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"Piazza, Mario"
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Fractional-Valued Modal Logic and Soft Bilateralism
In a recent paper, under the auspices of an unorthodox variety of bilateralism, we introduced a new kind of proof-theoretic semantics for the base modal logic \\(\\mathbf{K}\\), whose values lie in the closed interval \\([0,1]\\) of rational numbers [14]. In this paper, after clarifying our conception of bilateralism – dubbed “soft bilateralism” – we generalize the fractional method to encompass extensions and weakenings of \\(\\mathbf{K}\\). Specifically, we introduce well-behaved hypersequent calculi for the deontic logic \\(\\mathbf{D}\\) and the non-normal modal logics \\(\\mathbf{E}\\) and \\(\\mathbf{M}\\) and thoroughly investigate their structural properties.
Journal Article
Palazzo della civiltلa Italiana
In 2015, the storied fashion house Fendi moved its headquarters into the Palazzo della Civilta Italiana in Rome, a stark white cube perforated by symmetrical arches. Originally commissioned as part of an exhibition on Roman civilization for the 1942 world's fair, the architects took their cues from ancient history to create a building that was quintessentially Roman yet decidedly modern, earning its nickname the Square Colosseum. Because of its striking appearance and iconic status, the palazzo subsequently made appearances in a number of films by directors such as Roberto Rossellini, Federico Fellini, and Peter Greenaway. The building remained relatively abandoned throughout much of its existence, until its recent inhabitance by the always forward-thinking house of Fendi, an experience which Karl Lagerfeld has likened to being on a spaceship transported into the future.
Book
The Implicit Commitment of Arithmetical Theories and Its Semantic Core
According to the implicit commitment thesis, once accepting a mathematical formal system S, one is implicitly committed to additional resources not immediately available in S. Traditionally, this thesis has been understood as entailing that, in accepting S, we are bound to accept reflection principles for S and therefore claims in the language of S that are not derivable in S itself. It has recently become clear, however, that such reading of the implicit commitment thesis cannot be compatible with well-established positions in the foundations of mathematics which consider a specific theory S as self-justifying and doubt the legitimacy of any principle that is not derivable in S: examples are Tait's finitism and the role played in it by Primitive Recursive Arithmetic, Isaacson's thesis and Peano Arithmetic, Nelson's ultrafinitism and sub-exponential arithmetical systems. This casts doubts on the very adequacy of the implicit commitment thesis for arithmetical theories. In the paper we show that such foundational standpoints are nonetheless compatible with the implicit commitment thesis. We also show that they can even be compatible with genuine soundness extensions of S with suitable form of reflection. The analysis we propose is as follows: when accepting a system S, we are bound to accept a fixed set of principles extending S and expressing minimal soundness requirements for S, such as the fact that the non-logical axioms of S are true. We call this invariant component the semantic core of implicit commitment. But there is also a variable component of implicit commitment that crucially depends on the justification given for our acceptance of S in which, for instance, may or may not appear (proof-theoretic) reflection principles for S. We claim that the proposed framework regulates in a natural and uniform way our acceptance of different arithmetical theories.
Journal Article
FRACTIONAL SEMANTICS FOR CLASSICAL LOGIC
This article presents a new (multivalued) semantics for classical propositional logic. We begin by maximally extending the space of sequent proofs so as to admit proofs for any logical formula; then, we extract the new semantics by focusing on the axiomatic structure of proofs. In particular, the interpretation of a formula is given by the ratio between the number of identity axioms out of the total number of axioms occurring in any of its proofs. The outcome is an informational refinement of traditional Boolean semantics, obtained by breaking the symmetry between tautologies and contradictions.
Journal Article
Abduction as Deductive Saturation: a Proof-Theoretic Inquiry
Abductive reasoning involves finding the missing premise of an “unsaturated” deductive inference, thereby selecting a possible
explanans
for a conclusion based on a set of previously accepted premises. In this paper, we explore abductive reasoning from a structural proof-theory perspective. We present a hybrid sequent calculus for classical propositional logic that uses sequents and antisequents to define a procedure for identifying the set of analytic hypotheses that a rational agent would be expected to select as
explanans
when presented with an abductive problem. Specifically, we show that this set may not include the deductively minimal hypothesis due to the presence of redundant information. We also establish that the set of all analytic hypotheses exhausts all possible solutions to the given problem. Finally, we propose a deductive criterion for differentiating between the best
explanans
candidates and other hypotheses.
Journal Article
NON-CONTRACTIVE LOGICS, PARADOXES, AND MULTIPLICATIVE QUANTIFIERS
The paper investigates from a proof-theoretic perspective various non-contractive logical systems, which circumvent logical and semantic paradoxes. Until recently, such systems only displayed additive quantifiers (Grišin and Cantini). Systems with multiplicative quantifiers were proposed in the 2010s (Zardini), but they turned out to be inconsistent with the naive rules for truth or comprehension. We start by presenting a first-order system for disquotational truth with additive quantifiers and compare it with Grišin set theory. We then analyze the reasons behind the inconsistency phenomenon affecting multiplicative quantifiers. After interpreting the exponentials in affine logic as vacuous quantifiers, we show how such a logic can be simulated within a truth-free fragment of a system with multiplicative quantifiers. Finally, we establish that the logic for these multiplicative quantifiers (but without disquotational truth) is consistent, by proving that an infinitary version of the cut rule can be eliminated. This paves the way to a syntactic approach to the proof theory of infinitary logic with infinite sequents.
Journal Article
FRACTIONAL-VALUED MODAL LOGIC
This paper is dedicated to extending and adapting to modal logic the approach of fractional semantics to classical logic. This is a multi-valued semantics governed by pure proof-theoretic considerations, whose truth-values are the rational numbers in the closed interval
$[0,1]$
. Focusing on the modal logic K, the proposed methodology relies on three key components: bilateral sequent calculus, invertibility of the logical rules, and stability (proof-invariance). We show that our semantic analysis of K affords an informational refinement with respect to the standard Kripkean semantics (a new proof of Dugundji’s theorem is a case in point) and it raises the prospect of a proof-theoretic semantics for modal logic.
Journal Article
Possibilities regained: neo-Lewisian contextualism and ordinary life
According to David Lewis, the predicate ‘knows’ is context-sensitive in the sense that its truth conditions vary across conversational contexts, which stretch or compress the domain of error possibilities to be eliminated by the subject’s evidence (Lewis, Aust J Philos 74:549–567,
1996
; Lewis, J Philos Log 8:339–359,
1979
). Our concern in this paper is to thematize, assess, and overcome within a neo-Lewisian contextualist project two important mismatches between our use of ‘know’ in ordinary life and the use of ‘know’ by ‘Lewisian’ ordinary speakers. The first mismatch is that Lewisian contextualism still
overgenerates
the error possibilities which cannot be ignored in a given context, since it is oblivious to the distinction between ‘invented’ and ‘discovered’ possibilities. The second mismatch is a full-scale one: an adequate account of knowledge attribution is not exhausted by the subject’s
negative
capacity of pruning branches off the tree of counterpossibilities. We therefore introduce a new vector of value, which explains how ‘know’ comes in degrees: the satisfaction of ‘know better’ is made to depend on the capacity of
imagining
(actualized) possibilities connected in a relevant way with the subject’s (true) beliefs.
Journal Article
Molecular Biology Meets Logic: Context-Sensitiveness in Focus
Some real life processes, including molecular ones, are
context-sensitive
, in the sense that their outcome depends on side conditions that are most of the times difficult, or impossible, to express fully in advance. In this paper, we survey and discuss a logical account of context-sensitiveness in molecular processes, based on a kind of non-classical logic. This account also allows us to revisit the relationship between logic and philosophy of science (and philosophy of biology, in particular).
Journal Article