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Seminario di Logica e Filosofia della Scienza

Organizzazione Unità di ricerca LOG-LAB

 

 

Venerdì, 25 novembre 2022 ore 11

DILEF - Dipartimento di Lettere e Filosofia
via della Pergola 60, Firenze
Sala Altana


Giulio Fellin (Università di Verona)

Glivenko sequent classes and constructive cut elimination in geometric logics
(joint work with S. Negri, E. Orlandelli)

 

Abstract: A constructivisation of the cut-elimination proof for sequent calculi for classical, intuitionistic and minimal infinitary logics with geometric rules—given in earlier work by Sara Negri—is presented. This is achieved through a procedure where the non-constructive transfinite induction on the commutative sum of ordinals is replaced by two instances of Brouwer’s Bar Induction. The proof of admissibility of the structural rules is made ordinal-free by introducing a new well-founded parameter called proof embeddability. Additionally, conservativity for classical over intuitionistic/minimal logic for the seven (finitary) Glivenko sequent classes is here shown to hold also for the corresponding infinitary classes.

 

→ locandina


Venerdì, 11 novembre 2022 ore 11.00
Aula 6, Plesso didattico via Capponi 9, Firenze

 

Peter Schuster (Università di Verona)

Computing with ideal objects
(based on joint work with Ingo Blechschmidt, Universität Augsburg)

 

Abstract: Gödel's completeness theorem for arbitrary languages of first-order predicate logic is tied together with the Boolean ultrafilter theorem. In countable languages Lindenbaum's lemma works without transfinite methods: to construct a maximal consistent theory one  gradually extends any given consistent theory as long as consistency permits. Krivine could even do this without having to decide (in)consistency at every extension step; Berardi and Valentini have shown that similar strategies work in related contexts. While the latter doubted whether the resulting ideal objects would be of use for computation, Lombardi argues that their existence means no more than consistency of the given theory. We will address these issues, including the perspective of a working mathematician.

 

→ locandina


Venerdì, 28 ottobre 2022 ore 11.00

DILEF - Dipartimento di Lettere e Filosofia
via della Pergola 60, Firenze
Sala Altana

 

Laura Crosilla (University of Oslo)

On Weyl's predicative concept of set

 

Abstract: In the book Das Kontinuum (1918), Hermann Weyl presents a coherent and sophisticated approach to analysis from a predicativist perspective. In the first chapter of (Weyl 1918), Weyl introduces a predicative concept of set, according to which sets are built `from the bottom up' starting from the natural numbers. Weyl's concept of set is a variant of the traditional logical concept of set, for which a set is the extension of some concept. Weyl clearly contrasts this predicative concept of set with the concept of arbitrary set, which he finds wanting, especially when working with infinite sets.  In the second chapter of Das Kontinuum, he goes on to show that large portions of 19th century analysis can be developed on the basis of his predicative concept of set.
Das Kontinuum anticipated and inspired fundamental ideas in mathematical logic, especially the logical analysis of predicativity of the 1950-60's, Solomon Feferman’s work on predicativity and Errett Bishop’s constructive mathematics. The seeds of Das Kontinuum are already visible in the early (Weyl 1910), where Weyl, among other things, offers a clarification of Zermelo’s axiom schema of Separation.
In this talk, I examine Weyl's predicative concept of set in (Weyl 1918) and discuss its origins in (Weyl 1910).
References
- Weyl, H., 1910, Über die Definitionen der mathematischen Grundbegriffe, Mathematischnaturwissenschaftliche Blätter, 7, pp. 93–95 and pp. 109–113.
- Weyl, H., 1918, Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis, Veit, Leipzig. Translated in English, Dover Books on Mathematics, 2003.

 

locandina


 

Martedì, 17 maggio 2022 ore 15.30

Aula 7, plesso didattico di via Capponi 9

 

Marko Malink (New York University)

Boole, Peirce, and Schröder on Expressing Particular Propositions

Abstract:In his Laws of Thought, George Boole set out to represent Aristotle's syllogistic theory in a purely equational system of algebraic logic lacking propositional negation. In doing so, he faced the problem of how to express particular propositions of the form "Some A is B". Boole attempted to solve this problem by introducing what he called "indefinite symbols". Peirce and Schröder criticized this attempt and provided different solutions to the problem based on the theory of relations (Peirce) or propositional logic (Schröder). I will examine each of these approaches, and discuss how the debate about the representation of Aristotelian particular propositions shaped the development of algebraic logic in the second half of the 19th century. I will conclude by considering how the problem of expressing particular propositions is addressed in Alfred Tarski's calculus of relations.

 

Locandina


 

Giovedì, 12 maggio 2022 ore 16:00
Plesso didattico di via Capponi 9, Aula 6

 

Marko Malink (New York University)

The Origins of Conditional Logic: Theophrastus on Hypothetical Syllogisms

 

Abstract: Ancient Peripatetic logicians such as Theophrastus sought to establish the priority of categorical over propositional logic by reducing various modes of propositional reasoning to categorical form. To this end, Theophrastus proposed that the conditional "If φ then ψ" should be interpreted as a categorical proposition "A holds of all B", in which B corresponds to the antecedent φ, and A to the consequent ψ. Under this interpretation, Aristotle’s law of subalternation (A holds of all B, therefore A holds of some B) corresponds to a version of Boethius' Thesis (If φ then ψ, therefore not: If φ then not-ψ). Jonathan Barnes has argued that this consequence renders Theophrastus' program of reducing propositional to categorical logic inconsistent. In this paper, we will challenge Barnes's verdict. We will argue that the system of conditional logic developed by Theophrastus is both non-trivial and consistent. Theophrastus achieves such consistency by limiting the system to first-degree conditionals, in which both the antecedent and the consequent are simple categorical propositions.

 

Locandina


Venerdì, 6 maggio 2022 ore 11:00

Plesso didattico di via Capponi 9 - Aula 7

 

Paolo Maffezioli (Università di Torino)

Aristotle's arithmetical dictum

Abstract: In Prior Analytics Aristotle explains the universal predication by resorting to an intuitively clear, albeit quite elusive, mereological notion - that of being in the whole. Building on previous scholarly work on the mathematical origins of Aristotle's syllogistic, it is suggested that for Aristotle a universal affirmative proposition such as 'x is predicated of all y' is true if, and only if, y is a divisor of x. The adequacy of such an interpretation is assessed both theoretically and textually with respect to the other mereological interpretations of the dictum. This is a joint work with Riccardo Zanichelli.


Giovedì, 21 aprile 2022 ore 11:30

plesso didattico di via Capponi 9 - Aula 6

 

Jan Sprenger (Università di Torino)

Trivalent Logics for Indicative Conditionals

Abstract: There are two major research projects on indicative conditionals: the semantic project of determining their truth conditions, and the epistemological project of explaining how we should reason with them. This paper integrates both projects on the basis of a trivalent, truth-functional account of the truth conditions of conditionals. On the basis of these  semantics, we generate two logics for reasoning with conditionals: (i) a conditional logic of deductive inference (i.e., with certain premises) that generalizes classical logic; and (ii) a conditional logic of probabilistic inference (i.e., with uncertain premises) that generalizes Adams' logic to arbitrary compounds of conditionals. We then apply both logics to the controversies about the validity of Modus Ponens, Import-Export, and the various triviality results for conditionals. The end product is a unified theory of the truth conditions and probability of conditionals that makes sensible predictions for natural language reasoning.

The talk based on joint research with Paul Egré and Lorenzo Rossi. Preliminary work to this research has been published in the /Journal of Philosophical Logic/ :
https://link.springer.com/article/10.1007/s10992-020-09549-6 (open access). The talk will briefly summarize our work from that paper before presenting new results.


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Ultimo aggiornamento

16.11.2022

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