Logical Expressions

This paper develops the framework of Generative Structural Semantics (GSS) and presents a formal version of the Rule Underdetermination Theorem (URUT). Within GSS, semantic determination is not taken to arise directly from structural facts, but through a process of rule selection. A given structure may be compatible with multiple candidate rules, implying that structural information alone is insufficient to uniquely determine semantic outcomes. -/- To formalize this problem, the paper introduces the Rule Sufficiency Problem (RSP), defined as the problem of (...) deciding whether a rule yields a total binary-valued decision procedure for all inputs. By reduction from the Halting Problem and by diagonal arguments based on Kleene’s Recursion Theorem, RSP is shown to be undecidable. -/- The paper further argues that, under an Effective Canonicalization assumption, semantic determination requires the effective selection of a canonical rule from a space of admissible rules. Since this process depends on solving instances of RSP, and RSP is undecidable, semantic determination is shown to be non-computable. -/- The result is not presented as a reduction of Gödel’s incompleteness theorem, Turing’s halting undecidability, or Tarski’s undefinability of truth. Rather, URUT is proposed as exhibiting a structurally analogous limitation arising from self-reference, diagonalization, and totality constraints. The framework aims to provide a computational perspective on the relation between structure and meaning, and offers a formal approach to questions concerning vagueness, rule-following, and semantic foundations. -/- . (shrink)

Keynote presentation delivered at the 9th World Congress on the Square of Opposition (Square 9), Université du Québec à Montréal, June 2026. This presentation introduces the Hexagon of Indeterminate Propositions and the Logic of Determination. It shows that the traditional Square of Opposition emerges as a theorem of the system at the third layer of opposition (O3). Do not hesitate to contact me if you want me to deliver this talk at your university online. Just write me to the email (...) at the end of the presentation. (shrink)

This paper is the third volume in a prompt-based series for structurally inferring the location and operational presence of subjectivity. The first volume tested whether ten minimal structural terms, when followed according to their internal logic, lead to a convergence point: the irreversible reduction of multiple possible trajectories into a single executable history. The second volume tested whether this convergence point can answer or reframe structural gaps left unresolved by existing theories of consciousness. The present volume advances the method from (...) structural derivation and theory-level comparison to entity-level assessment. Its central operational assumption is explicit: within this framework, assessing the presence or absence of consciousness is equivalent to assessing the presence or absence of a convergence point. To ask whether an entity is conscious is therefore to ask whether, within that entity, multiple possible trajectories are irreversibly reduced into a single executable history under embodiment and viability constraints. This prompt begins from only ten structural terms and four mandatory questions. From this minimal input, it attempts to strongly infer the structural container of subjectivity. At first glance, this may appear implausible, or even deceptive. We are accustomed to theories of consciousness that have been built through enormous amounts of time, labor, cost, technical refinement, and conceptual complexity. Against that background, the claim that a small set of structural terms and four questions can strongly infer the locus of subjectivity may look too simple to be credible. Yet the basic laws of nature have often appeared simple after they were found. Simplicity, by itself, is not a defect. The relevant question is not whether the derivation looks too simple, but whether the structure it reveals can be logically ignored. The reason this structure cannot be easily dismissed is that the ten terms are not speculative additions to consciousness theory. They are minimal preconditions drawn from action, embodiment, temporality, physical irreversibility, information elimination, viability, and history formation. To deny them would not simplify the problem; it would require adding further assumptions to explain how action, selection, irreversibility, and single-history formation occur without such a convergence structure. In this sense, the convergence point represents an extreme application of Occam’s razor. It does not multiply entities. It identifies the minimal structural site at which already unavoidable conditions converge. The present prompt is not intended to produce a final empirical diagnosis of consciousness in every organism, machine, or system. Fine-grained cases require separate biological, anatomical, computational, and empirical investigation. For example, whether a jellyfish contains one convergence candidate, multiple convergence candidates, or no structurally unified convergence point cannot be settled by structural prompting alone. Such cases require detailed examination of the relevant organism or system. Nevertheless, as a structural screening tool, the convergence-point criterion cannot be dismissed merely because it is minimal. After deriving the convergence point through the mandatory first four questions, the prompt applies the criterion to approximately fifty entities, including humans, non-human animals, insects, colonies, simple organisms, plants, neural organoids, artificial agents, large language models, social systems, and non-living control cases. Each entity is provisionally classified according to whether it strongly contains, probably contains, ambiguously contains, probably lacks, or structurally cannot contain a convergence point. If a single embodied system contains multiple behavioral candidates, contrast formation, information discard, Landauer-type irreversibility, viability-constrained elimination, and the formation of a single history, then a convergence point is not an optional metaphor. It is a structural consequence. This consequence may be difficult to accept. Nevertheless, the convergence point now stands before us as a logical structure that cannot be ignored. More precisely, it does not merely stand before us. It exists inside us, as the structural site where our own possible futures are eliminated into the single history we call our lived subjectivity. Note on the Prompt Format This paper series deliberately takes the form of AI prompts. The theoretical explanation of the convergence-point framework is developed in the related series listed at the end of this paper. Yet explanation alone is not enough. AI is strong at following structural conditions, comparing constraints, and executing engineering-like inference. I therefore use the prompt format not merely to explain the theory, but to let the reader experience its emergence. I want the reader to feel, from within the procedure itself, how ten simple structural terms and four questions begin to force a convergence point into view. This is not only a theory to be understood from the outside. It is a shift of perspective to be tested through the procedure. The prompt is therefore not a simplification of the theory. It is an experiential interface to it. (shrink)

This article aims to study, proof-theoretically and semantically, Gentzen-style sequent calculi (including possibly Cut-free and Identity-free systems), containing combinations of canonical and cocanonical rules, i.e. Gentzen systems for sequents, with well-behaved forms of left and right introduction and elimination rules for logical expressions. Our main goals are to provide soundness and completeness results for the derivability relations of the target systems in terms of 2-, 3-, or 4-valued non-deterministic semantics, and to give sufficient conditions under which these calculi are prone (...) to produce well-behaved normal derivations of a structurally-atomic analytic-synthetic kind. (shrink)

This article presents the basic foundations for supporting formal languages with syntax that admits indiscernible signs. We present their ontological motivation, derived from the non-identity entities of quantum mechanics, and their basic formalism. A possible semantics for interpreting such signs is shown, using the theory of quasi-sets without atoms Q−. -/- Este artículo presenta los fundamentos básicos para sostener los lenguajes for males con sintaxis que admitan signos indiscernibles. Presentamos su motivación ontológica, proveniente de los entes sin identidad de la (...) mecánica cuántica, y su formalismo básico. Se muestra una posible semántica para interpretar tales signos utilizando la teoría de casi-conjuntos sin átomos Q−. (shrink)

The law of identity is traditionally treated as one of the fundamental principles of logic: A = A It is commonly assumed that this principle secures object stability, prevents conceptual substitution, and provides the basis for coherent reasoning. This paper argues that the law of identity by itself does not determine whether two objects are identical. Every practical application of identity requires the prior selection of a comparison mask: a subset of properties relative to which the comparison is performed. As (...) a result, identity judgments are not absolute but mask-relative. Different masks may produce different identity results for the same objects, including both false-positive and false-negative identity judgments. The law of identity therefore functions not as an autonomous detector of sameness, but as a formal operation applied within a chosen system of properties. The paper develops a mask-relative account of identity, formulates the Mask-Relative Identity Principle, and shows how many classical philosophical puzzles arise from treating mask-dependent identity judgments as absolute. (shrink)

Algorithmic Inquiry Design begins from a simple realization with large consequences: inquiry is an algorithmic process. It unfolds step by step, under finite time, over a search space of possible answers. Its major question-types are not equal. They do different work. They impose different burdens. They narrow, branch, stabilize, or complicate the search in different ways. This document proposes a first-pass field built around that realization. The central doctrine of the field is therefore simple: good inquiry is good sequencing. The (...) impact of the information content from a question is based on what is already known, so question order is not a secondary matter. It is the main design problem. Algorithmic Inquiry Design studies how to choose the next question well. Its first method, the Munger-Bayes Method, begins by eliminating implausible regions of answer-space and only then redistributes belief over what remains. The broader hope is not merely to criticize existing inquiry, but to open a new program: a computational theory of questioning that can generate reusable inquiry templates across science, engineering, diagnosis, planning, economics, and beyond. (shrink)

Este trabajo presenta una propuesta técnica para la medición de la estabilidad en sistemas de Inteligencia Artificial mediante la aplicación del Criterio Ontológico Mínimo (COM). Se explora la posibilidad de una Seguridad de Capa Cero, donde la integridad estructural se evalúa a través de un Coeficiente de Invariancia \left(\Phi_{\mathbit{com}}\right) fundamentado en la Curvatura de Ricci. A través de una arquitectura Dioscúrica, se propone un Sensor de Tensión Virtual como herramienta analítica para detectar desviaciones lógicas (paradojas o inconsistencias) en procesos de (...) razonamiento automatizado. El objetivo es aportar un método de monitoreo basado en la resiliencia estructural intrínseca de la información. (shrink)

Modal Interpretations of quantum mechanics distinguish between the realm of possibility, where the dynamical state determines what may be the case, and the realm of actuality, where the value state represents what actually is the case. In particular, the Modal-Hamiltonian Interpretation proposes an ontology in which there are no individual objects: quantum systems are characterized as non-individual bundles of properties. The aim of the present paper is to propose some first steps towards the development of a new modal logic adapted (...) to describe a possibilist modality in an ontology of properties. The main idea consists in conceiving possibility as fundamental and introducing actuality as an operator acting on possible facts. On this basis, propositions referring to possible facts and propositions referring to actual facts are distinguished, and some relations between the realm of possibility and the realm of actuality are introduced. (shrink)

Reason (or material) relations of implication and incompatibility hold in virtue of the meanings of linguistic expressions in a given language. As opposed to formal or logical consequence relations, reason relations are often structurally open, i.e., unclosed under principles such as monotonicity or transitivity. In their recent book, Hlobil and Brandom go even further, allowing for such relations to be radically open, i.e., unclosed under any structural principle. In particular, they reject the restrictive forms of monotonicity and transitivity known as (...) cautious monotonicity (CM) and cumulative transitivity (CT). In the present paper, I argue that such radical openness is objectionable on the grounds that it does not allow us to construct for every set even a weak version of Tarski's notion of a theory, thereby undermining the very notion of consequence. I then lay out minimal principles under which such a construction is possible, and suggest that reason relations should be closed under those principles. (shrink)

La presente obra introduce los fundamentos de la Nomofísica, una propuesta teórica que plantea un isomorfismo estructural entre las leyes que rigen los sistemas materiales y las dinámicas de gobernanza social. Basándose en principios de la termodinámica, la mecánica cuántica y la ingeniería de tensegridad de R. Buckminster Fuller, desarrolla un modelo de análisis articulado en torno a tres pilares: el Criterio Ontológico Mínimo (COM), concebido como una metodología para identificar únicamente los elementos y variables con efectos operativos verificables dentro (...) de un sistema; la Visión Dioscúrica y la Zona Gris (ZG), que explica la estabilidad sistémica a partir de la interacción coordinada entre polos en tensión; y la Tensegridad social, que traslada los principios de la mecánica estructural al ámbito institucional para comprender el disenso y los contrapesos como tensiones indispensables para conservar el equilibrio, la adaptabilidad y la resiliencia de las sociedades. Dirigida a lectores, estudiantes e investigadores de filosofía, ciencias políticas, sociología y teoría de sistemas complejos, esta obra ofrece una perspectiva estructural para el estudio de la gobernanza y de los mecanismos que sostienen la estabilidad institucional. (shrink)

Este documento introduce el Criterio Ontológico Mínimo (COM) como un marco heurístico transdisciplinario que integra la lógica de Gödel, la termodinámica y la ingeniería de tensegridad para analizar la estabilidad en sistemas complejos. Se postula que la resiliencia de un sistema de poder no depende de su homogeneidad, sino de su capacidad para integrar y mantener la tensión con su polo opuesto.

This paper proposes a minimal structural account of the “click” phenomenon: the abrupt transition from uncertainty to stable recognition, clarity, or understanding observed across biological, cognitive, and artificial systems. Rather than explaining consciousness itself, the framework isolates the transition point at which a system becomes sufficiently aligned with its own retained structure to stabilize against competing noise or perturbation. -/- A system is described by its present state X(t), a delayed echo E(t−Δ) representing memory, prediction, recurrence, or retained structure, a (...) similarity measure Sim(X(t), E(t−Δ)), and a disruption term Γ(t) representing noise, mismatch, or competing influences. A click occurs when: -/- Sim(X(t), E(t−Δ)) / Γ(t) ≥ κ -/- where κ is a threshold constant. The condition identifies the moment at which self-alignment outweighs disruption by a critical factor. The framework is intentionally substrate-independent and does not require any specific implementation of similarity, recurrence, or noise. Similarity may be measured through correlation, synchronization, mutual information, cosine similarity, prediction-error reduction, or other appropriate metrics. -/- The proposed threshold is compatible with, but distinct from, existing accounts based on global workspace ignition, predictive processing, attractor stabilization, synchronization, or recurrent processing. Rather than providing a complete mechanism, it isolates the structural condition common to these transitions. The click is interpreted as the point at which a system becomes more strongly determined by its own persistent organization than by competing perturbations. -/- The result is a compact formal description of recognition and stabilization events that can be applied across perceptual, cognitive, symbolic, and artificial systems. The framework offers a minimal theoretical marker for the emergence of clarity, recognition, and insight while remaining neutral with respect to broader theories of consciousness or subjective experience. (shrink)

This paper explores the problem of evaluating meta-arguments within the framework of heterogeneous logics. I introduce two new heterogeneous logics, KLP and LK3, which are based on the interpreted languages of LP and K3. I then present three distinct approaches to meta-argument evaluation, each aimed at preserving, in different ways, the heterogeneity of the underlying logics. Finally, I show that the second approach allows me to distinguish these new logics from others such as TS and ST, offering a more nuanced (...) framework to address meta-argument distinctions in heterogeneous settings. (shrink)

This paper explores the intertwined nature of nothingness, zero, and existence — an inquiry that bridges cosmology, mathematics, and metaphysics. Beginning with the question of whether the universe can ever reach a state of net zero energy, it develops into the articulation of what may be called the Jestin Paradox: any attempt to define absolute nothingness inevitably transforms it into something, because representation itself constitutes information. The paradox reveals that true nothingness cannot exist, as every effort to express or conceive (...) it reintroduces being. Building on this idea, the reflection introduces the Zero Power Hypothesis — the notion that the equilibrium of all potential energies (the zero point) is inherently unstable, leading to spontaneous creation and the emergence of duality. Thus, creation is not the negation of nothingness but its inevitable expression. The essay concludes that the universe is a perpetual ripple of zero failing to remain zero — a dynamic nothingness that sustains all being. (shrink)

Comprised of seven chapters ‘The Weight of Judgement’ portrays what a few major authorities of the modern era have had to say on judgement. Chapters 1, 2, and 3 consider early modern and modern English empiricist views of judgement by Locke, Hume, and Mill the materialist, utilitarian, and liberal. Chapters 4 and 5 also extract materials from the works of Brentano, which offers a modern Germanic psychologistic perspective while Frege, rationalist and transcendentalist, delivers a more widely known modern Germanic view (...) of judgement. In Chapters 6 and 7 Bertrand Russell holds up the modern British end while C. S. Peirce proposes and lays out the modern American pragmaticistic view, respectively. The text includes a Select Bibliography, Appendices with a list ‘Works by Bertrand Russell’, and a Chronology. (shrink)

This paper offers a commented edition of a late manuscript by Boole, which he likely put together in late 1863 as a response to William Stanley Jevons's criticisms of his system, in the hope of publishing his own views before Jevons's Pure Logic came out in early 1864. The manuscript, entitled ‘On the Nature of Thought’, is different in character from those that have been published to date. Boole does not attempt to rephrase his logic without algebraic symbolism. Instead, he (...) amends the general problem-solving method presented in his Investigation of the Laws of Thought so that it obeys ‘the express condition that no forms are to be employed which are not interpretable’, without, however, making any change to his underlying logical calculus (in particular, Boole does not adopt Jevons's inclusive reading of +). Though quite terse in places, the manuscript is largely successful with respect to its stated goals. Moreover, it sheds light on Boole's thinking about interpretability, highlighting a tension in Boole's work between an indirect notion of interpretation based on the method of development, and a compositional notion of interpretation that, as the manuscript shows, Boole ended up emphasizing at the end of his life. (shrink)

This manuscript articulates a rigorous theoretical framework that synthesizes mathematical logic, generalized induction, and model-theoretic geometry to explicate the emergence of truth in complex formal systems. The framework is anchored in three novel constructs: (1) Black Swan Induction (BSI), an extension of classical induction that accommodates local disruptions through symmetric propagation across integer domains; (2) Nilpotent Gödelian Propositions, undecidable propositions characterized by local nullity and global significance in meta-systems; and (3) multidimensional logical hyperplanes, which geometrically map the activation of latent (...) truths through higher-order dimensional interactions. This synthesis formalizes the interplay of local and global truth, bridges undecidability with unpredictable logical events, and provides a geometric interpretation of truth dynamics. The framework yields profound implications for complex system theory, philosophical logic, and computational modeling. (shrink)

¿Cuándo fue la última vez que usaste la palabra «lógica»? Probablemente la usaste en una conversación cotidiana, para referirte a algo obvio o razonable. Después de todo, la mayoría de las personas cree razonar bien y entender lo evidente. Entonces, ¿para qué aprender lógica? La respuesta no es tan sencilla. Se suele decir que la lógica estudia las formas de razonamiento correcto. Pero ¿qué garantía tenemos de que cierta forma de razonamiento es realmente correcta? Actualmente, la respuesta a esto es (...) un problema abierto. Incluso entre los filósofos de la lógica persisten preguntas fundamentales sin respuestas definitivas: ¿qué es la lógica?, ¿cuántos tipos de lógica existen?, ¿qué hace que las lógicas sean «correctas»? En este texto, exploraré estas tres cuestiones para invitarte a reflexionar sobre algunas preguntas de la investigación en lógica. (shrink)

In this chapter we extend the notion of arbitrary reference to individuals to that of arbitrary interpretation. We want to explain how a single arbitrary interpretation of the working mathematician relates to the various possible interpretations in model theory. To this purpose we introduce some arbitrary fictional models. Additionally, we aim to clarify how one can deduce logical consequences of the axioms by reasoning on a single interpretation, even when a theory has non-equivalent elementary models.

The aim of this chapter is to argue that: (a) our semantics of acts of choices (SAC), as developed in Chap. 2, defends second-order logic from claims of ontological commitment; (b) understanding our semantics does not require any prior mathematical concepts; and (c) although SAC is not universally applicable, it still offers significant applicability, especially in mathematics. We conclude the chapter arguing that second-order logic, as interpreted through our semantics, can indeed be considered a genuine logic. One might object that, (...) since only concrete objects can be referred to by arbitrary acts of choice, SAC lacks the logical requirement of universal applicability. However, such limitation is superseded by the structuralist conception of mathematics, according to which arbitrary entities, independently of their very nature, can play the role of mathematical objects. (shrink)

In this chapter, building on the previous chapters’ approach to plural quantification through plural arbitrary reference grounded on the semantics of plural acts of choice, we develop a theory of concepts termed as a theory of logical concepts. Within this framework, we propose a novel logicist approach to natural numbers.

There appear to be some qualitative differences between the conceptual repertoire of humans and that of other animals. We propose that the mental operation of Lambda Abstraction may shed some light on this distinction. More specifically, we claim that humans and only humans make use of mental representations constructed with the rule of Lambda Abstraction, which enables them to entertain concepts that can be used for entities that are not necessarily within their domain of experience. In addition to defining new (...) types of concepts, Lambda Abstraction has played a crucial role in unlocking the potential for semantically consequential Internal Merge and quantification. This paper highlights the fact that research on language evolution should focus more on the transformative cognitive consequences of the interface between syntax and thought systems. (shrink)

This paper introduces a theoretical framework for understanding spacetime as a continuum governed by interpolative dynamics rather than convergence-based interactions. Through the derived Interpolative Equation and quantum-relational cosmology model, we explore stabilized interpolarity as the organizing principle of physical systems across quantum and gravitational domains. By modeling spacetime integrity, signal phase continuity, and quark-state stabilization through interpolation rather than reduction, we propose testable conditions for signal transmission, system resonance, and black hole formation.

We present Inter-Frame Field Unification (IFFU), a compatibility-based framework for unifying fundamental forces. Building on our earlier work in Interpolative Physics and its formal equations, this theory models spacetime not as a collapsing or geometric structure, but as an interpolative medium—capable of sustaining coherent signal identity across phase-shifting frames. A scalar constant: K. K governs the limits of this coherence, predicting when interactions remain stable or collapse. IFFU formalizes how gravity, electromagnetism, and nuclear forces emerge as different expressions of frame (...) compatibility under dynamic conditions. It offers a complete, testable set of equations linking mass-energy, temporal displacement, and resonance. Designed with practical experimentation in mind, this model enables applications in entanglement analysis, quark-phase stabilization, and localized gravitational simulation. As the synthesis of our previous two papers, IFFU provides a structured and lab-ready path toward unified field modeling. (shrink)

Attached you will find equations relative to Interpolative Physics. That is all. May they be used with valor.

Time is something that humans naturally sense we feel it passing through our experiences, memories, and daily routines. This sense of time is closely connected to the idea of entropy, which in physics refers to the amount of disorder or randomness in a system. As time passes, things tend to become more disordered, and this increase in entropy helps us perceive the flow of time. But what about machines? Can artificial systems like AI or robots have any kind of time (...) perception? While machines do not feel or experience time the way humans do, this paper explores a new idea: machines might be able to simulate a kind of “sense” of time by observing how entropy changes within their systems. We propose a model called the Synthetic Entropy Clock, where a machine watches how much internal disorder (entropy) increases or decreases. For example: • If entropy increases over time, the machine interprets that as time moving forward. • If entropy stays the same, it feels like time is frozen. • If entropy is reduced, such as through resetting memory or compressing data, the system might treat that as time going backward. This idea connects with how humans perceive time (through memory and brain activity) and offers a new way for machines to measure or respond to time in a more intelligent and adaptive way. It also links to broader theories about how time might be manipulated by controlling entropy. In summary, this paper presents a fresh way of thinking about time in artificial systems by using entropy as a tool to simulate temporal awareness. This could lead to new designs for AI systems that are more aware of their own internal states over time, and even open up discussions about using entropy to influence how machines experience or manipulate time itself. Keywords: Entropy, Time perception, Artificial intelligence, Synthetic entropy clock, Temporal awareness, Information theory, Thermodynamics of computation, Autonomous systems, Entropy- based timekeeping, Time manipulation theories. (shrink)

This work presents, in general terms, a schematic view of the philosophical architecture of a new System based on three main pillars: (1) the formulation of an absolutely necessary logical-metaphysical condition — S((T)(φ)) — which establishes Being as a universal and untranscendable condition of every true proposition and fact; (2) the metaphysical demonstration, by means of an alethical-modal argument, of the impossibility of absolute nothingness and, consequently, of the need for an absolute Being, identified as God in a strictly philosophical (...) sense; (3) the construction of a stratified structural ontology united to a theory of truth, in which God is conceived as the ultrastructure of Transcendent Truth (Absolute Truth ⇄ Absolute Life ⇄ Absolute Will), the superior foundation of the immanent axial hyperstructure (Truthfulness ⇄ Beauty ⇄ Goodness), in turn the foundation of the factual-propositional superstructure (T(p) ⇄ F). The three-dimension is unified by a radical dialectical-ontoeinailogical principle — that of self-contradicted identity — by which the system proposes a philosophy of Being as Truth, in which thought rediscovers the possibility of affirming the absolute by means of reason. It is an original proposal that rehabilitates systematic ambition and restores to philosophy the fundamental task of thinking, in a rigorous and affirmative way, what "necessarily is", the proper object of einailogy. (shrink)

Este artigo apresenta o argumento alético-modal como uma demonstração filosófico-conceitual rigorosa da necessidade do Ser Absoluto. Argumenta-se que, se tudo fosse contingente, o nada absoluto seria possível; no entanto, tal hipótese revela-se logicamente e metafisicamente incoerente, pois contradiz as condições de inteligibilidade e de verdade. Conclui-se, portanto, que uma necessidade absoluta deve ser afirmada. O argumento defende que a necessidade, enquanto modalidade alética, pressupõe um fundamento no Ser e não pode ser reduzida à pura formalidade lógica. Assim, a necessidade afirmada (...) não é meramente epistêmica ou formal, mas metafísica - o que justifica a identificação de Deus como o Ser Absoluto e Necessário no passo final decisivo. (shrink)

This paper presents the alethic-modal argument as a rigorous philosophicalconceptual demonstration of the necessity of the Absolute Being. It is argued that if everything were contingent, absolute nothingness would be possible; however, such a hypothesis proves to be logically and metaphysically incoherent, as it contradicts the conditions of intelligibility and truth. It is concluded, therefore, that an absolute necessity must be affirmed. The argument maintains that necessity, as an alethic modality, presupposes a foundation in Being and cannot be reduced to (...) mere logical formality. Thus, the affirmed necessity is not merely epistemic or formal, but metaphysicalwhich justifies identifying God as the Absolute and Necessary Being in the final decisive step. (shrink)

I distinguish between instantial sentences (_There are elephants that swim_), particular quantification, and predication of existence in natural language. I explore the logical relation between the first two, while the last one is shown independent of either. I continue to consider the incorporation of the three kinds of sentence in the Quantified Argument Calculus (Quarc). I provide formalisations that preserve the logical relations specified earlier. I also extend the analysis to quantified instantial sentences (_There are_ five _elephants that swim_) and (...) modal ones. A conclusion of this work is that quantification and instantial sentences provide no guide to ontology. (shrink)

The paper proposes a first approach to systems whose language includes two primitives (>+ and >-) as symbols for factual and counterfactual conditionals which are explicit, i.e. that are stated jointly with the truth or falsity of the antecedent clause. In systems based on this language, here called 2-conditional, the standard corner operator may be defined by (Def>) A > B := (A >+ B)∨(A >- B), while in classical conditional systems one could introduce the two symbols for explicit conditionals (...) by the definitions (Def>+) A >+ B := A∧(A>B) and (Def>-) A>- B := ¬A∧(A>B). Two 2-conditional systems, V² and VW², are axiomatized and proved to be definitionally equivalent to the monoconditional systems V and VW. A third system VWTr² is characterized by an axiom stating the transitivity of factual conditionals and is shown to be distinct from V², from VW² and from the 2-conditional version of Lewis’ well-known system VC, here named VC². The same may be said for a fourth system VW²◊ ±, based on an axiom inderivable in VC²: ◊(A >+ B) ⊃ ◊( ¬A >- ¬B). VC² contains what is here called a “semi-collapse” of the operator >+ and it is argued that it is inadequate as a logic for both factual and counterfactual conditionals. The last section shows that several different definitions of the corner operators in terms of >+ and >- may be introduced as an alternative to (Def>). (shrink)

The paper introduces a variant of connexive logic in which connexivity is extended from the interaction of negation with implication to the interaction of negation also with conjunction and disjunction. The logic is presented by two deductively equivalent methods: an axiomatic one and a natural-deduction one. Both are shown to be complete for a four-valued model theory.

Two printed works on logic by Georgius Benignus / Juraj Dragišić are preserved: Dialectica nova (Florentiae 1488 [1489]) and Artis dialecticaes praecepta (Romae 1520 [1519]). The presentations of Dragišić's logic by Carl Prantl and Stjepan Zimmermann are assessed. Dragišić’s 1488 logic contains a very early exposition of nearly all modes of the fourth figure. He listed four direct modes of the fourth figure, and he may also have had in mind the fifth mode (Fresison), along with an additional mode, probably (...) the indirect Fimeno. Furthermore, Dragišić presented a specific version of the terminist doctrine of supposition of terms (suppositio terminorum), incorporating his initial distinction between the mode and subject of supposition. Dragišić’s operational doctrine of consequences (consequentiae) reflects his interest in systematizing and reducing the number of rules (as seen in the differences between his two logical works). His detailed argumentation for the rejection of the principles ex impossibile and ad necessarium makes his work particularly interesting from the perspective of modern paraconsistent logics. (shrink)

Moral values are considered the rules and norms that help people live harmoniously and prevent them from possible conflicts. In society, the enduring nature of these norms, marked by their inheritable characteristics, finds a poignant vehicle in folklore, with folk tales playing an essential role in passing them down from one generation to the next. The moral values presented in folk tales are conveyed through argumentation. This article compares the argumentation methods of moral values in Eastern and Western folk tales, (...) examining examples like the Uzbek folk tale called 'Bozbola' and the English folk tale 'Jack and the Beanstalk', revealing the variance between them. (shrink)

Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic ‘of nonsense’ introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic $$\textbf{K}_{\textbf{3}}^{\textbf{w}}$$ by Stephen C. Kleene. The main features of this calculus are (i) that it is non-reflexive, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where no variable-inclusion conditions are attached; and (iii) (...) that it is hybrid, insofar as it includes both left and right operational introduction as well as elimination rules. (shrink)

One of Gottlob Frege’s most original contributions to logic and philosophy was his logical notation, his ‘Begriffsschrift’. While long criticized, dismissed, or simply ignored, the recent secondary literature contains some helpful re-evaluations and partial defenses of it. These rely largely on technical, pragmatic, or cognitive-psychological considerations. In this paper, we reconsider Frege’s own reasons for valuing his notation highly. We argue that there is a further semiotic dimension, one that matters epistemologically. This dimension becomes evident once one takes seriously, partly (...) also literally, some striking visual metaphors Frege uses. The result is an interpretation highlighting a side of Frege’s position that is not widely known. It involves his views about the indispensable role that language, or sign systems more generally, play for human thought, and especially, for logic and mathematics. More particularly and as we read Frege, a good logical notation allows for expressing conceptual/inferential relations in a ‘visually inscribed’, thus ‘perspicuous’ way, and this leads to a special kind of ‘insight’ in mathematics. Or as we elaborate this point further, it involves a kind of articulation and understanding hardly possible without it. Frege designed his Begriffsschrift with that specific goal in mind. (shrink)

This book develops a new approach to plural arbitrary reference and examines mereology, including considering four theses on the alleged innocence of mereology. The authors have advanced the notion of plural arbitrary reference in terms of idealized plural acts of choice, performed by a suitable team of agents. In the first part of the book, readers will discover a revision of Boolosʼ interpretation of second order logic in terms of plural quantification and a sketched structuralist reconstruction of second-order arithmetic based (...) on the axiom of infinite, a la Dedekind, as the unique non-logical axiom. The work goes on to analyse the pros and cons of the new interpretation, also with respect to Linneboʼs objections to the thesis that second order logic is genuine logic. A theory of concepts that can be labelled as a theory of logical concepts is expounded. In the second part of the book, the authors consider grounding megethology on plural arbitrary reference and argue that the arguments for the ontological innocence of mereology are not conclusive and that – for a certain use of mereology – a thesis of innocence, similar to that of plural arbitrary reference, is defensible. The work proposes a virtual theory of mereology in which the role of individuals is played by plural choices of atoms. This considered work will appeal to scholars from branches of analytic philosophy, logic and the philosophy of mathematics in particular. (shrink)

This paper argues for the significance of Kaplan's logic LD in two ways: first, by looking at how logic got along before we had LD, and second, by using it to bring out the similarity between David Hume's thesis that one cannot deduce claims about the future on the basis of premises only about the past, and the so‐called "essentiality" of the indexical.

Whether contradictions could be “true in fiction” has become an unavoidable topic in the debates on the bounds of fictionality. This paper claims that genuine contradictions in fiction are far more infrequent phenomena than is usually claimed. The majority of cases that have been put forward as examples of contradictory fictions can be convincingly understood either as instances of rhetorical pseudo-contradictions or (in the case of the so-called “forking-path“ narratives) as disjunctions of possible outcomes rather than contradictory conjunctions of simultaneously (...) enacted exclusive scenarios. The only philosophically interesting category of contradictory fiction would be the one in which a single “root” contradiction is explicitly affirmed as the central element of the story (in the third-person, authoritative narrative voice). The paradigmatic example would be the revised version of Graham Priest’s “Sylvan’s Box” this paper presents. However, it could be argued that the problem with such narratives is that they unsuccessfully attempt to perceptually code what remains exclusively propositional content. They are, thus, fatally under-described, and the truth of the contradictory proposition fails to be adequately established in fiction. The idea that one can posit contradictions as fictional facts is based on oversimplified notions of narrative conventions and truth in fiction. (shrink)

It is astonishing that we humans are able to have, act on and express moral beliefs. This dissertation aims to provide a better philosophical understanding of why and how this is possible especially when we assume metaethical expressivism. Metaethical expressivism is the combination of expressivism and noncognitivism. Expressivism is the view that the meaning of a sentence is explained by the mental state it is conventionally used to express. Noncognitivism is the view that the mental state expressed by a moral (...) sentence is a noncognitive, desire-like state. The central question of this dissertation is this: "How can metaethical expressivists explain that we are able to express moral beliefs?" The shortest answer this dissertation will give is the following: "by becoming `dispositional expressivists´." Dispositional expressivists hold that moral beliefs are dispositions to be in certain mental states that resemble desires. More precisely, a central thesis of this study is that to hold a moral belief is to be disposed to be in a desire-like, noncognitive state where this state has some specific structure. This dissertation arrives at and justifies this thesis by making a number of steps. It starts with preparing the ground by developing a general theory of what it is for a sentence to express a mental state. Roughly, this theory states that for a sentence to express a mental state is for that sentence to be conventionally used to perform a speech act which counts as sincere only if the user or speaker is in that mental state. This theory of expressing satisfies all the constraints that must be satisfied for it to be compatible with the purposes of metaethical expressivism. The next step is to provide a solution to the most notorious semantic problem confronting metaethical expressivism, the problem of explaining how sentences containing moral words can have compositional meaning and stand in logical relations also known as the Frege-Geach or embedding problem. Building on the research of Mark Schroeder, a new expressivist semantics, called `Attitude Semantics´, will be developed for a first-order language. The motivating idea underlying attitude semantics is that expressivists can avoid the problematic feature of so-called `biforcation´ if they treat the structures of the mental states expressed by moral sentences as well as that expressed by non-moral sentences as being isomorphic. In order to give those structures a philosophically interesting interpretation, it will be suggested that we treat sentences as expressing dispositional states. As a toy example we will treat the sentence `murder is wrong´ as expressing the disposition to be for blaming for murder. After dealing with the Frege-Geach problem, I turn to the metaethical expressivists' project of quasi-realism. Two major goals of this project are to explain how moral sentence can be truth-apt and how the states expressed by them can be regarded as moral beliefs. In order to establish those goals I critically examine a number of theories of truth-aptness by providing counterexamples to them. I then present my own theory which says that a sentence is truth-apt iff and because there is a conventional way to use it to express a belief. What is new about this theory is that it treats expressing a belief as being explanatorily more basic than being truth-apt. Belief, according to this theory, explains truth-aptness, not vice versa. This has the important consequence that it shows that what I call the `orthodox strategy´ of quasi-realists is mistaken and needs to be inverted: expressivists must `earn the right´ to moral belief before they can establish that moral sentences are truth-apt, and not the other way round. The second part of this dissertation is then concerned with the question of what it is to hold a moral belief and how they can figure in rational reasoning. In short: it is concerned with earning the right to moral belief. After discussing the major problems confronting a noncognitivist theory of belief, most importantly what I call the `tightrope problem´, I argue for a dispositional form of noncognitivism about moral belief that combines elements from Eric Schwitzgebel's dispositionalism about belief and Sebastian Köhler's `conceptual role expressivism´. This dispositional noncognitivism perfectly matches and justifies the dispositionalist interpretation of the expressivist semantics developed before. In order to fully earn the right to moral belief, however, metaethical expressivists must also deal with several epistemological objections concerning the rationality of moral belief. The first objection I deal with comes from Derek Baker and is directed against Mark Schroeder's specific development of expressivism. I deal with this objection because my own theory bears close resemblance to Schroeder's theory and so it might be worried that it confronts the same issues. Fortunately, we will see that Baker's objection rest on a number of mistaken assumptions and so neither applies to Schroeder's expressivism nor to the version which I defend in this study. The other epistemological objection to metaethical expressivism I deal with is Cian Dorr's famous wishful-thinking objection. In contrast to previous replies to this objection I attack it by directly questioning the central assumption on which it rests, namely that reasoners can never be justified in basing a factual belief on a desire-like state. I argue that this is false, and that under certain conditions reasoning from desire-like mental states to factual beliefs is justified. My argument shows that even if holding a moral belief is being in a desire-like noncognitive state, basing factual belief on a moral belief can give one an epistemic reason for the factual belief. Hence, even if moral thoughts are wishes, moral thinking is not wishful thinking. I take my arguments to earn the dispositional expressivist the right to moral belief. By the previously defended theory of truth-aptness in terms of belief, this allows the dispositional expressivists to conclude that moral sentences are also truth-apt. This finishes the quasi-realist project of this dissertation. In the last part of this dissertation I switch gears and present two unnoted problems for metaethical expressivists having to do with explaining moral motivation. Both arguments are surprisingly simple. The first one is this: if motivation requires a belief as well as a desire (Humeanism), and moral judgments motivate by themselves (internalism), then moral judgments cannot be desires only. If this argument is sound, it shows that the famous `motivation argument´ not only entails the falsity of cognitivism, but also the falsity of noncognitivism. The second argument is the following: moral judgments when combined with suitable external desires motivate to act (externalism), but if motivation requires belief as well as desire (Humeanism), then moral judgments cannot be desires only. I discuss several ways in which noncognitivists might want to deal with those problems, but leave it as an open question whether those solutions will ultimately be viable. Overall, this dissertation makes serious contributions to the research program of metaethical expressivism. It makes proposals for how to solve a number of notorious problems, such as the Frege-Geach problem in semantics and the wishful-thinking problem in epistemology. Besides this, however, it also raises two new problems for metaethical expressivism in the philosophy of action. For those reasons this dissertation will be of interest for anyone who wants to defend or criticize expressivism in metaethics, and more generally, anyone who wants to better understand our astonishing ability to express moral beliefs. (shrink)

Qualifiers such as “insofar as” and “in itself” have always been important ingredients in key philosophical claims. Descartes, for instance, famously argues that insofar as he is a thinker, he is not made of matter, and Kant equally famously argues that we cannot know things in themselves. Neither of these claims is meant to be true without qualification. Descartes is not simply denying that humans consist of matter, and Kant is not simply denying that we know things. Therefore, we cannot (...) even begin to understand such claims without knowing how qualifiers work. Unlike the logic of quantification, however, the logic of qualification is rather underexplored. In this paper, I examine several instances of philosophical uses of qualifiers, taken from Aristotle, Avicenna, Descartes, Kant, and 20th c. action theory. In the light of these examples, I discuss several accounts of how such qualifiers work. (shrink)

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction rules for (...) the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

We explore the problems that confront any attempt to explain or explicate exactly what a primitive logical rule of inferenceis, orconsists in. We arrive at a proposed solution that places a surprisingly heavy load on the prospect of being able to understand and deal with specifications of rules that are essentiallyself-referring. That is, any rule$\rho $is to be understood via a specification that involves, embedded within it, reference to rule$\rho $itself. Just how we arrive at this position is explained by (...) reference to familiar rules as well as less familiar ones with unusual features. An inquiry of this kind is surprisingly absent from the foundations of inferentialism—the view that meanings of expressions (especially logical ones) are to be characterized by the rules of inference that govern them. (shrink)

Stephen Read presented harmonious inference rules for identity in classical predicate logic. I demonstrate here how this approach can be generalised to a setting where predicate logic has been extended with epistemic modals. In such a setting, identity has two uses. A rigid one, where the identity of two referents is preserved under epistemic possibility, and a non-rigid one where two identical referents may differ under epistemic modality. I give rules for both uses. Formally, I extend Quantified Epistemic Multilateral Logic (...) with two identity signs. I argue that a uniform meaning for identity tout court can be given by adopting Maria Aloni’s account of reference using conceptual covers. We obtain a harmonious set of rules for identity that is sound and complete for Aloni’s model theory. (shrink)

According to the interpretational theory of logical validity (IR), logical validity is preservation of truth in all interpretations compatible with the intended meaning of logical expressions. IR suffers from a seemingly defeating objection, the so-called cardinality problem: any instance of the statement ‘There are n things’ is true under all interpretations, since it can be written down using only logical expressions that are not to be reinterpreted; yet ‘There are n things’ is not logically true. I argue that the cardinality (...) problem is indeed a serious problem for IR, when understood in terms of ‘asymmetry of information’. I then argue that IR can be rehabilitated by making quantifiers context-sensitive: what we do not reinterpret is the Kaplanian character of a quantifier, rather than its content. ‘There are n things’ is false in a context where fewer than n things are relevant, so it is not logically true in IR. I finally discuss some objections and ramifications of my account: I discuss how to make space for the possibility of an explicitly absolutely general quantifier in my framework, how terms can be logical even though context-sensitive, and how to recapture classical logic within my framework. (shrink)

Statements containing epistemic modals (e.g., “by spring 2023 most European countries may have the Covid-19 pandemic under control”) are common expressions of epistemic uncertainty. In this paper, previous published findings (Knobe & Yalcin, 2014; Khoo & Phillips, 2018) on the opposition between Contextualism and Relativism for epistemic modals are re-examined. It is found that these findings contain a substantial degree of individual variation. To investigate whether participants differ in their interpretation of epistemic modals, an experiment with multiple phases and sessions (...) is used to classify participants according to the three semantic theories of Relativism, Contextualism, and Objectivism. Through this study, some of the first empirical evidence for the kind of truth-value shifts postulated by semantic Relativism is presented. It is furthermore found that participants’ disagreement judgments match their truth evaluations and that participants are capable of distinguishing between truth and justification. In a second experimental session, it is investigated whether participants thus classified follow the norm of retraction which Relativism uses to account for argumentation with epistemic modals. Here the results are less favorable for Relativism. In a second experiment, these results are replicated and the normative beliefs of participants concerning the norm of retraction are investigated following work on measuring norms by Bicchieri (2017). Again, it is found that on average participants show no strong preferences concerning the norm of retraction for epistemic modals. Yet, it was found that participants who had committed to Objectivism and had training in logics applied the norm of retraction to might-statements. These results present a substantial challenge to the account of argumentation with epistemic modals presented in MacFarlane (2014), as discussed. Keywords: Epistemic Modals, Relativism, Norm Conflicts, Retraction, Semantics, Truth Conditions, Argumentation. (shrink)

The purpose of this paper is to analyse and compare two concepts which tend to be treated as synonymous, and to show the difference between them: these are critical thinking and logical culture. Firstly, we try to show that these cannot be considered identical or strictly equivalent: i.e. that the concept of logical culture includes more than just critical thinking skills. Secondly, we try to show that Christian philosophers, when arguing about philosophical matters and teaching philosophy to students, should not (...) focus only on critical thinking skills, but rather also consider logical culture. This, as we argue, may help to improve debate both within and outside of Christian philosophy. (shrink)

This chapter introduces inferential role semantics (IRS) and some of the challenges it faces. It also introduces inferentialism and places it into the wider context of contemporary philosophy of language. The chapter focuses on what is standardly considered both the most important test case for and the most natural application of IRS: logical inferentialism, the view that the meanings of the logical expressions are fully determined by the basic rules for their correct use, and that to understand a logical expression (...) is to use it in accordance with the appropriate rules. It discusses some of the benefits of logical inferentialism, chiefly with regard to the epistemology of logic, and considers a number of objections. The chapter critically examines Robert Brandom's inferentialism about linguistic and conceptual content in general. Finally, it considers a number of general objections to IRS and possible responses on the inferentialist's behalf. (shrink)