesteps the notion of satisfaction, there are some drawbacks:

  1. The definition only applies to “rich” models, while models in which some elements are not denoted by constants are also very natural. Moreover, we are forced to consider truth of sentence \varphi in a model having a different signature than \varphi itself.
  2. The language changes with the model: for each model M, the definition applies only to the expanded language \mathcal{L}_M introducing new constants for each member of M.
  3. The definition forces us to consider uncountable languages, in order to assess truth in uncountable models.
  4. The expanded signature affects the available automorphism of a model. For example, the model (\mathbb{Z}, <) has infinitely many automorphisms in the signature with the relation < only, but only one if we add the constant 0.

So here is a way of combining these two approaches in such a way that:

  1. There is no detour through satisfaction.
  2. The language is countable and independent of the model in which a sentence is evaluated for truth (although the language will be expanded, just not in a way determined by the model).

Fix a basic language \mathcal{L}, and let c_1,c_2,\ldots be a countable set of new individual constants. For each n\ge 0 let \mathcal{L}_n = \mathcal{L} \cup \{c_1,\ldots,c_n\}  (so that \mathcal{L}_0 = \mathcal{L}). We define the notion of truth by induction on \mathcal{L}_n-sentences, simultaneously for every n. The quantifier case:

  • An \mathcal{L}_n-sentence \forall y \, \varphi(c_1,\ldots,c_n,y) is true in the \mathcal{L}_n-structure M if and only if the \mathcal{L}_{n+1}-sentence \varphi(c_1,\ldots,c_n,c_{n+1}) is true in every \mathcal{L}_{n+1}-expansion of M (i.e., if  \varphi(c_1,\ldots,c_n,c_{n+1}) is true in every \mathcal{L}_{n+1}-structure M' that differs from M only in that it assigns a denotation to c_{n+1}).

So, when \varphi is an \mathcal{L}-sentence, the definition gives us a direct account of the truth or falsity of \varphi is a model having the same signature.

 Posted by at 12:30