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97486.89044
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.
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347853.890555
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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347853.890571
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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514098.890581
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. …
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965613.890592
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.
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1121097.890604
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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1541166.890614
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.
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1830576.890625
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.
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2110919.890635
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.
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2110939.89065
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.
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2285136.890666
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.
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2285157.890676
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.
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2465043.890687
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.
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3256987.8907
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.
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3257009.89071
The present article aims at generalizing the approach to connexive logic that was initiated in [27], by following the work by Paul Egré and Guy Politzer. To this end, a variant of the connexive modal logic CK is introduced and some basic results including soundness and completeness results are established. A tableau calculus is also presented in an appendix.
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3316861.890721
This paper investigates two forms of the Routley star operation, one in Routley & Routley 1972 and the other in Leitgeb 2019. We use object theory (OT) to define both forms and show that in OT’s hyperintensional logic, (a) the two forms aren’t equivalent, but (b) become equivalent under certain conditions. We verify our definitions by showing that the principles governing both forms become derivable and need not be stipulated. Since no mathematics is assumed in OT, the existence of the Routley star image s of a situation s is therefore guaranteed not by set theory but by a theory of abstract objects. The work in the paper integrates Routley star into a more general theory of (partial) situations that has previously been used to develop the theory of possible worlds and impossible worlds.
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3330408.890731
In this paper, we provide an axiom system for the relevant logic of equivalence relation frames and prove completeness for it. This provides a partial answer to the longstanding open problem of axiomatizing frames for relevant modal logics where the modal accessibility relation is symmetric. Following this, we show that the logic enjoys Hallden completeness and that a related logic enjoys the disjunction property.
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3654950.89074
It is shown that one common formulation of Stalnaker’s semantics for conditionals is incomplete: it has no sound and (strongly) complete proof system. At first, this seems to conflict with well-known completeness results for this semantics (e.g., Stalnaker and Thomason 1967, Stalnaker 1970 and Lewis 1973, ch. 6). As it turns out, it does not: these completeness results rely on another closely-related formulation of the semantics that is provably complete. Specifically, the difference comes down to how the Limit Assumption is stated. I close with some remarks about what this means for the logic of conditionals.
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4013930.89075
1. Strong and weak notions of erasure are distinguished according to whether the single erasure procedure does or does not leave the environment in the same state independently of the pre-erasure state. 2. Purely thermodynamic considerations show that strong erasure cannot be dissipationless. 3. The main source of entropy creation in erasure processes at molecular scales is the entropy that must be created to suppress thermal fluctuations (“noise”). 4. A phase space analysis recovers no minimum entropy cost for weak erasure and a positive minimum entropy cost for strong erasure. 5. An information entropy term has been attributed mistakenly to pre-erasure states in the Gibbs formalism through the neglect of an additive constant in the “–k sum p log p” Gibbs entropy formula.
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4063616.890761
The inference pattern known as disjunctive syllogism (DS) appears as a derived rule in Gentzen’s natural deduction calculi NI and NK. This is a paradoxical feature of Gentzen’s calculi in so far as DS is sometimes thought of as appearing intuitively more elementary than the rules ∨E, ¬E, and EFQ that figure in its derivation. For this reason, many contemporary presentations of natural deduction depart from Gentzen and include DS as a primitive rule. However, such departures violate the spirit of natural deduction, according to which primitive rules are meant to relationally define logical connectives via universal properties (§2). This situation raises the question: Can disjunction be relationally defined with DS instead of with Gentzen’s ∨I and ∨E rules? We answer this question in the affirmative and explore the duality between Gentzen’s definition and our own (§3). We argue further that the two universal characterizations, rather than provide competing relational definitions of a single disjunction operator, disambiguate natural language’s “or” (§4). Finally, this disambiguation is shown to correspond exactly with the additive and multiplicative disjunctions of linear logic (§5). The hope is that this analysis sheds new light on the latter connective, so often deemed mysterious in writing about linear logic.
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4080248.89077
If the philosophy of mathematics wants to be rigorous, the concept of infinity must stop being equivocal (both potential and actual) as it currently is. The conception of infinity as actual is responsible for all the paradoxes that compromise the very foundation of mathematics and is also the basis on which Cantor's argument is based on the non-countability of R, and the existence of infinite cardinals of different magnitude. Here we present proof that all infinite sets (in a potential sense) are countable and that there are no infinite cardinals.
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4137917.890782
The philosophical literature on mathematical structuralism and its history has focused on the emergence of structuralism in the 19th century. Yet modern abstractionist accounts cannot provide an historical account for the abstraction process. This paper will examine the role of relations in the history of mathematics, focusing on three main epochs where relational abstraction is most prominent: ancient Greek, 17th and 19th centuries, to provide a philosophical account for the abstraction of structures. Though these structures emerged in the 19th century with definitional axioms, the need for such axioms in the abstraction process comes about, as this paper will show, after a series of relational abstractions without a suitable basis.
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4253405.890794
This article concerns various foundational aspects of the periodic system of the elements. These issues include the dual nature of the concept of an “element” to include element as a “basic substance” and as a “simple substance.” We will discuss the question of whether there is an optimal form of the periodic table, including whether the left-step table fulfils this role. We will also discuss the derivation or explanation of the [n ⫹ ᐉ , n] or Madelung rule for electron-shell filling and whether indeed it is important to attempt to derive this rule from first principles. In particular, we examine the views of two chemists, Henry Bent and Eugen Schwarz, who have independently addressed many of these issues.
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4830414.890809
This paper is about a problem which arose in mathematics but is now widely considered by mathematicians to be a matter “merely” for philosophy. I want to show what philosophy can contribute to solving the problem by returning it to mathematics, and I will do that by elucidating what it is to be a solution to a mathematical problem at all.
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4942622.890818
Angelic visitations in our world are at best rare, and at worst they never occur at all. Not so in Neil Fisk’s world. There, angelic visitations are common – and often deadly. Neil lost his wife to such a visitation, and he’s hated God ever since. The problem with this hatred is that Neil is quite sure his wife is in heaven, as he saw her soul ascending and has never seen her walking around in hell during the frequent glimpses the living are given of the underworld. Since Neil thinks he cannot willingly become devout, he must rely on a divine glitch; those who are caught in heaven’s light during an angelic visitation involuntarily become devout, and thus go to heaven. Luckily for Neil, he drives into a beam of heaven’s light, loses his sight, and becomes devout. Unluckily for Neil, God sends him to hell anyway.
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4945728.89083
The article summarizes the present state of research into the conceptual foundations of the periodic table. We give a brief historical account of the development of the periodic table and periodic system, including the impact of modern physics due to the discoveries of Moseley, Bohr, modern quantum mechanics etc. The role of the periodic table in the debate over the reduction of chemistry is discussed, including the attempts to derive the Madelung rule from first principles. Other current debates concern the concept of an “element” and its dual role as simple substance and elementary substance and the question of whether elements and groups of elements constitute natural kinds. The second of these issues bears on the question of further debates concerning the placement of certain elements like H, He, La and Ac in the periodic table.
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4947211.890845
Discussions on the compositionality of inferential roles concentrate on extralogical vocabulary. However, there are nontrivial problems concerning the compositionality of sentences formed by the standard constants of propositional logic. For example, is the inferential role of AB uniquely determined by those of A and B? And how is it determined? This paper investigates such questions. We also show that these issues raise matters of more significance than may prima facie appear.
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5265849.890855
laying down a program for this study. It is written for everyone who is curious about the world of symbols that surrounds us, in particular researchers and students in philosophy, history, cognitive science, and mathematics education. The main characteristics of mathematical notations are introduced and discussed in relation to the intended subject matter, the language in which the notations are verbalized, the cognitive resources needed for learning and understanding them, the tasks that they are used for, their material basis, and the historical context in which they are situated. Specific criteria for the design and assessment of notations are discussed, as well as ontological, epistemological, and methodological questions that arise from the study of mathematical notations and of their use in mathematical practice.
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5348592.890868
While the traditional conception of inductive logic is Carnapian, I develop a Peircean alternative and use it to unify formal learning theory, statistics, and a significant part of machine learning: supervised learning. Some crucial standards for evaluating non-deductive inferences have been assumed separately in those areas, but can actually be justified by a unifying principle.
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5402790.890878
Incurvati and Schlöder (Journal of Philosophical Logic, 51(6), 1549–1582, 2022) have recently proposed to define supervaluationist logic in a multilateral framework, and claimed that this defuses well-known objections concerning supervaluationism’s apparent departures from classical logic. However, we note that the unconventional multilateral syntax prevents a straightforward comparison of inference rules of different levels, across multi- and unilateral languages. This leaves it unclear how the supervaluationist multilateral logics actually relate to classical logic, and raises questions about Incurvati and Schlöder’s response to the objections. We overcome the obstacle, by developing a general method for comparisons of strength between multi-and unilateral logics. We apply it to establish precisely on which inferential levels the supervaluationist multilateral logics defined by Incurvati and Schlöder are classical. Furthermore, we prove general limits on how classical a multilateral logic can be while remaining supervaluationistically acceptable. Multilateral supervaluationism leads to sentential logic being classical on the levels of theorems and regular inferences, but necessarily strictly weaker on meta- and higher-levels, while in a first-order language with identity, even some classical theorems and inferences must be forfeited. Moreover, the results allow us to fill in the gaps of Incurvati and Schlöder’s strategy for defusing the relevant objections.