Logic and Philosophy of Logic

We investigate Steel’s conjecture in ‘The Core Model Iterability Problem’ [10], that if $\mathcal {W}$ and $\mathcal {R}$ are $\Omega +1$ -iterable, $1$ -small weasels, then $\mathcal {W}\leq ^{*}\mathcal {R}$ iff there is a club $C\subset \Omega $ such that for all $\alpha \in C$, if $\alpha $ is regular, then $\alpha ^{+\mathcal {W}}\leq \alpha ^{+\mathcal {R}}$. We will show that the conjecture fails, assuming that there is an iterable premouse M which models KP and which has a -Woodin cardinal. (...) On the other hand, we show that assuming there is no transitive model of KP with a Woodin cardinal the conjecture holds. In the course of this we will also show that if M is a premouse which models KP with a largest, regular, uncountable cardinal $\delta $, and $\mathbb {P} \in M$ is a forcing poset such that $M\models "\mathbb {P}\text { has the }\delta \text {-c.c.}"$, and $g\subset \mathbb {P}$ is M-generic, then $M[g]\models \text {KP}$. Additionally, we study the preservation of admissibility under iteration maps. At last, we will prove a fact about the closure of the set of ordinals at which a weasel has the S-hull property. This answers another question implicit in remarks in [10]. (shrink)

We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper, ${}^\omega \omega $ -bounding forcing notions, 2) the class of subproper, T-preserving forcing notions (where T is a fixed Souslin tree) and 3) the class of subproper, $[T]$ -preserving forcing notions (where T is an $\omega _1$ -tree) are iterable with revised countable support. In the (...) second part, we adopt Miyamoto’s theory of nice iterations, rather than revised countable support. We show that this approach allows us to drop a technical condition in the definitions of subcompleteness and subproperness, still resulting in forcing classes that are iterable in this way, preserve $\omega _1$, and, in the case of subcompleteness, don’t add reals. Further, we show that the analogs of the iteration theorems proved in the first part for RCS iterations hold for nice iterations as well. (shrink)

In this paper we study logical bilateralism understood as a theory of two primitive derivability relations, namely provability and refutability, in a language devoid of a primitive strong negation and without a falsum constant, $\bot $, and a verum constant, $\top $. There is thus no negation that toggles between provability and refutability, and there are no primitive constants that are used to define an “implies falsity” negation and a “co-implies truth” co-negation. This reduction of expressive power notwithstanding, there remains (...) some interaction between provability and refutability due to the presence of (i) a conditional and the refutability condition of conditionals and (ii) a co-implication and the provability condition of co-implications. Moreover, assuming a hyperconnexive understanding of refuting conditionals and a dual understanding of proving co-implications, neither non-trivial negation inconsistency nor hyperconnexivity is lost for unary negation connectives definable by means of certain surrogates of falsum and verum. Whilst a critical attitude towards $\bot $ and $\top $ can be justified by problematic aspects of the Brouwer-Heyting-Kolmogorov interpretation of the logical operations for these constants, the aim to reduce the availability of a toggling negation and observations on undefinability may also give further reasons to abandon $\bot $ and $\top $. (shrink)

Homotopy type theory (HoTT) enables reasoning about groups directly as the types of symmetries (automorphisms) of mathematical structures. The HoTT approach to groups—first put forward by Buchholtz, van Doorn, and Rijke—identifies a group with the type of objects of which it is the symmetries. This type is called the “delooping” of the group, taking a term from algebraic topology. This approach naturally extends the group theory to higher groups which have symmetries between symmetries, and so on. In this paper, we (...) formulate and prove a higher version of Schreier’s classification of all group extensions of a given group. Specifically, we prove that extensions of a group G by a group K are classified by actions of G on a delooping of K. Our proof is formalized in Cubical Agda, a dependently typed programming language and proof assistant which implements HoTT. (shrink)

We discuss how to formulate and understand contradiction. After stressing the importance of a correct formulation for a notion as important as the notion of contradiction, we present a variety of formulations of the proposition corresponding to “p and ¬p”, which is often considered as expressing contradiction. We then discuss the standard example of contradiction in classical logic and the way Wittgenstein de.

Avicenna's logic has gathered much interest in recent decades. However, one significant point that has hitherto been frequently neglected is what logic means to Avicenna in the first place. Considering this issue, the main aim of this paper is to investigate Avicenna's account of logic. This is done by focusing on the logical sections of five of his comprehensive works, namely, the Kitāb al-Madkhal (of the Kitāb al-Shifāʾ, the Cure), al-Najāt (the Salvation), Dānishnāmi-yi ʿAlāʾ ī (Philosophy for ʿAlāʾ-ad-Dawla), al-Mashriqiyyūn (the (...) Easterners) and al-Ishārāt wa-t-tanbīhāt (the Pointers and Reminders). The reason for choosing these five works is fourfold, namely, logico-philosophical, linguistic, chronological and diversity of intended audience. Each work is analyzed based on the three fundamental elements which Avicenna links to the notion of logic, i.e. its definition, its status and its goal. The results of this analysis are then compared. Overall, Avicenna seems to have held a very stable account of logic across these works. (shrink)

George Boole published his account on hypotheticals in his pamphlet The Mathematical Analysis of Logic in 1847. Hypothetical deductions were not as developed as categorical ones by Boole’s time. It was still common practice to reduce hypotheticals to categoricals. Boole innovated by proposing an algebraic method to derive (equations expressing) the conclusions of hypotheticals. He had developed a calculus of classes for categoricals in his first pamphlet chapters and seemingly intended extending it to hypotheticals. Nonetheless, propositions can be only true (...) or false. In contradistinction with his approach for categorical propositions, here it was not possible to consider things or classes of things but only the cases and conjunctures of circumstances in which the propositions are true. Some criticisms have been raised concerning Boole’s view of logic as the solution of algebraic equations. In order to circumvent such criticisms, we construct a deductive system based on Boole's algebraic treatment and apply it to deductions of those hypotheticals analyzed by Boole. We found that Barbara translated in our notation is useful to eliminate symbols in such deductions. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to hypotheticals in The Mathematical Analysis of Logic. (shrink)

Reasoning with diagrams is considered to be a peculiar form of reasoning. Diagrams are often associated with imagistic representations conveyed by spatial arrangements of lines, points, figures, or letters that can be manipulated to obtain knowledge on a subject matter. Reasoning with diagrams is not just ‘peculiar’ because reasoners use spatially arranged characters to obtain knowledge – diagrams apparently have cognitive surplus: they enable a quasi-intuitive form of knowledge. The present paper analyses the issue of diagrams’ cognitive value by enquiring (...) into the tradition of symbolic cognition developed by Leibniz, Lambert, and Kant. The proposal resulting from this enquiry is to question the idea that the cognitive value of diagrams lies solely in allowing evidence for inferences. The imaginative dimension of diagrams connects reasoning to doxastic attitudes of meditation and enquiry. (shrink)

Frege’s Caesar problem is best understood as a multidimensional problem that touches on semantical, epistemological as well as metaphysical issues. The aim of this paper is to shed light on a dimension that has been largely ignored in the literature. It is argued that Frege’s own presentation of the problem strongly suggests that the problem consists primarily in providing the concept of number with a ‘sharp delimitation’. The demand for the ‘completeness of definitions’ was already made by Frege in Grundlagen (...) and not just in his mature period, as is generally assumed. It derives from the Euclidean goal of Grundlagen to transform the arithmetic of his time into an ‘exact science’ in the proper sense. The metaphysical problem as well as the epistemological and the semantical problem that are involved in the Caesar problem only arise because the concept of number is not sharply delimited. The latter problem must therefore be regarded as the fundamental one. Moreover, it can be shown that Frege’s definition of number presupposes that all concepts of his system are sharply delimited. This explains why he considered the solution of the Caesar problem as an integral part of his logical foundation of arithmetic. (shrink)

In his Tractatus, Wittgenstein proposed a method for calculating probability using truth tables, which served as inspiration for Carnap and Ramsey's work on probability. Despite this, Wittgenstein's idea was not widely considered in the literature. This method involves comparing two propositions, where the first is considered only in true instances, while the other is analyzed only when the first is true. This approach is not dissimilar from Makinson's supraclassical logic, despite the use of different methods. The aim of this work (...) is to shed light on Wittgenstein's method, exploring its foundational aspects and demonstrating the relationship between Wittgenstein's probability and Makinson's supraclassical logic. By doing so, we argue that Wittgenstein anticipated some modern developments in logic, proposing one of the earliest systems capable of incorporating beliefs within a formal calculus. In the final section, we will discuss how Wittgenstein's approach resolves (or, better, dissolves) the Lottery Paradox, showing that within this framework, the paradox ceases to exist. (shrink)

This paper offers an account of the ship of Theseus paradox along the lines of the so-called nonstandard primitivism about vagueness. This account is inspired by a model of the ship of Theseus paradox offered by Dinis that considers near-equality, in the context of Nonstandard Analysis, as the proper way to model the `same as' relation. The output is a class of models which unifies the semantic account of vague gradable adjectives recently proposed by Dinis and Jacinto with that of (...) the `'same as' relation. It does so by taking both paradoxes to arise from a confusion between relations of marginal difference between vague degrees and ``"small'' precise relations between the things that have those degrees. (shrink)

Karl Llewellyn’s critique of the canons of statutory interpretation led to a decline in their use for several decades. His critique, however, faced sustained resistance from some corners of the academy and the judiciary. Although this resistance has had only a selective uptake, it animated a gradual revival of the canons and brought the state of scholarship to an impasse that is for the most part partisan. In this article, I examine the semantic canons from a deeper level and argue (...) that a universal assumption about them is false. Said assumption is that, although not dispositive, the semantic canons at least offer some reasons in favour of or against a candidate interpretation. Inclinations to rely on the semantic canons are also based on this assumption, although it is an assumption that the critics of the canons also share. I argue that this assumption is false because the semantic canons are a class of rules that are by nature not reason-giving. This provides a new ground against giving the semantic canons deliberative weight in questions of statutory interpretation. (shrink)

Questions regarding the nature and acquisition of mathematical knowledge are perhaps as old as mathematical thinking itself, while fundamental issues of mathematical ontology and epistemology have direct bearing on mathematical cognition. Several original contributions to logic and mathematics made by the American polymath, Charles Sanders Peirce, are of direct relevance to these fundamental issues. This chapter explores scientific reasoning as it relates to abduction, a name that Peirce coined for educated “guessing” of hypotheses, which he took to be “the first (...) step of scientific reasoning” and the only creative one. Yet he also argued that all deductive reasoning is mathematical and that all mathematical reasoning is diagrammatic. Representation, especially in the form of a diagrammatic system of logic that Peirce developed, is explored here along with his logic of inquiry, most notably in terms of its manifestation as the logic of ingenuity. Originating in the field of engineering, here the diagram of a problem serves as a heuristic substitute for evaluating the actual situation, an approach that can be extended to other forms of practical reasoning such as ethical deliberation. This chapter also touches upon such diverse but related subjects as non-Euclidean geometry and nonclassical logic, with additional examples that help to elucidate cognitive elements of mathematical knowledge. (shrink)