Logic and Philosophy of Logic

Lampert and Waldrop have recently presented a puzzle about moral responsibility and logical truth, in which they derive a contradiction from three apparently plausible principles: (A) no one is responsible for any logical truth; (B) if no one is responsible for something, then no one is responsible for what it strictly implies; and (C) someone is responsible for something. They argue that, in response, we must give up (B)—a principle that plays a key role in arguments for incompatibilism. In this (...) paper, I argue that this response is mistaken, and that the lesson to be drawn lies elsewhere. For the puzzle also rests on a conception of logical truth according to which there are contingent logical truths. And, given this conception, (A) is false—for reasons independent of (B). So the import here concerns, not a connection between responsibility and the debate over incompatibilism, but a connection between responsibility and the debate over the correct conception of logical truth: one can hold on to (A), or hold on to the relevant conception of logical truth, but not both. (shrink)

We investigate the tower spectrum in the generalized Baire space, i.e., the set of lengths of towers in $\kappa ^\kappa $. We show that both small and large tower spectra at all regular cardinals simultaneously are consistent. Furthermore, based on previous work by Bağ, the first author and Friedman, we prove that globally, a small tower spectrum is consistent with an arbitrarily large spectrum of maximal almost disjoint families. Finally, we show that any non-trivial upper bound on the tower spectrum (...) in $\kappa ^\kappa $ is consistent. (shrink)

In a recent paper, Nils Kürbis argues that bilateral natural deduction systems in which assertions and denials figure as hypothetical assumptions are unintelligible. In this paper, I respond to this claim on two counts. First, I argue that, if we think of bilateralism as a tool for articulating discursive norms, then supposition of assertions and denials in the context of bilateral natural deduction systems is perfectly intelligible. Second, I show that, by transposing such systems into sequent notation, one can make (...) perfect sense of them without talking about supposition at all, just talking in terms of relations of committive consequence. I conclude by providing some motivation for adopting this normative interpretation of bilateralism on which this response to Kürbis’s argument is based. (shrink)

Temporal Type Theory (TTT) has been recently introduced as a topos-theoretic approach to understanding the behaviour of systems over time. A truly innovative point of TTT is that it makes truth inherently dependent on time; this is to be contrasted with the classical approach in which past, present and future are related via logical operators. Further on this line of research, the notion of truth is substituted by the ‘time duration’ over which a proposition is true, giving rise to the (...) logic of temporal landscapes. In this paper, we embed the main notions of the Calendar and Time Type System (a framework that was built for reasoning about calendric notions in the context of the Web and is centred around the notion of time granularity) in Temporal Type Theory, attempting to address a common requirement in the practical use of temporal formalisms: the ability to speak about certain time moments and appropriate time windows in (past and future) time, in the form of hours, dates, holidays, etc. With a view to concrete applications, we provide examples on how calendar types can be combined with temporal operators of TTT to augment the expressive power of this framework. In that respect, the addition of calendar types genuinely adds to the range of Temporal Type Theory as a powerful specification language. (shrink)

It is shown that one common formulation of Stalnaker's semantics for conditionals is incomplete: it has no sound and (strongly) complete proof system. At first, this seems to conflict with well-known completeness results for this semantics (e.g., Stalnaker and Thomason 1967; Stalnaker 1970 and Lewis 1973, ch. 6). As it turns out, it does not: these completeness results rely on another closely-related formulation of the semantics that is provably complete. Specifically, the difference comes down to how the Limit Assumption is (...) stated. I close with some remarks about what this means for the logic of conditionals. (shrink)

We determine the strictly positive fragment \(\textsf{QPL}^+(\textsf{HA})\) of the quantified provability logic \(\textsf{QPL}(\textsf{HA})\) of Heyting Arithmetic. We show that \(\textsf{QPL}^+(\textsf{HA})\) is decidable and that it coincides with \(\textsf{QPL}^+(\textsf{PA})\), which is the strictly positive fragment of the quantified provability logic of of Peano Arithmetic. This positively resolves a previous conjecture of the authors described in [ 14 ]. On our way to proving these results, we carve out the strictly positive fragment \(\textsf{PL}^+(\textsf{HA})\) of the provability logic \(\textsf{PL}(\textsf{HA})\) of Heyting Arithmetic, provide (...) a simple axiomatization, and prove it to be sound and complete for two types of arithmetical interpretations. The simple fragments presented in this paper should be contrasted with a recent result by Mojtahedi [ 43 ], where an axiomatization for \(\textsf{PL}(\textsf{HA})\) is provided. This axiomatization, although decidable, is of considerable complexity. (shrink)

While sets are combinatorial collections, defined by their elements, classes are logical collections, defined by their membership conditions. We develop, in a potentialist setting, a predicative approach to (logical) classes of (combinatorial) sets. Some reasons emerge to adopt a stricter form of potentialism, which insists, not only that each object is generated at some stage of an incompletable process, but also that each truth is “made true” at some such stage. The natural logic of this strict form of potentialism is (...) semi-intuitionistic: where each set-sized domain is classical, the domain of all sets or all classes is intuitionistic. (shrink)

Two criticisms of the virtue-theoretic approach to argument appraisal are as follows. First, it is inadequate as argument cogency is conceptually independent of the characteristics of arguers (Bowell and Kingsbury 2013). Second, it is unmotivated since the viability of virtue argumentation theory (VAT) doesn’t require a virtue-theoretic approach to argument appraisal. This deflates the first criticism as an evaluation of VAT (Gascon 2016, Paglieri 2015). I consider each and explain why it is misguided highlighting the connection between the general concept (...) of good argument and associated criteria of goodness, and the connection between good arguments and good arguing. Résumé: L’approche fondée sur la théorie de la vertu pour l’évaluation des arguments fait l’objet de deux critiques. Premièrement, elle est inadéquate, car la force de l’argument est conceptuellement indépendante des caractéristiques des argumentateurs (Bowell et Kingsbury 2013). Deuxièmement, elle est dénuée de motivation, car la viabilité de la théorie de l’argumentation fondée sur la vertu (AFV) ne nécessite pas une approche fondée sur la théorie de la vertu pour l’évaluation des arguments. Cela dévalorise la première critique en tant qu’évaluation de la AFV (Gascon 2016, Paglieri 2015). J’examine chacune de ces critiques et j’explique pourquoi elles sont erronées en soulignant le lien entre le concept général de bon argument et les critères de bonté qui lui sont associés, et le lien entre les bons arguments et la bonne argumentation. (shrink)

In Memoriam - John Hayden Woods (1937-2024).

In the paper The Distinction Between False Dilemma and False Disjunctive Syllogism, by Taeda Tomić, Informal Logic vol. 41, No 4 (2021), pp. 607-639, there are typos in Figure 12, on page 630.

The paper presents an extended scheme for the inference to the best explanation (IBE). The scheme precisely treats the epistemic modifiers (“hypothetically,” “plausibly,” “presumably”) of the inference, acknowledges its contrastive nature, clarifies the logical support between premises and conclusions (linked, convergent, and serial support), and introduces additional premises essential for inferring justified conclusions (especially those related to causal explanations and more demanding standards of proof). Overall, it advances the existing schemes for IBE in argumentation theory and treats IBE as a (...) par excellence argumentative, rather than explanatory, form of reasoning. Résumé: L’article présente en détail un schéma pour l’inférence vers la meilleure explication (IME). Le schéma traite précisément les modificateurs épistémiques (« hypothétiquement », « plausiblement », « vraisemblablement ») de l’inférence, reconnaît sa nature qui fait contraste, clarifie l’appui logique entre les prémisses et les conclusions (l’appui lié, convergent et sériel) et introduit des prémisses supplémentaires essentielles pour inférer des conclusions justifiées (en particulier celles liées aux explications causales et aux normes de preuve plus exigeantes). Dans l’ensemble, l’article fait progresser les schémas existants pour l’IME dans la théorie de l’argumentation et traite l’IME comme une forme de raisonnement argumentatif par excellence, plutôt qu’explicatif. (shrink)

I am most grateful to the editors of Informal Logic for their willingness to publish my absurdly long paper (Siegel 2023a) in its entirety, and for organizing the four commentaries published along with it. I am grateful as well to Bart Garssen, Andrew Aberdein, Paula Olmos and Christoph Lumer for their insightful and challenging discussions. In what follows I respond to their criticisms and suggestions in the order in which they appear in the journal. Résumé: Je suis très reconnaissant aux (...) éditeurs d’Informal Logic pour leur volonté de publier mon article absurdement long (Siegel 2023a) dans son intégralité et pour avoir organisé les quatre commentaires publiés avec lui. Je suis également reconnaissant à Bart Garssen, Andrew Aberdein, Paula Olmos et Christoph Lumer pour leurs discussions perspicaces et stimulantes. Dans ce qui suit, je réponds à leurs critiques et suggestions dans l’ordre dans lequel elles apparaissent dans la revue. (shrink)

The scholarship on deep disagreements presents us with a considerable number of seemingly disparate characterizations concerning the nature of these disputes. This paper is motivated by the desire to grasp what these characterizations are. An answer is provided through the method of reconstructive analysis. Two ideal and paradigmatic models of deep disagreements are defined initially. Then, individual characterizations found in the scholarship are examined against the background of such models. Special attention is given to Fogelin’s paper, the work that initiates (...) modern discussion on deep disagreements. According to the interpretation provided in the following paper, both models inadvertently coexist in this seminal work. Résumé: Les études sur les désaccords profonds nous présentent un nombre considérable de propriétés apparemment disparates concernant la nature de ces conflits. La rédaction de cet article est motivée par le désir de comprendre quelles sont ces propriétés. Une réponse est fournie par la méthode de l'analyse reconstructive. Deux modèles idéaux et paradigmatiques des désaccords profonds sont d'abord définis. Ensuite, les propriétés individuelles trouvées dans les études sont examinées dans le contexte de ces modèles. Une attention particulière est accordée à l'article de Fogelin, l'ouvrage qui initie la discussion moderne sur les désaccords profonds. Selon l'interprétation fournie dans l'article suivant, les deux modèles coexistent par inadvertance dans cet ouvrage fondateur. (shrink)

Yu and Zenker (2022) argue that the oft-made distinction between convergent and linked argument structure is problematic. If their account holds, the linked/convergent distinction, at least as I have characterized it (Freeman 2011), seems to violate the dictum that structural analysis should come before evaluation. In this Reply I defend the position that we do not need to estimate or determine argument strength to determine whether the premises of an argument are linked or convergent. Résumé: Yu et Zenker (2022) soutiennent (...) que la distinction souvent faite entre structure d’argumentation convergente et structure d’argumentation liée est problématique. Si leur explication est valable, la distinction liée/convergente, du moins telle que je l’ai caractérisée (Freeman 2011), semble violer le dicton selon lequel l’analyse structurelle doit précéder l’évaluation. Dans cette réponse, je défends la position selon laquelle nous n’avons pas besoin d’estimer ou de déterminer la force de l’argument pour déterminer si les prémisses d’un argument sont liées ou convergentes. (shrink)

We argue that Conces’ and Walters’ tool FM2.0 is valuable but does not show that false dilemma is a formal fallacy. The FM2.0 assumes an ambiguous use of the term ‘formal fallacy’, different from how the term is used in logic, and may show that any argument is a formal fallacy. Moreover, the FM2.0 is developed by using one type of false disjunctive syllogism. However, the adequate application of FM2.0 on false dilemma does not lead to an invalid augmented argument, (...) contrary to what is expected in FM2.0, even assuming the ambiguous use of the term ‘formal fallacy’. Résumé: Nous soutenons que l’outil FM2.0 de Conces et de Walters est utile mais ne démontre pas que le faux dilemme est un sophisme formel. Le FM2.0 suppose une utilisation ambiguë du terme « sophisme formel », différente de son utilisation en logique, et peut montrer que tout argument est un sophisme formel. De plus, le FM2.0 est développé en adoptant un type de faux syllogisme disjonctif. Cependant, l’application adéquate du FM2.0 au faux dilemme ne conduit pas à un argument augmenté non valide, contrairement à ce qui est attendu dans le FM2.0, même en supposant l’utilisation ambiguë du terme « sophisme formel ». (shrink)

In this paper, I argue that inductive inferences can be reinterpreted as deductive through a metaphysical lens of modal necessity. Instead of viewing deduction as a limit case of induction where the conclusion is certain rather than probable, we can rethinking inductive inferences as a particular type of deductive inferences where the conclusion is necessitated by premises in specific modal ranges.

Soames attributes to Kripke the theory of epistemic possibility that uses metaphysical impossibilities in explaining necessary a posteriori truths. I attribute to Kripke a theory from epistemic counterparthood. I develop an epistemic accessibility based on Kripke’s appeal to Lewis’ counterpart theory that is reflexive, non-transitive, and non-symmetric. I also propose an epistemic counterpart function and a description function.

The philosophy of superdeterminism is based on a single scientific fact about the universe, namely that cause and effect in physics are not real. In 2020, accomplished Swedish theoretical physicist, Dr. Johan Hansson published a physics proof using Albert Einstein’s Theory of Special Relativity that our universe is superdeterministic meaning a predetermined static block universe without cause and effect in physics. The unity of our universe originates from its creation from the same nothingness under the zero energy universe theory. However, (...) nothingness cannot actually cause a universe to exist on its own under superdeterminism due to the absence of cause and effect in physics. Moreover, nothingness is the absence of anything real and cannot logically be the reality underlying our universe. Consequently, our universe must exist from the reality of the logic or Logos underlying our physical reality resulting from the origination of our universe from the same nothingness or common creation. (shrink)

The concept of location—of something being located at, or occupying, a region or a place—has been an important topic of philosophical investigation over the past fifteen years or so. Yet all the theories of location that have been put forward so far are unsatisfactory, because they fail to have the conceptual resources to describe certain basic locational phenomena. I introduce and partly develop a novel theory of location that does better in this respect than its predecessors. Its most distinctive feature (...) is the way in which it handles so-called cases of multilocation: according to the theory, objects capable of being multilocated occupy regions via proxies, and—in a slogan, and very roughly—to be multilocated is to have several proxies located at different regions. (shrink)

Proof-theoretic semantics (P-tS) is an innovative approach to grounding logical meaning in terms of proofs rather than traditional truth-conditional semantics. The point is not that one provides a proof system, but rather that one articulates meaning in terms of proofs and provability. To elucidate this paradigm shift, we commence with an introduction that contrasts the fundamental tenets of P-tS with the more prevalent model-theoretic approach to semantics. The contribution of this paper is a P-tS for a substructural logic, intuitionistic multiplicative (...) linear logic (IMLL). Specifically, we meticulously examine and refine the established P-tS for intuitionistic propositional logic. Subsequently, we present two novel and comprehensive forms of P-tS for IMLL. Notably, the semantics for IMLL in this paper embodies its resource interpretation through its number-of-uses reading (restricted to atoms). This stands in contrast to the conventional model-theoretic semantics of the logic, underscoring the value that P-tS brings to substructural logics. (shrink)

Many logic handbooks allude to the obvious connection between propositional logic and logic circuits. Truth functions in logic can be represented by logic circuits in which the high or low voltage levels of the circuits correspond to the true and false logic values, respectively. At the propositional logic level, the logical connectives of propositions can be simulated by logic circuits as follows: the true or false logical evaluation of atomic propositions corresponds to the high or low level of the circuit (...) input and the truth value of compound propositions corresponds to the circuit output state. A high circuit output signifies that the compound sentence is evaluated as true, whereas a low output indicates that the compound sentence is false. It is well known that in the world of logic circuits, the AND connective in logic corresponds to the AND gate, the OR connective to the OR gate and the negation operation to the inverter. The output of a circuit equivalent to contradiction is always low and that of a circuit corresponding to tautology is always high irrespective of the input state. The remaining compound formulas correspond to logic circuits with a high output level for some inputs and low output level for other inputs. However, what logical circuit can model a circular sentence? (shrink)

We present a logical framework that enables us to define a formal theory of computational trust in which this notion is analysed in terms of epistemic attitudes towards the possible objects of trust and in relation to existing evidence in favour of the trustworthiness of these objects. The framework is based on a quantified epistemic and justification logic featuring a non-standard handling of identities. Thus, the theory is able to account for the hyperintensional nature of computational trust. We present a (...) proof system and a frame semantics for the logic, we prove soundness and completeness results and we introduce the syntactical machinery required to define a theory of trust. (shrink)

I argue that discussions of Frege’s conception of elucidation have suffered from a conflation of two distinct issues: elucidation of primitive scientific terms, and elucidation of the logical categories. The former seeks to bring us to grasp the Bedeutung of terms that stand at the beginning of the chain of definitions of a scientific system. The latter cannot be understood on the model of securing agreement in Bedeutung at all. I show how existing discussions of Fregean elucidation insufficiently take this (...) difference into account. I adumbrate what I take to be a more accurate understanding of Fregean elucidation of the logical categories, starting from the observation that Frege, when engaged in such elucidation, consistently reverts to talking about signs. Frege, I argue, takes signs to possess logical features, and it is these logical features which his elucidations are meant to help us to grasp. For Frege, the nature of the logical categories lies in the signs. In this way, I argue, Frege’s approach to elucidating the logical categories is incompatible with a realist conception according to which there is a logical order of reality that is prior to the logical order of language. (shrink)

Theory of Mind (ToM), the cognitive capacity to attribute internal mental states to oneself and others, is a crucial component of social skills. Its formal study has become important, witness recent research on reasoning and information update by intelligent agents, and some proposals for its formal modelling have put forward settings based on Epistemic Logic (EL). Still, due to intrinsic idealisations, it is questionable whether EL can be used to model the high-order cognition of ‘real’ agents. This manuscript proposes a (...) framework that takes visibility, memory, perspective and communication as the main factors underpinning mental attributions. This is more in-line with findings in cognitive science and allows us to model well-known False-Belief Tasks that are typically used in testing people's ToM. We discuss some of the framework's technical features, arguing why it does justice to empirical observations. (shrink)

Desde los trabajos de von Neumann y Birkhoff hasta la actualidad, el estudio de distintas estructuras algebraicas asociadas al formalismo cuántico ha dado lugar a interesantes desarrollos. A modo de ejemplo, el teorema de Kochen-Specker ha tenido una fuerte repercusión en los fundamentos e interpretación de la teoría cuántica. En este trabajo, prestaremos especial atención al abordaje lógico-algebraico iniciado por von Neumann y Birkhoff (aunque también discutiremos otros formalismos, tales como la lógica de la superposición de Tzouvaras). Se presenta el (...) formalismo de Nmatrices como una semántica adecuada para el retículo de proyectores cuánticas, logrando una novedosa caracterización de los estados cuánticos como valuaciones de cierta Nmatriz cuántica. (shrink)

Several recent results bring into focus the superintuitionistic nature of most notions of proof-theoretic validity, but little work has been done evaluating the consequences of these results. Proof-theoretic validity claims to offer a formal explication of how inferences follow from the definitions of logic connectives (which are defined by their introduction rules). This paper explores whether the new results undermine this claim. It is argued that, while the formal results are worrying, superintuitionistic inferences are valid because the treatments of atomic (...) formulas are insufficiently general, and a resolution to this issue is proposed. (shrink)

Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can be defined as a sheaf for the coherent topology on a certain category of compact Hausdorff spaces. In this case, the sheaf condition has a fairly simple explicit description, which arises from studying the relationship between the coherent, regular, and extensive topologies. In this (...) paper, we establish this relationship under minimal assumptions on the category, going beyond the case of compact Hausdorff spaces. Along the way, we also provide a characterization of sheaves and covering sieves for these categories. All results in this paper have been fully formalized in the Lean proof assistant. (shrink)

The modality of normality distinguishes states of affairs which are normal from those which are abnormal. Existing work on the modality of normality assumes that it is a restriction of metaphysical modality. In this paper, we argue that this assumption is inappropriate and explore the consequences of abandoning it. -/- After preliminary discussion (§1), we introduce the dominant framework for reasoning about normality (§2) and argue that it ascribes implausibly strong structural properties to the modality. In its place, we propose (...) a new framework, which avoids this commitment (§§3-5). This account has a number of interesting features, which we explore in both an informal and formal setting. If correct, it implies that the modality of normality occupies a distinctive place in the space of modalities. Before concluding, we consider some of the wider implications of our account (§6), focusing on the role normality has played in epistemic theorizing. (shrink)

Reconciling the many “faces” of Peirce – the Scientist, Philosopher, and Metaphysician - helps to make sense of the open-endedness and versatility of semiotics. Semiosis, for Peirce, knows no rigid hermeneutic or disciplinary bounds. It thus forces us to be open to interdisciplinary and holistic inquiries. The pragmatic maxim sets limits on metaphysical speculation, but it also legitimates the extension of the experimentalist method into cosmological, metaphysical, and even religious domains. Although Peirce's religious speculations are ultimately unsatisfactory, understanding why Peirce (...) expanded his thinking to such domains reveals the expansionist logic of Peircean semiotics beyond the mere idiosyncracies of the man. The triadic, dynamic logic of pragmatist inquiry seeks to expand to all social processes of fixing belief and forming habits, however quixotic; but it remains dubious whether such expansion can fully comport to the scientific, experimentalist method that Peirce preferred. (shrink)

In this article, we will consider the application of intuitionistic tense logics to construct logical systems in which the thesis of logical determinism cannot be expressed.

I apply truthmaker semantics to the logic of conditional imperatives.

In constructional ontology, entities emerge by construction, that is, from the application of constructors to objects. We explore this approach to ontology, focusing on three modules: the constructors, the inputs to the constructors, and the constructional process. Our aim is to identify and assess some key theoretical choices arising in an ontology of this kind.