Radical Anti-Realism and Substructural Logics

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three dogmas of proof-theoretic semantics are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence.

Computability is one of the fundamental notions of mathematics and computer science, trying to capture the effective content of mathematics and its applications. Computability Theory explores the frontiers and limits of effectiveness and algorithmic methods. It has its origins in Godel's Incompleteness Theorems and the formalization of computability by Turing and others, which later led to the emergence of computer science as we know it today. Computability Theory is strongly connected to other areas of mathematics and theoretical computer science. The core of this theory is the analysis of relative computability and the induced degrees of unsolvability; its applications are mainly to Kolmogorov complexity and randomness as well as mathematical logic, analysis and algebra. Current research in computability theory stresses these applications and focuses on algorithmic randomness, computable analysis, computable model theory, and reverse mathematics (proof theory). Recent advances in these research directions have revealed some deep interactions not only among these areas but also with the core parts of computability theory. The goal of this Dagstuhl Seminar is to bring together researchers from all parts of computability theory and related areas in order to discuss advances in the individual areas and the interactions among those.

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

Uncomputability and undecidability in economic theory

General proof theory studies how proofs are structured and how they relate to each other, and not primarily what can be proved in particular formal systems. It has been developed within the framework of Gentzen-style proof theory, as well as in categorial proof theory. As Dag Prawitz's monograph "Natural Deduction" (1965) paved the way for this development (he also proposed the term "General Proof Theory"), it is most appropriate to use this topic to celebrate 50 years of this work. The conference took place 27-29 November, 2015 in Tübingen at the Department of Philosophy. The proceedings collect abstracts, slides and papers of the presentations given, as well as contributions from two speakers who were unable to attend.

The theory of computational complexity is based on the tradeoff between two computational resources, memory space and computer time. This paper investigates the physical counterparts of these resources. Memory space is the number of bits of information required by a computer over the course of a computation. Physically, information is intimately connected with entropy: entropy can be thought of as random bits of information. The physical analogue of the amount of memory space required for a computation, Ф- SPACE, is the difference between the maximum entropy Smax of the computing system and its actual entropy, S.Ф-SPACE = S - Smax is the thermodynamic depth of the computation. Ф-SPACE is the maximum number of bits of memory space available for a physical system to perform computation. In conventional computational complexity theory, computer time is identified as the number of elementary logical operations (AND, OR, NOT, COPY) required to perform the computation. The physical analogue of time computational complexity is simply the time required for the physical computing system to perform the computation. In order to obtain a number that measures the number of physical operations that can be performed by the system over time t, we define Ф-TIME to be proportional to the computation time multiplied by the energy E of the computing system over the course of the computation: Ф-TIME = 2Et/πħ. Here, the proportionality factor 2/πħ is chosen to match the time it takes a computing system with energy E to flip a bit in time t. Ф-TIME is thus the number of bit flips that can take place in an information processing system. In addition, we propose a novel measure of physical complexity that combines these measures. The free energy F = E - TS of a system is the amount of energy that is available to do useful work - as opposed to the energy bound up in the thermal fluctuations of heat. We define the free complexity to be the free energy × time product, 2Ft/πħ required by a physical system to perform a computation. Physical spatial and temporal complexity provide upper bounds on the amount of computation that can be performed by any physical system. We discuss the implications of physical complexity for the computational complexity hierarchy, in particular, we show that in the presence of errors in the computation, Ф-SPACE is proportional Ф-TIME. We apply the resulting theory of physical computational complexity to analyze the computational power of black holes: exactly half of the energy of a black hole is available for physical computation.

Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this article, we will survey the use ofstructural proof theoryas an alternative foundation. Researchers have been using this foundation for the past 35 years to elevate logic programming from its roots in first-order classical logic into higher-order versions of intuitionistic and linear logic. These more expressive logic programming languages allow for capturing stateful computations and rich forms of abstractions, including higher-order programming, modularity, and abstract data types. Term-level bindings are another kind of abstraction, and these are given an elegant and direct treatment within both proof theory and these extended logic programming languages. Logic programming has also inspired new results in proof theory, such as those involving polarity and focused proofs. These recent results provide a high-level means for presenting the differences between forward-chaining and backward-chaining style inferences. Anchoring logic programming in proof theory has also helped identify its connections and differences with functional programming, deductive databases, and model checking.

Reprint of: A more general general proof theory

Models of complex systems are widely used in the physical and social sciences, and the concept of layering, typically building upon graph-theoretic structure, is a common feature. We describe an intuitionistic substructural logic called ILGL that gives an account of layering. The logic is a bunched system, combining the usual intuitionistic connectives, together with a non-commutative, non-associative conjunction (used to capture layering) and its associated implications. We give soundness and completeness theorems for a labelled tableaux system with respect to a Kripke semantics on graphs. We then give an equivalent relational semantics, itself proven equivalent to an algebraic semantics via a representation theorem. We utilise this result in two ways. First, we prove decidability of the logic by showing the finite embeddability property holds for the algebraic semantics. Second, we prove a Stone-type duality theorem for the logic. By introducing the notions of ILGL hyperdoctrine and indexed layered frame we are able to extend this result to a predicate version of the logic and prove soundness and completeness theorems for an extension of the layered graph semantics . We indicate the utility of predicate ILGL with a resource-labelled bigraph model.

This paper characterizes the admissible rules for six interesting intuitionistic modal logics: iCK4, iCS4≡IPC, strong Löb logic iSL, modalized Heyting calculus mHC, Kuznetsov-Muravitsky logic KM, and propositional lax logic PLL. Admissible rules are rules that can be added to a logic without changing the set of theorems of the logic. We provide a Gentzen-style proof theory for admissibility that combines methods known for intuitionistic propositional logic and classical modal logic. From this proof theory, we extract bases for the admissible rules, i.e., sets of admissible rules that derive all other admissible rules. In addition, we show that admissibility is decidable for these logics.

We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O'Hearn and Pym's logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBI-formulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the proof-theoretic level, a very natural formalism for CBI is provided by a display calculus \`a la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics.

With help of a compact Prolog-based theorem prover for Intuitionistic Propositional Logic, we synthesize minimal assumptions under which a given formula formula becomes a theorem. After applying our synthesis algorithm to cover basic abductive reasoning mechanisms, we synthesize conjunctions of literals that mimic rows of truth tables in classical or intermediate logics and we abduce conditional hypotheses that turn the theorems of classical or intermediate logics into theorems in intuitionistic logic. One step further, we generalize our abductive reasoning mechanism to synthesize more expressive sequent premises using a minimal set of canonical formulas, to which arbitrary formulas in the calculus can be reduced while preserving their provability. Organized as a self-contained literate Prolog program, the paper supports interactive exploration of its content and ensures full replicability of our results.

Prawitz proposed certain notions of proof-theoretic validity and conjectured that intuitionistic logic is complete for them [11, 12]. Considering propositional logic, we present a general framework of five abstract conditions which any proof-theoretic semantics should obey. Then we formulate several more specific conditions under which the intuitionistic propositional calculus (IPC) turns out to be semantically incomplete. Here a crucial role is played by the generalized disjunction principle. Turning to concrete semantics, we show that prominent proposals, including Prawitz’s, satisfy at least one of these conditions, thus rendering IPC semantically incomplete for them. Only for Goldfarb’s [1] proof-theoretic semantics, which deviates from standard approaches, IPC turns out to be complete. Overall, these results show that basic ideas of proof-theoretic semantics for propositional logic are not captured by IPC.

Kauffman nets, also known as N-K models, have been studied extensively because their dynamics can be used to model a variety of interesting dynamical processes. This paper investigates the properties of the problem of determining whether or not a given configuration of a Kauffman net has a predecessor. Here it is shown that when the parameter K that governs the number of connections grows as ln(N) , where N is the number of elements, the problem of finding a solution is extremely sensitive to small changes in the problem statement. This result has implications for studies of the physics of random systems and also may have applications for questions in computational complexity theory.

Relational semantics for full linear logic