Frames and Theories II

Tarski's 1936 account of logical consequence is at the heart of the discussion I want to try to have. In particular, I want to highlight a discussion that happened (for the most part) in the 1980s and 1990s about what exactly Tarski's account *is*. A good gateway into this debate is Greg Ray's 1996 paper.

As with any moderately interesting philosophical debate, there are a lot of subtleties and technicalities that play a central role. As is standard in any moderately interesting summary of a philosophical debate, I will ignore most of these. I will also focus on the easier case of theorems rather than on logical consequence. This is for simplicity as well, but I admit that in this case the sacrifice is a bit harder to swallow.

Given these simplifications the (yet more oversimplified) core issue in the Ray-Etchemendy debate was which of the following was the actual Tarskian account of logical consequence:

1. Let U be the universal domain -- the class of all objects. Then s is a theorem just if no matter how we assign members of U to our variables, sets of members of U to predicates, etc., s always turns out true.

2. As above, but rather than restricting U to be the unique universal domain, we allow it to be any class of objects at all.

The question, that is, is about whether Tarski allowed domain relative assignments. Etchemendy's Tarski endorses the first account, rejecting domain relative assignments. Ray's Tarski endorses the second, accepting domain relative assignments. I find Ray's arguments convincing. But I'm not a historian, so take that with a grain of salt.

The framework we're using here is obviously quite different from the Tarskian framework being debated in the Etchemenday-Ray episode. But we are going to have to ask a similar question. This is because there are two natural ways we could define logical consequence. The first involves varying frames; the second involves a universal frame. To be more explicit, let's first define truth for the zero-order language.

Let [F,D,v] be a model, and let t be a theory in F. The recursive definition of truth-in-F-at-t is as follows:

  1. Pa1...an is true in F at t just if (recall our assumptions from before!) [a1,...,an] is in the valuation of P at t.

  2. 'A and B' is true in F at t just if A is true in F at t and B is true in F at t.

  3. 'A or B' is true in F at t just if A is true in F at t or B is true in F at t.

  4. 'not A' is true in F at t just if A fails to be true at the complement of any prime theory extending t.

  5. 'if A then B' is true in F at t just if A is true in F at u only if B is true in F at the closure of u under t.

Dang. We're already over 500 words. I thought we'd be able to get further. To keep things manageable I'll stop there. Next time, I'll give some motivation for these clauses and return to the thread.