Abstract. The practice of first-order logic is replete with meta-level concepts. Most notably there are meta-variables ranging over formulae, variables, and terms, and properties of syntax such as alpha-equivalence, capture-avoiding substitution and assumptions about freshness of variables with respect to metavariables. We present one-and-a-halfth-order logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of one-and-a-halfth-order logic derivability, show them equivalent, show that the derivations satisfy cut-elimination, and prove correctness of an interpretation of first-order logic within it. We discuss the technicalities in a wider context as a case-study for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation

1 Motivation Since Modal Logics are an extension of Propositional Logic, they provide Boolean operators for constructing complex formulae. However, most Modal Logics do not admit Boolean operators for constructing complex modal parameters to be used in the box and diamond operators. This asymmetry is not present in Boolean Modal Logics, in which box and diamond quantify over arbitrary Boolean combinations of atomic modal parameters [9]. Boolean Modal Logics have been considered in various forms and contexts: 1. "Pure " Boolean Modal Logic has been studied in [9]. Negation and intersection of modal parameters occur in some variants of Propositional Dynamic Logic, see, e.g., [7, 16, 22].

The formalization of the “part – of ” relationship goes back to the mereology of S. Le´sniewski, subsequently taken up by Leonard & Goodman (1940), and Clarke (1981). In this paper we investigate relation algebras obtained from different notions of “part–of”, respectively, “connectedness” in various domains. We obtain minimal models for the relational part of mereology in a general setting, and when the underlying set is an atomless Boolean algebra. 1

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Countable relation algebras with homogeneous representations are characterised by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is !-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another twoplayer game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this ap...

1.1 Contradictoriness and inconsistency, consistency and non-contradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory

A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.

We explore the relation--algebraic aspects of the region connection calculus (RCC) of Randell et al. (1992a). In particular, we present a refinement of the RCC8 table which shows that the axioms provide for more relations than are listed in the present table. We also show that each RCC model leads to a Boolean algebra. Finally, we prove that a refined version of the RCC5 table has as models all atomless Boolean algebras B with the natural ordering as the "part -- of" relation, and that the table is closed under first order definable relations iff B is homogeneous. 1 Introduction Qualitative reasoning (QR) has its origins in the exploration of properties of physical systems when numerical information is not sufficient -- or not present -- to explain the situation at hand (Weld and Kleer, 1990). Furthermore, it is a tool to represent the abstractions of researchers who are constructing numerical systems which model the physical world. Thus, it fills a gap in data modeling which often l...

this paper for a discussion on the merits or otherwise of Kripke semantics and its "sufficiency" extension. Just as Kripke frames are dual to a class of Boolean algebras with modal operators [18, 24], one can build a duality for frames and Boolean algebras with sufficiency operators. Mixed structures occur when modal and sufficiency operators arise from the same accessibility relation. In this paper we introduce the classes of sufficiency algebras and that of mixed algebras which include both a modal and a sufficiency operator, and study representation and duality theory for these classes of algebras. We also give examples for classes of first-order definable frames, where such operators are required for a "modal-style" axiomatisation. 2 Why sufficiency and mixed algebras?

Introduction The combined investigation of mathematical logic and relativity theory is not at all new, as follows. Direction (1): Already at the beginnings, i.e. around 1920, Einstein's friend Reichenbach set to himself the goal of building up relativity as a logical theory, i.e. a theory purely in rst-order logic, cf. [30]. Similarly, Carnap [8, 9] pursued and advocated the same goal using a more sophisticated version of mathematical logic. De nability theory, one of the most important branches of modern logic, was brought to existence in the relativity book [30] because of the special needs of relativity theory. (This eld reached maturity via extensive work by Tarski on de nability as indicated in the dissertation.) Direction (2): The foundations of logic presuppose a kind of world-view (Weltanschaung) which has an eect on the \structure" of the theory of logic. In this sense the latest developments of relativity and related areas (e.g. Godel space-time, Kerr space-time) provid

Abstract. In informal mathematical usage we often reason using languages with binding. We usually find ourselves placing capture-avoidance constraints on where variables can and cannot occur free. We describe a logical derivation system which allows a direct formalisation of such assertions, along with a direct formalisation of their constraints. We base our logic on equality, probably the simplest available judgement form. In spite of this, we can axiomatise systems of logic and computation such as first-order logic or the lambda-calculus in a very direct and natural way. We investigate the theory of derivations, prove a suitable semantics sound and complete, and discuss existing and future research. 1