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17623.750325
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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75345.750432
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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248600.750442
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.
-
409600.750448
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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453939.750454
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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768303.750459
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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819471.750464
“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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856732.75047
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.
-
1340029.750476
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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1434640.750485
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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1436166.750505
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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1865287.750516
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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2076183.750525
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.
-
2172359.750534
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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2516642.750544
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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2673149.750553
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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3070225.750562
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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3076646.75057
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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3129408.750581
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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3438496.750591
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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3468523.750601
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.
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3784794.750616
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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3837029.750627
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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3870623.750641
[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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3894583.750651
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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4047611.750661
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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4104051.750669
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.
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4250338.750678
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. …
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4528588.750687
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.
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4650071.750701
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.