Implication, Equivalence, and Negation

A system $HCL_{\overset{\neg}{\leftrightarrow}}$ in the language of {$ \neg, \leftrightarrow $} is obtained by adding a single negation-less axiom schema to $HLL_{\overset{\neg}{\leftrightarrow}}$ (the standard Hilbert-type system for multiplicative linear logic without propositional constants), and changing $ \rightarrow $ to $\leftrightarrow$. $HCL_{\overset{\neg}{\leftrightarrow}}$ is weakly, but not strongly, sound and complete for ${\bf CL}_{\overset{\neg}{\leftrightarrow}}$ (the {$ \neg,\leftrightarrow$} – fragment of classical logic). By adding the Ex Falso rule to $HCL_{\overset{\neg}{\leftrightarrow}}$ we get a system with is strongly sound and complete for ${\bf CL}_ {\overset{\neg}{\leftrightarrow}}$ . It is shown that the use of a new rule cannot be replaced by the addition of axiom schemas. A simple semantics for which $HCL_{\overset{\neg}{\leftrightarrow}}$ itself is strongly sound and complete is given. It is also shown that $L_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ , the logic induced by $HCL_{\overset{\neg}{\leftrightarrow}}$ , has a single non-trivial proper axiomatic extension, that this extension and ${\bf CL}_{\overset{\neg}{\leftrightarrow}}$ are the only proper extensions in the language of { $\neg$, $\leftrightarrow$ } of $ {\bf L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ , and that $ {\bf L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ and its single axiomatic extension are the only logics in {$ \neg, \leftrightarrow$ } which have a connective with the relevant deduction property, but are not equivalent $\neg$ to an axiomatic extension of ${\bf R}_{\overset{\neg}{\leftrightarrow}}$ (the intensional fragment of the relevant logic ${\bf R}$). Finally, we discuss the question whether $ {\bf L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ can be taken as a paraconsistent logic.

Dynamical Phenomena and Their Models: Truth and Empirical Correctness

In the epistemological tradition, there are two main interpretations of the semantic relation that an empirical theory may bear to the real world. According to realism, the theory-world relationship should be conceived as truth; according to instrumentalism, instead, it should be limited to empirical adequacy. Then, depending on how empirical theories are conceived, either syntactically as a class of sentences, or semantically as a class of models, the concepts of truth and empirical adequacy assume different and specific forms. In this paper, we review two main conceptions of truth (one sentence-based and one model-based) and two of empirical adequacy (one sentence-based and one model-based), we point out their respective difficulties, and we give a first formulation of a new general view of the theory-world relationship, which we call Methodological Constructive Realism (MCR). We then show how the content of MCR can be further specified and expressed in a definite and precise form. The bulk of the paper shows in detail how it is possible to accomplish this goal for the special case of deterministic dynamical phenomena and their correlated deterministic models. This special version of MCR is formulated as an axiomatic extension of set theory, whose specific axioms constitute a formal ontology that provides an adequate framework for analyzing the two semantic relations of truth and empirical correctness, as well as their connections.

Four-valued expansions of Dunn-Belnap's logic (I): Basic characterizations

Basic results of the paper are that any four-valued expansion L4 of Dunn-Belnap's logic DB4 is defined by a unique (up to isomorphism) conjunctive matrix with exactly two distinguished values over an expansion 𝔄4 of a De Morgan non-Boolean four-valued diamond, but by no matrix with either less than four values or a single [non-]distinguished value, and has no proper extension satisfying Variable Sharing Property (VSP). We then characterize L4's having a theorem / inconsistent formula, satisfying VSP and being [inferentially] maximal / subclassical / maximally paraconsistent, in particular, algebraically through ℳ4|𝔄4's (not) having certain submatrices|subalebras. Likewise, [providing 𝔄4 is regular / has no three-element subalgebra] L4 has a proper consistent axiomatic extension if[f] ℳ4 has a proper paraconsistent / two-valued submatrix [in which case the logic of this submatrix is the only proper consistent axiomatic extension of L4 and is relatively axiomatized by the Excluded Middle law axiom]. As a generic tool (applicable, in particular, to both classically-negative and implicative expansionsof DB4), we also prove that the lattice of axiomatic extensions of the logic of an implicative matrix ℳ with equality determinant is dual to the distributive lattice of lower cones of the set of all submatrices of ℳ with non-distinguished values.

First-order swap structures semantics for some logics of formal inconsistency

The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (i.e. logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous approaches to quantified LFIs presented in the literature. The case of QmbC, the simpler quantified LFI expanding classical logic, will be analyzed in detail. An axiomatic extension of QmbC called $\textbf{QLFI1}_\circ $ is also studied, which is equivalent to the quantified version of da Costa and D’Ottaviano 3-valued logic J3. The semantical structures for this logic turn out to be Tarkian structures based on twist structures. The expansion of QmbC and $\textbf{QLFI1}_\circ $ with a standard equality predicate is also considered.

PROOF SYSTEMS FOR VARIOUS FDE-BASED MODAL LOGICS

We present novel proof systems for various FDE-based modal logics. Among the systems considered are a number of Belnapian modal logics introduced in Odintsov & Wansing (2010) and Odintsov & Wansing (2017), as well as the modal logic KN4 with strong implication introduced in Goble (2006). In particular, we provide a Hilbert-style axiom system for the logic $BK^{\square - } $ and characterize the logic BK as an axiomatic extension of the system $BK^{FS} $. For KN4 we provide both an FDE-style axiom system and a decidable sequent calculus for which a contraction elimination and a cut elimination result are shown.

An Extension of the Solidarity Value for Environments with Externalities

In this paper, we propose an axiomatic extension for the Solidarity value of Nowak and Radzik [1994] A solidarity value for [Formula: see text]-person transferable utility games, Int. J. Game Theor. 23, 43–48] to the class of games with externalities. This value is characterized as the unique function that satisfies linearity, symmetry, efficiency and average nullity. In this context, we discuss a key subject of how to extend the concept of average marginal contribution to settings where externalities are present.

Discriminator logics (Research announcement)

A discriminator logic is the 1-assertional logic of a discriminator variety V having two constant terms 0 and 1 such that V ⊨ 0 1 iff every member of V is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system SBPC, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator logic S can be presented (up to definitional equivalence) as an axiomatic extension of SBPC by a set of extensional logical connectives taken from the language of S. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work.

THE CONSENSUS VALUE FOR GAMES IN PARTITION FUNCTION FORM

This paper studies a procedural and axiomatic extension of the consensus value [cf. Ju et al. (2007)] to the class of partition function form games. This value is characterized as the unique function that satisfies efficiency, complete symmetry, the quasi-null player property and additivity. By means of the transfer property, a second characterization is provided. Moreover, it is shown that the consensus value satisfies individual rationality under a superadditivity condition, and well balances the tradeoff between coalitional effects and externality effects. In this respect, explicit differences with other solution concepts are indicated.