1,720,989 research outputs found
Eugenio Orlandelli e Giovanna Corsi, Corso di logica modale proposizionale, Carocci editore, Roma 2019, pp. 193.
No full textIn questa Lettura Critica viene analizzato il volume di E. Orlandelli e G. Corsi "Corso di Logica Modale Proposizionale", trattandone il contenuto anche in relazione ad altri manuali di testo sull’argomento. Merito di questo testo è quello di unire ad una parte “standard”, in cui i temi classici delle logiche modali sono introdotti in modo particolarmente chiaro (semantiche relazionali, calcoli alla Hilbert, modelli canonici, decidibilità via metodo delle filtrazioni), una parte più "innovativa", nella quale sono trattate efficaci tecniche di dimostrazione per le logiche modali (calcoli dei sequenti etichettati e metodo dei diagrammi)
Paradigmi di computazione per i numeri reali
No full textIn questo articolo analizziamo e compariamo i diversi modelli di computazione per i numeri reali presentati in letteratura
La matematica della mente: il pensiero come calcolo in Hobbes e Boole
No full textIn questo articolo confrontiamo la trattazione della logica in Hobbes e Boole
Alan Turing and the foundations of computable analysis
No full textWe investigate Turing’s contributions to computability theory for real numbers
and real functions presented in [22, 24, 26]. In particular, it is shown how two fundamental
approaches to computable analysis, the so-called ‘Type-2 Theory of Effectivity’ (TTE) and
the ‘realRAM machine’ model, have their foundations in Turing’s work, in spite of the two
incompatible notions of computability they involve. It is also shown, by contrast, how the
modern conceptual tools provided by these two paradigms allow a systematic interpretation
of Turing’s pioneering work in the subject
Borel complexity of topological operations on computable metric spaces
No full textWe study the Borel complexity of topological operations on
closed subsets of computable metric spaces. The investigated operations
include set theoretic operations as union and intersection, but also typical
topological operations such as the closure of the complement, the
closure of the interior, the boundary and the derivative of a set. These
operations are studied with respect to different computability structures
on the hyperspace of closed subsets. These structures include positive
or negative information on the represented closed subsets. Topologically,
they correspond to the lower or upper Fell topology, respectively, and
the induced computability concepts generalize the classical notions of
r.e. or co-r.e. subsets, respectively. The operations are classified with respect
to effective measurability in the Borel hierarchy and it turns out
that most operations can be located in the first three levels of the hierarchy,
or they are not even Borel measurable at all. In some cases the
effective Borel measurability depends on further properties of the underlying
metric spaces, such as effective local compactness and effective
local connectedness
Non-Normal Super-Strict Implications
Full text linkThis paper introduces the logics of super-strict implications that are based on C.I. Lewis’ non-normal modal logics S2 and S3. The semantics of these logics is based on Kripke’s semantics for non-normal modal logics. This solves a question we left open in a previous paper by showing that these logics are weakly connexive
Computability and incomputability of differential equations
No full textThe paper analyzes the fundamental results about the computational properties of differential equations, in particular about the apparent incomputability of the wave equation system
Weihrauch degrees, omniscience principles and weak computability
No full textIn this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more
precisely, a natural extension for multi-valued functions on represented spaces. We call the corresponding
equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower
semi-lattice. It turns out that parallelization is a closure operator for this semi-lattice and that the par-
allelized Weihrauch degrees even form a lattice into which the Medvedev lattice and the Turing degrees
can be embedded. The importance of Weihrauch degrees is based on the fact that multi-valued functions
on represented spaces can be considered as realizers of mathematical theorems in a very natural way and
studying the Weihrauch reductions between theorems in this sense means to ask which theorems can be
transformed continuously or computably into each other. As crucial corner points of this classification
scheme the limited principle of omniscience LPO, the lesser limited principle of omniscience LLPO and
their parallelizations are studied. It is proved that parallelized LLPO is equivalent to Weak König’s Lemma
and hence to the Hahn–Banach Theorem in this new and very strong sense. We call amulti-valued function
weakly computable if it is reducible to the Weihrauch degree of parallelized LLPO and we present a new
proof, based on a computational version of Kleene’s ternary logic, that the class of weakly computable
operations is closed under composition. Moreover, weakly computable operations on computable metric
spaces are characterized as operations that admit upper semi-computable compact-valued selectors and it
is proved that any single-valued weakly computable operation is already computable in the ordinary sense
Interpolation in Extensions of First-Order Logic
Full text linkWe prove a generalization of Maehara’s lemma to show that the extensions
of classical and intuitionistic first-order logic with a special type of geometric axioms,
called singular geometric axioms, have Craig’s interpolation property. As a corollary, we
obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with
identity, as well as interpolation for several mathematical theories, including the theory
of equivalence relations, (strict) partial and linear orders, and various intuitionistic order
theories such as apartness and positive partial and linear orders
Super-Strict Implications
Full text linkThis paper introduces the logics of super-strict implications, where a super-strictimplication is a strengthening of C.I. Lewis’ strict implication that avoids not onlythe paradoxes of material implication but also those of strict implication. Thesemantics of super-strict implications is obtained by strengthening the (normal)relational semantics for strict implication. We consider all logics of super-strictimplications that are based on relational frames for modal logics in the modalcube. it is shown that all logics of super-strict implications are connexive logicsin that they validate Aristotle’s Theses and (weak) Boethius’s Theses. A proof-theoretic characterisation of logics of super-strict implications is given by meansof G3-style labelled calculi, and it is proved that the structural rules of inferenceare admissible in these calculi. It is also shown that validity in theS5-basedlogic of super-strict implications is equivalent to validity in G. Priest’s negation-as-cancellation-based logic. Hence, we also give a cut-free calculus for Priest’slogic
- …
