This paper exhibits a general and uniform method to prove axiomatic completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ, where x occurs only positively in γ, we obtain the flat modal fixpoint language by adding to the language of polymodal logic a connective γ for each γΓ. The term γ is meant to be interpreted as the least fixed point of the functional interpretation of the term γ. We consider the following (...) problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language on Kripke structures. We prove two results that solve this problem.First, let be the logic obtained from the basic polymodal by adding a Kozen–Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective γ. Provided that each indexing formula γ satisfies a certain syntactic criterion, we prove this axiom system to be complete.Second, addressing the general case, we prove the soundness and completeness of an extension of . This extension is obtained via an effective procedure that, given an indexing formula γ as input, returns a finite set of axioms and derivation rules for γ, of size bounded by the length of γ. Thus the axiom system is finite whenever Γ is finite. (shrink)

A μ-algebra is a model of a first-order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms where μx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications.Standard μ-algebras are complete meaning that their lattice reduct is a complete lattice. We prove that any nontrivial quasivariety of μ-algebras contains a μ-algebra that has no embedding into a complete μ-algebra.We then focus on modal μ-algebras, i.e. algebraic (...) models of the propositional modal μ-calculus. We prove that free modal μ-algebras satisfy a condition–reminiscent of Whitman’s condition for free lattices–which allows us to prove that modal operators are adjoints on free modal μ-algebras, least prefixed points of Σ1-operations satisfy the constructive relation μx.f=logical and operatorn≥0fn. These properties imply the following statement: the MacNeille–Dedekind completion of a free modal μ-algebra is a complete modal μ-algebra and moreover the canonical embedding preserves all the operations in the class image of the fixed point alternation hierarchy. (shrink)

Parity games are combinatorial representations of closed Boolean μ-terms. By adding to them draw positions, they have been organized by Arnold and Santocanale [3] and [27] into a μ-calculus whose standard interpretation is over the class of all complete lattices. As done by Berwanger et al. [8] and [9] for the propositional modal μ-calculus, it is possible to classify parity games into levels of a hierarchy according to the number of fixed-point variables. We ask whether this hierarchy collapses w.r.t. the (...) standard interpretation. We answer this question negatively by providing, for each n≥1, a parity game Gn with these properties: it unravels to a μ-term built up with n fixed-point variables, it is not semantically equivalent to any game with strictly less than n−2 fixed-point variables. (shrink)

This book focuses on analyzing data on current initiatives and best practices in citizen engagement and education programs across various disciplines by highlighting emergent research and application techniques within citizen science initiatives.

This article discusses the logic of the Epimenides Paradox using a graphical representation of the logical operators involved. The graphical representation of the Logical Operators is the one used in digital electronics. This method of investigation is very effective in revealing the hidden error in the reasoning that leads to the contradiction. The article explains the vicious circle that the mind goes through when trying to analyze the Paradox. Finally, the article proposes a solution of the Paradox with final conclusions (...) that are not paradoxical but simply counterintuitive. The endless deduction is generated by subsequent evaluations that take place on different levels. In its evaluation the Mind alternates a level of logical evaluation with a level of semantic evaluation. (shrink)

"Results presented in two international seminars held in Palermo and Milan in 2012"--Page 1.

The present book is a collection of 9 essays, focusing on the ethical, religious, and political aspects of Wittgenstein's thought, illustrated and investigated with reference to their complex interaction with Wittgenstein's philosophical method and with his conception of language and human agency.

Principi e ordinamento romano : una riflessione sulle orme di Fritz Schulz -- Homo liber e homo sacer : due archetipi dell'appartenenza -- L'humanitas tra diritto romano e totalitarismo hitleriano -- Sull'eccezione di dolo generale -- L'arbitraggio sul prezzo -- Diritto romano e scienza del diritto -- Roma e i suoi giuristi nel pensiero di Nicolás Gómez Dávila -- Tradizione e misura umana del diritto: considerazioni sulla figura del giurista a partire da Giuseppe Grosso.

The interest in Andronikos Kallistos, a leading personality among the Greek émigrés who participated in Italian Humanism, arose at the end of the nineteenth century within the frame of the studies on Byzantine scholars of the Renaissance. Researchers have only glimpsed the depth of Kallistos' erudite personality. To date, nearly 130 manuscripts have been found bearing evidence of his work as a copyist and philologist. However, research into both his scribal and scholarly activity remains fragmented into many isolated contributions, mainly (...) concerning specific chapters of the manuscript tradition of classical Greek authors. Adopting a synergistic approach to historical, philological, codicological, and paleographic data within this framework, this monograph study aims to fulfil the following tasks: outlining an updated biography; defining Kallistos' scribal activity better by means of a thorough examination of all surviving manuscript sources; attempting to reconstruct the development of his book collection; acknowledging Kallistos' scholarly activity both as a teacher and philologist; making an inventory of all the manuscripts which bear traces of his writing; and, finally, publishing Kallistos' works. (shrink)