Distributing intersection and union types with splits and duality (functional pearl)

Subtyping with intersection and union types is nowadays common in many programming languages. From the perspective of logic, the subtyping problem is essentially the problem of determining logical entailment : does a logical statement follow from another one? Unfortunately, algorithms for deciding subtyping and logical entailment with intersections, unions and various distributivity laws can be highly non-trivial. This functional pearl presents a novel algorithmic formulation for subtyping (and logical entailment) in the presence of various distributivity rules between intersections, unions and implications (i.e. function types). Unlike many existing algorithms which first normalize types and then apply a subtyping algorithm on the normalized types, our new subtyping algorithm works directly on source types. Our algorithm is based on two recent ideas: a generalization of subtyping based on the duality of language constructs called duotyping ; and splittable types , which characterize types that decompose into two simpler types. We show that our algorithm is sound, complete and decidable with respect to a declarative formulation of subtyping based on the minimal relevant logic B + . Moreover, it leads to a simple and compact implementation in under 50 lines of functional code.

Arithmetic Formulated in a Logic of Meaning Containment

We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in Brady [7]. We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic. We also contrast the two approaches to classical recapture, again favouring our approach in [7]. We then consider our previous development of Peano arithmetic including primitive recursive functions, finally extending this work to that of general recursion.

Arithmetic Formulated Relevantly

The purpose of this paper is to formulate first-order Peano arithmetic within the resources of relevant logic, and to demonstrate certain properties of the system thus formulated. Striking among these properties are the facts that (1) it is trivial that relevant arithmetic is absolutely consistent, but (2) classical first-order Peano arithmetic is straightforwardly contained in relevant arithmetic. Under (1), I shall show in particular that 0 = 1 is a non-theorem of relevant arithmetic; this, of course, is exactly the formula whose unprovability was sought in the Hilbert program for proving arithmetic consistent. Under (2), I shall exhibit the requisite translation, drawing some Goedelian conclusions therefrom. Left open, however, is the critical problem whether Ackermann’s rule γ is admissible for theories of relevant arithmetic. The particular system of relevant Peano arithmetic featured in this paper shall be called R♯. Its logical base shall be the system R of relevant implication, taken in its first-order form RQ. Among other Peano arithmetics we shall consider here in particular the systems C♯, J♯, and RM3♯; these are based respectively on the classical logic C, the intuitionistic logic J, and the Sobocinski-Dunn semi-relevant logic RM3. And another feature of the paper will be the presentation of a system of natural deduction for R♯, along lines valid for first-order relevant theories in general. This formulation of R♯ makes it possible to construct relevantly valid arithmetical deductions in an easy and natural way; it is based on, but is in some respects more convenient than, the natural deduction formulations for relevant logics developed by Anderson and Belnap in Entailment.

Inconsistent Models for Relevant Arithmetics

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle. The resulting model-theoretic structure (called in this paper RM32) is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3i here). We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete (whether or not consistent and recursively enumerable). In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show (unlike classical quantum arithmetic) that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest.

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.

Relevance and Verification

A. J. Ayer’s empiricist criterion of meaning was supposed to have sorted all statements into nonsense on the one hand, and tautologies or genuinely factual statements on the other. Unfortunately for Ayer, it follows from classical logic that his criterion is trivial—it classifies all statements as either tautologies or genuinely factual, but none as nonsense. However, in this paper, I argue that Ayer’s criterion of meaning can be defended from classical proofs of its triviality by the adoption of a relevant logic—an idea which is motivated because, according to Ayer, the genuinely factual statements are those which observation is relevant to.

Boolean negation and non-conservativity II: The variable-sharing property

Many relevant logics are conservatively extended by Boolean negation. Not all, however. This paper shows an acute form of non-conservativeness, namely that the Boolean-free fragment of the Boolean extension of a relevant logic need not always satisfy the variable-sharing property. In fact, it is shown that such an extension can in fact yield classical logic. For a vast range of relevant logic, however, it is shown that the variable-sharing property, restricted to the Boolean-free fragment, still holds for the Boolean extended logic.