-
50224.015827
This paper reconsiders the metaphysical implication of Einstein algebras, prompted by the recent objections of Chen (2024) on Rosenstock et al. (2015)’s conclusion. Rosenstock et al.’s duality theorem of smooth manifolds and smooth algebras supports a conventional wisdom which states that the Einstein algebra formalism is not more “relationalist” than the standard manifold formalism. Nevertheless, as Chen points out, smooth algebras are different from the relevant algebraic structure of an Einstein algebra. It is therefore questionable if Rosenstock et al.’s duality theorem can support the conventional wisdom. After a re-visit of John Earman’s classic works on the program of Leibniz algebras, I formalize the program in category theory and propose a new formal criterion to determine whether an algebraic formalism is more “relationalist” than the standard manifold formalism or not. Based on the new formal criterion, I show that the conventional wisdom is still true, though supported by a new technical result. I also show that Rosenstock et al. (2015)’s insight can be re-casted as a corollary of the new result. Finally, I provide a justification of the new formal criterion with a discussion of Sikorski algebras and differential spaces. The paper therefore provides a new perspective for formally investigating the metaphysical implication of an algebraic formalism for the theory of space and time.
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102459.015891
Some authors maintain that we can use causal Bayes nets to infer whether X → Y or X ← Y by consulting a probability distribution defined over some exogenous source of variation for X or Y . We raise a problem for this approach. Specifically, we point out that there are cases where an exogenous cause of X (Ex) has no probabilistic influence on Y no matter the direction of causation — namely, cases where Ex → X → Y and Ex → X ← Y are probabilistically indistinguishable. We then assess the philosophical significance of this problem and discuss some potential solutions.
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136053.015906
[Editor’s Note: The following new entry by Klaas Kraay replaces the
former entry on this topic by the previous author.]
The topic of divine freedom concerns the extent to which a divine
being — in particular, the supreme divine being, God — can
be free. There are, of course, many different conceptions of who or
what God is. This entry will focus on one enormously important and
influential model, according to which God is a personal being who
exists necessarily, who is essentially omnipotent, omniscient,
perfectly good, and perfectly rational, and who is the creator and
sustainer of all that contingently
exists.[ 1 ]
(For more discussion of these attributes, see the entries on
omnipotence,
omniscience,
perfect goodness,
and
creation and conservation.)
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160013.015913
Canonical is a solver for type inhabitation in dependent type theory, that is, the problem of producing a term of a given type. We present a Lean tactic which invokes Canonical to generate proof terms and synthesize programs. The tactic supports higher-order and dependently-typed goals, structural recursion over indexed inductive types, and definitional equality. Canonical finds proofs for 84% of Natural Number Game problems in 51 seconds total.
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313041.01592
The desirable gambles framework provides a rigorous foundation for imprecise probability theory but relies heavily on linear utility via its coherence axioms. In our related work, we introduced function-coherent gambles to accommodate nonlinear utility. However, when repeated gambles are played over time—especially in intertemporal choice where rewards compound multiplicatively— the standard additive combination axiom fails to capture the appropriate long-run evaluation. In this paper we extend the framework by relaxing the additive combination axiom and introducing a nonlinear combination operator that effectively aggregates repeated gambles in the log-domain. This operator preserves the time-average (geometric) growth rate and addresses the ergodicity problem. We prove the key algebraic properties of the operator, discuss its impact on coherence, risk assessment, and representation, and provide a series of illustrative examples. Our approach bridges the gap between expectation values and time averages and unifies normative theory with empirically observed non-stationary reward dynamics. Keywords. Desirability, non-linear utility, ergodicity, intertemporal choice, non-additive dynamics, function-coherent gambles, risk measures.
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369481.015925
A firm wishes to persuade a patient to take a drug by making either positive statements like “if you take our drug, you will be cured”, or negative statements like “anyone who was not cured did not take our drug”. Patients are neither Bayesian nor strategic: They use a decision procedure based on sampling past cases. We characterize the firm’s optimal statement, and analyze competition between firms making either positive statements about themselves or negative statements about their rivals. The model highlights that logically equivalent statements can differ in effectiveness and identifies circumstances favoring negative ads over positive ones.
-
515768.015931
Causal Finitism—the thesis that nothing can have an infinite causal history—implies that there is a first cause, and our best hypothesis for what a first cause would be is God. Thus:
- If Causal Finitism is true, God exists. …
-
794018.015938
Indicative conditional antecedents appear to be remarkably scopeless: they are scopeless with respect to the truth-functional connectives, scopeless with respect to epistemic modals, and scopeless with respect to each other (i.e., commutative). This pervasive scopelessness is a basic explanandum for any theory of indicatives, and the subject of much recent work. In this paper I revisit the theory of McGee [1989], which already comes surprisingly close to delivering a simple and powerful account of all of this scopelessness. I reformulate the theory as information-sensitive in the contemporary sense, and extend it with epistemic modals. On the resulting theory, epistemic modals become in e!ect quantifiers over choice functions, and their scopeless interaction with indicative antecedents drops out naturally. I give McGee’s logic a new axiomatization, and show that if his Import-Export axiom is replaced with a weaker Commutativity axiom stating that indicative antecedents commute, then Import-Export can be derived. I explain how the issue of commutativity interacts with the question how to extend information-sensitive theories of the indicative to modal antecedents. Along the way I add to the collapse results of both McGee [1985] and Mandelkern [2021], showing that under weak assumptions, Commutativity is in tension with Modus Ponens and (more generally) with the principle Mandelkern calls Ad Falsum. I convict Ad Falsum, and refine the case against Modus Ponens.
-
915501.015945
Bell’s theorem states that no model that respects Local Causality and Statistical Independence can account for the correlations predicted by quantum mechanics via entangled states. This paper proposes a new approach, using backward-in-time conditional probabilities, which relaxes conventional assumptions of temporal ordering while preserving Statistical Independence as a “fine-tuning” condition. It is shown how such models can account for EPR/Bell correlations and, analogously, the GHZ predictions while nevertheless forbidding superluminal signalling.
-
1434439.015952
Deng, Hani, and Ma [arXiv:2503.01800] claim to resolve Hilbert’s Sixth Problem by deriving the Navier-Stokes-Fourier equations from Newtonian mechanics via an iterated limit: a Boltzmann-Grad limit (ε → 0, N εd−1 = α fixed) yielding the Boltzmann equation, followed by a hydrodynamic limit (α → ∞) to obtain fluid dynamics. Though mathematically rigorous, their approach harbors two critical physical flaws. First, the vanishing volume fraction (N ε → 0) confines the system to a dilute gas, incapable of embodying dense fluid properties even as α scales, rendering the resulting equations a rescaled gas model rather than a true continuum. Second, the Boltzmann equation’s reliance on molecular chaos collapses in fluid-like regimes, where recollisions and correlations invalidate its derivation from Newtonian dynamics. These inconsistencies expose a disconnect between the formalism and the physical essence of fluids, failing to capture emergent, density-driven phenomena central to Hilbert’s vision. We contend that the Sixth Problem remains open, urging a rethink of classical kinetic theory’s limits and the exploration of alternative frameworks to unify microscale mechanics with macroscale fluid behavior.
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1535448.015958
We argue that logicism, the thesis that mathematics is reducible to logic and analytic truths, is true. We do so by (a) developing a formal framework with comprehension and abstraction principles, (b) giving reasons for thinking that this framework is part of logic, (c) showing how the denotations for predicates and individual terms of an arbitrary mathematical theory can be viewed as logical objects that exist in the framework, and (d) showing how each theorem of a mathematical theory can be given an analytically true reading in the logical framework.
-
1693955.015966
When I was trying to work out my intuitions about causal paradoxes of infinity, which eventually led to my formulating the thesis of causal finitism (CF)—that nothing can have an infinite causal history—I toyed with views that involved information. …
-
1820896.015973
We characterize Martin-Lof randomness and Schnorr randomness in terms of the merging of opinions, along the lines of the Blackwell-Dubins Theorem [BD62]. After setting up a general framework for defining notions of merging randomness, we focus on finite horizon events, that is, on weak merging in the sense of Kalai-Lehrer [KL94]. In contrast to Blackwell-Dubins and Kalai-Lehrer, we consider not only the total variational distance but also the Hellinger distance and the Kullback-Leibler divergence. Our main result is a characterization of Martin-Lof randomness and Schnorr randomness in terms of weak merging and the summable Kullback-Leibler divergence. The main proof idea is that the Kullback-Leibler divergence between µ and ν, at a given stage of the learning process, is exactly the incremental growth, at that stage, of the predictable process of the Doob decomposition of the ν-submartingale L(σ) = − ln µ(σ) ν(σ) . These characterizations of algorithmic randomness notions in terms of the Kullback-Leibler divergence can be viewed as global analogues of Vovk’s theorem [Vov87] on what transpires locally with individual Martin- Lof µ- and ν-random points and the Hellinger distance between µ, ν.
-
1953406.015978
In this brief article I respond to Seifert’s recent views on the periodic law and the periodic table in connection with the views of philosophers regarding laws of nature. I argue that the author makes some factual as well as conceptual errors which are in conflict with some generally held views regarding the periodic law and the periodic table.
-
1953583.015985
This article describes confirmation of the proposition that numbers are identified with operators in the following three steps. 1. The set of operators to construct finite cardinals satisfies Peano Axioms. 2. Accordingly, the natural numbers can be identified with these operators. 3. From the operators, five kinds of operators are derived, and on the basis of the step 2, the integers, the fractions, the real numbers, the complex numbers and the quaternions are identified with the five kinds of operators respectively. These operators stand in a sequential inclusion relationship, in contrast to the embedding relationship between those kinds of numbers defined as sets.
-
1953607.015991
Inconsistencies! What do they mean? Can we support them? With this paper, we hope to contribute to the claim that we can tolerate inconsistencies in certain situations even without considering any logic that may enable us to do that, say some paraconsistent logic. We argue that in many cases where we apply reason, we work in domains where inconsistencies appear, and even so, we neither get them out (but ‘support’ them) nor modify the underlying logic (such as classical logic) to avoid logical troubles. To make things more precise, we distinguish between inconsistency, anomaly, and contradiction. Our thesis is that we can reason sensibly with classical logic even in the presence of inconsistencies once (as we explain) we either ‘do not go there’ or make things so that the inconsistent sentences cannot be joined to arrive at a contradiction. Some sample cases are given to motivate the discussion.
-
2123724.015996
Free choice sequences play a key role in the Brouwerian continuum. Using recent modal analysis of potential infinity, we can make sense of free choice sequences as potentially infinite sequences of natural numbers without adopting Brouwer’s distinctive idealistic metaphysics. This provides classicists with a means to make sense of intuitionistic ideas from their own classical perspective. I develop a modal-potentialist theory of real numbers that suffices to capture the most distinctive features of intuitionistic analysis, such as Brouwer’s continuity theorem, the existence of a sequence that is monotone, bounded, and non-convergent, and the inability to decompose the continuum non-trivially.
-
2374091.016005
Sunwin chính chủ sở hữu bộ core game cùng hệ thống chăm sóc khách hàng vô địch. Sunwin hiện nay giả mạo rất nhiều anh em chú ý check kĩ uy tín đường link để đảm bảo an toàn và trải nghiệm game đỉnh cao duy nhất. …
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2374091.01601
Sunwin chính chủ sở hữu bộ core game cùng hệ thống chăm sóc khách hàng vô địch. Sunwin hiện nay giả mạo rất nhiều anh em chú ý check kĩ uy tín đường link để đảm bảo an toàn và trải nghiệm game đỉnh cao duy nhất. …
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2540336.016016
This is a bit of a shaggy dog story, but I think it’s fun, and there’s a moral about the nature of mathematical research. Act 1
Once I was interested in the McGee graph, nicely animated here by Mamouka Jibladze:
This is the unique (3,7)-cage, meaning a graph such that each vertex has 3 neighbors and the shortest cycle has length 7. …
-
2991851.016022
In operational quantum mechanics two measurements are called operationally equivalent if they yield the same distribution of outcomes in every quantum state and hence are represented by the same operator. In this paper, I will show that the ontological models for quantum mechanics and, more generally, for any operational theory sensitively depend on which measurement we choose from the class of operationally equivalent measurements, or more precisely, which of the chosen measurements can be performed simultaneously. To this goal, I will take first three examples—a classical theory, the EPR-Bell scenario and the Popescu-Rochlich box; then realize each example by two operationally equivalent but different operational theories—one with a trivial and another with a non-trivial compatibility structure; and finally show that the ontological models for the different theories will be different with respect to their causal structure, contextuality, and fine-tuning.
-
3147335.016029
High speed store required: 947 words. No of bits in a word: 64 Is the program overlaid? No No. of magnetic tapes required: None What other peripherals are used? Card Reader; Line Printer No. of cards in combined program and test deck: 112 Card punching code: EBCDIC Keywords: Atomic, Molecular, Nuclear, Rotation Matrix, Rotation Group, Representation, Euler Angle, Symmetry, Helicity, Correlation.
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3567404.016035
Brian Leftow’s 2022 book, Anselm’s Argument: Divine Necessity is an impressively thorough discussion of Anselmian modal metaphysics, centred around what he takes to be Anselm’s strongest “argument from perfection” (Leftow’s preferred term for an Ontological Argument). This is not the famous argument from Proslogion 2, nor even the modal argument that some have claimed to find in Proslogion 3, but rather, an argument from Anselm’s Reply to Gaunilo, expressed in the following quotation: “If … something than which no greater can be thought … existed, neither actually nor in the mind could it not exist. Otherwise it would not be something than which no greater can be thought. But whatever can be thought to exist and does not exist, if it existed, would be able actually or in the mind not to exist. For this reason, if it can be thought, it cannot not exist.” (p. 66) Before turning to this argument, Leftow offers an extended and closely-argued case for understanding Anselm’s modality in terms of absolute necessity and possibility, with a metaphysical foundation on powers as argued for at length (575 pages) in his 2012 book God and Necessity. After presenting this interpretation in Chapter 1, Leftow’s second chapter discusses various theological applications (such as the fixity of the past, God’s veracity, and immortality), addressing them in a way that both expounds and defends what he takes to be Anselm’s approach. Then in Chapter 3 Leftow addresses certain problems, for both his philosophical and interpretative claims, while Chapter 4 spells out the key Anselmian argument, together with Leftow’s suggested improvements. Chapter 5 explains how the argument depends on Brouwer’s system of modal logic, and defends this while also endorsing the more standard and comprehensive system S5.
-
3856814.016041
trices. The main aim is to construct a system of Nmatrices by substituting standard sets by quasets. Since QST is a conservative extension of ZFA (the Zermelo-Fraenkel set theory with Atoms), it is possible to obtain generalized Nmatrices (Q-Nmatrices). Since the original formulation of QST is not completely adequate for the developments we advance here, some possible amendments to the theory are also considered. One of the most interesting traits of such an extension is the existence of complementary quasets which admit elements with undetermined membership. Such elements can be interpreted as quantum systems in superposed states. We also present a relationship of QST with the theory of Rough Sets RST, which grants the existence of models for QST formed by rough sets. Some consequences of the given formalism for the relation of logical consequence are also analysed.
-
4137157.016046
Two problems are investigated. Why is it that in his solutions to logical problems, Boole’s logical/numerical operations can be difficult to pin down, and why did his late manuscript attempt to get rid of division by zero fall short of that goal? It is suggested that the former is due to different readings that he gives to the operations according to the stage of the solution routine, and the latter is due to a strict confinement to equational reasoning.
-
4137177.016055
Ancient formulations of the distinction between continuous and separative hypotheticals, made by Peripatetics working under Stoic influence, can be vague and confusing. Perhaps the clearest expositor of the matter was Galen. We review his account, provide two formal articulations of it, verify their equivalence, and show that for what we call ‘simple’ hypotheticals, the formal line of demarcation is independent of whether or not modality is taken into account.
-
4311374.016064
We furnish a core-logical development of the Gödel numbering framework that allows metamathematicians to attain limitative results about arithmetical truth without incorporating a genuine truth predicate into the language in a way that would lead to semantic closure. We show how Tarski’s celebrated theorem on the arithmetical undefinability of arithmetical truth can be established using only core logic in both the object language and the metalanguage. We do so at a high level of abstraction, by augmenting the usual first-order language of arithmetic with a primitive predicate Tr and then showing how it cannot be a truth predicate for the augmented language. McGee established an important result about consistent theories that are in the language of arithmetic augmented by such a “truth predicate” Tr and that use Gödel numbering to refer to expressions of the augmented language. Given the nature of his sought result, he was forced to use classical reasoning at the meta level. He did so, however, on the additional and tacit presupposition that the arithmetical theories in question (in the object language) would be closed under classical logic. That left open the dialectical possibility that a constructivist (or intuitionist) could claim not to be discomfited by the results, even if they were to “give a pass” on the unavoidably classical reasoning at the meta level. In this study we “constructivize” McGee’s result, by presuming only core logic for the object language. This shows that the perplexity induced by McGee’s result will confront the constructivist (or intuitionist) as well.
-
4311395.01607
Berry’s Paradox, like Russell’s Paradox, is a ‘paradox’ in name only. It differs from genuine logico-semantic paradoxes such as the Liar Paradox, Grelling’s Paradox, the Postcard Paradox, Yablo’s Paradox, the Knower Paradox, Prior’s Intensional Paradoxes, and their ilk. These latter arise from semantic closure. Their genuine paradoxicality manifests itself as the non-normalizability of the formal proofs or disproofs associated with them. The Russell, the Berry, and the Burali-Forti ‘paradoxes’, by contrast, simply reveal the straightforward inconsistency of their respective existential claims—that the Russell set exists; that the Berry number exists; and that the ordinal of the well-ordering of all ordinals exists. The disproofs of these existential claims are in free logic and are in normal form. They show that certain complex singular terms do not—indeed, cannot—denote. All this counsels reconsideration of Ramsey’s famous division of paradoxes and contradictions into his Group A and Group B. The proof-theoretic criterion of genuine paradoxicality formally explicates an informal and occasionally confused notion. The criterion should be allowed to reform our intuitions about what makes for genuine paradoxicality, as opposed to straightforward (albeit surprising) inconsistency.
-
4491281.016076
A critique is given of the attempt by Hettema and Kuipers to formalize the periodic table. In particular I dispute their notions of identifying a naïve periodic table with tables having a constant periodicity of eight elements and their views on the different conceptions of the atom by chemists and physicists. The views of Hettema and Kuipers on the reduction of the periodic system to atomic physics are also considered critically.
-
5283225.016083
In this short note, which is the final chapter of the volume 60 Years of Connexive Logic, we list ten open problems. Some of these problems are technical and precisely stated, while others are less technical and even speculative. We hope that the list inspires some readers to contribute to the field by tackling one or many of the problems.