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176792.086699
Achilles and the tortoise compete in a race where the beginning (the start) is at point O and end (the finish) is at point P. At all times the tortoise can run at a speed that is a fraction of Achilles' speed at most (with being a positive real number lower than 1, 0 < < 1), and both start the race at t = 0 at O. If the trajectory joining O with P is a straight line, Achilles will obviously win every time. It is easy to prove that there is a trajectory joining O and P along which the tortoise has a strategy to win every time, reaching the finish before Achilles.
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771440.086842
The paper studies class theory over the logic HYPE recently introduced by Hannes Leitgeb. We formulate suitable abstraction principles and show their consistency by displaying a class of fixed-point (term) models. By adapting a classical result by Brady, we show their inconsistency with standard extensionality principles, as well as the incompatibility of our semantics with weak extensionality principles introduced in the literature. We then formulate our version of weak extensionality (appropriate to the behaviour of the conditional in HYPE) and show its consistency with one of the abstraction principles previously introduced. We conclude with observations and examples supporting the claim that, although arithmetical axioms over HYPE are as strong as classical arithmetical axioms, the behaviour of classes over HYPE is akin to the one displayed by classes in other nonclassical class theories.
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771461.086865
The paper studies classical, type-free theories of truth and determinateness. Recently, Volker Halbach and Kentaro Fujimoto proposed a novel approach to classical determinate truth, in which determinateness is axiomatized by a primitive predicate. In the paper we propose a different strategy to develop theories of classical determinate truth in Halbach and Fujimoto’s sense featuring a defined determinateness predicate. This puts our theories of classical determinate truth in continuity with a standard approach to determinateness by authors such as Feferman and Reinhardt. The theories entail all principles of Fujimoto and Halbach’s theories, and are proof-theoretically equivalent to Halbach and Fujimoto’s CD . They will be shown to be logically equivalent to a class of natural theories of truth, the classical closures of Kripke-Feferman truth. The analysis of the proposed theories will also provide new insights on Fujimoto and Halbach’s theories: we show that the latter cannot prove most of the axioms of the classical closures of Kripke-Feferman truth. This entails that, unlike what happens in our theories of truth and determinateness, Fujimoto and Halbach’s inner theories – the sentences living under two layers of truth – cannot be closed under standard logical rules of inference.
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771483.086881
Supervaluational fixed-point semantics for truth cannot be axiomatized because of its recursion-theoretic complexity. Johannes Stern (Supervaluation-Style Truth Without Supervaluations, Journal of Philosophical Logic, 2018) proposed a new strategy (supervaluational-style truth) to capture the essential aspects of the supervaluational evaluation schema whilst limiting its recursion-theoretic complexity, hence resulting in ( -categorical) axiomatizations. Unfortunately, as we show in the paper, this strategy was not fully realized in Stern’s original work: in fact, we provide counterexamples to some of Stern’s key claims. However, we also vindicate Stern’s project by providing different semantic incarnations of the idea and corresponding -categorical axiomatizations. The results provide a deeper picture of the relationships between standard supervaluationism and supervaluational-style truth.
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831072.086895
The concept of infinity has long occupied a central place at the intersection of mathematics and philosophy. This paper explores the multifaceted concept of infinity, beginning with its mathematical foundations, distinguishing between potential and actual infinity and outlining the revolutionary insights of Cantorian set theory. The paper then explores paradoxes such as Hilbert’s Hotel, the St. Petersburg Paradox, and Thomson’s Lamp, each of which reveals tensions between mathematical formalism and basic human intuition. Adopting a philosophical approach, the paper analyzes how five major frameworks—Platonism, formalism, constructivism, structuralism, and intuitionism—each grapple with the metaphysical and epistemological implications of infinity. While each framework provides unique insights, none fully resolves the many paradoxes inherent in infinite mathematical objects. Ultimately, this paper argues that infinity serves not as a problem to be conclusively solved, but as a generative lens through which to ask deeper questions about the nature of mathematics, knowledge, and reality itself.
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831095.086913
The Recursive Ontological Calculus (ROC) furnishes a complete, machine-verifiable axiomatisation of symbolic identity, curvature, and semantic recursion. Building directly on C. S. Peirce’s triadic conception of the sign, ROC links category-theoretic morphology with information-geometric entropy bounds. We present formal schemas, a sequent calculus equipped with an infinitary Master Recursion Equation, eleven core theorems (T1–T11), and cross-framework embeddings into ordinary category theory, ZFC, and Homotopy Type Theory. Worked examples demonstrate numeric curvature computation, gauge-orbit quantisation, and prime-gate symbolic statistics.
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1004006.08693
This paper’s first aim is to prove a modernized Occam’s razor beyond a reasonable doubt. To summarize the main argument in one sentence: If we consider all possible, intelligible, scientific models of ever-higher complexity, democratically, the predictions most favored by these complex models will agree with the predictions of the simplest models. This fact can be proven mathematically, thereby validating Occam’s razor.
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1276338.086947
Simple games in partition function form are used to model voting situations where a coalition being winning or losing might depend on the way players outside that coalition organize themselves. Such a game is called a plurality voting game if in every partition there is at least one winning coalition. In the present paper, we introduce an equal impact power index for this class of voting games and provide an axiomatic characterization. This power index is based on equal weight for every partition, equal weight for every winning coalition in a partition, and equal weight for each player in a winning coalition. Since some of the axioms we develop are conditioned on the power impact of losing coalitions becoming winning in a partition, our characterization heavily depends on a new result showing the existence of such elementary transitions between plurality voting games in terms of single embedded winning coalitions. The axioms restrict then the impact of such elementary transitions on the power of different types of players.
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1465556.086962
This paper introduces the conceptual foundations of the Ontomorphic Peircean Calculus, a first-order formal system constructed from Charles Sanders Peirce’s triadic logic and recast in categorical, topological, and algebraic terms. Identity, inference, and modality are defined as consequences of recursive morphism closure over a non-metric symbolic manifold Φ. Presence arises from symbolic saturation governed by the compression functional I(p). This system unifies logic, physics, and ontology through symbolic recursion and curvature, replacing metric assumptions with recursive cost topology. All structures—identity, mass, time, causality—emerge from the self-coherence of morphic braids in a purely symbolic substrate, thereby replacing metric foundations with compression-curvature dynamics that computationally bridge the essential logical architecture of the theoretical and practical sciences simultaneously.
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1812241.086978
What’s the probability that God exists? Here, we’re talking about the God of traditional theism, the O3 world-creator (omniscient, omnipotent, omnibenevolent), or a supremely perfect being. This question recently came up in a debate between Matt Dillahunty and Matthew Adelstein. …
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1869963.086992
Phil Dowe’s Conserved Quantity Theory (CQT) is based on the following theses: (a) CQT is the result of an empirical analysis and not a conceptual one, (b) CQT is metaphysically contingent, and (c) CQT is refutable. I argue, on the one hand, that theses (a), (b), and (c) are not only problematic in themselves, but also they are incompatible with each other and, on the other, that the choice of these theses is explained by the particular position that the author embraces regarding the relationship between metaphysics and physics.
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2043218.087006
We develop a classification of general Carrollian structures, permitting affine connections with both torsion and non-metricity. We compare with a recent classification of general Galilean structures in order to present a unified perspective on both. Moreover, we demonstrate how both sets of structures emerge from the most general possible Lorentzian structures in their respective limits, and we highlight the role of global hyperbolicity in constraining both structures. We then leverage this work in order to construct for the first time an ultra-relativistic geometric trinity of gravitational theories, and consider connections which are simultaneously compatible with Galilean and Carrollian structures. We close by outlining a number of open questions and future prospects.
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2204218.08704
The dialogical stance on meaning in the Lorenzen-Lorenz tradition is dynamic, as it is based on interaction between players, and contextual, as meaning depends on the set of rules adopted for the dialogical justification of claims including those implicit in linguistic practice. Grasping the meaning of an expression or an action amounts to identifying the rationale behind our verbal and behavioural practices. This knowledge is informed by the collective intelligence embodied within public criticism Different aspects of meaning are made explicit within the game rules: particle rules for the meaning of logical constants, the Socratic rule for non-logical constants and structural rules that set contextual meaning by shaping the development of a play. The level of plays is governed by these meaning-determining rules, and validity (or proof) is built from the plays. The result is a framework that grounds language and logic in the dynamics of dialogical meaning, and which has proven fruitful for studying frameworks for the logical analysis of language, modern and ancient.
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2248557.087067
Many physicalists nowadays, and Bigelow for one, stand ready to carry metaphysical baggage when they find it worth the weight. This physicalist’s philosophy of mathematics is premised on selective, a posteriori realism about immanent universals. Bigelow’s universals, like D. M. Armstrong’s, are recurrent elements of the physical world; and mathematical objects are universals. The result is a thoroughgoing threefold realism: mathematical realism, scientific realism, and the realism that stands opposed to nominalism.
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2562921.087081
Since Meyer and Dunn showed that the rule γ is admissible in E, relevantists have produced new proofs of the admissibility of γ for an ever more expansive list of relevant logics. We show in this paper that this is not cause to think that this is the norm; rather γ fails to be admissible in a wide variety of relevant logics. As an upshot, we suggest that the proper view of γ-admissibility is as a coherence criterion, and thus as a selection criterion for logical theory choice.
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2614089.087095
“Degenerate Case” Dialetheism
Motivation: trouble with even the most sophisticated and beautiful gappy approaches e.g. Kripke - the ‘not true’ and samesaying. Priest’s view really is better in a way. A resting place. …
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2651350.08711
In this three-part essay, I investigate Frege’s platonist and anti-creationist position in Grundgesetze der Arithmetik and to some extent also in Die Grundlagen der Arithmetik. In Sect. 1.1, I analyze his arithmetical and logical platonism in Grundgesetze. I argue that the reference-fixing strategy for value-range names—and indirectly also for numerical singular terms—that Frege pursues in Grundgesetze I gives rise to a conflict with the supposed mind- and language-independent existence of numbers and logical objects in general. In Sect. 1.2 and 1.3, I discuss the non-creativity of Frege’s definitions in Grundgesetze and the case of what I call weakly creative definitions.
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3134647.087124
Proof automation is crucial to large-scale formal mathematics and software/hardware verification projects in ITPs. Sophisticated tools called hammers have been developed to provide general-purpose proof automation in ITPs such as Coq and Isabelle, leveraging the power of ATPs. An important component of a hammer is the translation algorithm from the ITP’s logical system to the ATP’s logical system. In this paper, we propose a novel translation algorithm for ITPs based on dependent type theory. The algorithm is implemented in Lean 4 under the name Lean-auto. When combined with ATPs, Lean-auto provides general-purpose, ATP-based proof automation in Lean 4 for the first time. Soundness of the main translation procedure is guaranteed, and experimental results suggest that our algorithm is sufficiently complete to automate the proof of many problems that arise in practical uses of Lean 4. We also find that Lean-auto solves more problems than existing tools on Lean 4’s math library Mathlib4.
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3229258.087137
We propose a framework for the analysis of choice behaviour when the latter is made explicitly in chronological order. We relate this framework to the traditional choice theoretic setting from which the chronological aspect is absent, and compare it to other frameworks that extend this traditional setting. Then, we use this framework to analyse various models of preference discovery. We characterise, via simple revealed preference tests, several models that differ in terms of (1) the priors that the decision-maker holds about alternatives and (2) whether the decision-maker chooses period by period or uses her knowledge about future menus to inform her present choices. These results provide novel testable implications for the preference discovery process of myopic and forward-looking agents.
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3230784.087154
Consultant Statistician
Edinburgh
Relevant significance? Be careful what you wish for
Despised and Rejected
Scarcely a good word can be had for statistical significance these days. We are admonished (as if we did not know) that just because a null hypothesis has been ‘rejected’ by some statistical test, it does not mean it is not true and thus it does not follow that significance implies a genuine effect of treatment. …
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3659905.087171
We introduce a projection-based semantic interpretation of differentiation within the Universal Theory of Differentiation (UTD), reframing acts of distinction as structured projections of relational patterns. Building on UTD’s categorical and topos-theoretic foundations, we extend the formalism with a recursive theory of differentiational convergence. We define Stable Differentiational Identities (SDIs) as the terminal forms of recursive differentiation, prove their uniqueness and hierarchical organization, and derive a transparency theorem showing that systems capable of stable recursion can reflect upon their own structure. These results support an ontological model in which complexity, identity, and semantic expressibility emerge from structured difference. Applications span logic, semantics, quantum mechanics, and machine learning, with experiments validating the structural and computational power of the framework.
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3870801.087185
The article offers a novel reconstruction of Hilbert’s early metatheory of formal axiomatics. His foundational work from the turn of the last century is often regarded as a central contribution to a “model-theoretic” viewpoint in modern logic and mathematics. The article will re-assess Hilbert’s role in the development of model theory by focusing on two aspects of his contributions to the axiomatic foundations of geometry and analysis. First, we examine Hilbert’s conception of mathematical theories and their interpretations; in particular, we argue that his early semantic views can be understood in terms of a notion of translational isomorphism between models of an axiomatic theory. Second, we offer a logical reconstruction of his consistency and independence results in geometry in terms of the notion of interpretability between theories.
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3966977.087199
Arithmetical truth-value realists hold that any proposition in the language of arithmetic has a fully determined truth value. Arithmetical truth-value necessists add that this truth value is necessary rather than merely contingent. …
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4311260.087214
Given a time t and a world w, possible or not, say that w is t-possible if and only if there is a possible world wt that matches w in all atemporal respects as well as with respect to all that happens up to and including time t. For instance, a world just like ours but where in 2027 a square circle appears is 2026-possible but not 2028-possible. …
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4467767.087228
Philosophical discussion of the Two-Envelope Paradox has suffered from a lack of formal precision. I discuss various versions of the paradoxical argument using modern probability theory, which helps to make diagnoses that are simpler, more insightful, and provably correct. Paradoxical arguments are revealed to be fallacious for one of three reasons: (1) the argument makes a formal mistake such as an equivocation fallacy; (2) the argument disregards relevant uncertainty about or variability in a unit of measurement; (3) the argument uses an invalid decision rule. I improve upon various existing diagnoses and discuss what kind of philosophical and decision-theoretic import the paradox has.
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4864843.087242
We will be using classical sentential (viz., truth-functional/Boolean) logic as our background, deductive logical theory. This theory (viz., the truth-table method we will be using to reason, semantically, about it) traces back to Peirce [8] (and, later, Wittgenstein [17]). The basic units of analysis in sentential logic are atomic sentences. These are meant to be declarative sentences which contain no (sentential) logical connectives. We will use capital letters: A, B, C , . . . to denote atomic sentences. The only other elements of the language of sentential logic (LSL) are the (sentential) logical connectives (hereafter, the connectives) themselves. The meanings of the connectives are given by the following truth-table definitions.
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4871264.087255
At present, there are at least two set theories motivated by quantum ontology: Décio Krause’s quasi-set theory (Q) and Maria Dalla Chiara and Giuliano Toraldo di Francia’s quasi-set theory (QST). Recent work [Jorge-Holik-Krause, 2023] has established certain links between QST and Pawlak’s rough set theory (RST), showing that both are strong candidates for providing a non-deterministic semantics of N matrices that generalizes those based on ZF. In this work, we show that the new atomless quasi-set theory Q , recently introduced to account for a quantum property ontology [Krause-Jorge, 2024], has strong structural similarities with QST and RST. We study the level of extensionality that each theory presents, its relation to the Leibniz principle and the rigidity property. We believe that developing common features among these three theories can motivate common fields of research. By revealing shared structures, the developments of each theory can have a positive impact on the others.
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4924026.08727
Truth-value realism about (first-order) arithmetic is the thesis that for any first-order logic sentence in the language of arithmetic (i.e., using the successor, addition and multiplication functions along with the name “0”), there is a definite truth value, either true or false. …
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5233114.087284
I introduce Inevitable Actualization (IA), an ontological modality: if (1) the universe’s future time involves an unbounded sequence of causal trials (H∞) and (2) a state S has a non-zero physical probability Pn > in trial n such that the sum n=1 Pn diverges, then S is guaranteed to occur with probability one. IA is developed through a rigorous measure-theoretic foundation, probabilistic modeling with dependence (under standard mixing conditions) and absorbing-state exceptions, contrasting IA with classical modalities and modern multiverse theories. Positioned as a distinct third category alongside necessity and contingency, IA’s unique grounding rests on temporal structure and probability. I address objections (Boltzmann brains, the measure problem, and identity duplication) and illustrate IA’s implications for ethics, cosmology, and personal identity, acknowledging formal challenges.
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5263141.087298
In the topic-sensitive theory of the logic of imagination due to Berto [3], the topic of the imaginative output must be contained within the imaginative input. That is, imaginative episodes can never expand what they are about. We argue, with Badura [2], that this constraint is implausible from a psychological point of view, and it wrongly predicts the falsehood of true reports of imagination. Thus the constraint should be relaxed; but how? A number of direct approaches to relaxing the controversial content-inclusion constraint are explored in this paper. The core idea is to consider adding an expansion operator to the mereology of topics. The logic that results depends on the formal constraints placed on topic expansion, the choice of which are subject to philosophical dispute. The first semantics we explore is a topological approach using a closure operator, and we show that the resulting logic is the same as Berto’s own system. The second approach uses an inclusive and monotone increasing operator, and we give a sound and complete axiomatiation for its logic. The third approach uses an inclusive and additive operator, and we show that the associated logic is strictly weaker than the previous two systems, and additivity is not definable in the language. The latter result suggests that involved techniques or a more expressive language is required for a complete axiomatization of the system, which is left as an open question. All three systems are simple tweaks on Berto’s system in that the language remains propositional, and the underlying theory of topics is unchanged.