Observational Advantages and Occurrence Referentiality
Francesco Bellucci and Jim Burton
Abstract.
Logical diagrams are known to have certain advantages over sentential notations for particular reasoning tasks: using a diagram may make logical consequences directly evident, when these consequences are “hidden†in an equivalently expressive sentential language. This phenomenon is known as a “free ride†or “observational advantage.†Where does this advantage come from, and why? We answer this question by distinguishing two general kinds of logical languages: occurrence-referential languages, in which sameness of reference (for sentential and predicate variables) is determined by the sameness of variable occurrence, and type-referential languages, in which sameness of variable type that determines sameness of reference. We explain that it is the occurrence-referential nature of some languages that explains for their observational advantage over equivalently expressive type-referential languages.
I found your paper to be very interesting. Are you able to share any ideas for how you see this research progressing? Do you have plans to apply your theory to other systems to further demonstrate the correctness of your ideas? If so, what systems do you think would be suitable candidates?
Thanks for your appreciation Gem! I think the next step would be a systematic evaluation of the difference between unitary and non-unitary types of Euler-based diagrams. It seems to me that this is a crucial difference. Also, I think we should address the question of occurrence-ref in Venn-based diagrams: it was not clear to us when we wrote the paper whether these are occurrence-ref or type-ref. We also planned to investigate another kind of occurrence-referentiality, which we only mention in passing in this paper and which has to do with individuals rather than classes or predicates. So an investigation in this direction would involve a study of systems that are occ-ref with respect to individuals and type-ref with respect to predicates, like Peirce’s graphs. Ideally, this would enable us to construct a typology of systems (we had some ideas but too primitive to be included in this paper).