This paper presents a way of formalising definite descriptions with a binary quantifier ι, where ιx[F, G] is read as ‘The F is G’. Introduction and elimination rules for ι in a system of intuitionist negative free logic are formulated. Procedures for removing maximal formulas of the form ιx[F, G] are given, and it is shown that deductions in the system can be brought into normal form.

(2012). Modal logics for reasoning about infinite unions and intersections of binary relations. Journal of Applied Non-Classical Logics: Vol. 22, No. 4, pp. 275-294. doi: 10.1080/11663081.2012.705960.

This paper presents rules in sequent calculus for a binary quantifier I to formalise definite descriptions: Ix[F, G] means ‘The F is G’. The rules are suitable to be added to a system of positive free logic. The paper extends the proof of a cut elimination theorem for this system by Indrzejczak by proving the cases for the rules of I. There are also brief comparisons of the present approach to the more common one that formalises definite descriptions (...) with a term forming operator. In the final section rules for I for negative free and classical logic are also mentioned. (shrink)

Standard unary modal logic and binary modal logic, i.e. modal logic with one binary operator, are shown to be definitional extensions of one another when an additional axiom |$U$| is added to the basic axiomatization of the binary side. This is a strengthening of our previous results. It follows that all unary modal logics extending Classical Modal Logic, in other words all unary modal logics with a neighborhood semantics, can equivalently be seen as (...) binary modal logics. This in particular applies to standard modal logics, which can be given simple natural axiomatizations in binary form. We illustrate this in the logic K. We call such logics binary expansions of the unary modal logics. There are many more such binary expansions than the ones given by the axiom |$U$|⁠. We initiate an investigation of the properties of these expansions and in particular of the maximal binary expansions of a logic. Our results directly imply that all sub- and superintuitionistic logics with a standard modal companion also have binary modal companions. The latter also applies to the weak subintuitionistic logic WF of our previous papers. This logic doesn’t seem to have a unary modal companion. (shrink)

We consider the problem of induction over languages containing binary relations and outline a way of interpreting and constructing a class of probability functions on the sentences of such a language. Some principles of inductive reasoning satisfied by these probability functions are discussed, leading in turn to a representation theorem for a more general class of probability functions satisfying these principles.

We give an overview of decidability results for modal logics having a binary modality. We put an emphasis on the demonstration of proof-techniques, and hope that this will also help in finding the borderlines between decidable and undecidable fragments of usual first-order logic.

ABSTRACT We characterize those binary ramified frames for which propositional modal logic is as expressive as the corresponding first-order logic.

I show that the logic $\textsf {TJK}^{d+}$, one of the strongest logics currently known to support the naive theory of truth, is obtained from the Kripke semantics for constant domain intuitionistic logic by dropping the requirement that the accessibility relation is reflexive and only allowing reflexive worlds to serve as counterexamples to logical consequence. In addition, I provide a simplified natural deduction system for $\textsf {TJK}^{d+}$, in which a restricted form of conditional proof is used to establish conditionals.

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations.

G3 is Gödelian 3-valued logic, G3\(_\text{Ł}^\leq\) is its paraconsistent counterpart and G3\(_\text{Ł}^1\) is a strong extension of G3\(_\text{Ł}^\leq\). The aim of this paper is to endow each one of the logics just mentioned with a 2 set-up binary Routley semantics.

B. Kooi and A. Tamminga present a correspondence analysis for extensions of G. Priest’s logic of paradox. Each unary or binary extension is characterizable by a special operator and analyzable via a sound and complete natural deduction system. The present paper develops a sound and complete proof searching technique for the binary extensions of the logic of paradox.

Neural networks are widely used in systems of artificial intelligence, but due to their black box nature, they have so far evaded formal analysis to certify that they satisfy desirable properties, mainly when they perform critical tasks. In this work, we introduce methods for the formal analysis of reachability and robustness of neural networks that are modeled as rational McNaughton functions by, first, stating such properties in the language of Łukasiewicz infinitely-valued logic and, then, using the reasoning techniques of (...) such logical system. We also present a case study where we employ the proposed techniques in an actual neural network that we trained to predict whether it will rain tomorrow in Australia. (shrink)

A classical result by Słupecki states that a logic L is functionally complete for the 3-element set of truth-values THREE if, in addition to functionally including Łukasiewicz’s 3-valued logic Ł3, what he names the ‘$T$-function’ is definable in L. By leaning upon this classical result, we prove a general theorem for defining binary expansions of Kleene’s strong logic that are functionally complete for THREE.

It is shown that both classical and intuitionistic propositional logics of an associative binary modality are undecidable. The proof is based on the deduction theorem for these logics.

We compute the domination monoid in the theory of dense meet‐trees. In order to show that this monoid is well‐defined, we prove weak binarity of and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory from [7]. We then describe the domination monoids of such expansions in terms of those of the expanding relations.

We investigate the formal theory of binary quantifiers, that is, quantifiers that take seriously the surface structure of natural language quantifier phrases. We show how to develop a natural deduction system for logics of this sort and demonstrate soundness and completeness results.

Working in the weakening of constructive Zermelo-Fraenkel set theory in which the subset collection scheme is omitted, we show that the binary refinement principle implies all the instances of the exponentiation axiom in which the basis is a discrete set. In particular binary refinement implies that the class of detachable subsets of a set form a set. Binary refinement was originally extracted from the fullness axiom, an equivalent of subset collection, as a principle that was sufficient to (...) prove that the Dedekind reals form a set. Here we show that the Cauchy reals also form a set. More generally, binary refinement ensures that one remains in the realm of sets when one starts from discrete sets and one applies the operations of exponentiation and binary product a finite number of times. (shrink)

Orthogonality of all families of pairwise weakly orthogonal 1-types for ℵ0-categorical weakly o-minimal theories of finite convexity rank has been proved in 6. Here we prove orthogonality of all such families for binary 1-types in an arbitrary ℵ0-categorical weakly o-minimal theory and give an extended criterion for binarity of ℵ0-categorical weakly o-minimal theories . © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

A case has been made by various authors that the normative and processual notion of personhood found in African philosophy is discriminatory: it has been labelled as sexist, ableist and anti-queer. Within the anti-queer critique, one area that has not been specifically addressed in the literature is whether this notion of personhood is biased against people who identify as non-binary with respect to gender. This includes people who are gender fluid and gender neutral, among others. In this article, we (...) argue that the binary conception of gender is deeply connected to a colonial agenda, and does not accurately fit extra-colonial gender conceptions. Through an analysis of extra-colonial language and logic, read together with the flexibility inherent in the processual notion of personhood (on our reading), we argue that binaries do not sit comfortably with extra-colonial African thinking, and as such, processual and normative personhood can be understood in a way that embraces non-binary individuals. Further, the standards for achieving personhood are set by and achieved within the community, so a community which recognises and celebrates non-binary individuals can set up a framework for them to achieve personhood. Accordingly, normative personhood in African thought is found not to exclude those of non-binary gender. This reading saves the normative approach from some of the critiques levelled against it, as well as: i) enhancing consistency in an African world view; ii) strengthening the decolonial agenda; and iii) showing normative personhood to be fairer and more inclusive. (shrink)

We introduce the notion of a convex tree. We show that the binary expansion for real numbers in the unit interval ) is equivalent to weak König lemma ) for trees having at most two nodes at each level, and we prove that the intermediate value theorem is equivalent to \ for convex trees, in the framework of constructive reverse mathematics.

In this paper, we explore the relation between an agent’s doxastic attitude and her arguments in support of a given claim. Our main contribution is the design of a logical setting that allows us reason about binary arguments which are either in favour or against a certain claim. This is a setting in which arguments and propositions are the basic building blocks so that the concept of argument-based belief emerges in a straightforward way. We work against the background of (...) Dung’s abstract argumentation framework, extending it to a new setting in which we can study the formal properties of binary arguments as well as the larger structures they establish. This paper introduces a formal ‘two-dimensional’ language to talk about propositions and arguments, for which a sound and complete axiom system is provided. (shrink)

We provide a natural extension of the intuitionistic natural deduction system NJ with at most two consequences, which we call a binary conclusion natural deduction system. The classical natural deduction system NK is defined as NJ together with the law of the excluded middle or the double negation principle. However, in contrast to NK, the new BCN system has no such classical rule, but has two kinds of rules managing two consequences. One is to eliminate one consequence and the (...) other is to introduce another consequence. Except on this point, BCN proofs are derived in exactly the same way as those in NJ. It is shown that with respect to the propositional fragment of NK, the BCN system is sound and sufficiently complete to prove classical propositional theorems. Moreover, as a corollary we obtain a general form of Glivenko's theorem and disjunction property for one of the two consequences. We also show the normalization theorem for BCN. Finally, we study some extensions of BCN to classical predicate logic. (shrink)

What role, if any, does formal logic play in characterizing epistemically rational belief? Traditionally, belief is seen in a binary way - either one believes a proposition, or one doesn't. Given this picture, it is attractive to impose certain deductive constraints on rational belief: that one's beliefs be logically consistent, and that one believe the logical consequences of one's beliefs. A less popular picture sees belief as a graded phenomenon.

In this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.

We shall prove here that any binary relation on a base E with cardinality n > 6 is reconstructible from its restrictions of cardinality 2, 3, 4 and . This proof needs results of part I of this paper where we characterize any pair of relations R, R' which are 2-, 3- and 4-hypomorphic. As a corollary we obtain that any binary relation is -reconstructible.

We discuss some new properties of the natural Galois connection among set relation algebras, permutation groups, and first order logic. In particular, we exhibit infinitely many permutational relation algebras without a Galois closed representation, and we also show that every relation algebra on a set with at most six elements is Galois closed and essentially unique. Thus, we obtain the surprising result that on such sets, logic with three variables is as powerful in expression as full first order (...) logic. (shrink)

Let ( * X, * T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in * X is defined as m(x) = μ T (st(x)). Consider the relation $R_{\mathrm{ns}} = \{\langle x, y \rangle \mid x, y \in \mathrm{ns} (^\ast X) \text{and} y \in m(x)\}.$ Frank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of R ns to the whole of * X. Wattenberg's (...) extensions of R ns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of R ns to * X should be possible in any topological space. A study of such extensions of R ns is the purpose of the present paper. We call a binary relation $W \subseteq ^\ast X \times ^\ast X$ an infinitesimal on * X if it is monadic and reflexive on * X. We prove, among other things, that the existence of an infinitesimal on * X that extends R ns is equivalent to the condition that the space (X, T) be regular. (shrink)

Logics of binary relations corresponding, among others, to the class RRA of representable relation algebras and the class FRA of full relation algebras are presented together with the proof systems in the style of dual tableaux. Next, the logics are extended with relational constants interpreted as point relations. Applications of these logics to reasoning in non-classical logics are recalled. An example is given of a dual tableau proof of an equation which is RRA-valid, while not RA-valid.

Logics of binary relations corresponding, among others, to the class RRA of representable relation algebras and the class FRA of full relation algebras are presented together with the proof systems in the style of dual tableaux. Next, the logics are extended with relational constants interpreted as point relations. Applications of these logics to reasoning in non-classical logics are recalled. An example is given of a dual tableau proof of an equation which is RRA-valid, while not RA-valid.

We prove that there are no binary homogeneous pseudoplanes.

A deductive system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{S}$$ \end{document} (in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T, i.e., the binary relation between formulas\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\{ \left\langle {\alpha,\beta } \right\rangle :T,\alpha \vdash s \beta and T,\beta \vdash s \alpha \},$$ \end{document}is a congruence relation on the formula algebra. The multiterm deduction-detachment theorem is a natural generalization of (...) the deduction theorem of the classical and intuitionistic propositional calculi (IPC) in which a finite system of possibly compound formulas collectively plays the role of the implication connective of IPC. We investigate the deductive structure of Fregean deductive systems with the multiterm deduction-detachment theorem within the framework of abstract algebraic logic. It is shown that each deductive system of this kind has a deductive structure very close to that of the implicational fragment of IPC. Moreover, it is algebraizable and the algebraic structure of its equivalent quasivariety is very close to that of the variety of Hilbert algebras. The equivalent quasivariety is however not in general a variety. This gives an example of a relatively point-regular, congruence-orderable, and congruence-distributive quasivariety that fails to be a variety, and provides what apparently is the first evidence of a significant difference between the multiterm deduction-detachment theorem and the more familiar form of the theorem where there is a single implication connective. (shrink)

The paper describes the isomorphic lattices of quasivarieties of commutative quasigroup modes and of cancellative commutative binary modes. Each quasivariety is characterised by providing a quasi-equational basis. A structural description is also given. Both lattices are uncountable and distributive.

The paper describes the isomorphic lattices of quasivarieties of commutative quasigroup modes and of cancellative commutative binary modes. Each quasivariety is characterised by providing a quasi-equational basis. A structural description is also given. Both lattices are uncountable and distributive.

A similarity relation is a reflexive and symmetric binary relation between objects. Similarity is relative: it depends on the set of properties of objects used in determining their similarity or dissimilarity. A multi-modal logical language for reasoning about relative similarities is presented. The modalities correspond semantically to the upper and lower approximations of a set of objects by similarity relations corresponding to all subsets of a given set of properties of objects. A complete deduction system for the language is (...) presented. (shrink)