We generalize the notion of consequence relation standard in abstract treatments of logic to accommodate intuitions of relevance. The guiding idea follows the use criterion, according to which in order for some premises to have some conclusion(s) as consequence(s), the premises must each be used in some way to obtain the conclusion(s). This relevance intuition turns out to require not just a failure of monotonicity, but also a move to considering consequence relations as obtaining between multisets. We motivate and state basic definitions of relevant consequence relations, both in single conclusion (asymmetric) and multiple conclusion (symmetric) settings, as well as derivations and theories, guided by the use intuitions, and prove a number of results indicating that the definitions capture the desired results (at least in many cases).

Neo-Fregeanism aims to provide a possible route to knowledge of arithmetic via Hume’s principle, but this is of only limited significance if it cannot account for how the vast majority of arithmetic knowledge, accrued by ordinary people, is obtained. I argue that Hume’s principle does not capture what is ordinarily meant by numerical identity, but that we can do much better by buttressing plural logic with plural versions of the ancestral operator, obtaining natural and plausible characterizations of various key arithmetic concepts, including finiteness, equinumerosity and addition and multiplication of cardinality—revealing these to be logical concepts, and obtaining much of ordinary arithmetic knowledge as logical knowledge. Supplementing this with an abstraction principle and a simple axiom of infinity (known either empirically or modally) we obtain a full interpretation of arithmetic.

Rybakov (1984a) proved that the admissible rules of $\mathsf {IPC}$ are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility.

Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic ${\cal L}$ is associated with a matrix semantics $Mo{d^{\rm{*}}}{\cal L}$. This article is a contribution to the systematic study of the so-called truth sets of the matrices in $Mo{d^{\rm{*}}}{\cal L}$. In particular, we show that the fact that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ can be defined by means of equations with universally quantified parameters is captured by an order-theoretic property of the Leibniz operator restricted to deductive filters of ${\cal L}$. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ are implicitly definable if and only if the Leibniz operator is injective on deductive filters of ${\cal L}$ over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of ${\cal L}$ to its deductive filters over arbitrary algebras. We show that this is the case for logics expressed in a countable language, and that it need not be true in general. Finally we consider an intermediate condition on the truth sets in $Mo{d^{\rm{*}}}{\cal L}$ that corresponds to the order-reflection of the Leibniz operator.

A one-premiss rule is said to be archetypal for a consequence relation when not only is the conclusion of any application of the rule a consequence (according to that relation) of the premiss, but whenever one formula has another as a consequence, these formulas are respectively equivalent to a premiss and a conclusion of some application of the rule. We are concerned here with the consequence relation of classical propositional logic and with the task of extending the above notion of archetypality to rules with more than one premiss, and providing an informative characterization of the set of rules falling under the more general notion.

The form of information presented can influence its utility for the conveying of knowledge by affecting an interpreter’s ability to reason with the information. There are distinct types of representational systems (for example, symbolic versus diagrammatic logics), various sub-systems (for example, propositional versus predicate logics), and even within a single representational system there may be different means of expressing the same piece of information content. Thus, to display information, choices must be made between its different representations, depending upon many factors such as: the context, the reasoning tasks to be considered, user preferences or desires (for example, for short symbolic sentences or minimal clutter within diagrammatic systems). The identification of all equivalent representations with the same information content is a sensible precursor to attempts to minimise a metric over this class. We posit that defining notions of semantic redundancy and identifying the syntactic properties that encapsulate redundancy can help in achieving the goal of completely identifying equivalences within a single notational system or across multiple systems, but that care must be taken when extending systems, since refinements of redundancy conditions may be necessary even for conservative system extensions. We demonstrate this theory within two diagrammatic systems, which are Euler-diagram-based notations. Such notations can be used to represent logical information and have applications including visualisation of database queries, social network visualisation, statistical data visualisation, and as the basis of more expressive diagrammatic logics such as constraint languages used in software specification and reasoning. The development of the new associated machinery and concepts required is important in its own right since it increases the growing body of knowledge on diagrammatic logics. In particular, we consider Euler diagrams with shading, and then we conservatively extend the system to include projections, which allow for a much greater degree of flexibility of representation. We give syntactic properties that encapsulate semantic equivalence in both systems, whilst observing that the same semantic concept of redundancy is significantly more difficult to realise as syntactic properties in the extended system with projections.