Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle (...) and I apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions. (shrink)

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle (...) and I apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions. (shrink)

In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity).We show that hybrid logic offers a genuinely first-order perspective on Kripke semantics: it is possible to define base logics which extend automatically to a (...) wide variety of frame classes and to prove completeness using the Henkin method. In the weaker languages, this requires the use of non-orthodox rules. We discuss these rules in detail and prove non-eliminability and eliminability results. We also show how another type of rule, which reflects the structure of the strong Priorean language, can be employed to give an even wider coverage of frame classes. We show that this deductive apparatus gets progressively simpler as we work our way up the expressivity hierarchy, and conclude the paper by showing that the approach transfers to first-order hybrid logic. (shrink)

In the present note we make use of some information given in [2]. Also, the terminology and notation do not dier from those accepted in [2]; in particular this concerns the formalism for the predicate calculus. Let A be a model of a rst-order language L. We say that A realizes a set of formulas Fla i A j= [a] for some valuation a in A and all 2 . We say that A omits i A does not realize . (...) A formula 2 Fla is consistent with a theory T in i there is a model of T which realizes fg. (shrink)

This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere (...) in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included. (shrink)

By generalizing Kreisel’s proof of the Second Incompleteness Theorem of G¨odel I extract a general principle which can also be used for otherpurely model-theoretical proofs of that theorem.

Translation of: Kritik der reinen Vernunft.

Sequent calculi constitute an interesting and important category of proof systems. They are much less known than axiomatic systems or natural deduction systems are, and they are much less known than they should be. Sequent calculi were designed as a theoretical framework for investigations of logical consequence, and they live up to the expectations completely as an abundant source of meta-logical results. The goal of this book is to provide a fairly comprehensive view of sequent calculi -- including a (...) wide range of variations. The focus is on sequent calculi for various non-classical logics, from intuitionistic logic to relevance logic, through linear and modal logics. A particular version of sequent calculi, the so-called consecution calculi, have seen important new developments in the last decade or so. The invention of new consecution calculi for various relevance logics allowed the last major open problem in the area of relevance logic to be solved positively: pure ticket entailment is decidable. An exposition of this result is included in chapter 9 together with further new decidability results (for less famous systems). A series of other results that were obtained by J. M. Dunn and me, or by me in the last decade or so, are also presented in various places in the book. Some of these results are slightly improved in their current presentation. Obviously, many calculi and several important theorems are not new. They are included here to ensure the completeness of the picture; their original formulations may be found in the referenced publications. This book contains very little about semantics, in general, and about the semantics of non-classical logic in particular. (shrink)

In the 18th and 19th centuries two transitions took place in the development of mathematical analysis: a shift from the geometric approach to the formula-centered approach, followed by a shift from the formula-centered approach to the concept-centered approach. We identify, on the basis of Bolzano's Purely Analytic Proof [Bolzano 1817], the ways in which Bolzano's approach can be said to be concept-centered. Moreover, we conclude that Bolzano's attitude towards the geometric approach on the one hand and the formula-centered approach (...) on the other were of a different nature; the for­mer being one of rejection, the latter of non-participation. Bolzano supports his concept-centered methodology by philosophical views, which were partially shared by mathematicians with a formula-centered approach to analysis. (shrink)

This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and (...) elementary geometry. The aim is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians. (shrink)

I am interested in the use Kant makes of the pure intuition of space, and of properties and principles of space and spaces (i.e. figures, like spheres and lines), in the special metaphysical project of MAN. This is a large topic, so I will focus here on an aspect of it: the role of these things in his treatment of some of the laws of matter treated in the Dynamics and Mechanics Chapters. In MAN and other texts, Kant speaks (...) of space as the “ground,” “condition,” and “basis” of various laws, including the inverse-square and inverse-cube laws of attractive and repulsive force, and the Third Law of Mechanics. Moreover, in his proofs of all the laws just mentioned, the language of “construction” figures prominently, which suggests that Kant’s proofs (somehow) rest on or involve mathematical construction in his technical sense. Such claims give rise to a number of questions. How do properties and principles of space and spaces serve to ground this particular set of laws? Which spatial properties and principles is Kant appealing to? What, if anything, does the spatial grounding of the inverse-square and inverse-cube laws of diffusion (treated in the Dynamics Chapter) have in common with that of the Third treated in the Mechanics Chapter)? What role—if any—does mathematical construction play in Kant’s proofs of these laws? Finally, how if at all, are Kant’s grounding claims consistent with his other commitments—for example, how are they consistent with his notorious denial in Prolegomena §38 that there are any laws that “lie in space” (Prol 4:321)? I offer answers to these questions. (shrink)

In this paper, we show the equivalence between the provability of a proof system of basic hybrid logic and that of translated formulas of the classical predicate logic with equality and explicit substitution by a purely proof–theoretic method. Then we show the equivalence of two groups of proof systems of hybrid logic: the group of labelled deduction systems and the group of modal logic-based systems.

It is well known that the modal logic S5 can be embedded in the classical predicate logic by interpreting the modal operator in terms of a quantifier. Wajsberg [10] proved this fact in a syntactic way. Mints [7] extended this result to the quantified version of S5; using a purely proof-theoretic method he showed that the quantified S5 corresponds to the classical predicate logic with one-sorted variable. In this paper we extend Mints' result to the basic modal logic S4; (...) we investigate the correspondence between the quantified versions of S4 (with and without the Barcan formula) and the classical predicate logic (with one-sorted variable). We present a purely proof-theoretic proof-transformation method, reducing an LK-proof of an interpreted formula to a modal proof. (shrink)

Writing strategic documents is a major practice of many actors striving to see their educational ideas realised in the curriculum. In these documents, arguments are systematically developed to create the legitimacy of a new educational goal and competence to make claims about it. Through a qualitative analysis of the writing strategies used in these texts, I show how two of the main actors in the Czech educational discourse have developed a proof that a new educational goal is needed. I (...) draw on the connection of the relational approach in the sociology of education with Lyotard’s analytical semantics of instances in the event. The comparison of the writing strategies in the two documents reveals differences in the formation of a particular pattern of justification. In one case the texts function as a herald of pure reality, and in the other case as a messenger of other witnesses. This reveals different regimens of proof, although both of them were written as prescriptive directives – normative models of the educational world. (shrink)

It is well known that the modal logic S5 can be embedded in the classical predicate logic by interpreting the modal operator in terms of a quantifier. Wajsberg proved this fact in a syntactic way. Mints extended this result to the quantified version of S5; using a purely proof-theoretic method he showed that the quantified S5 corresponds to the classical predicate logic with one-sorted variable. In this paper we extend Mints’ result to the basic modal logic S4; we investigate (...) the correspondence between the quantified versions of S4 and the classical predicate logic. We present a purely proof-theoretic proof-transformation method, reducing an LK-proof of an interpreted formula to a modal proof. (shrink)

This book provides a detailed, updated exposition and defense of five of the historically most important (but in recent years largely neglected) philosophical proofs of God’s existence: the Aristotelian, the Neo-Platonic, the Augustinian, the Thomistic, and the Rationalist. It also offers a thorough treatment of each of the key divine attributes—unity, simplicity, eternity, omnipotence, omniscience, perfect goodness, and so forth—showing that they must be possessed by the God whose existence is demonstrated by the proofs. Finally, it answers at length all (...) of the objections that have been leveled against these proofs. This work provides as ambitious and complete a defense of traditional natural theology as is currently in print. Its aim is to vindicate the view of the greatest philosophers of the past— thinkers like Aristotle, Plotinus, Augustine, Aquinas, Leibniz, and many others— that the existence of God can be established with certainty by way of purely rational arguments. It thereby serves as a refutation both of atheism and of the fideism that gives aid and comfort to atheism. (shrink)

Is science getting at the truth? The sceptics – those who spread doubt about science – often employ a simple argument: scientists were sure in the past, and then they ended up being wrong. Such sceptics draw on dramatic quotes from eminent scientists such as Lord Kelvin, who reportedly stated at the turn of the 20th century “There is nothing new to be discovered in physics now,” shortly before physics was dramatically transformed. They ask: given the history of science, wouldn’t (...) it be naïve to think that current scientific theories reveal ‘the truth’, and will never be discarded in favour of other theories? Through a combination of historical investigation and philosophical-sociological analysis, Identifying Future-Proof Science defends science against such potentially dangerous scepticism. It is argued that we can confidently identify many scientific claims that are future-proof: they will last forever, so long as science continues. How do we identify future-proof claims? This appears to be a new question for science scholars, and not an unimportant one. It is argued that the best way to identify future-proof science is to avoid any attempt to analyse the relevant first-order scientific evidence, instead focusing purely on second-order evidence. Specifically, a scientific claim is future-proof when the relevant scientific community is large, international, and diverse, and at least 95% of that community would describe the claim as a ‘scientific fact’. In the entire history of science, no claim meeting these criteria has ever been overturned, despite enormous opportunity. (shrink)

We discuss the problem raised by Miller to re-prove the well-known equivalences of some Lindenbaum theorems for deductive systems without an application of the Axiom of Choice. We present five special constructions of deductive systems, each of them providing some partial solutions to the mathematical problem. We conclude with a short discussion of the underlying philosophical problem of deciding, whether a given proof satisfies our demand that the Axiom of Choice is not applied.

The paper contains proof-theoretic investigation on extensions of Kripke-Platek set theory, KP, which accommodate first-order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Пn reflection rules. This leads to consistency proofs for the theories KP+Пn reflection using a small amount of arithmetic and the well-foundedness of a certain ordinal system with respect to primitive decending sequences. Regarding future work, we intend to avail ourselves of these new (...) cut elimination techniques to attain an ordinal analysis of П12 comprehension by approaching П12 comprehension through transfinite levels of reflection. (shrink)

In the constructive theory of uniform spaces there occurs a technique of proof in which the application of a weak form of the law of excluded middle is circumvented by purely analytic means. The essence of this proof—technique is extracted and then applied in several different situations.

Miller, D., G. Nadathur, F. Pfenning and A. Scedrov, Uniform proofs as a foundation for logic programming, Annals of Pure and Applied Logic 51 125–157. A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search (...) instructions. The operational semantics is formalized by the identification of a class of cut-free sequent proofs called uniform proofs. A uniform proof is one that can be found by a goal-directed search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that first-order and higher-order versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance of the various generalizations to first-order Horn clauses is briefly discussed. (shrink)

A common framing has it that any adequate treatment of quotation has to abandon one of the following three principles: (i) The quoted expression is a syntactic constituent of the quote phrase; (ii) If two expressions are derived by applying the same syntactic rule to a sequence of synonymous expressions, then they are synonymous; (iii) The language contains synonymous but distinct expressions. In the following, a formal syntax and semantics will be provided for a quotational language which adheres to all (...) three principles. The point here is not merely to provide a “possibility proof”, but to reveal the hard constraints on the theory of quotation, and to highlight certain assumptions at the syntax/semantics/post-semantics interfaces. (shrink)

This paper deals with a proof theory for a theory T22 of recursively Mahlo ordinals in the form of Π2-reflecting on Π2-reflecting ordinals using a subsystem Od of the system O of ordinal diagrams in Arai 353). This paper is the first published one in which a proof-theoretic analysis à la Gentzen–Takeuti of recursively large ordinals is expounded.

Kant wrote two versions of the Transcendental Deduction, the first, “A-”Deduction in 1781, and the second, “B-”Deduction in 1787. Since Henrich's “The Proof Structure of Kant's Transcendental Deduction”, most work on the Transcendental Deduction attempts to make sense of the B-Deduction's two-step argument structure. Though the A-Deduction has suffered comparative neglect, it has received some attention from interpreters who take its extended treatment of the “subjective” side of cognition to amount to a brand of proto-functionalism. Whatever the merits and (...) demerits of these proto-functionalist approaches, they tend to deemphasize the two arguments that constitute the “objective” side of the A-Deduction, the “argument from above” and then the “argument from below”. Since Kant himself refers to this objective side of the A-Deduction as the “Deduction of the Pure Concepts of the Understanding”, it is surprising that the structure of these arguments has not received closer scrutiny. This is doubly true since Kant actually claims that his revisions for the 1787 version of the Deduction impacted only the “presentation” of it. Any lessons learned from the central arguments of the A-Deduction should help clarify the structure of its younger and more closely studied brother. (shrink)

A logic of grounding where what is grounded can be a collection of truths is a “many-many” logic of ground. The idea that grounding might be irreducibly many-many has recently been suggested by Dasgupta. In this paper I present a range of novel philosophical and logical reasons for being interested in many-many logics of ground. I then show how Fine’s State-Space semantics for the Pure Logic of Ground can be extended to the many-many case, giving rise to the (...) class='Hi'>Pure Logic of Many-Many Ground. In the second, more technical, part of the paper, I do two things. First, I present an alternative formalization of plg; this allows us to simplify Fine’s completeness proof for plg. Second, I formalize plmmg using an infinitary sequent calculus and prove that this formalization is sound and complete. (shrink)

We present a setting in which the search for a proof of B or a refutation of B can be carried out simultaneously: in contrast, the usual approach in automated deduction views proving B or proving ¬B as two, possibly unrelated, activities. Our approach to proof and refutation is described as a two-player game in which each player follows the same rules. A winning strategy translates to a proof of the formula and a counter-winning strategy translates to (...) a refutation of the formula. The game is described for multiplicative and additive linear logic . A game theoretic treatment of the multiplicative connectives is intricate and our approach to it involves two important ingredients. First, labeled graph structures are used to represent positions in a game and, second, the game playing must deal with the failure of a given player and with an appropriate resumption of play. This latter ingredient accounts for the fact that neither player might win. (shrink)

In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations of it in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic. Proof theory for GL has been studied intensively up to the present (...) day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay. In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut-elimination and some standard consequences of it: the interpolation and definability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call Gödel sentences, using a purely proof-theoretical method. (shrink)

We introduce an extension of natural deduction that is suitable for dealing with modal operators and induction. We provide a proof reduction system and we prove a strong normalization theorem for an intuitionistic calculus. As a consequence we obtain a purely syntactic proof of consistency. We also present a classical calculus and we relate provability in the two calculi by means of an adequate formula translation.

In this paper we calibrate the exact proof-theoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction.

In order to perform certain actions – such as incarcerating a person or revoking parental rights – the state must establish certain facts to a particular standard of proof. These standards – such as preponderance of evidence and beyond reasonable doubt – are often interpreted as likelihoods or epistemic confidences. Many theorists construe them numerically; beyond reasonable doubt, for example, is often construed as 90 to 95% confidence in the guilt of the defendant. -/- A family of influential cases (...) suggests standards of proof should not be interpreted numerically. These ‘proof paradoxes’ illustrate that purely statistical evidence can warrant high credence in a disputed fact without satisfying the relevant legal standard. In this essay I evaluate three influential attempts to explain why merely statistical evidence cannot satisfy legal standards. (shrink)

Our aim is to express in exact terms the old idea of solving problems by pure questioning. We consider the problem of derivability: "Is A derivable from Δ by classical propositional logic?". We develop a calculus of questions E*; a proof (called a Socratic proof) is a sequence of questions ending with a question whose affirmative answer is, in a sense, evident. The calculus is sound and complete with respect to classical propositional logic. A Socratic proof (...) in E* can be transformed into a Gentzen-style proof in some sequent calculi. Next we develop a calculus of questions E**; Socratic proofs in E** can be transformed into analytic tableaux. We show that Socratic proofs can be grounded in Inferential Erotetic Logic. After a slight modification, the analyzed systems can also be viewed as hypersequent calculi. (shrink)

First-order logic is formalized by means of tools taken from the logic of questions. A calculus of questions which is a counterpart of the Pure Calculus of Quantifiers is presented. A direct proof of completeness of the calculus is given.

In the constructive theory of uniform spaces there occurs a technique of proof in which the application of a weak form of the law of excluded middle is circumvented by purely analytic means. The essence of this proof-technique is extracted and then applied in several different situations.