Axiomatic Extensions of IMT3 Logic

In this paper we characterize, classify and axiomatize all axiomatic extensions of the IMT3 logic. This logic is the axiomatic extension of the involutive monoidal t-norm logic given by ¬φ³ ∨ φ. For our purpose we study the lattice of all subvarieties of the class IMT3, which is the variety of IMTL-algebras given by the equation ¬(x³) ∨ x ≈ ?, and it is the algebraic counterpart of IMT3 logic. Since every subvariety of IMT3 is generated by their totally ordered members, we study the structure of all IMT3-chains in order to determine the lattice of all subvarieties of IMT3. Given a family of IMT3-chains the number of elements of the largest odd finite subalgebra in the family and the number of elements of the largest even finite subalgebra in the family turns out to be a complete classifier of the variety generated. We obtain a canonical set of generators and a finite equational axiomatization for each subvariety and, for each corresponding logic, a finite set of characteristic matrices and a finite set of axioms.     

Interpolation in Extensions of First-Order Logic

We 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.

     

On the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics.     

First-Order Dialogical Games and Tableaux

We present a new proof of soundness/completeness of tableaux with respect to dialogical games in Classical First-Order Logic. As far as we know it is the first thorough result for dialogical games where finiteness of plays is guaranteed by means of what we call repetition ranks.     

Axiomatic quantum mechanics and radioactive decay

This paper explores the consequences of the orthodox resolution of the measurement problem for the axiomatic base of non-relativistic elementary quantum mechanics. It is argued that the standard resolution of the measurement problem generates a paradox whose dissolution may be achieved through an enrichment of the axiomatic foundations of quantum mechanics. These results are also linked to some recent creative proposals by Nancy Cartwright concerning the nature of the so-called reduction of the wave packet.     

An Axiomatic Theory of Law

This paper presents in outline Luigi Ferrajoli’s axiomatic and general theory of law, as developed in his lifelong work Principia Iuris. The first section focuses on the three main aspects of the theory: the methodological, the theoretical and the pragmatic, which respectively represent the theory’s syntax, semantics and its pragmatics. Ferrajoli identifies three deontic gaps of norms: firstly, the one between their validity and efficacy; secondly, the one between their justice and validity; and finally, and most importantly, the one between validity and existence (i.e. normative force). The presence of such gaps is, according to Ferrajoli, the extraordinary innovation that entrenched constitutions have brought into modern legal systems, by establishing norms that are superior to statutes and case law. In this sense, all normative phenomena (except for the constitution itself) can be conceived both as norms and as facts. In the second section the role of juridical science is briefly discussed.     

Equality and Monodic First-Order Temporal Logic

It has been shown recently that monodic first-order temporal logic without functional symbols but with equality is incomplete, i.e., the set of the valid formulae of this logic is not recursively enumerable. In this paper we show that an even simpler fragment consisting of monodic monadic two-variable formulae is not recursively enumerable.     

First-Order Theories of Orthogonality Structures

We investigate certain aspects of the first-order theory oforthogonality structures - structures consisting of a domainof lines subject to a binary orthogonality relation. In particular,we establish definitions of various geometric and algebraicnotions in terms of orthogonality, describe the constructionof extremal subspaces using orthogonality, and show that thefirst-order theory of line orthogonality in the Euclidean n-spaceis not ₀-categorical for n 3.     

Axiomatic Theories of Partial Ground II

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our theory is a proof-theoretically conservative extension of the ramified theory of positive truth up to ?? ₀ and thus is consistent. We argue that this theory provides a natural solution to Fine’s “puzzle of ground” about the interaction of truth and ground. Finally, we show that if we apply the truth-predicate to sentences involving our ground-predicate, we run into paradoxes similar to the semantic paradoxes: we get ground-theoretical paradoxes of self-reference.     

Axiomatic Theories of Partial Ground I

This is part one of a two-part paper, in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. This allows us to connect theories of partial ground with axiomatic theories of truth. In this part of the paper, we develop an axiomatization of the relation of partial ground over the truths of arithmetic and show that the theory is a proof-theoretically conservative extension of the theory PT of positive truth. We construct models for the theory and draw some conclusions for the semantics of conceptualist ground.     

Embedding Friendly First-Order Paradefinite and Connexive Logics

Journal of Philosophical Logic - First-order intuitionistic and classical Nelson–Wansing and Arieli–Avron–Zamansky logics, which are regarded as paradefinite and connexive logics,...     

Remarks on Compositionality and Weak Axiomatic Theories of Truth

The paper draws attention to an important, but apparently neglected distinction relating to axiomatic theories of truth, viz. the distinction between weakly and strongly truth-compositional theories of truth. The paper argues that the distinction might be helpful in classifying weak axiomatic theories of truth and examines some of them with respect to it.     

The First-Order Theories of Dedekind Algebras

A Dedekind Algebra is an ordered pair (B,h) where B is a non-empty set and h is an injective unary function on B. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called configurations of the Dedekind algebra. There are N₀ isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on omega called its configuration signature. The configuration signature of a Dedekind algebra counts the number of configurations in the decomposition of the algebra in each isomorphism type.The configuration signature of a Dedekind algebra encodes the structure of that algebra in the sense that two Dedekind algebras are isomorphic iff their configuration signatures are identical. Configuration signatures are used to establish various results in the first-order model theory of Dedekind algebras. These include categoricity results for the first-order theories of Dedekind algebras and existence and uniqueness results for homogeneous, universal and saturated Dedekind algebras. Fundamental to these results is a condition on configuration signatures that is necessary and sufficient for elementary equivalence.     

Variations and Extensions of Filial Therapy

Various procedures for training parents to be play therapists for their own young children are described. In their play encounters with the child, not only is each of the parents learning ways of being with the child that are indicative of empathy and caring, but they are also learning that the way each acts toward the child in the playroom (and in the home) is reflective of their own conflicts as persons and as spouses.