Galileo [2] famously asked whether there are more natural numbers than square numbers. On the one hand, every square number is a natural number, while the converse is false, suggesting that there are strictly more natural numbers than squares. On the other hand, there is an obvious way to define a one-to-one correspondence between the two collections, obtained by mapping any natural number to its square, suggesting that they are, in fact, equinumerous. Galileo’s paradox is a striking example of a clash between two intuitive principles about sizes of infinite collections [4]. According to the Part-Whole Principle (PW), any proper subcollection of a collection \(A\) has size strictly less than the size of \(A\). According to the Bijection Principle (BP), any two collections have the same size if and only if there is a one-to-one correspondence between them. While the modern notion of cardinality obeys both principles in the finite, Cantor famously adopted (BP) as the foundation of his transfinite arithmetic, thus rejecting (PW). However, the recently developed theory of numerosities [1] has been presented as a viable alternative to the Cantorian picture that is based on (PW) rather than on (BP).

In this talk, I will focus on numerosities for countable sets. I will argue that the current theory of numerosities faces some major issues because it mixes two distinct intuitions. The first one is the Euclidean Intuition that the whole is always greater than any of its proper parts. The second, which I call the Density Intuition, is the intuition that the size of a set of natural numbers is determined by the frequency with which its elements appear in the sequence of natural numbers. While the first intuition is compatible with a very natural invariance condition that one would want to impose on any adequate notion of size for sets, the second intuition (and therefore also the standard theory of numerosities) is not. I will propose an alternative theory that is based purely on the Euclidean Intuition and is, in some precise axiomatic sense, the best way to combine part-whole intuitions with invariance criteria. Time permitting, I will also give a semantic intuition for this theory of Purely Euclidean Numerosities that is based on possibility semantics for classical first-order logic [3].

Bibliography

1. Benci, Vieri and Di Nasso, Mauro,Numerosities of labelled sets: a new way of counting,Advances in Mathematics,vol. 173 (2003), pp. 50–67.
2. Galilei, Galileo_Discorsi e dimostrazioni matematiche intorno a due nuove scienze_,Boringhieri, 1958.
3. Holliday, Wesley H.,Possibility Semantics, Selected Topics from Contemporary Logics, vol. 2 (2021).
4. Mancosu, Paolo, Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable?,The Review of Symbolic Logic,vol. 2 (2009), no. 4, pp. 612–646.

Ever since Russell attacked him, Bergson is commonly seen as an author alien to analytical tradition and even as an irrationalist. I want to join some of the efforts to defend Bergson against such charges, this time focusing on his view of logic. While Bergson indeed seems to consider logic as more of a problem than a solution, we should understand that his remarks pertain only to a very specific notion of logic. And this notion is being overcome by the development logic underwent since Bergson´s times. Today, logic is marked by great plurality of logical systems. I argue, though, that rather than pointing to the mere multiplicity of logics, we should read this development as pointing to a more dynamic conception of logic. Instead of the popular logical pluralism, one should therefore speak of logical dynamism. With this understanding, we can reconcile both Bergson´s seemingly irationalist pronouncements about logic and his effort to carefully craft his arguments. And the dynamics of logic is best understood by means of Bergsonian philosophy. Bergson ultimately offers the analytical tradition a new understanding of logic and possibly even of rigour, a notion so dear to this tradition.

Bibliography

1. Pavel Arazim,Beyond Logical Pluralism and Logical Monism,Logica Universalis,vol. 14 (2020), no. 2, pp. 151–174.
2. Bertrand Russell,The Philosophy of Bergson,The Monist,vol. 22 (1912), no. 3, pp. 321–347.
3. Gillian Russell,Logical nihilism: Could there be no logic?,Philosophical Issues,,vol. 28 (2018), no. 1, pp. 308–324.
4. Henri Bergson,Time and free will: An essay on the immediate data of consciousness.,London: George Allen and Unwin Ltd.,1910.
5. J. Beall, G. Restall,Logical Pluralism,Oxford: Oxford University Press,2006.

This work presents a novel version of the multiverse theory of sets called hierarchical pluralism by introducing the desideretum of ‘degrees of intentionality’ of theories. The presented view is articulated for the purpose of reconciling, in the philosophy of set theory, epistemological realism and the multiverse theory of sets so as to preserve a considerable amount of epistemic objectivity when working with the multiverse theory. I give some arguments in favour of a hierarchical picture of the multiverse in which theories or models are thought to be ordered with respect to their plausibility, as a manifestation of endorsing the idea that some set theories are more plausible than others. The proposed multiverse account settles the pluralist’s dilemma, the dichotomy that there is a trade-off between the richness of mathematical ontology and the objectivity of mathematical truth. The view also extends and serves as an alternative position to Balaguer’s intention-based Platonism from which he claims that a certain version of mathematical pluralism follows.

We consider the problems arising in figuring out the right counterparts to the basic conditional logic \(\mathsf{CK}\) when the propositional basis of the logic is no longer assumed to be classical. We argue that, as long as the new underlying logic is constructive, this problem shows essential resemblance to the problem of figuring out the right intuitionistic counterparts to the well-known classical modal logics as addressed, e.g. in [3], where the famous set of six requirements was put forward.

Among these requirements, the last and the most important one demands an explanation of the semantics of conditionals/modalities in terms of the first-order version of the underlying non-classical logic, and we fundamentally agree with A. Simpson’s intepretation of this explanation as the faithfulness of the embedding into the first-order version of the underlying logic provided for the candidate conditional/modal logic by the standard translation borrowed from the classical case.

However, both the choice of the underlying non-classical logic and the peculiar features of the conditional logic may pose additional challenges.

We illustrate these challenges by the examples \(\mathsf{IntCK}\) and \(\mathsf{N4CK}\), the two recently proposed analogues of \(\mathsf{CK}\) (see [1] and [2]) based on the intuitionistic propositional logic and on the paraconsistent variant of Nelson’s logic of strong negation, respectively.

Bibliography

1. G. Olkhovikov. An intuitionistically complete system of basic intuitionistic conditional logic. Submitted, preprint available at (2023).
2. G. Olkhovikov. A basic system of paraconsistent Nelsonian logic of conditionals. Submitted, preprint available at (2023).
3. A. Simpson. The Proof Theory and Semantics of Intuitionistic modal Logic, PhD Thesis, University of Edinburgh (1994).

Carnap [4] demonstrated that the usual axiomatisations of classical propositional and first-order logic fall short of ‘fully formalising’ these systems. In particular, while there is a tight correspondence between syntactic, proof-theoretic, and semantic, model-theoretic, explications of the notion of consequence for these systems, a similarly adequate correspondence between inferential and model-theoretic aspects of the meanings of their logical expressions is lacking. More precisely, Carnap showed that there is a significant mismatch between the intended model-theoretic values of the logical constants and the model-theoretic values actually determined by the usual rules of inference. The standard axiomatisations of classical propositional and first-order logic are, thus, not categorical for the intended model-theoretic values of the logical constants of these systems. This is Carnap’s (categoricity) problem.

Carnap’s problem has significant repercussions for a range of projects and positions in the philosophy of logic, language and mathematics. Although Carnap’s original considerations focused on classical propositional and first-order logic, its consequences have since been investigated for other classes of logical expressions, including intuitionistic connectives [1], modal operators [2], as well as generalized [3] and higher-order quantifiers [7]. Furthermore, a variety of different solution-strategies have been advanced in the literature: from modifying the language or format of the consequence relation [8,9], over re-interpretations of the notion of an inference rule [5,6], towards adopting additional constraints on the space of admissible meanings [1,3].

The goal of this talk is to survey these different solution-strategies with an eye towards their ability of solving Carnap’s problem in full generality and their resulting conception of (logical) meaning. An interesting upshot of this investigation concerns the, sometimes implicitly, adopted meaning-theoretic constraints by different ways of resolving Carnapian underdetermination. Moreover, Carnap’s problem raises interesting questions about what it takes to have completely grasped or characterised a logical notion, resembling a similar discussion in the philosophy of mathematics. This parallel will be explored further in the talk.

Bibliography

1. D. Bonnay, D. Westerståhl, Compositionality Solves Carnap’s Problem,Erkenntnis,vol. 81 (2016), no. 4, pp. 721–739.
2. \bysame_Carnap’s Problem for Modal Logic_,Review of Symbolic Logic,vol. 16 (2023), no. 2, pp. 578–602.
3. D. Bonnay, S.G.W. Speitel,The Ways of Logicality: Invariance and Categoricity,The Semantic Conception of Logic. Essays on Consequence, Invariance, and Meaning(G. Sagi and J. Woods, editors),Cambridge University Press,2021,pp. 55–79.
4. R. Carnap,Formalization of Logic,Harvard University Press,1943.
5. J.W. Garson,What Logics Mean: From Proof Theory to Model-Theoretic Semantics,Cambridge University Press,2013.
6. V. McGee,Everything,Between Logic and Intuition: Essays in Honor of Charles Parsons(G. Sher and R. Tieszen, editors),Cambridge University Press,2000,pp. 54–79.
7. J. Murzi, B. Topey, Categoricity by Convention,Philosophical Studies,vol. 178 (2021), no. 10, pp. 3391–3420.
8. I. Rumfitt, ‘Yes’ and ‘No’,Mind,vol. 109 (2000), no. 436, pp. 781–823.
9. D.J. Shoesmith, T. Smiley,Multiple Conclusion Logic,Cambridge University Press,1978.