Invited talks of the 3rd Filomena workshop
"Logical Realisms"

Logical realism is a view about the metaphysical status of logic, but it comes in many forms. Common to most if not all the views captured by the label “logical realism” is that logical facts are mind and languageindependent. But that does not entail anything about the nature of logical facts or about our epistemic access to them. Another open question is whether logical realism entails logical monism, the view that there is one true logic, or whether it is compatible with some forms of logical pluralism. The goal of this paper is to outline and systematize the different ways that logical realism could be entertained and to examine some of the challenges that these views face.

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Sequent systems usually do not reflect explicitly semantical properties. Hence, proving soundness and completeness of a sequent system w.r.t. its semantics can be cumbersome. In this work we explore the connections between linear nested sequent calculi (LNS) and semantics of various logics. Commencing with intuitionistic logic, we start by presenting Maehara’s mLJ [4], a multiple conclusion intuitionistic sequent system. Then we consider an extension of the sequent framework called nested systems [1,2], establishing some proof theoretical results for it. We show that the nestings in intuitionistic logic satisfies the following properties:
1. although nestings are independent and can be created in parallel, provability of only one of them is enough for proving the nested sequent; 2. all rules can be restricted so that to be applied at the last two levels of a nesting. This allows a simplification on systems, by restricting the tree structure of nested sequents to that of a line, with rules restricted to its endactive version [3]. We then show how to automatically label the linear nested systems and how to relate these systems with the usual Kripke semantics for various logics. Finally, we move to (classical) multimodal logics, relating (general) frames with labelled simply dependent multimodal logics. References 1. Brunnler, K.: Deep sequent systems for modal logic. Arch. Math. Log. 48, 551–577 (2009), http://link.springer.com/article/10.1007/s0015300901373 2. Fitting, M.: Nested sequents for intuitionistic logics. Notre Dame Journal of Formal Logic 55(1), 41–61 (2014), http://dx.doi.org/10.1215/002945272377869 3. Lellmann, B., Pimentel, E.: Proof search in nested sequent calculi. In: Logic for Programming, Artificial Intelligence, and Reasoning  20th International Conference, LPAR20 2015, Suva, Fiji, November 2428, 2015, Proceedings. pp. 558–574 (2015), http://dx.doi.org/10.1007/9783662488997_39 4. Maehara, S.: Eine darstellung der intuitionistischen logik in der klassischen. Nagoya Mathe matical Journal pp. 45–64 (1954) 
"A semantical view of Linear Nested Systems"

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"Bridging the Gap Between Science and Metaphysics, with some Help from Quantum Mechanics"

One of the greatest challenges for analytic metaphysicians concerns the relation of the output of their work with contemporary science. At least for those working under the umbrella of socalled “naturalistic metaphysics”, it is expected that metaphysical theories relate profitably with science. However, it is not even clear how such a relation is to be spelled out in details. We shall begin by enlightening this issue, and separating some distinct kinds of relations that metaphysics may bear with science. As we shall see, one of the greatest expectations comes from the supposed justification a metaphysical theory may derive by being somehow associated with a scientific theory; it is expected that we bridge the gap between metaphysics and epistemology (as encapsulated in science). We shall propose that under a reasonable understanding of the task of metaphysics, there is no way to avoid some metaphysical underdetermination, unless more than mere association with a scientific theory is required. Typically, it is claimed that a decision between competing metaphysical theories should be made based on their theoretical virtues: simplicity, economy in primitive notions, elegance, perhaps continuity with (part of) common sense, among others. Instead of discussing how theoretical virtues may help metaphysicians, we propose that the obtaining of metaphysical underdetermination is not as easily as it seems when we come to metaphysics associated with science. In fact, it results that it is not always so easy to advance a metaphysical theory consistent with empirical science. As a result, science may help us eliminate theories from the logical space of possibilities; that is, theories inconsistent with science should not be considered as real options for the naturalistic metaphysician, and so, unable to generate more metaphysical underdetermination. In this sense, even though metaphysical theories may not be justified, they may be ruled out by science. This is not as good as it was hoped for, but it is still a kind of improvement of our situation, and, besides, is not far from the situation in science itself. We illustrate how the thesis works with examples from the discussion about individuality in orthodox quantum mechanics.

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In this talk we will offer a novel picture of mathematics, where the theory of speech acts ([4]) plays a constitutive role of mathematical reality. Our starting point consists in the analysis of the debate between mathematical realists and antirealists; in particular the discussion whether or not mathematical objects exist, and if so in which sense. We will try to undermine the incompatibility of these two opposite positions, arguing that the goal of mathematics is not the study of abstract objects (but that they are only a useful means). Toward this end we will make use of the distinction between propositional content and forms of representation, arguing that what is commonly understood as a mathematical object is, only, part of a form of representation. The main argument for the application of a theory of speech acts to mathematics will be offered by the semantic homogeneity between mathematical and natural language (see [1]), sustained by the realists. Given this homogeneity it will therefore sufficient to show that the use of speech acts is a fundamental component of mathematical discourse. We will therefore offer a taxonomy of speech acts in mathematics ([2], [3]).
In the end, as an application of the image of mathematics offered, we will offer a new definition of abstract object and we will outline a response to an indispensability argument à la QuinePutnam, showing the ontological indeterminateness of its outcome. [1] P. Benacerraf, Mathematical truth, The Journal of Philosophy, 70(19), pp. 661–679, 1973. [2] L. San Mauro, G. Venturi, Towards a theory of speech acts in mathematics: the case of naturalness, submitted. [3] L. San Mauro, G. Venturi, Speech acts in mathematics: a manifesto, in preparation. [4] J. R. Searle, Speech acts: An essay in the philosophy of language, volume 626. Cambridge University Press, 1969. 
"Speech Acts in Mathematics"
