# A Strongly Differing Opinion on Proof-Theoretic Semantics?

**Abstract** Responding to an invitation from Peter Schroeder-Heister, the paper reacts to some criticisms of 'model theory' voiced among proof theorists interested in prooftheoretic semantics. It argues that the criticisms are poorly targeted: they conflate model theory with model-theoretic semantics and with the model-theoretic definition of logical consequence, which are three largely unrelated areas of study. On defining the meanings of logical constants, and of natural language expressions in general, the paper lays out some methodological requirements that any satisfactory definitions would need to meet, for example about generalisability from one context of use to other contexts. On defining logical consequence, the paper argues that some points made recently by Schroeder-Heister and Kosta Došen are largely sound and probably uncontroversial if clearly stated, but their impact is blurred by some question-begging formulations.

It was very kind of Peter Schroeder-Heister to invite me to contribute to this meaty conference. He said:

. . . you would fit very well into this meeting, even though (or perhaps because) you have opinions that strongly differ from [those] of the majority

of people at the conference. Perhaps you can give a talk in defence of model theory, as far as the foundations of logic are concerned.

(1)

That's a fantastic invitation, and I went to the meeting resolved to disagree with as many people as possible.

In the event it was not so easy. Partly there was serious research being done in proof theory, and I am not a proof theorist. Partly there were a good number of entirely sensible and friendly people. But also I often found it hard to see what the issues were. I think this was not entirely my fault. Straw men were being set up and knocked down. I could see this most clearly when the straw men were described as model theorists, because I do know something about model theory, and some of the views being attributed to model theorists were not ones I recognised. This impression was strengthened when I read a recent paper of Peter's in *Synthese* [14].

So I had plenty to disagree with, but not in a very satisfactory way. It's more edifying to discuss substantive issues than to clear away misunderstandings. But the clearance work has to be done first. I will try to keep it both brief and profitable.

I thank Peter Schroeder-Heister and Kosta Došen for some valuable discussions.

**1 Straw Model Theory**

A good place to start will be an elegant paper of Dag Prawitz [11] from 1974. There is a lot that I agree with in the paper, but I was pulled up sharp when he said:

In model theory, one concentrates on questions like what sentences are logically valid and what sentences follow logically from other sentences [11, p. 66].

I can say with absolute confidence that I never met a model theorist who 'concentrates on questions like what sentences are logically valid and what sentences follow logically from other sentences'. On his next page Prawitz discusses Alfred Tarski's proposal for defining logical consequence, from his paper of 1936 [17]. So it seems likely that Prawitz reached the view stated in (2) by assuming that Tarski's 1936 paper is of interest to model theorists. This is not in fact the case. Nothing in the paper is of any interest to model theorists, except perhaps those with an interest in the prehistory of their subject.

Peter Schroeder-Heister adds another ingredient to the mix in his recent paper [14], namely model-theoretic semantics. This is a discipline concerned with describing meanings, so Peter rightly connects it with questions about how one should describe the meanings of logical constants. But its origins are quite different from those of the model-theoretic truth definition, and it belongs to a different research community. Model theorists don't do model-theoretic semantics either. I do know one person who contributes to model-theoretic semantics using techniques of model theory, namely Dag Westerståhl; but there are not many of him. In short, the three areas of research—model theory, the definition of logical consequence and model-theoretic semantics—are quite different and they have hardly anything in common beyond a connection with models in the sense associated with Alfred Tarski.

So now let me unpick the historical relations between these areas. (All the comments on Tarski below draw out material from my [10].)

There are some other research areas that connect with Tarski's notion of models but not with each other. One is mental model theory as pursued by the cognitive

scientist Ruth Byrne [2], and another is the model-theoretic syntax advocated by the linguists Geoffrey Pullum and Barbara Scholz [12].