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208262.828339
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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299631.82841
Given the extreme importance that Wittgenstein attached to the
aesthetic dimension of life, it is in one sense surprising that he
wrote so little on the subject. It is true that we have the notes
assembled from his lectures on aesthetics given to a small group of
students in private rooms in Cambridge in the summer of 1938
(Wittgenstein 1966, henceforth LA) and we have G. E. Moore’s
record of some of Wittgenstein’s lectures in the period
1930–33 (Moore 1972). Of Wittgenstein’s own writings, we
find remarks on literature, poetry, architecture, the visual arts, and
especially music and the philosophy of culture more broadly scattered
throughout his writings on the philosophies of language, mind,
mathematics, and philosophical method, as well as in his more personal
notebooks; a number of these are collected in Culture and
Value (Wittgenstein 1980a).
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842267.828418
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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1112427.828431
An adequate theory of representation should distinguish between the structure of a representation and the structure of what it represents. I argue that the simplest sorts of transformers (the architecture that underlies most familiar Large Language Models) have only a very lightweight structure for their representations: insofar as they work with the structure of language, they represent it but do not use it. In addition to being interesting in its own right, this also shows how we may use high-level invariants at the computational level to place constraints on representational formats at the algorithmic level.
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1397901.828437
In his paper, Cian Dorr presents formal development of the view that higher-order entities such as properties, relations, and propositions act not just act as semantic values of predicates and sentences, but also as referents of referential noun phrases (NPs), generally considered singular terms. Dorr’s paper focuses on properties; thus, wise as in Socrates is wise is taken to stand for the very same entity, a property, as the NPs wisdom and the property of being wise. The view entails that lots of expressions now would apply to entities of different types: some, the, is interesting now apply to entities of the type of individuals as well as the type of properties. Moreover, quantifiers like everything will now be able to range over both individuals and properties, and in fact over both individuals and properties at once (Everything is interesting). These problems are dealt with by imposing type ambiguities on the relevant expressions and allowing quantifiers like everything to be specified for sum types, roughly, a disjunctive specification of types.
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1583697.828443
We argue that logicism, the thesis that mathematics is reducible to logic and analytic truths, is true. We do so by (a) developing a formal framework with comprehension and abstraction principles, (b) giving reasons for thinking that this framework is part of logic, (c) showing how the denotations for predicates and individual terms of an arbitrary mathematical theory can be viewed as logical objects that exist in the framework, and (d) showing how each theorem of a mathematical theory can be given an analytically true reading in the logical framework.
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2422340.828451
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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2422340.828456
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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2580780.828462
The philosopher Joseph S. Ullian died late last year. He is probably best-known for an introduction to epistemology co-authored with W. V. Quine, that is very much of its time. But what caught my eye in the obits was his reputation as a baseball fanatic. …
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2588585.828468
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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3675602.828473
At the start of the pandemic, Peter Singer and I argued that our top priority should be to learn more, fast. I feel similarly about AI, today. I’m far from an expert on the topic, so the main things I want to do in this post are to (i) share some resources that I’ve found helpful as a novice starting to learn more about the topic over the past couple months, and (ii) invite others to do likewise! …
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3905063.828479
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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4137282.828484
A. I guess because I'm exploring the format in some of my own writing. Q. A. It's not ready to show to anyone. In fact the project is more notional than actual—a few notes in a plain text file, which I peek at from time to time. …
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4185426.82849
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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4281309.828498
Many philosophers and linguists agree that there are two kinds of conversational implicatures: there are not only the well-known paradigm examples of conversational implicatures that are not entailed by the sentences that are used to bring them about; there are also less-often discussed conversational implicatures that are entailed by the sentences in question. In this paper, I take a closer look by examining classical candidates as well as novel contenders for entailed conversational implicatures. I argue that one might rightly classify some of these cases as conversational implicatures but show that doing so has so far unnoticed consequences.
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4359623.828504
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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4359644.82851
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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4759073.828515
Philosophers of language have tended to treat names merely as tools for talking about individuals, either directly or as part of a denoting phrase. We argue that names are every bit as much tools for tracking, maintaining, and performatively updating our positions in social space, as well as projecting a linguistic persona. This pushes us towards a revised picture of the meanings of names, one which incorporates what we shall call a ‘social sense’.
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5331474.828523
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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5331496.828529
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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5346854.828534
In Family Values, Harry Brighouse and Adam Smith ask whether children need parents. That inquiry seems a wild project, but then philosophers are supposed to question everything and follow the argument where it leads. …
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5391348.82854
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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5404895.828545
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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5440320.82855
Through a series of empirical studies involving native speakers of English, German, and Chinese, this paper reveals that the predicate “true” is inherently ambiguous in the empirical domain. Truth statements such as “It is true that Tom is at the party” seem to be ambivalent between two readings. On the first reading, the statement means “Reality is such that Tom is at the party.” On the second reading, the statement means “According to what X believes, Tom is at the party.” While there appear to exist some cross-cultural differences in the interpretation of the statements, the overall findings robustly indicate that “true” has multiple meanings in the realm of empirical matters.
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5440340.828555
Semantic features are components of concepts. In philosophy, there is a predominant focus on those features that are necessary (and jointly sufficient) for the application of a concept. Consequently, the method of cases has been the paradigm tool among philosophers, including experimental philosophers. However, whether a feature is salient is often far more important for cognitive processes like memory, categorization, recognition and even decision-making than whether it is necessary. The primary objective of this paper is to emphasize the significance of researching salient features of concepts. I thereby advocate the use of semantic feature production tasks, which not only enable researchers to determine whether a feature is salient, but also provide a complementary method for studying ordinary language use. I will discuss empirical data on three concepts, conspiracy theory, female/male professor, and life, to illustrate that semantic feature production tasks can help philosophers (a) identify those salient features that play a central role in our reasoning about and with concepts, (b) examine socially relevant stereotypes, and (c) investigate the structure of concepts.
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5564062.82856
Jeremy Kuhn, Carlo Geraci, Philippe Schlenker, Brent Strickland. Boundaries in space and time: Iconic biases across modalities. Cognition, 2021, 210, pp.104596. �10.1016/j.cognition.2021.104596�.
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5808744.828565
The nature of branching in the many-worlds interpretation (MWI) of quantum mechanics remains an open question, particularly regarding its locality and compatibility with special relativity. This paper challenges the conventional view that branching is either global or local, demonstrating instead that it is nonlocal for entangled systems. Through a new analysis of the EPR-Bohm experiment, I argue that global branching has several potential issues and can hardly be justified. At the same time, I argue that branching cannot be entirely local, as entangled particles exhibit simultaneous, spacelike-separated branching, manifesting an apparent action at a distance within individual worlds. However, while nonlocal branching suggests the emergence of a preferred Lorentz frame within each world, the multiverse as a whole retains full Lorentz invariance, ensuring no superluminal signaling. By refining the ontology of branching and resolving tensions between MWI and relativistic constraints, this analysis may help advance our understanding of quantum nonlocality and also strengthen MWI’s standing as a viable interpretation of quantum mechanics.
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6138103.82857
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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6265980.828576
This paper argues for a unified account of semantic and pragmatic infelicity. It is argued that an utterance is infelicitous when it communicates an inconsistent set of propositions, given the context. In cases of semantic infelicity the relevant utterance expresses a set of inconsistent propositions, whereas pragmatic infelicity is a matter of the utterance conflicting with contextual expectations or assumptions. We spell out this view within the standard framework according to which a central aim of communication is to update a body of information shared among the participants. We show that this account explains different kinds of infelicity for both declarative and non-declarative utterances. Further, the account is seen to make correct predictions for a range of cases involving irony, joking, and related non-assertoric utterances.
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6376132.828581
Until the ‘70s, the received view in the theory of reference was that the referent of a term was identified by certain descriptions that competent speakers associated with the term; for example, the referent of the proper name ‘Aristotle’ was determined by its association with a description like ‘the pupil of Plato and teacher of Alexander the Great’; the reference of the natural kind term ‘tiger’, by a description like ‘large feline with yellow and black stripes and a white belly’. But then came the revolution in the theory of reference, stemming particularly from the works of Kripke (1980) and Putnam (1975). It was argued that this “Description Theory” was fundamentally wrong for many terms, including ‘Aristotle’ and ‘tiger’. “Ignorance and error” arguments were particularly influential. People are often too ignorant to supply descriptions that would uniquely identify the referents of their terms. Most of us refer successfully with ‘elm’, but could not come close to describing those trees well enough to distinguish them from other trees like beeches. Speakers can also associate erroneous descriptions with a term; some who use ‘Einstein’ to refer successfully to the famous physicist wrongly think he invented the atomic bomb.