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# Cardinality, Worlds and Paradox

**Research Project: **History and Philosophy of Logic and Mathematics

## 7th June 2016 - 8th June 2016

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A joint workshop between the Arché Research Centre and the Department of Philosophy at the University of Oslo.

Confirmed speakers:

Josh Dever (Texas at Austin and Arché)

Peter Fritz (Oslo)

Øystein Linnebo (Oslo)

Lavinia Picollo (Munich)

Graham Priest (CUNY Graduate Center)

Agustin Rayo (MIT and Oslo)

Stewart Shapiro (OSU)

Gabriel Uzquiano (USC and Arché)

Various relatives of Russell’s paradox have been thought to have important ramifications in logic and metaphysics. It has been thought, for example, that there are strictly more propositions than possible worlds, and that this poses a serious problem for possible worlds semantics. And it has similarly been argued that the Russellian paradoxes set non-trivial constraints on the cardinality of the universe. Moreover, it has been thought that variants of the Russellian arguments place strict limits on the granularity of properties and propositions. The arguments, however, are not airtight and they invite questions such as the following:

### Abstracts

Josh Dever, “Quantity Without Quantities”

I explore the prospects for a contingentist combination of ZFCU with an axiom of uelement set existence with a strong plenitude principle that it is possible for the urelements to be absolutely infinite in number, with the combination depending crucially on the thought that the size of the set-theoertic universe is modally variable. The resulting picture connects the iterative conception of set to a grounding story on which varying numbers of urelements can drive varying heights to the iteration. Along the way I consider some points about the content of large cardinal hypotheses across such modal variation, and dabble a bit in multisets and sequences as alternative focuses of grounding principles and quantity representatives.

Peter Fritz, “A Purely Recombinatorial Puzzle”

A new puzzle of modal recombination is presented which relies purely on resources of first-order modal logic. It shows that naive recombinatorial reasoning, which has previously been shown to be inconsistent with various assumptions concerning propositions, sets and classes, leads to inconsistency by itself. The context sensitivity of modal expressions is suggested as the source of the puzzle, and it is argued that it gives us reason to reconsider the assumption that the notion of metaphysical necessity is in good standing.

Øystein Linnebo, “Cantor countered”

By a higher-order generalization of Cantor’s theorem, we appear able to prove that any given plurality has more subpluralities than it has members. After outlining some puzzles to which the generalization gives rise, I explore the prospects for blocking the generalization by restricting the higher-order comprehension schemes. The restrictions in question are motivated by a view of the universe as “open-ended”— in a sense that generalizes the ancient notion of potential infinity.

Lavinia Picollo, “The expressive function of truth”

It is often said that the truth predicate serves a logico-expressive function, namely, it allows for the expression of so-called `infinite conjunctions’. This function prompts the formulation of logics or formal theories of truth. We argue that what principles these systems should validate depends on what it means for an infinite conjunction to express or stand in for all its `conjuncts’. We examine two accounts for this phenomenon that are available in the literature and show them to be substantially flawed. We put forward a new approach and discuss whether classical or non-classical logics are to be preferred as basis for theories of truth. We also propose a reconceptualisation of deflationism, according to which the meaning of truth in natural language is largely irrelevant for the deflationist’s choice of a truth theory. Furthermore, we discuss a similar approach to the class-theoretic paradoxes and suggest that we should be deflationists about classes as well.

Graham Priest, “Paradoxical Propositions”

This paper concerns two paradoxes involving propositions. The first is Russell’s paradox from Appendix B of The Principles of Mathematics, a version of which was later given by Myhill. The second is a paradox in the framework of possible worlds, given by Kaplan. This paper shows a number of things about these paradoxes. First, we will see that, though the Russell/Myhill paradox and the Kaplan paradox might appear somewhat different, they are really just variants of the same phenomenon. Though they do this in different ways, the core of each paradox is to use the notion of a proposition to construct a function, f, from the power set of some set into the set itself. Next we will see how this paradox fits into the Inclosure Schema. Finally, I will provide a model of the paradox in question, showing its results to be non-trivial, though inconsistent.

Agustin Rayo, “Quantification and Realism”

I argue for a facts-first conception of reality, according to which there are many different ways of carving logical space into objects.

Stewart Shapiro, “Ontology in mathematics (and beyond): the case of points” (joint work with Geoffrey Hellman)

It is probably safe to say that the prevailing views on ontology and ontological commitment trace to Quine’s “On what there is”. The slogan is “To exist is to be the value of a bound variable”. One can thus speak of the ontology of a mathematical theory, and the commitments of a mathematician, once we get straight on what it is to assert, adopt, accept, … a mathematical theory. The purpose of the present exercise is to see how the broadly Quinean themes play themselves out, given some plausible assumptions about mathematics—a broadly structuralist perspective. There are some ramifications concerning the more mathematized (or structural) aspects of ordinary, scientific theories. The main case study concerns points in space or space-time.

Gabriel Uzquiano (USC/St Andrews), “A puzzle for Cantorian reasoning”

Cantorian reasoning is often supposed to establish certain cardinal inequalities in logic and metaphysics. It tells us that there are more propositions than possible worlds, more Fregean concepts than objects, more mereological fusions than atoms, etc. Some of these cardinal inequalities compare objects of different sorts, e.g., there are more mereological fusions than atoms. Other inequalities, however, concern cross-categorical cardinal comparisons, e.g., there are more Fregean concepts than objects and there are more pluralities than objects – given more than one object. The talk will be particularly concerned with the latter uses of Cantorian reasoning. A close look at them may suggest some distance between the purely formal statement of the relevant generalization of Cantor’s theorem on which they rely and the informal gloss they receive when deployed in Cantorian reasoning. The purpose of this talk is to bridge the gap between the two.

### Other events in History and Philosophy of Logic and Mathematics

May 2015: Wyclif and the Realist Tradition in 14th-Century Logic

Nov 2015: Inferentialism