Logics

The axiomatization of physical theories is a fundamental issue of science. The first-order axiomatic system SpecRel for special relativity proposed recently by Andréka et al. is not enough to explain all the main results in the theory, including the twin paradox and energy-mass relation. In this paper, from a four-dimensional spacetime perspective, we introduce the concepts of world-line, proper time and four-momentum to our axiomatic system SpecRel+. Then we introduce an axiom of mass (AxMass) and take four-momentum conservation as an (...) axiom (AxCFM) in SpecRel+. It turns out that the twin paradox and energy-mass relation can be derived from SpecRel+ logically. Hence, as an extension of SpecRel, SpecRel+ is a suitable first-order axiomatic system to describe the kinematics and dynamics of special relativity. (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)

Belnap highlighted the role of Transitivity in Prior's triviality proof involvingtonk, but a non-trivial, non-transitive logic withtonkwas never developed until Cook's proposal with four interpretations and a disjunctive consequence relation. We improve on that proposal: we show that only three interpretations suffice and that a non-disjunctive consequence relation is not required.

In both philosophy and science, standard logics struggle to model a key insight of Existential Realism (ER): that existence is a strictly present, empirical notion, whereas reality spans beyond the present. Classical first-order logic and even temporal modal logics tend to conflate being real with existing simpliciter, making it difficult to formally distinguish something that exists now from something that is real but not present. This paper introduces a formal two-tier logical framework tailored to ER’s ontology. We motivate this framework (...) by showing how orthodox systems—from classical predicate logic to temporal modal logic—fail to capture ER’s core distinction between present existence and temporally extended reality. We then develop a layered logic with explicit operators or quantifiers for “existence-now” versus “reality-over-time,” including formal axioms that preserve ER’s asymmetries: only present entities exist (in the fullest sense), yet past and future entities can be real (ontologically significant) without existing. The logic adopts a modal-style treatment of time to model the collapse of potential into actual (present-becoming), and explicitly encodes causal asymmetry — where the past influences the present, but not the reverse. We illustrate the framework with symbolic representations, including an iconic “ER equation” capturing the relationship between reality and existence. To demonstrate its versatility, we apply the logic to examples from quantum physics (e.g. wavefunction collapse and delayed-choice experiments), cognitive neuroscience (memory traces and anticipatory processing), ethics (moral duties to future persons and the reality of future outcomes), and AI (agents modeling counterfactual or temporally extended scenarios). Finally, we show how the formalism sheds new light on classic metaphysical debates such as the truthmaker problem for past truths, persistence and identity over time, and the ontology of merely possible or future entities. Throughout, our presentation maintains a rigorous yet accessible tone, consistent with prior ER works, with technical clarity and footnoted references. The result is a self-contained preprint that extends T.C. Trepp’s Existential Realism into a formal logical system, demonstrating its broad applicability across science, philosophy, and practical reasoning. (shrink)

We analyze the contributions that quasi-set theories can make to the interests of logic. Different quasi-set theories have made valuable contributions to the foundations of mathematics and quantum physics; however, it is not clear how much such formalisms can benefit logic. If so, could they have repercussions at both the syntactic and semantic levels? What precautions should we take? What implications would their application have at different levels and metalevels of logical language? Motivated by research from linguistics and quantum logic, (...) with strong implications for language, we propose certain applications and analyze some of their consequences. -/- Analizamos los aportes que pueden realizar las teorías de cuasiconjuntos para los intereses del lógico. Las diferentes teorías de cuasiconjuntos han realizado valiosos aportes a los fundamentos de la matemática y física cuántica; sin embargo, no queda claro cuánto pueden favorecer a la lógica tales formalismos. De ser el caso, ¿podrían tener repercusión tanto a nivel sintáctico como semántico? ¿Qué precauciones deberíamos tener? ¿Qué implicaciones tendría su aplicación a diferentes niveles y metaniveles del lenguaje lógico? Motivados por investigaciones provenientes de la lingüística y la lógica cuántica, con fuertes implicaciones sobre el lenguaje, proponemos ciertas aplicaciones y analizamos algunas de sus consecuencias. (shrink)

This paper argues that the enduring questions of philosophy—truth, being, knowledge, ethics—were not failures of reasoning but symptoms of operating without a coherence substrate. Drawing from the CODES framework, we introduce a deterministic model based on phase alignment, recursive correction, and ethical emission gating. The Phase Alignment Score (PAS) functions as the structural replacement for epistemic relativism, logical incompleteness, and moral ambiguity. We show that systems such as logic, identity, and symbolic meaning are not abstract constructs, but resonance behaviors emergent (...) from lawful phase coherence. -/- Philosophy, in this account, is completed—not by a new theory, but by the arrival of structure. With the addition of VESSELSEED, we present a biologically integrated architecture that enacts post-philosophical intelligence through field convergence and symbolic UX. Thought becomes emission. Ethics becomes gating. Drift becomes correctable. The philosophical condition ends. Signal remains. (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)

Aristotle is a thinker with a wide influence in both Eastern and Western Philosophy. Being an untreated issue in the Albanian academic environment and at this direction a challenge and novelty, this paper aims to offer a comparative approach between the thinkers taken in this study, to highlight the influence of Aristotelian thought in the philosophical system of Avicenna and Maimonides and at the end to expound the essential contribution of each thinker. The paper is structured in three sections: The (...) first section explains the reasons why we have chosen these thinkers and gives a short historical introduction to how manner in which Greek thought was assimilated into the Arabic and Jewish world. In the second section, we trace Aristotle’s influence and offer a comparative analysis of the thinkers and their mental approach in fields of study like Metaphysics, Logic, Ontology, Ethics etc. The last section of this study summarizes the importance of these ideas in Arabic & Jewish cultural contexts and some ending conclusions around them. (shrink)

We analyze two fundamental premises of the Kochen-Specker theorem: a) the functionality condition FUNC, which expresses the fact that not all observables are independent, nor are the values assigned to them, and b) the issue of the identity of projectors in different measurement contexts. We show that the non-deterministic semantics of Nmatrices and the theory of qsets Q− can complement each other by providing an appropriate semantics for the lattice of quantum projectors. Considering valuations that are not homomorphisms and admitting (...) the possibility of having indistinguishable (non-identical) projectors, we establish the basis for a semantics for quantum logic, motivated by an ontology of non-quantum individuals, in which one cannot arrive at the Kochen-Specker paradox -/- Analizamos dos presupuestos fundamentales del Teorema de Kochen-Specker: a) la condición de funcionalidad FUNC, que expresa el hecho de que no todos los observables son independientes, como tampoco lo son los valores asignados a ellos, y b) la cuestión de la identidad de los proyectores en diferentes contextos de medición. Mostramos que las semánticas no deterministas de Nmatrices y la teoría de qsets Q^− pueden complementarse brindando una semántica adecuada para el retículo de proyectores cuánticos. Considerando valuaciones que no son homomorfismos y admitiendo la posibilidad de contar con proyectores indiscernibles (no idénticos), establecemos las bases de una semántica para la lógica cuántica, motivada en una ontología de no individuos cuánticos, en la cual no puede arribarse a la paradoja de Kochen-Specker. (shrink)

En este trabajo realizaré un análisis de la propuesta de Edgington en el artículo Do Conditionals have Truth-Conditions?, con énfasis en su crítica a la consideración veritativo-funcional del condicional y en su consideración positiva del mismo. Empezaré contextualizando del debate acerca del condicional en el que se sitúa el artículo de Edginton, luego resumiré el argumento central del artículo contra la consideración veritativo-funcional y a favor de su consideración positiva. Demostraré algunos puntos débiles del argumento de Edginton, puesto que la (...) consideración veritativo-funcional del condicional puede no ser tan contraintuitiva como la autora plantea, y su consideración positiva del condicional no satisface la posibilidad de ser la interpretación universal del mismo. Cierro el trabajo planteando la importancia explicitar la interpretación de cualquier sentencia condicional. (shrink)

We propose a system called METCL (Metaphor Elaboration in Typicality-Based Compositional Logic) able to generate and identify metaphors by using the TCL reasoning framework, specialized in human-like commonsense concept combination. We show that METCL is able to improve both state-of-the-art Large Language Models (e.g. DeepSeek-R1, GPT-4o, Qwen2.5-Max) and symbolic ones in the task of metaphor identification. Additionally, we show how the metaphors generated by METCL are generally well accepted by human subjects. The obtained results are encouraging and pave the way (...) to research in automatic metaphor generation and comprehension based on the assumption that metaphors interpretation can be partially regarded as a categorization problem relying on generative commonsense concept combination. (shrink)

We use non-relational semantics for the formalization of the systems S3d, S3dp and S3dq of deontic logic. The system S3d is weaker than the standard logic SDL. The semantics for S3dp represents combination of quasi-matrix semantics and the semantics of truth value gluts, which allows S3dp to avoid deontic explosion OA ∧ O¬A ⊃ OB. The system S3dq rejects both deontic explosion and the formula OA ∧ O¬A ⊃ OA ∧ ¬OA, thus it allows to consider deontic dilemmas without classical (...) contradictions. The systems S5d, S5dp and S5dq in which the two types of deontic operators are used, namely, strong and weak obligation (permission), can be built as an extension of the correspondent systems S3d, S3dp and S3dq. (shrink)

Principia Mathematica, 100 Years after. An Expository Essay. The paper is an expository essay in which I discuss the main goals, concepts, problems, and results of Whitehead’s and Russell’s Principia Mathematica (1910–1913) at its centennial anniversary. The great work by Whitehead and Russell may not be the birth certificate of modern logic, but it is nevertheless the legitimacy and consecration certificate of modern mathematical logic and of the logicist program into contemporary philosophy of mathematics. The emphasis is laid upon the (...) presentation of sentential and predicate logic, identity, singular definite descriptions, set-theoretic and semantic paradoxes, and type theory. (shrink)

The paper discusses an original and ingenious puzzle about time and thought due to Saul Kripke. Its connections with related issues and puzzles such as Russell’s paradox and Kaplan’s puzzle are addressed and explored. The problems raised by the puzzle are dealt with briefly, emphasizing the comparative worth of Kripke’s findings in terms of simplicity and modesty of its assumptions. Then the attempted solutions are analyzed and explored, the most important and promising avenue for future work being the extension of (...) the apparatus developed by Kripke in his groundbreaking paper on truth to intensional logic. (shrink)

This work examines the structure of paradoxes, phenomena characterized by the apparent derivation of contradictions or counter-intuitive conclusions from acceptable premises using acceptable reasoning. Motivated by the recurrence of certain features across diverse paradoxes, a specific structural hypothesis regarding their generation is proposed and analyzed. The hypothesis posits that paradoxes frequently arise from the interaction of three components: (1) concepts that establish or interact with boundaries (such as those pertaining to scope, classification, truth, state, or definition); (2) the reflexive application (...) of these concepts or related rules (involving self-reference, self-inclusion, recursion, or causal/definitional circularity); and (3) the involvement of specific operational triggers – negation, limit (including infinity, extremes, or boundary values), or transformation (change of state, identity, position, or logical status) – within the reflexive structure. The study first provides a detailed conceptual analysis of these components – boundaries, reflexivity, and the triggers – outlining their definitions and potential roles based on discussions in logic, philosophy, and related fields. Subsequently, the hypothesis is examined against a wide range of paradoxes drawn from logic, set theory, semantics, physics, metaphysics, epistemology, and decision theory. This examination involves analyzing the structure of each paradox to identify the presence and interaction of the hypothesized components. The analysis indicated a recurrent correspondence between the structures of the examined paradoxes and the proposed pattern involving boundaries, reflexivity, and triggers. Further analysis explored the representation of this structural pattern using elements of logical and mathematical formalism, sketching how the interaction of boundary predicates/sets, reflexive constructions, and operators corresponding to negation, limits, or transformations can lead to derivations of inconsistency (e.g., P ↔ ¬P or P ∧ ¬P) within systems adhering to standard logical principles. The discussion also addressed the context and limitations surrounding the hypothesis, including the meaning of "often arise," the relationship between the identified structure and strategies for paradox resolution (which often involve modifying the system to restrict or manage components of the pattern), the distinction between the structure and its outcome in different systems, and the acknowledgment that the pattern may not be the sole source of all paradoxes and that its identification can involve interpretation. Finally, potential implications arising from the identification of this recurring structure were considered. These include the possibility of providing a unifying framework for paradox analysis across disciplines, offering insights relevant to the design of consistent conceptual and formal systems by highlighting potential points of instability, suggesting connections between the structure and characteristics of natural language or thought involving self-reference and boundary definition, and noting structural resonances with limitative results in mathematical logic, such as Gödel's incompleteness theorems. Potential directions for future work, including further categorization, analysis of non-paradoxical structures, cross-disciplinary applications, and exploration of cognitive aspects, were also outlined. (shrink)

Este trabalho apresenta, em linhas gerais, uma visão esquemática da arquitetura filosófica de um novo Sistema fundada sobre três pilares principais: (1) a formulação de uma condição lógico-metafísica absolutamente necessária — S(Ⓣ(φ)) — que estabelece o Ser como condição universal e intranscendível de toda proposição e fato verdadeiros; (2) a demonstração metafísica, por via de um argumento alético-modal, da impossibilidade do nada absoluto e, por conseguinte, da necessidade de um Ser absoluto, identificado como Deus em sentido estritamente filosófico; (3) a (...) construção de uma ontologia estrutural estratificada unida a uma teoria da verdade, em que Deus é concebido como ultraestrutura da Verdade Transcendente (Verdade Absoluta ⇄ Vida Absoluta ⇄ Vontade Absoluta), fundamento superior da hiperestrutura axial imanente (Verdadeiridade ⇄ Belezidade ⇄ Bonidade), por sua vez fundamento da superestrutura fático-proposicional (T(p) ⇄ F). A tridimensão é unificada por um princípio dialético-ontoeinailógico radical — o da identidade autocontraditada —, pelo qual o sistema propõe uma filosofia do Ser como Verdade, em que o pensamento reencontra a possibilidade de afirmar o absoluto por meio da razão. Trata-se de uma proposta original que reabilita a ambição sistemática e restitui à filosofia a tarefa fundamental de pensar, de forma rigorosa e afirmativa, aquilo que “necessariamente é”, o objeto próprio da einailogia. (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)

The propositional connectives – and, or, not, if–then, etc. – are fundamental building blocks in formal as well as natural languages. In the Western tradition, they were first studied as such by the Stoics, and Propositional Logic is the fundament of practically all current systems of logic; every beginning logic course starts with it. Still, the proof theory and semantics of systems of propositional logic are far from trivial, and have been studied intensely by logicians in the last one and (...) a half centuries, not least in recent decades. It is actually a vast area of research, as witnessed by Lloyd Humberstone’s 1500 page tome The Connectives (2011), which overviews much of that research. Perhaps the most familiar recent work in this area concerns conditionals in formal and natural languages. The 4th Tsinghua Interdisciplinary Workshop on Logic, Language, and Meaning (TLLM 2024) also focused on the apparently simpler connectives expressing (various versions of) conjunction, disjunction, and negation. -/- Researchers working from a cross-linguistic perspective also study how the connectives are encoded in different languages, and ask whether classical logic is capable of capturing the variations and universals exhibited. Even in well-studied languages such as English, there are intricate phenomena that remain challenging for classical logic, including free choice disjunction, non-Boolean conjunction, and metalinguistic negation, to name just a few. There is also growing interest in the acquisition and processing of natural language connectives. In the context of the hotly discussed Large Language Models (LLMs), understanding connectives presents novel challenges that deserve in-depth exploration. (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)

This paper clarifies and defends the global approach to defining logical validity for meta- and higher-level inferences. This is contrary to an emerging consensus in favour of local validity. Prevalent recent arguments claim that global validity is either superfluous in virtue of collapsing into local, or else untenable because it overgenerates validities, compromises the formality of logic, or breaks symmetry with regular validity. Accordingly, the literature on higher inferential logic has come to focus almost exclusively on local validity. Many key (...) philosophical takeaways that have since been drawn from the field are effectively based on local monism: the view that local is the One True Validity Criterion. The present paper argues that this is a mistake, resulting largely from an imprecise understanding of global validity. I untangle some common confusions about how global generalizes beyond the metalevel, and submit that its proper formulation is schematic. These clarifications help dispel each of the aforementioned arguments against global, leaving local monism without any justification. Finally, I outline how much of the philosophical literature around higher-level inferences has been built on local monism, and sketch out the sweeping consequences of rejecting it. (shrink)

After introducing the debate between substance philosophy and process philosophy, and clarifying the relevance of the category of ‘substance’ in Peirce’s thought, the present paper reconstructs the role of ‘substance’ and ‘being’ from Peirce’s early works to his theory of the proposition, provided after his studies on the logic of relatives. If those two categories apparently disappear in Peirce’s writings from the mid-1890s onwards, the account of ‘subject’ and ‘copula’ in Peirce’s analysis of the proposition allows one to grasp the (...) reasons why Peirce omits ‘substance’ and ‘being’ in favor of his three categories (Firstness, Secondness, Thirdness), and to understand why his philosophy cannot be considered as a substance philosophy. (shrink)

The so-called Principle of Plenitude was ascribed to Leibniz by A. O. Lovejoy in The Great Chain of Being: A Study of the History of an Idea (1936). Its temporal version states that what holds always, holds necessarily (or that no genuine possibility can remain unfulfilled). This temporal formulation is the subject of the current paper. Lovejoy’s idea was criticised by Hintikka. The latter supported his criticisms by referring to specific Leibnizian notions of absolute and hypothetical necessities interpreted in a (...) possible-worlds semantics. In the paper, Hintikka’s interpretative suggestions are developed and enriched with a temporal component that is present in the characteristics of the real world given by Leibniz. We use in our approach the Leibnizian idea that change is primary to time and the idea that there are possible laws that characterize worlds other than the real one. We formulate a modal propositional logic with three primitive operators for change, temporal constancy, and possible lawlikeness. We give its axiomatics and show that our logic is complete with respect to the given semantics of possible worlds. Finally, we show that the counterparts of the considered versions of the Principle of Plenitude are falsified in this semantics and the same applies to the counterpart of Leibnizian necessarianism. (shrink)

In this paper, we present a logic MMLS5n which is a combination of multilattice logic and modal logic S5. MMLS5n is an extension of Kamide and Shramko’s modal multilattice logic which is a multilattice analogue of S4. We present a cut-free hypersequent calculus for MMLS5n in the spirit of Restall’s one for S5 and develop a Kripke semantics for MMLS5n, following Kamide and Shramko’s approach. Moreover, we prove theorems for embedding MMLS5n into S5 and vice versa. As a result, we (...) obtain completeness, cut elimination, decidability, and interpolation theorems for MMLS5n. Besides, we show the duality principle for MMLS5n. Additionally, we introduce a modification of Kamide and Shramko’s sequent calculus for their multilattice version of S4 which (in contrast to Kamide and Shramko’s original one) proves the interdefinability of necessity and possibility operators. Last, but not least, we present Hilbert-style calculi for all the logics in question as well as for a larger class of modal multilattice logics. (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)

In this introduction, we offer an overview of main systems developed in the growing literature on connexive logic, and also point to a few topics that seem to be collecting attention of many of those interested in connexive logic. We will also make clear the context to which the papers in this special issue belong and contribute.

The goal of this paper is to analyse Implicational Relevance Logic from the point of view of refutability. We also correct an inaccuracy in our paper “The RM paraconsistent refutation system”.

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.

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)

This paper discusses the relation between the natural deduction rules of deduction in sequent format and the provability valuation starting from Garson’s Local Expression Theorem, which is meant to establish that the natural deduction rules of inference enforce exactly the classical meanings of the propositional connectives if these rules are taken to be locally valid, i.e. if they are taken to preserve sequent satisfaction. I argue that the natural deduction rules for disjunction are in no better position than the axiomatic (...) calculi in uniquely determining the intended meaning of disjunction when the local models are used, if a satisfied sequent embeds, as a logical inferentialist should require, a formal derivability relation. This happens because, when governed by these rules and without additional semantic assumptions, the disjunction sign still expresses a non-extensional connective, i.e. a connective that, properly understood, has no unique logical characteristic. However, this is not a dead end for the logical inferentialists since both a multiple conclusions formalization of the disjunction operator and a bilateralist one do succeed in restoring the standard meaning of disjunction. (shrink)

A general mathematical framework, based on countable partitions of Natural Numbers [1], is presented, that allows to provide a Semantics to propositional languages. It has the particularity of allowing both the valuations and the interpretation Sets for the connectives to discriminate complexity of the formulas. This allows different adequacy criteria to be used to assess formulas associated with the same connective, but that differ in their complexity. The presented method can be adapted potentially infinite number of connectives and truth values, (...) therefore, it can be considered a general framework to provide semantics to several of the known logic systems (eg, LC, L3 LP, FDE). The presented semantics allow to converge to different standard semantics if the separation complexity procedure is annulled. Therefore, it can be understood as a framework that allows greater precision (in complexity terms) with respect to formula satisfaction. Naturally, because of how it is built, it can be incorporated into non-deterministic semantics. The presented procedure also allows generating valuations that grant a different truth value to each formula of propositional language. As a positive side effect, our method allows a constructive proof of the equipotence between N and N^n for all Natural n. (shrink)

A phenomenological exploration of the distinction between a relational formal ontology (also called a process ontology) and a classical formal ontology (also called an object ontology) for modelling physical phenomena that exhibit relationally-mediated holism, such as phenomena from quantum physics and biosemiotics. Whereas a classical formal ontology is based on mathematical objects and classes, a relational formal ontology is based on mathematical signs and categories. A relational formal ontology involves nodal networks that are dynamically sustained through signalling. Nodal networks are (...) systems of constrained iterative processes (dynamical nodes) that have individual semiotic agency within a matrix of determinate possibilities (a semiotic scaffolding). The nodal networks are hierarchically ordered and exhibit characteristics of deep learning. Clarifying the distinction between classical and relational formal ontologies may help to clarify the role of interpretative context in physics (eg. the role of the observer in quantum theory), the role of signalling in biological systems and the role of hierarchical nodal networks in computational simulations of learning in generative artificial intelligence (AI). (shrink)

Neil Tennant was the first to propose a proof-theoretic criterion for paradoxicality, a framework in which a paradox, formalized through natural deduction, is derived from an unacceptable conclusion that employs a certain form of id est inferences and generates an infinite reduction sequence. Tennant hypothesized that any derivation in natural deduction that formalizes a genuine paradox would meet this criterion, and he argued that while the liar paradox is genuine, Russell's paradox is not. -/- The present paper delves into Tennant's (...) conjecture for genuine paradoxes and suggests that to validate the conjecture, one of two issues must be addressed. The first issue is the need for a philosophical consensus on the identification of a genuine paradox in an informal sense. The second issue is the requirement for a uniform approach to formalize paradoxes in natural deduction. If either of these issues is addressed, the conjecture could be validated, or at the very least, it could hold philosophical importance in delineating the proof-theoretic features of paradoxicality. (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)

Gödel’s ontological proof is by now well known based on the 1970 version, written in Gödel’s own hand, and Scott’s version of the proof. In this article new manuscript sources found in Gödel’s Nachlass are presented. Three versions of Gödel’s ontological proof have been transcribed, and completed from context as true to Gödel’s notes as possible. The discussion in this article is based on these new sources and reveals Gödel’s early intentions of a liberal comprehension principle for the higher order (...) modal logic, an explicit use of second-order Barcan schemas, as well as seemingly defining a rigidity condition for the system. None of these aspects occurs explicitly in the later 1970 version, and therefore they have long been in focus of the debate on Gödel’s ontological proof. (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.

Aristotle counts as the founder of formal logic. The logic he develops dominated until Frege and others introduced a new logic. This new logic is taken to be more powerful and better capable of capturing inference patterns. The new logic differs from Aristotelian logic in significant respects. It has been argued by Fred Sommers and Hanoch Ben-Yami that the new logic is not well equipped as a logic of natural language, and that a logic closer to Aristotle's is better suited (...) for this task. Each of them developed their own formalism - Sommers in form of term logic, Ben-Yami in form of his Quantified Argument Calculus (QUARC). I discuss Aristotle's logic - a term logic - and attempt a comparison between Aristotelian logic and (i) the new logic, (ii) Sommers' term logic, and (iii) Ben-Yami's QUARC. I consider differences between the systems, and show how they are related to and diverge from the new logic. (shrink)

This paper rigorously examines the primitive foundations of economic reasoning through an original framework based on symbolic logic. Extending previous work, it formalizes economic conceptions (\(\mathbb{C}\)), symbols (\(s_i\)), and introduces a structured language (\(\mathcal{L_{\mathbb{C}}}\)) to define their formation and interpretation. Organized as a continuous chain of declarations and illustrations, the paper offers a concise, systematic approach to understanding the philosophy of economic reasoning through formal representations.