Bilateral Inversion Principles

We develop rules for coalgebras in type theory, and give meaning explanations for them. We show that elements of coalgebras are determined by their elimination rules, whereas the introduction rules can be considered as derived. This is in contrast with algebraic data types, for which the opposite is true: elements are determined by their introduction rules, and the elimination rules can be considered as derived. In this sense, the function type from the logical framework is more like a coalgebraic data type, the elements of which are determined by the elimination rule. We illustrate why the simplest form of guarded recursion is nothing but the introduction rule originating from the formulation of coalgebras in category theory. We discuss restrictions needed in order to preserve decidability of equality.

The idea of an ‘inversion principle’, and the name itself, originated in the work of Paul Lorenzen in the 1950s, as a method to generate new admissible rules within a certain syntactic context. Some fifteen years later, the idea was taken up by Dag Prawitz to devise a strategy of normalization for natural deduction calculi (this being an analogue of Gentzen's cut-elimination theorem for sequent calculi). Later, Prawitz used the inversion principle again, attributing it with a semantic role. Still working in natural deduction calculi, he formulated a general type of schematic introduction rules to be matched – thanks to the idea supporting the inversion principle – by a corresponding general schematic Elimination rule. This was an attempt to provide a solution to the problem suggested by the often quoted note of Gentzen. According to Gentzen ‘it should be possible to display the elimination rules as unique functions of the corresponding introduction rules on the basis of certain requirements’. Many people have since worked on this topic, which can be appropriately seen as the birthplace of what are now referred to as “general elimination rules”, recently studied thoroughly by Sara Negri and Jan von Plato. In this study, we retrace the main threads of this chapter of proof-theoretical investigation, using Lorenzen's original framework as a general guide.

The basic idea behind that account is that we need to distinguish straightforward inconsistency, or self-contradiction, of a set of assumptions, from paradoxicality. Both involve proofs of absurdity (1). But the proofs of absurdity in connection with straightforward contradictions are normalizable, whereas those in connection with paradoxes are not. Proofs are normalizable when they can be brought into normal form by a finite sequence of applications of reduction procedures. These reduction procedures are designed to get rid of unnecessary prolixity. Such prolixity can arise, most importantly, by applying an introduction rule for a logical operator and then immediately applying the corresponding elimination rule. The result is a sentence occurrence within the proof standing as the conclusion of an application of the introduction rule and as the major premiss of an application of the corresponding elimination rule. Reductions get rid of such 'maximal' sentence occurrences, which stand as unwanted 'knuckles' in the proof. The reduction procedures for the logical operators are designed to eliminate such unnecessary detours within proofs.1

The behavior of languages such as C++, Java, Smalltalk and Eiffel and the modeling features such classes, inheritance, polymorphism, message passing are discussed. The important theoretical approaches such as subtyping, F-bounds, matching and the object calculus are also discussed. A mathematical model was developed, that show the possibility of motivating the existence of something as relatively sophisticated as an object with a state and method. The two kind of type rules include: introduction rules that describes the formation of functions and records and elimination rules, describing the decomposition of constructions.

We explicate the different ways that a first-order sentence can be true (resp., false) in a model M, as formal objects, called (M-relative) truth-makers (resp., falsity-makers). M-relative truth-makers and falsity-makers are co-inductively definable, by appeal to the “atomic facts” in M, and to certain rules of verification and of falsification, collectively called rules of evaluation. Each logical operator has a rule of verification, much like an introduction rule; and a rule of falsification, much like an elimination rule. Applications of the rules (∀) and (∃) involve infinite furcation when the domain of M is infinite. But even in the infinite case, truth-makers and falsity-makers are tree-like objects whose branches are at most finitely long. A sentence φ is true (resp., false) in a model M (in the sense of Tarski) if and only if there existsπ such that π is an M-relative truth-maker (resp., falsity-maker) for φ. With “ways of being true” explicated as these logical truthmakers, one can re-conceive logical consequence between given premises and a conclusion. It obtains just in case there is a suitable method for transforming M-relative truthmakers for the premises into an M-relative truthmaker for the conclusion, whatever the model M may be.

In the Trinity Term of 1977 I had the privilege of hearing Michael Dummett lecture in the Old Library at All Souls on the justification and criticism of logical laws, and subsequently I also assisted Mark Helme in the production of some unauthorised, not altogether successful, notes on these lectures. I can vividly recall my surprise when Prof. Dummett stated, in connection with the idea with which he had up till then been mainly concerned, namely the idea that meaning is primarily determined in terms of the Gentzen introduction rules, that my old tutor, the Swedish logician Per Martin-Lof, in correspondence, held that “the meanings of the logical constants CAN IN NO WAY be given by the introduction rules, rather they are given by the elimination rules”.1 This was, as far as I knew from tutorials and conversations with Martin-Lof, not a position he would defend at the time and certainly not today. In fact, he has steadfastly held by the introduction-rule orthodoxy in all relevant publications from 1979 onwards.2 Taken jointly those publications constitute, as far as I know, the only sustained effort towards a realisation of a constructivist theory of meaning for a sizeable interpreted language serving the needs of pure mathematics on a scale comparable to that of Frege’s Grundgesetze. From this circumstance alone it seems clear that much of Dummettian interest can be found in the works of Martin-Lof.

A general formulation of inductive and recursive definitions in Martin-Löf's type theory is presented. It extends Backhouse's ‘Do-It-Yourself Type Theory’ to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. The formulation is in natural deduction and is intended to be a natural generalisation to type theory of Martin-Löf's theory of iterated inductive definitions in predicate logic.Formal criteria are given for correct formation and introduction rules of a new set former capturing definition by strictly positive, iterated, generalised induction. Moreover, there is an inversion principle for deriving elimination and equality rules from the formation and introduction rules. Finally, there is an alternative schematic presentation of definition by recursion.The resulting theory is a flexible and powerful language for programming and constructive mathematics. We hint at the wealth of possible applications by showing several basic examples: predicate logic, generalised induction, and a formalisation of the untyped lambda calculus.

Prawitz proved a theorem, formalising ‘harmony’ in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, then we can extend the theorem to the classical case. Properly arranged there is a thoroughgoing ‘harmony’, in the classical rules. Indeed, as we shall see, they are, all together, far more ‘harmonious’ in the general sense than has been commonly observed. As this paper will show, the appearance of disharmony has only arisen because of the illogical way in which natural deduction rules for Classical Logic have been presented.

Parallelized elimination rules in natural deduction correspond to Left rules in the sequent calculus; and introduction rules correspond to Right rules. These rules may be construed as inductive clauses in the inductive definition of the notion of sequent proof. There is a natural isomorphism between natural deductions in Core Logic and the sequent proofs that correspond to them. We examine the relations, between sequents, of concentration and dilution; and describe what it is for one sequent to strengthen another. We examine some possible global restrictions on proof-formation, designed to prevent proofs from proving dilutions of sequents already proved by a subproof. We establish the important result that the sequent rules of Core Logic maintain concentration, and we explain its importance for automated proof-search.