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Live from Trends in Logic VII (day 2)

Today we had four quite exciting talks. The first one, given by Oystein Linnebo (Bristol) was devoted to A Partial Defense of Frege's Basic Law V. Oystein started off with the intuitions that there is some pressure to accept Frege's BLV (which says that extensions of two concepts are identical iff exactly the same objects fall under that concept). After criticizing the limitation-of-size approach to restricted versions of the comprehension principle, he went modal-and-iterative about BLV. That is, BLV was used to capture how new sets are formed at new stages using the objects already existing in previous stages, and modal operators were thrown in to express the intuition that we're talking about the possible ways our set-formation process can go. This gives a fairly intuitive criterion for a plurality determining a set: it has to have the same elements across possible worlds. Proof-theoretically, once you take S4.2 as the underlying modal logic, throw in some trans-world extensionality principles for pluralities and sets and introduce the potential plural collapse ("it is necessary that for any xx it is possible that there is a y such that y is the set of xx's"), you can get (a reinterpretation of) Zermelo set theory minus infinity and foundation.

The second talk, by Leon Horsten, was devoted to the relation between numbers and counting systems. Leon defended and described the view dubbed computational structuralism: it's kinda like structuralism, but you take arithmetic to be about the structure of arithmetical notational systems. The basic idea is that if one has a recursively introduced notational systems (so that the symbol denoting the successor is computable), and the addition function is also computable, the system is isomorphic to the intended omega-sequences.

Michael Resnik and Stewart Shapiro both talked about qualms that arise around identity conditions for structures and their elements (or positions in them). Resnik, roughly, was arguing that in certain contexts identity claims (and questions about identity) of certain structures doesn't make sense, whereas Shapiro was rather inclined to say that it's not as much the identity questions that are misled, but rather that certain terms may seem and behave like singular terms, despite referring indeterminately to many objects.