Normative pragmatics and proof theory: unfinished notes on Restall
How can proof theory give us a normative account of communication? This post is an exposition of (what I think is) Restall’s answer.
These are my (unfinished) notes on a talk by Restall on proofs and what they’re good for: very roughly, this is my recap of the part on normative pragmatics and proof theory. I originally wrote them up for a friend and the notes stop abruptly; I leave them here as a quick primer for those who may be interested. Further detail can be found in Restall’s paper on multiple conclusions.Â
Notation: I use X and Y for sets, A and B for propositions, and CAPITALS WITH ITALICS for logical connectives.
I. What does proof have to do with communication?
People assert things. They also deny things. These assertions and denials are part of communication. There are norms on these assertions and denials. We can be more precise by representing assertions and denials as [X:Y], where X is a set of assertions and Y is a set of denials.
It is out of bounds (intuitively, not legitimate) to assert and deny certain things at the same time. For example, we have the norm of Identity.
Identity: [A:A] is out of bounds. We can’t assert A and deny A at the same time.
Weakening: If [X:Y] is out of bounds, then so is [X, A:Y] and [X:Y, A]. If we can’t assert X and deny Y, then adding anything to the assertions or the denials doesn’t change that. (This doesn’t hold in cases of nonmonotonicity - like with typicality assumptions.)
Cut: If [X, A:Y] and [X: A,Y] is out of bounds, then so is [X:Y]. If asserting A or denying A is out of bounds, then the original position must have been out of bounds anyway.
These norms are named that way because they mimic the familiar structural rules of the sequent calculus. Replace “:” with the turnstile. Identity is then the sequent calculus structural rule of identity, and so on.
Our rules of communication give us our structural rules for proof.Â
II. How do we define logical connectives?
What does it mean to define words? Here’s a take on it: something is square iff it is a rectangle and has all sides equal. On this picture, definitions are more like abbreviations. Notice that we’re using logical vocabulary for our definitions: square are things which are rectangles and have all sides equal. This means that we use logic in our definitions. How do we define logical words, then?Â
We can define logical connectives in terms of other connectives. How can we define A AND B? Well, we can define it in terms of NOTÂ and OR, such that A AND BÂ is defined as NOTÂ A OR NOT B. How do we define NOTÂ A? Well, we can take it to be an abbreviation for IFÂ A THEN FALSUM. And so on. But this process need never end.
Another way of defining logical terms is by how we use them. Take AND. When can we assert A AND B? We can give the following rule: [X, A AND B: Y] iff [X, A, B:Y]. Take that as definitional. When do we deny A AND B? Well, we put A AND B on the right of the turnstile (since that’s where denials go), and grind through the definitions until we get our answer. (Restall’s slides use Cut.)
