This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary (...) for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed. (shrink)

Models are constructed for a variety of systems of quantified relevance logic with identity. Models are given for systems with different principles governing the transitivity of identity and substitution, and the relative merits of these principles are discussed. The models in this paper are all extensions of the semantics of Fine's Semantics for Quantified Relevance Logic (Journal of Philosophical Logic 17 (1988)).

We begin to fill a lacuna in the relevance logic enterprise by providing a foundational analysis of identity in relevance logic. We consider rival interpretations of identity in this context, settling on the relevant indiscernibility interpretation, an interpretation related to Dunn's relevant predication project. We propose a general test for the stability of an axiomatisation of identity, relative to this interpretation, and we put various axiomatisations to this test. We fill our discussion out with both formal and philosophical remarks on (...) identity in relevance logic. (shrink)

This paper reformulates and decides a certain conjecture in Dunn's 'Relevant Predication 1: The Formal Theory' (Journal of Philosophical Logic 16, 347-381, 1987). This conjecture of Dunn's relates his object-language characterisation of a property's being relevant in a variable x to certain grammatical characterisations of relevance, analogous to some given by Helman, in 'Relevant Implication and Relevant Functions' (to appear in Entailment: The Logic of Relevance and Necessity, vol. 2, by Alan Ross Anderson, Nuel Belnap, and J. Michael Dunn et (...) al.) In the course of the investigation this paper also investigates Kit Fine's semantics for quantified relevance logics, which appears in his appropriately titled 'Semantics for Quantified Relevance Logics'. (shrink)

Based on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. (...) The reason is interesting: if PA is slightly weakened to a subtheory P+, it admits the complex ring C as a model; thus QRF is chosen to be a theorem of PA but false in C. Inasmuch as all strictly positive theorems of R# are already theorems of P+, this nonconservativity result shows that QRF is also a nontheorem of R#. As a consequence, Ackermann's rule γ is inadmissible in R#. Accordingly, an extension of R# which retains its good features is desired. The system R##, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between R# and R##, which does formalize arithmetic on relevant principles, but also admits γ in a natural way? (shrink)

We pursue the idea that predicate logic is a “fibred algebra” while propositional logic is a single algebra; in the context of intuitionism, this algebraic understanding of predicate logic goes back to Lawvere, in particular his concept of hyperdoctrine. Here, we aim at demonstrating that the notion of monad-relativised hyperdoctrines, which are what we call fibred algebras, yields algebraisations of a wide variety of predicate logics. More specifically, we discuss a typed, first-order version of the non-commutative Full Lambek calculus, which (...) has extensively been studied in the past few decades, functioning as a unifying language for different sorts of logical systems (classical, intuitionistic, linear, fuzzy, relevant, etc.). Through the concept of Full Lambek hyperdoctrines, we establish both generic and set-theoretical completeness results for any extension of the base system; the latter arises from a dual adjunction, and is relevant to the tripos-to-topos construction and quantale-valued sets. Furthermore, we give a hyperdoctrinal account of Girard’s and Gödel’s translation. (shrink)

This paper defines a category of bounded distributive lattice-ordered grupoids with a left-residual operation that corresponds to a weak system in the family of relevant logics. Algebras corresponding to stronger systems are obtained by adding further postulates. A duality theoey piggy-backed on the Priestley duality theory for distributive lattices is developed for these algebras. The duality theory is then applied in providing characterizations of the dual spaces corresponding to stronger relevant logics.

We critically investigate and refine Dunn's relevant predication, his formalisation of the notion of a real property. We argue that Dunn's original dialectical moves presuppose some interpretation of relevant identity, though none is given. We then re-motivate the proposal in a broader context, considering the prospects for a classical formalisation of real properties, particularly of Geach's implicit distinction between real and ''Cambridge'' properties. After arguing against these prospects, we turn to relevance logic, re-motivating relevant predication with Geach's distinction in mind. (...) Finally we draw out some consequences of Dunn's proposal for the theory of identity in relevance logic. (shrink)

There is an intuition, notoriously difficult to formalise, that only some predicates express real properties. J. M. Dunn formalises this intuition with relevance logic, proposing a notion of relevant predication. For each first order formula Ax, Dunn specifies another formula that is intuitively interpreted as "Ax expresses a real property". Chapter I calls such an approach an object language approach, since the claim that Ax expresses a real property is rendered as a formula in the object language. On a metalanguage (...) approach, on the other hand, the claim that Ax expresses a real property would be metalinguistic, mentioning but not using the formula Ax. ;Our Introduction begins by investigating Dunn's motivation for relevant predication, and argues that it implicitly presupposes some interpretation of identity. Indeed, part of the dissertation's work is to use relevant predication as a key to a coherent account of relevant identity. ;Chapter I re-motivates relevant predication. We consider P. Geach's distinction between real and "Cambridge" change, and suggest that there is an underlying distinction between real and Cambridge predicates. We further argue that no classical object language approach can formalise this distinction. So we turn to relevance logic, remotivating relevant predication, with Geach's distinction in mind. ;In light of this new motivation, we argue that relevant prediction relies on a logically weak notion of identity, according to which "x = y" means "x and y share all relevant properties". Identity is taken up in a more technical setting in Chapters IV and V. ;Subsequent chapters investigate technical issues that flow from Chapter I. Chapter II investigates metalinguistic grammatical characterisations of relevant predication. Chapter III concerns the semantics of first order relevance logic. Chapter IV investigates the identity in R and its relationship to relevant predication. In particular, Chapter IV argues against introducing the axiom, $\to$ Ay), in relevance logic, since the resulting systems are unstable, in a sense motivated by the notion of relevant identity underlying Dunn's notion of relevant predication. Finally Chapter V investigates the possibility of a coherent theory of identity in R's modal relatives ${\bf R}\sp\square$ and E. (shrink)

The set of identities provable in primitive recursive arithmetic without induction (PRE), with or without standard quantifier free successor axioms, is recursive. A finite number of identities can be added to PRE such that the set of identities provable become complete r.e. (with or without successor axioms). If the successor axiom y≠0 → gx(S(x) = y) is added to PRE, then the set of identities provable become complete r.e. (with or without 1≠0). If PRE is augmented by definition by cases, (...) then the set of identities provable become complete r.e. (with or without successor axioms). Equivalents of primitive recursive arithmetic (PRA) are given, involving the rule of induction restricted to identities. (shrink)