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:
- Be a useable introduction to philosophical logic, accessible to someone who's done only an introductory course in logic, covering at least some models and proofs for propositional logic, and maybe a little bit of predicate logic.
- 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.)
- Develop an extended argument, showing how proof rules (of a certain shape) can be understood
as definitions.
- And finally, relate 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.
The book is in three parts.
- The Tools: in which core concepts from proof theory are introduced. Three chapters, one on natural deduction proofs (and normalisation), one on the sequent calculus (and cut elimination) and one on models and more, relating proofs and models, including soundness and completeness results, understood from a proof-first perspective..
- The 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.
- The Upshot: in which we see consequences for logic and language, epistemology and metaphysics, etc. Three chapters, one on connectives, discussing issues around conditionals and nenegation (and in particular, discussions of vagueness and the semantic paradoxes), one on quantifiers, modality and their interaction, and one on logic and meaning.
An early draft was discussed at
this symposium in
Buenos Aires in July 2018. Since then, I moved to St Andrews, and have updated the manuscript
based on feedback I received there, and talks I have given since then.
