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35416.754077
In a recent paper, S. Gao has claimed that, under the assumption that the initial state of the universe is a pure quantum state, only the many worlds interpretation can account for the observed arrow of time. We show that his argument is untenable and that if endorsed it potentially leads to undermine the search for a scientific explanation of certain phenomena.
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190797.754244
In my previous post, I argued against divine desire versions of divine command theory. Reflecting on that post, I saw that there is a simple variant of divine desire that helps with some of the problems in that post. …
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345400.754279
In this article, we discuss a simple argument that modal metaphysics is misconceived, and responses to it. Unlike Quine’s, this argument begins with the banal observation that there are different candidate interpretations of the predicate ‘could have been the case’. This is analogous to the observation that there are different candidate interpretations of the predicate ‘is a member of’. The argument then infers that the search for metaphysical necessities is misguided in much the way the ‘set-theoretic pluralist’ (Hamkins and Clarke-Doane [2017] claims that the search for the true axioms of set theory is. We show that the obvious responses to this argument fail.
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634334.754296
Call a quantifier ‘unrestricted’ if it ranges over absolutely all objects. Arguably, unrestricted quantification is often presupposed in philosophical inquiry. However, developing a semantic theory that vindicates unrestricted quantification proves rather difficult, at least as long as we formulate our semantic theory within a classical first-order language. It has been argued that using a type theory as framework for our semantic theory provides a resolution of this problem, at least if a broadly Fregean interpretation of type theory is assumed. However, the intelligibility of this interpretation has been questioned. In this paper I introduce a type-free theory of properties that can also be used to vindicate unrestricted quantification. This alternative emerges very naturally by reflecting on the features on which the type-theoretic solution of the problem of unrestricted quantification relies. Although this alternative theory is formulated in a non-classical logic, it preserves the deductive strength of classical strict type theory in a natural way. The ideas developed in this paper make crucial use of Russell’s notion of range of significance.
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865593.754311
In this article, we provide three generators of propositional formulae for arbitrary languages, which uniformly sample three different formulae spaces. They take the same three parameters as input, namely, a desired depth, a set of atomics and a set of logical constants (with specified arities). The first generator returns formulae of exactly the given depth, using all or some of the propositional letters. The second does the same but samples up-to the given depth. The third generator outputs formulae with exactly the desired depth and all the atomics in the set. To make the generators uniform (i.e. to make them return every formula in their space with the same probability), we will prove various cardinality results about those spaces.
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1098388.754325
We propose a model of incomplete twofold multiprior preferences, in which an act f is ranked above an act g only when f provides higher utility in a worst-case scenario than what g provides in a best-case scenario. The model explains failures of contingent reasoning, captured through a weakening of the state-by-state monotonicity (or dominance) axiom. Our model gives rise to rich comparative statics results, as well as extension exercises, and connections to choice theory. We present an application to second-price auctions.
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1435216.754339
Any intermediate propositional logic (i.e., a logic including intuitionistic logic and contained in classical logic) can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert’s ε-calculus. The first and second ε-theorems for classical logic establish conservativity of the ε-calculus over its classical base logic. It is well known that the second ε-theorem fails for the intuitionistic ε-calculus, as prenexation is impossible. The paper investigates the effect of adding critical ε- and τ -formulas and using the translation of quantifiers into ε- and τ -terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate ετ -calculi. The “extended” first ε-theorem holds if the base logic is finite-valued Godel-Dummett logic, fails otherwise, but holds for certain provable formulas in infinite-valued Godel logic. The second ε-theorem also holds for finite-valued first-order Godel logics. The methods used to prove the extended first ε-theorem for infinite-valued Godel logic suggest applications to theories of arithmetic.
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1628006.754367
This book is an introduction to the logic of partitions on a set as well as the (quantum) logic of partitions (direct-sum decompositions or DSDs) on a vector space. Partitions of a set are categorically dual to subsets of a set. Thus the logic of partitions is, in that sense, the dual to the Boolean logic of subsets (usually presented as the special case of propositional logic). Since partitions can be seen as the inverse image partitions of random variables or numerical attributes (without the actual values but retaining the information as to when the values are the same or different), partition logic is the logic of random variables or numerical attributes (abstracted from the actual values). On the lattice of partitions of an arbitrary unstructured set, there is a rich algebraic structure of dual operations of implication and co-implication–resembling a non-distributive version of Heyting and co-Heyting algebras. Subsets linearize to subspaces of a vector space and the usual quantum logic is the logic of the (closed) subspaces of the Hilbert spaces used in quantum mechanics (QM). Set partitions linearize to DSDs of a vector space so the logic of partitions linearizes to the logic of DSDs that can then be specialized to the Hilbert spaces of QM. Since each diagonalizable linear operator, e.g., the observables of QM, on a vector operator determines a DSD of eigenspaces so the quantum logic of DSDs is the logic of observables (abstracted from the actual eigenvalues).
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1737303.754385
It is widely accepted that necessity comes in different varieties, often called ‘kinds’: metaphysical necessity, logical necessity, natural necessity, conceptual necessity, moral necessity, to name but a few – and the same goes for the varieties of possibility. What is usually not fully appreciated, however, is that modal variety is not simply ‘unidimensional’: it does not only involve one main variable – kind, whose values are the particular kinds of necessity. Rather, I argue, it is ‘bidimensional’, involving two distinct variables – domain and strength. In the first part of the paper, I introduce and develop the proposed bidimensional picture of modal variety, defending it against the common, unidimensional one. In the second part, I consider how the main available accounts of necessities and their relations rely, at least to a significant extent, on the latter picture, pointing out important limitations that they face as a result. I also show how, accordingly, alternative accounts based on a clear and systematic distinction between domain and strength would overcome those limitations. I conclude that, beyond the particular bidimensionalist view defended, our understanding of the modal realm may benefit from more direct debate on whether and how it is multidimensional.
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1804064.754401
There are two dominant approaches to quantification: the Fregean and the Tarskian. While the Tarskian approach is standard and familiar, deep conceptual objections have been pressed against its employment of variables as genuine syntactic and semantic units. Because they do not explicitly rely on variables, Fregean approaches are held to avoid these worries. The apparent result is that the Fregean can deliver something that the Tarskian is unable to, namely a compositional semantic treatment of quantification centered on truth and reference. We argue that the Fregean approach faces the same choice: abandon compositionality or abandon the centrality of truth and reference to semantic theory. Indeed, we argue that developing a fully compositional semantics in the tradition of Frege leads to a typographic variant of the most radical of Tarskian views: variabilism, the view that names should be modeled as Tarskian variables. We conclude with the consequences of this result for Frege’s distinction between sense and reference.
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2409863.754415
Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability. I then consider whether diagrams can be essential to the proofs in which they appear.
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2929509.754429
Part 1 introduced the ‘binary octahedral group’. This time I just want to show you some more pictures related to this group. I’ll give just enough explanation to hint at what’s going on. For more details, check out this webpage:
• Greg Egan, Symmetries and the 24-cell. …
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2973400.754444
In this paper, I show how to incorporate insights from the model-theoretic semantics for negation (insights due the late J. Michael Dunn [4], among others [1, 12]), into a proof-first understanding of the semantics of negation. I then discuss the different ways a logical pluralist may understand the underlying accounts of proofs and their significance.
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3103423.754458
A conditional logic is incompatible with this thesis if we can derive a contradiction, or some dire restriction on P r, from the assumption that P r respects the logic (i.e. assigns probability 1 to every theorem of the logic) and is subject to Stalnaker’s thesis. Some conditional logics are incompatible with Stalnaker’s thesis – indeed, Stalnaker’s own preferred logic of conditionals is incompatible with Stalnaker’s thesis (see Stalnaker [?], Hajek and Hall [?].) Stalnaker himself rejected the thesis and not the logic; in this paper we investigate the alternative hypothesis. Namely: Which conditional logics are compatible with Stalnaker’s thesis?
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3110702.754471
Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between expressive power and meta-theoretic properties when comparing first-order and second-order logic.
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3590684.754485
Logic is usually presented as a tool of rational inquiry; however, many logicians in fact treat logic so that it does not serve us, but rather governs us – as rational beings we are subordinated to the logical laws we aspire to disclose. We denote the view that logic primarily serves us as logica serviens, while denoting the thesis that it primarily governs our reasoning as logica dominans. We argue that treating logic as logica dominans is misguided, for it leads to the idea of a “genuine” logic within a “genuine” language. Instead of this, we offer a naturalistic picture, according to which the only languages that exist are the natural languages and the artificial languages logicians have built. There is, we argue, no language beyond these, especially none that would be a wholesome vehicle of reasoning like the natural languages and yet be transparently rigorous like the artificial ones. Logic is a matter of using the artificial languages as idealized models of the natural ones, whereby we pinpoint the laws of logic by means of zooming in on a reflective equilibrium.
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3626023.754499
In this paper I apply the concept of inter-Model Inconsistency in Set Theory (MIST), introduced by Antos, to select positions in the current universe-multiverse debate in philosophy of set theory: I reinterpret H. Woodin’s Ultimate L, J. D. Hamkins’ multiverse, S.-D. Friedman’s hyperuniverse and the algebraic multiverse as normative strategies to deal with the situation of de facto inconsistency toleration in set theory as described by MIST. In particular, my aim is to situate these positions on the spectrum from inconsistency avoidance to inconsistency toleration. By doing so, I connect a debate in philosophy of set theory with a debate in philosophy of science about the role of inconsistencies in the natural sciences. While there are important differences, like the lack of threatening explosive inferences, I show how specific philosophical positions in the philosophy of set theory can be interpreted as reactions to a state of inconsistency similar to analogous reactions studied in the philosophy of science literature. My hope is that this transfer operation from philosophy of science to mathematics sheds a new light on the current discussion in philosophy of set theory; and that it can help to bring philosophy of mathematics and philosophy of science closer together.
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3626065.754513
According to Augustine, abstract objects are ideas in the Mind of God. Because numbers are a type of abstract object, it would follow that numbers are ideas in the Mind of God. Let us call such a view the Augustinian View of Numbers (AVN). In this paper, I present a formal theory for AVN. The theory stems from the symmetry conception of God as it appears in Studtmann (2021). I show that Robinson’s Arithmetic, Q, can be interpreted by the theory in Studtmann’s paper. The interpretation is made possible by identifying the set of natural numbers with God, 0 with Being, and the successor function with the essence function. The resulting theory can then be augmented to include Peano Arithmetic by adding a set-theoretic version of induction and a comprehension schema restricted to arithmetically definable properties. In addition to these formal matters, the paper provides a characterization of the mind of God. According to the characterization, the Being essences that constitute God’s mind act as both numbers and representations – each has all the properties of some number and encodes all the properties of that number’s predecessor. The conception of God that emerges by the end of the discussion is a conception of an infinite, ineffable, axiologically and metaphysically ultimate entity that contains objects that not only serve as numbers but also encode information about each other.
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3717112.754526
Alternative set theory was created by the Czech mathematician Petr Vopěnka in 1979 as an alternative to Cantor’s set theory. Vopěnka criticised Cantor’s approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vopěnka grasps the phenomena of vagueness. Infinite sets are defined as sets containing proper semisets, i.e. vague parts of sets limited by the horizon. The new interpretation extends the field of applicability of mathematics and simultaneously indicates its limits. Compared to strict finitism and other attempts at a reduction of the infinite to the finite Vopěnka’s theory reverses the process: he models the finite in the infinite.
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3779777.75454
Two recent arguments draw startling and puzzling conclusions about relations and 2nd-order logic (2OL). The first argument concludes that 2nd-order quantifiers can’t be interpreted as ranging over relations. This is puzzling because 2OL is traditionally understood as the formalism for quantifying over relations, and so the conclusion seems to imply we can’t quantify over relations full stop. The second argument, which concludes that unwelcome consequences arise if relations and relatedness are analyzed rather than taken as primitive, utilizes premises that imply that 2OL faces the very same consequences. This is puzzling because relations and predication are taken as primitive in 2OL, and so the latter should be immune to the problems raised for an analysis. I consider these two arguments in light of a precise theory of relations. In particular, I show that object theory (Zalta 1983, 1988), which is an extension of 2OL with identity, provides systematic existence and identity conditions for relations, properties, and states of affairs that forestall the two arguments.
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4126566.754554
We present a revenge argument for non-reflexive theories of semantic notions – theories which restrict the rule of assumption, or (equivalently) initial sequents of the form . Our strategy follows the general template articulated in Murzi and Rossi [ ]: we proceed via the definition of a notion of paradoxicality for non-reflexive theories which in turn breeds paradoxes that standard non-reflexive theories are unable to block.
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4126647.754568
The notion of grounding is usually conceived as an objective and explanatory relation. It connects two relata if one—the ground—determines or explains the other—the consequence. In the contemporary literature on grounding, much effort has been devoted to logically characterize the formal aspects of grounding, but a major hard problem remains: defining suitable grounding principles for universal and existential formulae. Indeed, several grounding principles for quantified formulae have been proposed, but all of them are exposed to paradoxes in some very natural contexts of application. We introduce in this paper a first-order formal system that captures the notion of grounding and avoids the paradoxes in a novel and nontrivial way. The system we present formally develops Bolzano’s ideas on grounding by employing Hilbert’s ε-terms and an adapted version of Fine’s theory of arbitrary objects.
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4147437.754582
This paper argues, first, that the information problem poses a foundational challenge to mainstream semantics. It proposes, second, to address this problem by drawing on notions from Kit Fine’s essentialist framework. More specifically, it claims that the information problem can be avoided by strengthening standard truth theories, employing an operator expressing the notion of a relative constitutive semantic requirement. As a result, the paper proposes to construe semantic theories as theories of semantic requirements, and semantic knowledge as knowledge of such requirements.
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4150998.754596
This article aims to present a Free Dialogic Logic [FDL] as a general framework for hypothesis generation in the practice of modelling in science. Our proposal is based on the idea that the inferential function that models fulfil during the modelling process (surrogate reasoning) should be carried out without ontological commitments. The starting point to achieve our objective is that the scientific consideration of models without a target is a symptom that, on the one hand, the Applicability of Logic should be considered among the conditions of adequacy that should take into account all modeling process and, on the other, that the inferential apparatus at the base of the surrogate reasoning process must be rid of realistic assumptions that lead to erroneous conclusions. In this sense, we propose as an alternative an ontologically neutral inferential system in the perspective of dialogical pragmatism.
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4225890.75461
The transition metals are more complicated than lighter elements. Why? Because they’re the first whose electron wavefunctions are described by quadratic functions of and — not just linear or constant. …
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4329668.754624
I’ve been thinking about chemistry lately. I’m always amazed by how far we can get in the study of multi-electron atoms using ideas from the hydrogen atom, which has just one electron. If we ignore the electron’s spin and special relativity, and use just Schrödinger’s equation, the hydrogen atom is exactly solvable—and a key part of any thorough introduction to quantum mechanics. …
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4373961.754638
Glue Semantics (Glue) is a general framework for semantic composition and the syntax–semantics interface. The framework grew out of an interdisciplinary collaboration at the intersection of formal linguistics, formal logic, and computer science. Glue assumes a separate level of syntax; this can be any syntactic framework in which syntactic structures have heads. Glue uses a fragment of linear logic for semantic composition. General linear logic terms in Glue meaning constructors are instantiated relative to a syntactic parse. The separation of the logic of composition from structural syntax distinguishes Glue from other theories of semantic composition and the syntax–semantics interface. It allows Glue to capture semantic ambiguity, such as quantifier scope ambiguity, without necessarily positing an ambiguity in the syntactic structure. Glue is introduced here in relation to four of its key properties, which are used as organizing themes: resource-sensitive composition, flexible composition, autonomy of syntax, and syntax/semantics non-isomorphism.
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4557852.754652
We live in the information age. Claude Shannon, as the father of the information age, gave us a theory of communications that quanti…ed an "amount of information," but, as he pointed out, "no concept of information itself was de…ned." Logical entropy provides that de…nition. Logical entropy is the natural measure of the notion of information based on distinctions, differences, distinguishability, and diversity. It is the (normalized) quantitative measure of the distinctions of a partition on a set–just as the Boole-Laplace logical probability is the normalized quantitative measure of the elements of a subset of a set. And partitions and subsets are mathematically dual concepts–so the logic of partitions is dual in that sense to the usual Boolean logic of subsets, and hence the name "logical entropy." The logical entropy of a partition has a simple interpretation as the probability that a distinction or dit (elements in different blocks) is obtained in two independent draws from the underlying set. The Shannon entropy is shown to also be based on this notion of information-as-distinctions; it is the average minimum number of binary partitions (bits) that need to be joined to make all the same distinctions of the given partition. Hence all the concepts of simple, joint, conditional, and mutual logical entropy can be transformed into the corresponding concepts of Shannon entropy by a uniform non-linear dit-bit transform. And …nally logical entropy linearizes naturally to the corresponding quantum concept. The quantum logical entropy of an observable applied to a state is the probability that two different eigenvalues are obtained in two independent projective measurements of that observable on that state.
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4570404.754666
Kreisel’s conjecture is the statement: if, for all n ∈ N, PA `k steps ϕ(n), then PA ` ∀x.ϕ(x). For a theory of arithmetic T , given a recursive function h, T `≤h ϕ holds if there is a proof of ϕ in T whose code is at most h(#ϕ). This notion depends on the underlying coding. P T (x) is a provability predicate for `≤h in T . It is shown that there exists a sentence ϕ and a total recursive function h such that T `≤h PrT(pPrT (pϕq) → ϕq), but T `≤h ϕ, where PrT stands for the standard provability predicate in T . This statement is related to a conjecture by Montagna. Also variants and weakenings of Kreisel’s conjecture are studied. By use of reflexion principles, one can obtain a theory T Γ that extends T such that a version of Kreisel’s conjecture holds: given a recursive function h and ϕ(x) a Γ- formula (where Γ is an arbitrarily fixed class of formulas) such that, for all n ∈ N, T `≤h ϕ(n), then T Γ ` ∀x.ϕ(x). Derivability conditions are studied for a theory to statisfy the following implication: if T ` ∀x.P T (pϕ(x)q), then T ` ∀x.ϕ(x). This corresponds to an arithmetization of Kreisel’s conjecture. It is shown that, for certain theories, there exists a function h such that ` ⊆ `≤h.
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4705093.754685
The logician, author, and sleight-of-hand magician Raymond Smullyan is one of the most celebrated puzzle makers of all time. In 1982, in The Two-Year College Mathematics Journal, he presented a series of puzzles involving a fictional Inspector Craig of Scotland Yard and some asylums in France [4]. The series was later reprinted in his anthology, The Lady or the Tiger? and Other Logic Puzzles. In the asylums, every inhabitant is either a doctor or a patient, and every inhabitant is either sane or insane. Moreover, the sane inhabitants are entirely sane and the insane inhabitants are entirely insane, in the following sense: for any proposition ?, a sane inhabitant believes ? if and only if ? is true and an insane inhabitant believes ? if and only if ? is false. In the next section, we reproduce the last of these puzzles. Smullyan’s solutions show that the hypotheses lead to a shocking conclusion: in this particular asylum, all the patients are sane and all the doctors are insane. The puzzle is thus an homage to a story by Edgar Allen Poe, “The System of Doctor Tarr and Professor Fether,” which features an asylum in which the patients have imprisoned the doctors and assumed their roles.