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2155.231823
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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114363.231962
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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117469.231987
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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118952.232
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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437590.232011
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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520333.232026
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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574531.232035
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.
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636660.232045
I would like to begin this review by stating that this is an absolutely wonderful book that is full of gems about the elements and the periodic table. In my own 2007 book on the periodic table I concluded that we should perhaps think of the variety of tables that have appeared as spanning a spectrum running from the most abstract and ‘perfect’ tables such as Janet’s left-step table representation, to the unruly tables that emphasize the uniqueness of elements. To illustrate the latter category, I featured an image of Rayner-Canham’s table that is also the table shown on the front cover of his new book now under review. Rayner Canham’s book is all about the individuality of elements and how so many of the commonly held trends in the periodic table are far more complicated than we normally acknowledge.
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636680.232057
In this paper, we introduce a concept of non-dependence of variables in formulas. A formula in first-order logic is non-dependent of a variable if the truth value of this formula does not depend on the value of that variable. This variable non-dependence can be subject to constraints on the value of some variables which appear in the formula, these constraints are expressed by another first-order formula. After investigating its basic properties, we apply this concept to simplify convoluted formulas by bringing out and discarding redundant nested quantifiers. Such convoluted formulas typically appear when one uses a translation function interpreting a theory into another.
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867436.232068
This paper introduces a digital method for analyzing propositional logical equivalences. It transforms the theorem-proof method from the complex statement-derivation method to a simple number-comparison method. By applying the digital calculation method and the expression-number lookup table, we can quickly and directly discover and prove logical equivalences based on the identical numbers, no additional operations are needed. This approach demonstrates significant advantages over the conventional methods in terms of simplicity and efficiency.
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976506.23208
It has been a long day and you are making your way through a paper related to your work. You suddenly come across the following remark: “. . . since ? and ? are eigenvectors of ? with distinct eigenvalues, they are linearly independent.” Wait—how does the proof go? You should really know this. Here ? and ? are nonzero elements of a vector space ? and ? ∶ ? → ? is a linear map. You force yourself to pick up a pen and write down the following argument: Let ?(?) = ?? and ?(?) = ?? with ? ≠ ?. Suppose ?? + ?? = 0. Applying ? and using linearity, we have ??? + ??? = 0. Multiplying the original equation by ?, we have ??? + ??? = 0. Subtracting the two yields (? − ?)?? = 0 and since ? − ? and ? are nonzero, we have ? = 0. The corresponding argument with ? and ? swapped yields ? = 0, so the only linear combination of ? and ? that yields is the trivial one.
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1038694.23209
This paper develops the model theory of normal modal logics based on partial “possibilities” instead of total “worlds,” following Humber-stone [1981] instead of Kripke [1963]. Possibility semantics can be seen as extending to modal logic the semantics for classical logic used in weak forcing in set theory, or as semanticizing a negative translation of classical modal logic into intuitionistic modal logic. Thus, possibility frames are based on posets with accessibility relations, like intuitionistic modal frames, but with the constraint that the interpretation of every formula is a regular open set in the Alexandrov topology on the poset. The standard world frames for modal logic are the special case of possibility frames wherein the poset is discrete. The analogues of classical Kripke frames, i.e., full world frames, are full possibility frames, in which propositional variables may be interpreted as any regular open sets. We develop the beginnings of duality theory, definability and correspondence theory, and completeness theory for possibility frames. The duality theory, relating possibility frames to Boolean algebras with operators (BAOs), shows the way in which full possibility frames are a generalization of Kripke frames. Whereas Thomason [1975a] Previous versions of this article circulated online as the working papers Holliday 2015, Holliday 2016, and Holliday 2018. The present version updates Holliday 2018 based on the review process for The Australasian Journal of Logic.
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1041998.2321
We consider the following question: how close to the ancestral root of a phylogenetic tree is the most recent common ancestor of k species randomly sampled from the tips of the tree? For trees having shapes predicted by the Yule–Harding model, it is known that the most recent common ancestor is likely to be close to (or equal to) the root of the full tree, even as n becomes large (for k fixed). However, this result does not extend to models of tree shape that more closely describe phylogenies encountered in evolutionary biology. We investigate the impact of tree shape (via the Aldous β−splitting model) to predict the number of edges that separate the most recent common ancestor of a random sample of k tip species and the root of the parent tree they are sampled from. Both exact and asymptotic results are presented. We also briefly consider a variation of the process in which a random number of tip species are sampled.
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1155858.232112
We rigorously describe the relation in which a credence function should stand to a set of chance functions in order for these to be compatible in the way mandated by the Principal Principle. This resolves an apparent contradiction in the literature, by means of providing a formal way of combining credences with modest chance functions so that the latter indeed serve as guides for the former. Along the way we note some problematic consequences of taking admissibility to imply requirements involving probabilistic independence. We also argue, contra [12], that the Principal Principle does not imply the Principal of Indifference.
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1199054.232123
We study Doob’s Consistency Theorem and Freedman’s Inconsistency Theorem from the vantage point of computable probability and algorithmic randomness. We show that the Schnorr random elements of the parameter space are computably consistent, when there is a map from the sample space to the parameter space satisfying many of the same properties as limiting relative frequencies. We show that the generic inconsistency in Freedman’s Theorem is effectively generic, which implies the existence of computable parameters which are not computably consistent. Taken together, this work provides a computability-theoretic solution to Diaconis and Freedman’s problem of “know[ing] for which [parameters] θ the rule [Bayes’ rule] is consistent” ([DF86, 4]), and it strengthens recent similar results of Takahashi [Tak23] on Martin-Lof randomness in Cantor space.
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1601779.232133
Discussive logic was introduced by Ja´skowski as a logic of discussion. In this note we show that some natural translation-based formalizations of discussive logic in modal logic do not yield a paraconsistent logic but rather classical logic. Some alternative modal formalizations of discussive logic that avoid the collapse into classical logic are put forward.
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1601864.232144
Stanisław Ja´skowski is known to be one of the modern founders of paraconsistent logic, together with Newton C. A. da Costa. The most important contribution of Ja´skowski is that he clearly distinguished two notions for a theory, namely a theory being contradictory (or inconsistent in [18]) and a theory being trivial (or overfilled in [18]). In addition to this distinction, he also presented a system of paraconsistent logic known as D2 which is often referred to as discursive logic or discussive logic (cf. [18, 19]). In this article, the disjunction-free fragment of Ja´skowski’s discussive logic is shown to be complete with respect to three- and four-valued semantics. Note here that D2 is known to be not complete with respect to any finitely many-valued semantics, which is proved by Jerzy Kotas in [20]. As a byproduct of the main result, a simple axiomatization of the disjunction-free fragment of Ja´skowski’s discussive logic in the language of classical logic is obtained. For the problem of axiomatization of D2, see [24].
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1674808.232155
The relative virtues of 8- and 18-column periodic tables are discussed, followed by a brief mention of a 32-column table. Next, the left-step periodic table, as first introduced by Janet, is presented, as are the various attractive features of this representation. The advantages include what is termed here as the regularization of atomic number triads and a better rationalization of first-member anomalies. The distinction between simple substance and element is also explained as is the significance of this issue to the left-step table. Finally, I respond to some recent criticisms of previous work that I have published on atomic number triads of elements. It is becoming increasingly acknowledged that the discovery of the periodic table took place at the hands of at least six individuals working independently in different parts of the world (Scerri, A Tale of Seven Scientists, Oxford University Press, New York, 2016). In the intervening 150 or so years since the most well known of these tables were published, by Dmitri Mendeleev, at least 1000 periodic systems have appeared either in print form (Van Spronsen, The Periodic System of Chemical Elements. A History of the First Hundred Years, Elsevier, New York, 1969; Mazurs, Graphic Representations of the Periodic System during One Hundred Years, University Alabama Press, Alabama, 1974) or more recently on the Internet (Leach, https:// www. meta- synth esis. com/ webbo ok/ 35_ pt/ pt_ datab ase. php).
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1712867.232173
This paper examines two approaches to presuppositions: one viewing them as inferences projecting from sentences under negation and other logical operators, and another defining them as admittance conditions of utterances. Neither approach fully accounts for the ‘proviso problem’, which arises when a sentence’s presuppositional inferences are logically stronger than its necessary admittance conditions. To address this challenge, we propose a calculus of a trivalent logic that formally distinguishes between admittance and projection, extending Karttunen’s dynamic, logical form-based analysis. The resulting framework enables a simple pragmatic strategy: presuppositional conclusions are accommodated unless overridden by a contextually likelier admittance condition. We provide evidence that this approach is empirically superior to methods that address the proviso problem using pragmatic strengthening.
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1775218.232185
Hotze is known now for his work on tense, and on the vagaries of pronouns, but I got to know him when he was working on questions. His work on the semantics of wh-questions — a lot of it with Sigrid Beck — heavily influenced my thinking about the syntax of movement. I blame him for my resulting, years-long, obsession with multidominance. His work was the first step in a long line of interesting work on the syntax and semantics of wh-questions that continues today. The immediate predecessor to this work was Hotze’s equally important dissertation: one of the first attempts to explain an island condition entirely from its semantics. It remains an important role model for the contemporary work on the semantics of islands, and opened my eyes to the wider possibilities of finding the source of islands. Thank you Hotze for starting me on a journey that has dominated my research life. But the reason I’m contributing to your volume is even more personal: it’s because the other thing I learned when I got to know you is how much I like you. In this note, I’ll sketch a few facts about wh-movement that expand on the view in Beck and Rullman (1998) and Rullmann and Beck (1998) that wh-phrases are interpreted in their underlying position, no matter where they show up in the surface representation. In addition to the semantic reasons for this conclusion, there are straightforward facts about anaphora that animate this view. A famous kind of example of this is (1).
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1775338.232196
I will look at the idea that personal pronouns and reflexives compete to express certain meanings. I’m going to ignore plurals (because the semantics is too hard) (3) and I’m going to adopt the view that reflexives can be divided into two classes: one in which the reflexive expresses a local anaphoric dependency, and another in which reflexives express other relations (“emphatic,” logophoric, etc.). (See Pollard and Sag 1992.) It is only the first class of reflexives that figure in my presentation.
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1869243.232207
I explore an unorthodox perspective on the logical foundations of English on which speakers apply their logical competence by building and composing alternative sets, or ‘menus’, of entities or states throughout the grammar. logical connectives are ‘menu constructors’: The conjunction is a collective operation for putting combinations of items ‘on’ a menu, disjunction contributes nondeterminism or choice between items, while negation renders items ‘off menu’ by introducing negative entities or states. The system allows for determiner phrases to be interpreted uniformly in a lower type as menus compiled of positive, negative, or hybrid entities, rather than in the higher-order type of generalized quantifiers. Through a new compositional method, the negation contributed by a non-positive entity is able to pass through a semantic derivation in a well-behaved manner. This approach enables a “non-Boolean” collective treatment of sentences involving determiner phrase conjunctions with non-upward entailing conjuncts, which have previously been considered one of the toughest challenges for the collective theory.
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1869313.232219
This paper presents a uniform analysis of free choice constructions in English that incorporates a mechanism of arbitrary variability directly into their meaning. I propose that speakers interpret the values of certain variables or discourse referents as ‘fungible’, such that they could equally have taken any other value within an appropriate range. The semantics tracks this fungibility or arbitrariness, which can project to the sentential level and generate free choice readings with conjunctive or universal force.
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2014413.23223
It is uncontroversial that definite plurals in natural language stand for pluralities and permit predicates that are inherently distributive as in (1a) as well as predicates that can apply both collectively or distributively, as in (1b): (1) a. The stones are grey. b. The stones are heavy.
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2058708.232241
This paper presents a novel approach: using a digital calculation method for propositional logical reasoning. The paper demonstrates how to discover the primitive numbers and the digital calculation formulas by analyzing the truth tables. Then it illustrates how to calculate and compare the truth values of various expressions by using the digital calculation method. As an enhanced alternative to existing approaches, the proposed method transforms the statement-based or table-based reasoning into number-based reasoning. Thereby, it eliminates the need for using truth tables, and obviates the need for applying theorems, rewriting statements, and changing symbols. It provides a more streamlined solution for a single reasoning, while demonstrating more efficiency for multiple reasonings in long-term use. It is suitable for manual calculation, large-scale computation, AI and automated reasoning.
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2102446.232254
How the semantic significance of numerical discourse gets determined is a metasemantic issue par excellence. At the sub-sentential level, the issue is riddled with difficulties due to the contested metaphysical status of the subject matter of numerical discourse, i.e. numbers and numerical properties and relations. I propose to set those difficulties aside and focus instead on the sentential level, specifically, on obvious affinities between whole numerical and non-numerical sentences and how their significance is determined. From such a perspective, Frege’s 1884 construction of number, while famously mathematically untenable, fares better than other approaches in the philosophy of mathematics. Despite the work’s foundational untenability, it is metasemantically superior to extant alternatives.
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2808576.232266
The article begins with a response to a recent contribution by Jensen, in which he has criticized several aspects of the use of triads of elements, including Döbereiner’s original introduction of the concept and the modern use of atomic number triads by some authors including myself. Such triads are groups of three elements, one of which has approximately the average atomic weight of the other two elements, as well as having intermediate chemical reactivity. I also examine Jensen’s attempted reconstruction Mendeleev’s use of triads in predicting the atomic weights of three hitherto unknown elements, that were subsequently named gallium, germanium and scandium. The present article then considers the use of atomic number triads, in conjunction with the phenomenon of first member anomaly, in order to offer support for Janet’s left-step periodic table, in which helium is relocated into group 2 of the table. Such a table features triads in which the 2nd and third elements of each group, without fail, fall into periods of equal length, a feature that is absent in the conventional 18-column or the conventional 32-column table. The dual sense of the term element, which is the source of much discussion in the philosophy of chemistry, is alluded to in further support of such a relocation of helium that may at first appear to contradict chemical intuition.
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2906461.232277
We consider a model of case-based planning, where a position is a vector of numbers, and a case is an edge in the directed graph of positions. The planner generates new plans by using cases that are similar to those she has observed in the past. In the benchmark model presented here, similarity is de…ned by equality of differences (between the target and the source position). We prove a complexity result that shows why planning requires imagination and is not easily done algorithmically. We put this result in the context of learning and expertise in case-based models, distinguishing among information, insight, and imagination.
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3684407.232289
We present a new uniform method for studying modal companions of superintuitionistic rule systems and related notions, based on the machinery of stable canonical rules. Using this method, we obtain alternative proofs of the Blok-Esakia theorem and of the Dummett-Lemmon conjecture for rule systems. Since stable canonical rules may be developed for any rule system admitting filtration, our method generalizes smoothly to richer signatures. Using essentially the same argument, we obtain a proof of an analogue of the Blok-Esakia theorem for bi-superintuitionistic and tense rule systems, and of the Kuznetsov-Muravitsky isomorphism between rule systems extending the modal intuitionistic logic KM and modal rule systems extending the provability logic GL. In addition, our proof of the Dummett-Lemmon conjecture also generalizes to the bi-superintuitionistic and tense cases.
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3747578.2323
Recall the age-old problem of whether a sitting man can walk. Latin logicians introduced in the first half of the 12th century the distinction between a de sensu or de dicto compounded reading (i.e. it is possible that a man walks while sitting) and a de re divided understanding (i.e. it is possible for the man who is now sitting to stand up and walk) in order to prevent the assumption that if a subject has one predicate or another, it possibly has both together.