Ansten Klev

The distinction between Church- and Curry-style typing is well established in type theory. In the Church style, the type of a term is intrinsic to it and determines with which other terms it may be combined in the construction of complex terms. In the Curry style, the type of a term is extrinsic to it and does not influence the rules of term formation. Martin-Löf’s meaning explanations for constructive type theory turn out to be sensitive to this distinction. We spell out a Church and a Curry interpretation of the meaning explanations and shed light on some philosophical differences between them.

The Curry–Howard correspondence, according to which propositions are types, suggests that every paradox formulable in natural deduction has a type-theoretical counterpart. I will give a purely type-theoretical formulation of Curry’s paradox. On the basis of the definition of a type, Γ(A), Curry’s reasoning can be adapted to show the existence of an object of the arbitrary type A. This is paradoxical for several reasons, among others that A might be an empty type. The solution to the paradox consists in seeing that Γ(A) is not a well-defined type.

This book honors the original and influential work by Göran Sundholm in the fields of the philosophy and history of logic and mathematics. Borne from two conferences held in Paris and Leiden on the occasion of Göran Sundholm’s retirement in 2019, the contributions collected in this volume represent work from leading logicians and philosophers. Reflecting Sundholm’s contributions to the history and philosophy of logic, this book is divided into two parts: the architecture and archaeology of logic. The essays collected in the ‘architecture’ section cover primarily the systematic approach to basic logical concepts taken by Sundholm, including type theory, epistemic assumptions, and notions of consequence. The ‘archaeology’ section includes contributions focused on Sundholm’s contributions to the history of philosophy and logic. Enclosing these two sections are, on the one end, autobiographical remarks of Sundholm's and, on the other, a paper on cooking and philosophy, reflecting another of Sundholm's passions in life. This book is of interest to logicians, philosophers, mathematicians, and computer scientists.

According to the iterative conception of set, sets are formed in stages. According to the purely iterative conception of set, sets are formed by iterated application of a set-of operation. The cumulative hierarchy is a mathematical realization of the iterative conception of set. A mathematical realization of the purely iterative conception can be found in Peter Aczel’s type-theoretic model of constructive set theory. I will explain Aczel’s model construction in a way that presupposes no previous familiarity with the theories on which it is based.

The aim here is to investigate assertion and inference as notions of logic. Assertion will be explained in terms of its purpose, which is to give interlocutors the right to request the assertor to do a certain task. The assertion is correct if, and only if, the assertor knows how to do this task. Inference will be explained as an assertion equipped with what I shall call a justification profile, a strategy for making good on the assertion. The inference is valid if, and only if, correctness is preserved from the premiss assertions to the conclusion assertion. Most of this is a spelling out of views on assertion and inference presented by Per Martin-Löf in lectures since 2015.

A solid foundation of modal logic requires a clear conception of the notion of modality. Modern modal logic treats modality as a propositional operator. I shall present an alternative according to which modality applies primarily to illocutionary force, that is, to the force, or mood, of a speech act. By a first step of internalization, modality applied at this level is pushed to the level of speech-act content. By a second step of internalization, we reach a propositional operator validating the modal logic S4. After a brief discussion of problematic modality and possibility, the article concludes with an extension of the account that identifies modality with illocutionary force. Throughout, close attention is paid to the intended interpretation of the formalism. All of the rules stipulated will be justified on the basis of this interpretation.

An inference is valid if it guarantees the transferability of knowledge from the premisses to the conclusion. If knowledge is here understood as demonstrative knowledge, and demonstration is explained as a chain of valid inferences, we are caught in an explanatory circle. In recent lectures, Per Martin-Löf has sought to avoid the circle by specifying the notion of knowledge appealed to in the explanation of the validity of inference as knowledge of a kind weaker than demonstrative knowledge. The resulting explanation is the main topic of this article.

A detailed argument is provided for the thesis that Dedekind was a logicist about arithmetic. The rules of inference employed in Dedekind's construction of arithmetic are, by his lights, all purely logical in character, and the definitions are all explicit; even the definition of the natural numbers as the abstract type of simply infinite systems can be seen to be explicit. The primitive concepts of the construction are logical in their being intrinsically tied to the functioning of the understanding.

The spiritus asper as used by Frege in a letter to Russell from 1904 bears resemblance to Church’s lambda. It is natural to ask how they relate to each other. An alternative approach to functional abstraction developed by Per Martin-Löf some thirty years ago allows us to describe the relationship precisely. Frege’s spiritus asper provides a way of restructuring a unary function name in Frege’s sense such that the argument place indicator occurs all the way to the right. Martin-Löf’s alternative approach shows that this is only half of what lambda does. The other half is the deletion of the argument place indicator, resulting in what Frege would have called an isolated function name.

There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down.

The logic of identity contains riches not seen through the coarse lens of predicate logic. This is one of several lessons to draw from the subtle treatment of identity in Martin‐Löf type theory, to which the reader will be introduced in this article. After a brief general introduction we shall mainly be concerned with the distinction between identity propositions and identity judgements. These differ from each other both in logical form and in logical strength. Along the way, connections to philosophical debates concerning identity are noted. Some use of logical symbolism is inevitable in any serious discussion of type theory, but the emphasis here is on basic ideas rather than technicalities.

This paper discusses some of the ways in which Martin-Löf type theory differs from set theory. The discussion concentrates on conceptual, rather than technical, differences. It revolves around four topics: sets versus types; syntax; functions; and identity. The difference between sets and types is spelt out as the difference between unified pluralities and kinds, or sorts. A detailed comparison is then offered of the syntax of the two languages. Emphasis is placed on the distinction between proposition and judgement, drawn by type theory, but not by set theory. Unlike set theory, type theory treats the notion of function as primitive. It is shown that certain inconveniences pertaining to function application that afflicts the set- theoretical account of functions are thus avoided. Finally, the distinction, drawn in type theory, between judgemental and propositional identity is discussed. It is argued that the criterion of identity for a domain cannot be formulated in terms of propositional identity. It follows that the axiom of extensionality cannot be taken as a statement of the criterion of identity for sets.

Cet article donne un aperçu des diverses traces de la doctrine des catégories dans les écrits de Carnap. Les notions de catégories jouent un rôle particulièrement important dans le livre Der logische Aufbau der Welt, mais on les retrouve également dans de nombreuses autres œuvres de Carnap. Sa thèse fait allusion à des catégories en plusieurs endroits. Son approche de la logique a été, pendant longtemps, fondée sur la théorie des types, incarnation de la doctrine des catégories dans la logique moderne. Et son point de vue sur la science unifiée est mieux compris à l’aide de la notion de catégorie.

Rudolf Carnap’s contributions to the Paris 1935 Congress for scientific philosophy signal three important changes in his philosophy: his semantic turn; what would later be called the “liberalization of empiricism”; and his adoption of the so-called thing language as a basis for the language of science. This paper examines this third change. In particular, it considers Carnap’s motivation for adopting the thing language as the protocol language of unified science and the virtues of the thing language in comparison with other types of protocol language.

An act’s form of apprehension determines whether it is a perception, an imagination, or a signitive act. It must be distinguished from the act’s quality, which determines whether the act is, for instance, assertoric, merely entertaining, wishing, or doubting. The notion of form of apprehension is explained by recourse to the so-called content-apprehension model ; it is characteristic of the Logical Investigations that in it all objectifying acts are analyzed in terms of that model. The distinction between intuitive and signitive acts is made, and the notion of saturation is described, by recourse to the notion of form of apprehension.

The eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Löf type theory. They are, however, justified by the meaning explanations, and a higher-order eta rule is part of that type theory. The main aim of this paper is to clarify this somewhat puzzling situation. It will be argued that lower-order eta rules do not, whereas the higher-order eta rule does, accord with the understanding of judgemental identity as definitional identity. A subsidiary aim is to clarify precisely what an eta rule is. This will involve showing how such rules relate to various other notions of type theory, proof theory, and category theory.

This monograph proposes a new way of implementing interaction in logic. It also provides an elementary introduction to Constructive Type Theory. The authors equally emphasize basic ideas and finer technical details. In addition, many worked out exercises and examples will help readers to better understand the concepts under discussion. One of the chief ideas animating this study is that the dialogical understanding of definitional equality and its execution provide both a simple and a direct way of implementing the CTT approach within a game-theoretical conception of meaning. In addition, the importance of the play level over the strategy level is stressed, binding together the matter of execution with that of equality and the finitary perspective on games constituting meaning. According to this perspective the emergence of concepts are not only games of giving and asking for reasons, they are also games that include moves establishing how it is that the reasons brought forward accomplish their explicative task. Thus, immanent reasoning games are dialogical games of Why and How.

Martin-Löf’s Constructive Type Theory (CTT) is a formal language developed in order to reason constructively about mathematics. It is thus a formal language conceived primarily as a tool to reason with rather than a formal language conceived primarily as a mathematical system to reason about. Constructive Type Theory is therefore much closer in spirit to Frege’s ideography and to the language of Russell and Whitehead’s Principia Mathematica than to the majority of logical systems (“logics”) studied by contemporary logicians. Since CTT is designed as a language to reason with, much attention is paid to the explanation of basic concepts. This is perhaps the main reason why the style of presentation of CTT differs somewhat from the style of presentation typically found in, for instance, ordinary logic textbooks. For those new to the system it might be useful to approach an introduction, such as the one given below, more as a language course than as a course in mathematics.

The standard natural deduction rules for the identity predicate have seemed to some not to be harmonious. Stephen Read has suggested an alternative introduction rule that restores harmony but presupposes second-order logic. Here it will be shown that the standard rules are in fact harmonious. To this end, natural deduction will be enriched with a theory of definitional identity. This leads to a novel conception of canonical derivation, on the basis of which the identity elimination rule can be justified in a proof-theoretical manner.