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301431.755659
We introduce a challenge designed to evaluate the capability of Large Language Models (LLMs) in performing mathematical induction proofs, with a particular focus on nested induction. Our task requires models to construct direct induction proofs in both formal and informal settings, without relying on any preexisting lemmas. Experimental results indicate that current models struggle with generating direct induction proofs, suggesting that there remains significant room for improvement.
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309586.755724
On its surface, a sentence like If Laura becomes a zombie, she wants you to shoot her looks like a plain conditional with the attitude want in its consequent. However, the most salient reading of this sentence is not about the desires of a hypothetical zombie- Laura. Rather, it asserts that the actual, non-zombie Laura has a certain restricted attitude: her present desires, when considering only possible states of affairs in which she becomes a zombie, are such that you shoot her. This can be contrasted with the shifted reading about zombie-desires that arises with conditional morphosyntax, e.g., If Laura became a zombie, she would want you to shoot her. Furthermore, as Blumberg and Holguín (J Semant 36(3):377–406, 2019) note, restricted attitude readings can also arise in disjunctive environments, as in Either a lot of people are on the deck outside, or I regret that I didn’t bring more friends. We provide a novel analysis of restricted and shifted readings in conditional and disjunctive environments, with a few crucial features. First, both restricted and shifted attitude conditionals are in fact “regular” conditionals with attitudes in their consequents, which accords with their surface-level appearance and contrasts with Pasternak’s (The mereology of attitudes, Ph.D. thesis, Stony Brook University, Stony Brook, NY, 2018) Kratzerian approach, in which the if -clause restricts the attitude directly. Second, whether the attitude is or is not shifted—i.e., zombie versus actual desires—is dependent on the presence or absence of conditional morphosyntax. And third, the restriction of the attitude is effected by means of aboutness, a concept for which we provide two potential Kai von Fintel and Robert Pasternak are listed alphabetically and share joint lead authorship of this work.
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424641.755733
Today I want to make a little digression into the quaternions. We won’t need this for anything later—it’s just for fun. But it’s quite beautiful. We saw in Part 8 that if we take the spin of the electron into account, we can think of bound states of the hydrogen atom as spinor-valued functions on the 3-sphere. …
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897975.75574
The paper proposes and studies new classical, type-free theories of truth and determinateness with unprecedented features. The theories are fully compositional, strongly classical (namely, their internal and external logics are both classical), and feature a defined determinateness predicate satisfying desirable and widely agreed principles. The theories capture a conception of truth and determinateness according to which the generalizing power associated with the classicality and full compositionality of truth is combined with the identification of a natural class of sentences – the determinate ones – for which clear-cut semantic rules are available. Our theories can also be seen as the classical closures of Kripke-Feferman truth: their ω-models, which we precisely pinned down, result from including in the extension of the truth predicate the sentences that are satisfied by a Kripkean closed-off fixed point model. The theories compare to recent theories proposed by Fujimoto and Halbach, featuring a primitive determinateness predicate. In the paper we show that our theories entail all principles of Fujimoto and Halbach’s theories, and are proof-theoretically equivalent to Fujimoto and Halbach’s CD . We also show establish some negative results on Fujimoto and Halbach’s theories: such results show that, unlike what happens in our theories, the primitive determinateness predicate prevents one from establishing clear and unrestricted semantic rules for the language with type-free truth.
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1204620.755745
In Part 4 we saw that the classical Kepler problem—the problem of a single classical particle in an inverse square force—has symmetry under the group of rotations of 4-dimensional space Since the Lie algebra of this group is
we must have conserved quantities
and
corresponding to these two copies of The physical meaning of these quantities is a bit obscure until we form linear combinations
Then is the angular momentum of the particle, while is a subtler conserved quantity: it’s the eccentricity vector of the particle divided by where the energy is negative for bound states (that is, elliptical orbits)
The advantage of working with and is that these quantities have very nice Poisson brackets:
This says they generate two commuting symmetries. …
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1524279.755751
Історія логіки – актуальний напрямок досліджень в царині сучасного логічного знання. Такі розвідки, по-перше, сприяють створенню загальної картини еволюції логіки, усвідомленню змін предмета, що їх вона зазнавала як наука і як навчальна дисципліна, а також змін у парадигмальних принципах її історичного розвитку, засадничих правилах побудови логічних теорій та інструментарієві останніх. По-друге, історикологічні дослідження надають можливість виявити те, як логічні концепції впливали на інші наукові дисципліни, передусім філософію та математику. По-третє, історико-логічний аналіз дозволяє розглянути логічну позицію певного автора в широкому історико-філософському контексті, показати, як філософські ідеї впливали на розвиток логічного знання. По-четверте, дослідження в царині історії логіки допомагають розглянути її в широкому історико-культурному контексті, з’ясувати взаємовплив різних логічних поглядів та певних культурних традицій і особливостей історичних епох.
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1524321.755757
We present a logic which deals with connexive exclusion. Exclusion (also called “co-implication”) is considered to be a propositional connective dual to the connective of implication. Similarly to implication, exclusion turns out to be non-connexive in both classical and intuitionistic logics, in the sense that it does not satisfy certain principles that express such connexivity. We formulate these principles for connexive exclusion, which are in some sense dual to the well-known Aristotle’s and Boethius’ theses for connexive implication. A logical system in a language containing exclusion and negation can be called a logic of connexive exclusion if and only if it obeys these principles, and, in addition, the connective of exclusion in it is asymmetric, thus being different from a simple mutual incompatibility of propositions. We will develop a certain approach to such a logic of connexive exclusion based on a semantic justification of the connective in question. Our paradigm logic of connexive implication will be the connexive logic C, and exactly like this logic the logic of connexive exclusion turns out to be contradictory though not trivial.
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1755062.755762
Levy’s Upward Theorem says that the conditional expectation of an integrable random variable converges with probability one to its true value with increasing information. In this paper, we use methods from effective probability theory to characterise the probability one set along which convergence to the truth occurs, and the rate at which the convergence occurs. We work within the setting of computable probability measures defined on computable Polish spaces and introduce a new general theory of effective disintegrations. We use this machinery to prove our main results, which (1) identify the points along which certain classes of effective random variables converge to the truth in terms of certain classes of algorithmically random points, and which further (2) identify when computable rates of convergence exist. Our convergence results significantly generalize earlier results within a unifying novel abstract framework, and there are no precursors of our results on computable rates of convergence. Finally, we make a case for the importance of our work for the foundations of Bayesian probability theory.
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1851461.755767
There are four well-known models of fundamental objective probabilistic reality: classical probability, comparative probability, non-Archimedean probability, and primitive conditional probability. I offer two desiderata for an account of fundamental objective probability, comprehensiveness and non-superfluity. It is plausible that classical probabilities lack comprehensiveness by not capturing some intuitively correct probability comparisons, such as that it is less likely that 0 = 1 than that a dart randomly thrown at a target will hit the exact center, even though both classically have probability zero. We thus want a comparison between probabilities with a higher resolution than we get from classical probabilities. Comparative and non-Archimedean probabilities have a hope of providing such a comparison, but for known reasons do not appear to satisfy our desiderata. The last approach to this problem is to employ primitive conditional probabilities, such as Popper functions, and then argue that P(0 = 1 | 0 = 1 or hit center) = < 1 = P (hit center | 0 = 1 or hit center). But now we have a technical question: How can we reconstruct a probability comparison, ideally satisfying the standard axioms of comparative probability, from a primitive conditional probability? I will prove that, given some plausible assumptions, it is impossible to perform this task: conditional probabilities just do not carry enough information to define a satisfactory comparative probability. The result is that of the models, no one satisfies our two desiderata. We end by briefly considering three paths forward.
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2016499.755773
This is probably an old thing that has been discussed to death, but I only now noticed it. Suppose an open future view on which future contingents cannot have truth value. What happens to entailments? …
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2260812.755779
We establish the equivalence of two much debated impartiality criteria for social welfare orders: Anonymity and Permutation Invariance. Informally, Anonymity says that, in order to determine whether one social welfare distribution w is at least as good as another distribution v, it suffices to know, for every welfare level, how many people have that welfare level according to w and how many people have that welfare level according to v. Permutation Invariance, by contrast, says that, to determine whether w is at least as good as v, it suffices to know, for every pair of welfare levels, how many people have that pair of welfare levels in w and v respectively.
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2456159.755784
A recent result from theoretical computer science provides for the verification of answers to the Halting Problem, even when there is no plausible means by which to derive those answers using a bottom-up approach. We argue that this result has profound implications for the existence of strongly emergent phenomena. In this work we develop a computer science-based framework for thinking about strong emergence and in doing so demonstrate the plausibility of strongly emergent phenomena existing in our universe. We identify six sufficient criteria for strong emergence and detail the actuality of five of the six criteria. Finally, we argue for the plausibility of the sixth criterion by analogy and a case study of Boltzmann brains (with additional case studies provided in the appendices.)
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2542582.755798
This paper proposes a dynamic temporal logic that is appropriate for modeling the dynamics of scientific knowledge (especially in historical sciences, such as Archaeology, Paleontology and Geology). For this formalization of historical knowledge, the work is divided into two topics: firstly, we define a temporal branching structure and define the terms for application in Philosophy of Science; Finally, we define a logical system that consists of a variation of Public Announcement Logic in terms of temporal logic, with appropriate rules in a tableaux method.
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2628887.755804
In this paper, I develop a “safety result” for applied mathematics. I show that whenever a theory in natural science entails some non-mathematical conclusion via an application of mathematics, there is a counterpart theory that carries no commitment to mathematical objects, entails the same conclusion, and the claims of which are true if the claims of the original theory are “correct”: roughly, true given the assumption that mathematical objects exist. The framework used for proving the safety result has some advantages over existing nominalistic accounts of applied mathematics. It also provides a nominalistic account of pure mathematics.
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3131631.755812
There is a longstanding puzzle about empty names. On the one hand, the principles of classical logic seem quite plausible. On the other hand, there would seem to be truths involving empty names that require rejecting certain classically valid principles.
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3131681.75582
Consider the property of being something that is identical to Hesperus. For short, call this the property of being Hesperus. What is the nature of this property? How does it relate to the property of being Phosphorus? And how do these properties relate to the purely haecceitistic property of being v—the unique thing that has the property of being Hesperus and the property of being Phosphorus?
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3209922.755828
Casajus (J Econ Theory 178, 2018, 105–123) provides a characterization of the class of positively weighted Shapley value for …nite games from an in…nite universe of players via three properties: e¢ ciency, the null player out property, and superweak differential marginality. The latter requires two players’payoffs to change in the same direction whenever only their joint productivity changes, that is, their individual productivities stay the same. Strengthening this property into (weak) differential marginality yields a characterization of the Shapley value. We suggest a relaxation of superweak differential marginality into two subproperties: (i) hyperweak differential marginality and (ii) superweak differential marginality for in…nite subdomains. The former (i) only rules out changes in the opposite direction. The latter (ii) requires changes in the same direction for players within certain in…nite subuniverses. Together with e¢ ciency and the null player out property, these properties characterize the class of weighted Shapley values.
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3231434.755834
In a system with identity, quotation, and an axiom predicate, a classical extension of the system yields a falsity. The result illustrates a novel form of instability in classical logic. Notably, the phenomenon arises without vocabulary such as ’true’ or ’provable’. Conservative extensions are safe expansions: They add expressive resources while proving the same theorems (or at most, terminological variants thereof). Conservative extensions are foundational for major developments, including the Lowenheim-Skolem theorems, precise comparisons of proof-theoretic strength (Simpson 2009), and the understanding of reflection principles in arithmetic and set theory (Feferman 1962). The purpose here is not to question these developments, but rather to advise caution for the future. Some extensions that appear quite conservative end up not being so. In a system with identity, quotation, and a metalinguistic singular term, a purely syntactic predicate for axioms can create instability under an innocent-looking extension.
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3231456.755839
It is known that some diagonal arguments, when formalized, do not demonstrate the impossibility of the diagonal object, but instead reveal a breakdown in definability or encoding. For example, in a formal setting, Richard’s paradox does not yield a contradiction; it instead reflects that one of the relevant sets is ill-defined. (For elaboration and other examples, see Simmons 1993, Chapter 2.) This invites the possibility that other diagonal arguments may reflect similar anomalies. The diagonal argument against a universal p.r. function is considered in this light. The impetus is an algorithm which appears to satisfy all standard criteria for being p.r. while simulating the computation of fi(i, n) for any index i of a binary p.r. function. The paper does not attempt to explain why this construction apparently survives the usual diagonal objection, but presents it in a form precise enough to support that analysis.
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3294397.755844
In logic and philosophy of logic, “formalization” covers a broad range of interrelated issues: some philosophers hold that logical systems are means to formalize theories and reasoning (Dutilh Novaes 2012), others seek to formalize semantical by syntactical systems (Carnap 1942/43), ask whether logical languages are formalizations of natural languages (Stokhof 2018), teach undergraduates to formalize arguments using elementary logic, debate how to formalize notions such as moral obligation (Hansson 2018), or develop formalizations of belief change processes (Rott 2001). This variety goes hand in hand with an equally broad range of general views about what logic and its role in philosophy is or should be – whether, for example, logic is first of all a tool for reasoning (Dutilh Novaes 2012), a mathematical theory of certain formal structures which can be used to model philosophically interesting phenomena (Hansson 2018; Sagi 2020a; Stokhof 2018), or a theory that studies inferential relations in natural language and enables us to show that certain ordinary-language arguments are valid (Peregrin/Svoboda 2017), to name just a few. More or less implicitly, these approaches contain views on what the target phenomena of formalizing are (languages, arguments, …), what kind of relation formalizations have to it (model, tool, …) and whether formalizing is an integral part of logic or an application of it.
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3406311.75585
In this paper we will try to provide a solid form of intrinsic set theoretical optimism. In other words, we will try to vindicate Gödel’s views on phenomenology as a method for arriving at new axioms of ZFC in order to decide independent statements such as CH. Since we have previously written on this very same subject [41, 43, 44], it is necessary to provide a justification for addressing it once again.
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3751966.755856
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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4346614.755863
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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4346635.755868
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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4346657.755874
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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4406246.755879
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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4406269.755885
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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4579180.755891
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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4851512.755898
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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5040730.755904
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.