Proof, Rules and Meaning

June 2021

Proof, Rules and Meaning: book manuscript in progress.

This is my next book-length writing project. I am writing a book which aims to do these things:

1. Be a useable introduction to philosophical logic, accessible to someone who's done only an introductory course in logic, covering at least some model theory and proof theory of propositional logic, and maybe a little bit of predicate logic.
2. Be a user-friendly, pedagogically useful and philosophically motivated presentation of cut-elimination, normalisation and conservative extension, both (a) why they're important to semantics and (b) how to actually prove them. (I don't think there are any books like this currently available, but I'd be happy to be shown wrong.)
3. Present the duality between model theory and proof theory in a philosophically illuminating and clear fashion.
4. And then apply these results to issues concerning meaning, epistemology and metaphysics, including issues of logical consequence and rationality, the problem of absolute generality, and the status of modality.

Here is an outline of the manuscript, showing how the parts hold together. At least so far — I'm still writing the third part.

The book is in three parts.

1. Tools: in which core concepts from proof theory are introduced.
2. The Core Argument: in which I motivate defining rules, and show how they answer Arthur Prior's challenge concerning when an inference rule defines a logical concept.
3. Insights: in which we see consequences for logic and language, epistemology and metaphysics, etc.

The book draft was discussed at this symposium in Buenos Aires in July 2018. I’m currently finalising the last section of the book, and updating the whole manuscript based on the feedback I got there from colleagues.