Complete Additivity and Modal Incompleteness

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem, `Syntactic aspects of modal incompleteness theorems,' and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness

Fermat’s last theorem proved in Hilbert arithmetic. I. From the proof by induction to the viewpoint of Hilbert arithmetic

In a previous paper, an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for “n = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, second part of the paper is outlined, getting directed to an eventual proof of the case “n=3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it

Self-referential probability

This thesis focuses on expressively rich languages that can formalise talk about probability. These languages have sentences that say something about probabilities of probabilities, but also sentences that say something about the probability of themselves. For example: (π): “The probability of the sentence labelled π is not greater than 1/2.” Such sentences lead to philosophical and technical challenges. For example seemingly harmless principles, such as an introspection principle, lead to inconsistencies with the axioms of probability in this framework. This thesis aims to answer two questions relevant to such frameworks, which correspond to the two parts of the thesis: “How can one develop a formal semantics for this framework?” and “What rational constraints are there on an agent once such expressive frameworks are considered?”. In this second part we are considering probability as measuring an agent’s degrees of belief. In fact that concept of probability will be the motivating one throughout the thesis. The first chapter of the thesis provides an introduction to the framework. The following four chapters, which make up Part I, focus on the question of how to provide a semantics for this expressively rich framework. In Chapter 2, we discuss some preliminaries and why developing semantics for such a framework is challenging. We will generally base our semantics on certain possible world structures that we call probabilistic modal structures. These immediately allow for a definition of a natural semantics in restrictive languages but not in the expressively rich languages that this thesis focuses on. The chapter also presents an overview of the strategy that will be used throughout this part of the thesis: we will generalise theories and semantics developed for the liar paradox, which is the sentence: (λ): “The sentence labelled λ is not true”. In Chapter 3, we will present a semantics that generalises a very influential theory of truth: a Kripke-style theory (Kripke, 1975) using a strong Kleene evaluation scheme. A feature of this semantics is that we can understand it as assigning sentences intervals as probability values instead of single numbers. Certain axioms of probability have to be dropped, for example “P ‘λ ∨ ¬λ’ = 1” is not satisfied in the construction, but the semantics can be seen as assigning non-classical probabilities. This semantics allows one to further understand the languages, for example the conflict with introspection, where one can see that the appropriate way to express the principle of introspection in this case is in fact to use a truth predicate in its formulation. We also develop an axiomatic system and show that it is complete in the presence of the ω-rule which allows one to fix the standard model of arithmetic. In Chapter 4, we will consider another Kripke-style semantics but now based on a supervaluational evaluation scheme. This variation is particularly interesting because it bears a close relationship to imprecise probabilities where agents’ credal states are taken to be sets of probability functions. In this chapter, we will also consider how to use this language to describe imprecise agents reasoning about one another. These considerations provide us with an argument for using imprecise probabilities that is very different from traditional justifications: by allowing agents to have imprecise probabilities one can easily extend a semantics to languages with sentences that talk about their own probability, whereas the traditional precise probabilist cannot directly apply his semantics to such languages. In Chapter 5, a revision theory of probability will be developed. In this one retains classical logic and traditional probability theory but the price to pay is that one obtains a transfinite sequence of interpretations of the language and identifying any particular interpretation as “correct” is problematic. In developing this we are particularly interested in finding limit stage interpretations that can themselves be used as good models for probability and truth. We will require that the limit stages “sum up” the previous stages, understood in a strong way. In this chapter two strategies for defining the successor stages are discussed. We first discuss defining (successor) probabilities by considering relative frequencies in the revision sequence up to that stage, extending ideas from Leitgeb (2012). The second strategy is to base the construction on a probabilistic modal structure and use the accessibility measure from that to determine the interpretation of probability. That concludes Part I and the development of semantics. In Part II, we consider rationality requirements on agents who have beliefs about self-referential probability sentences like π. For such sentences, a choice of the agent’s credences will affect which worlds are possible. Caie (2013) has argued that the accuracy and Dutch book arguments should be modified because the agent should only care about her inaccuracy or payoffs in the world(s) that could be actual if she adopted the considered credences. We consider this suggestion for the accuracy argument in Chapter 7 and the Dutch book argument in Chapter 8. Chapter 6 acts as an introduction to these considerations. We will show that these modified accuracy and Dutch book criteria lead to an agent being rationally required to be probabilistically incoherent, have negative credences, fail to be introspective and fail to assign the same credence to logically equivalent sentences. We will also show that this accuracy criterion depends on how inaccuracy is measured and that the accuracy criterion differs from the Dutch book criterion. We will in fact suggest rejecting Caie’s suggested modifications. For the accuracy argument, we suggest in Section 7.3 that the agent should consider how accurate the considered credences are from the perspective of her current credences. We will also consider how to generalise this version of the accuracy criterion and present ideas suggesting that it connects to the vi semantics developed in Part I. For the Dutch book argument, in Section 8.6 we suggest that this is a case where an agent should not bet with his credences

On counting problems in nonstandard models of Peano arithmetic with applications to groups

Coding devices in Peano arithmetic (PA) allow complicated finite objects such as groups to be encoded in a model $M$ ╞ PA. We call such coded objects $M$ -finite. This thesis concerns $M$ -finite abelian groups, and counting problems for $M$-finite groups. We define a notion of cardinality for non-$M$ -finite sets via the suprema and infima of appropriate $M$ -finite sets, if these agree we call the set $M$ -countable. We investigate properties of $M$ -countable sets and give examples which demonstrate marked differences to measure theory. Many of the pathologies are related to the arithmetic of cuts and we show what can be recovered in special cases. We propose a notion of measure that mimics the Carathéodory definition. We show that an $M$ -countable subgroup of any $M$ -finite group has an $M$ -countable transversal of appropriate cardinality. We look at $M$ -finite abelian groups. After discussing consequences of the basis theorem we concentrate on the case of a single $M$ -finite group $C$($p$$^k$) and investigate its external structure as an infinite abelian group. We prove that certain externally divisible subgroups of $C$($p$$^k$) have $M$ -countable complements. We generalize this result to show that $d$$G$, the divisible part of $G$, has an $M$ -countable complement for a general $M$ -finite abelian $G$

Nondeterminism in algebraic specifications and algebraic programs

"Nondeterminism in Algebraic Specifications and Algebraic Programs" presents a mathematical theory for the integration of three concepts: non-determinism, axiomatic specification and term rewriting. For non-deterministic programs, an algebraic specification language is provided which admits the application of automated tools based on term rewriting techniques. This general framework is used to explore connections between logic programming and algebraic programming. Examples from various areas of computer science are given, including results of computer experiments with a prototypical implementation. This book should be of interest to readers working within several fields of theoretical computer science, from algebraic specification theory to formal descriptions of distributed systems

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book