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66302.880759
In this brief article I respond to Seifert’s recent views on the periodic law and the periodic table in connection with the views of philosophers regarding laws of nature. I argue that the author makes some factual as well as conceptual errors which are in conflict with some generally held views regarding the periodic law and the periodic table.
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927979.880843
A sentential connective ⋆ is said to be univocal, relative to a formal system F for a sentential logic containing ⋆ iff any two connectives ⋆1 and ⋆2 which satisfy the same F rules (and axioms) as ⋆ are such that similar formulas involving ⋆ and ⋆2 are inter-derivable in F . To be more precise, suppose is a unary connective. Then is univocal relative to F iff for any ⋆1 and ⋆2 satisfying the same principles as ⋆ in F, we have ⋆1α ⊢F 2 . And, if is binary, then is univocal relative to F iff for any 1 and 2 satisfying the same principles as ⋆ in F , we have α ⋆1 ⊢F α ⋆2 . In order to illustrate this definition of univocity, it is helpful to begin with a simple historical example.
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987730.880863
We set up a general framework for higher order probabilities. A simple HOP (Higher Order Probability space) consists of a probability space and an operation PR, such that, for every event A and every real closed interval A, PR(A ,A) is the event that A’s "true" probability lies in A. (The "true" probability can be construed here either as the objective probability, or the probability assigned by an expert, or the one assigned eventually in a fuller state of knowledge.) In a general HOP the operation PR has also an additional argument ranging over an ordered set of time-points, or, more generally, over a partially ordered set of stages; PR({A,t,A) is the event that A's probability at stage ¢ lies in 4. First we investigate simple HOPs and then the general ones. Assuming some intuitively justified axioms, we derive the most general structure of such a space. We also indicate various connections with modal logic.
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1162234.880878
Theories of ‘actual causation’ aim to provide an informative guide for assessing which events cause which others in circumstances where almost everything else is known: which other events occurred or did not occur, and how (if at all) the occurrence or non-occurrence of a particular event (regarded as values of a variable) can depend on speci c other events (or their absence), also regarded as values of variables. The ultimate aim is a theory that can agreeably be applied in causally fraught circumstances of technology, the law, and everyday life, where the identi cation of relevant features is not immediate and judgements of causation are entwined with judgements of moral or legal responsibility. Joseph Halpern’s Actual Causality is the latest and most extensive addition to this e ort, carried out in a tradition that holds causation to be di erence making.
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1335773.880899
Halvorson has proposed an intriguing example of a pair of theories whose categories are equivalent but which are not themselves definitionally equivalent. Moreover, it seems obvious that these theories are not equivalent in any intuitive sense. We offer a new topological proof that these theories are not definitionally equivalent. However, the underlying theorem for this claim has a converse that shows a surprising collection of theories, which are superficially similar to those in Halvorson’s example, turn out to be definitionally equivalent after all. This offers some new insight into what is going “wrong” in the Halvorson example.
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1682258.880913
We argue that it is neither necessary nor sufficient for a mathematical proof to have epistemic value that it be “correct”, in the sense of formalizable in a formal proof system. We then present a view on the relationship between mathematics and logic that clarifies the role of formal correctness in mathematics. Finally, we discuss the significance of these arguments for recent discussions about automated theorem provers and applications of AI to mathematics.
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2025566.880926
Some learning strategies that work well when computational considerations are abstracted away from become severely limiting when such considerations are taken into account. We illustrate this phenomenon for agents who attempt to extrapolate patterns in binary data streams chosen from among a countable family of possibilities. If computational constraints are ignored, then two strategies that will always work are learning by enumeration (enumerate the possibilities—in order of simplicity, say—then search for the one earliest in the ordering that agrees with your data and use it to predict the next data point) and Bayesian learning. But there are many types of computable data streams that, although they can be successfully extrapolated by computable agents, cannot be handled by any computable learner by enumeration. And while there is a sense in which Bayesian learning is a fully general strategy for computable learners, the ability to mimic powerful learners comes at a price for Bayesians: they cannot, in general, become highly confident of their predictions in the limit of large data sets and they cannot, in general, use priors that incorporate all relevant background knowledge.
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2116677.880946
The existence of non-standard models of first-order Peano-Arithmetic (PA) threatens to undermine the claim of the moderate mathematical realist that non-mysterious access to the natural number structure is possible on the basis of our best arithmetical theories. The move to logics stronger than FOL is denied to the moderate realist on the grounds that it merely shifts the indeterminacy “one level up” into the meta- theory by, illegitimately, assuming the determinacy of the notions needed to formulate such logics. This paper argues that the challenge can be met. We show how the quantifier “there are infinitely many” can be uniquely determined in a naturalistically acceptable fashion and thus be used in the formulation of a theory of arithmetic. We compare the approach pursued here with Field’s justification of the same device and the popular strategy of invoking a second-order formalism, and argue that it is more robust than either of the alternative proposals.
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2252571.880959
Completely positive trace-preserving maps S, also known as quantum channels, arise in quantum physics as a description of how the density operator ρ of a system changes in a given time interval, allowing not only for unitary evolution but arbitrary operations including measurements or other interaction with an environment. It is known that if the Hilbert space H that ρ acts on is finite-dimensional, then every S must have a fixed point, i.e., a density operator ρ with S(ρ ) = ρ . In infinite dimension, S need not have a fixed point in general. However, we prove here the existence of a fixed point under a certain additional assumption which is, roughly speaking, that S leaves invariant a certain set of density operators with bounded “cost” of preparation. The proof is an application of the Schauder-Tychonoff fixed point theorem. Our motivation for this question comes from a proposal of Deutsch for how to define quantum theory in a space-time with closed timelike curves; our result supports the viability of Deutsch’s proposal.
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2560570.880971
Unger famously argues that he doesn’t exist, by claiming a contradiction between three claims (I am quoting (1) and (2) verbatim, but simplifying (3)):
I exist. If I exist, then I consist of many cells, but a finite number. …
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2720402.880985
The original architects of the representational theory of measurement interpreted their formalism operationally and explicitly acknowledged that some aspects of their representations are conventional. We argue that the conventional elements of the representations afforded by the theory require careful scrutiny as one moves toward a more metaphysically robust interpretation by showing that there is a sense in which the very number system one uses to represent a physical quantity such as mass or length is conventional. This result undermines inferences which impute structure from the numerical representational structure to the quantity it is used to represent.
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2769545.881002
Last century, Michael Dummett argued that the principles of intuitionistic logic are semantically neutral, and that classical logic involves a distinctive commitment to realism. The ensuing debate over realism and anti-realism and intuitionistic logic has now receded from view. The situation is reversed in mathematics: constructive reasoning has become more popular in the 21st century with the rise of proof assistants based on constructive type theory. In this paper, I revisit Dummett’s concerns in the light of these developments, arguing that both constructive and classical reasoning are recognisable and coherent assertoric and inferential practices.
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2991535.881015
We investigate whether ordinary quantification over objects is an extensional phenomenon, or rather creates non-extensional contexts; each claim having been propounded by prominent philosophers. It turns out that the question only makes sense relative to a background theory of syntax and semantics (here called a grammar) that goes well beyond the inductive definition of formulas and the recursive definition of satisfaction. Two schemas for building quantificational grammars are developed, one that invariably constructs extensional grammars (in which quantification, in particular, thus behaves extensionally) and another that only generates non-extensional grammars (and in which quantification is responsible for the failure of extensionality). We then ask whether there are reasons to favor one of these grammar schemas over the other, and examine an argument according to which the proper formalization of deictic utterances requires adoption of non-extensional grammars.
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2991565.881032
Given any set E of expressions freely generated from a set of atoms by syntactic operations, there exist trivially compositional functions on E (to wit, the injective and the constant functions), but also plenty of non-trivially compositional functions. Here we show that within the space of non-injective functions (and so a fortiori within the space of non-injective and non-constant functions), compositional functions are not sufficiently abundant in order to generate the consequence relation of every propositional logic. Logical consequence relations thus impose substantive constraints on the existence of compositional functions when coupled with the condition of noninjectivity (though not without it). We ask how the apriori exclusion of injective functions from the search space might be justified, and we discuss the prospects of claims to the effect that any function can be “encoded” in a compositional one.
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3009846.881045
The 2021 Nobel Prize in Economics recognized a theory of causal inference that warrants more attention from philosophers. To this end, I design a tutorial on that theory for philosophers and develop a dialectic that connects to a traditional debate in philosophy: the Lewis-Stalnaker debate on Conditional Excluded Middle (CEM). I first defend CEM, presenting a new Quine-Putnam indispensability argument based on the Nobel-winning application of the Rubin causal model (the potential outcome framework). Then, I switch sides to challenge this argument, introducing an updated version of the Rubin causal model that preserves the successful application while dispensing with CEM.
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3009901.881058
This paper explores the artificial intelligence (AI) containment problem, specifically addressing the challenge of creating effective safeguards for artificial general intelligence (AGI) and superintelligence. I argue that complete control—defined as full predictability of AI actions and total adherence to safety requirements—is unattainable. The paper reviews five key constraints: incompleteness, indeterminacy, unverifiability, incomputability, and incorrigibility. These limitations are grounded in logical, philosophical, mathematical, and computational theories, such as Gödel’s incompleteness theorem and the halting problem, which collectively prove the impossibility of AI containment. I argue that instead of pursuing complete AI containment, resources should be allocated to risk management strategies that acknowledge AI’s unpredictability and prioritize adaptive oversight mechanisms.
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3234007.881071
Today I’d like to dig a little deeper into some ideas from Part 2. I’ve been talking about causal loop diagrams. Very roughly speaking, a causal loop diagram is a graph with labeled edges. I showed how to ‘pull back’ and ‘push forward’ these labels along maps of graphs. …
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3277972.881084
Bisimulations are standard in modal logic and, more generally, in the theory of state-transition systems. The quotient structure of a Kripke model with respect to the bisimulation relation is called a bisimulation contraction. The bisimulation contraction is a minimal model bisimilar to the original model, and hence, for (image-)finite models, a minimal model modally equivalent to the original. Similar definitions exist for bounded bisimulations (k-bisimulations) and bounded bisimulation contractions. Two finite models are k-bisimilar if and only if they are modally equivalent up to modal depth k. However, the quotient structure with respect to the k-bisimulation relation does not guarantee a minimal model preserving modal equivalence to depth k. In this paper, we remedy this asymmetry to standard bisimulations and provide a novel definition of bounded contractions called rooted k-contractions. We prove that rooted k-contractions preserve k-bisimilarity and are minimal with this property. Finally, we show that rooted k-contractions can be exponentially more succinct than standard k-contractions.
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3353296.881096
Truthmaker semantics is a non-classical logical framework that has recently garnered significant interest in philosophy, logic, and natural language semantics. It redefines the propositional connectives and gives rise to more fine-grained entailment relations than classical logic. In its model theory, truth is not determined with respect to possible worlds, but with respect to truthmakers, such as states or events. Unlike possible worlds, these truthmakers may be partial; they may be either coherent or incoherent; and they are understood to be exactly or wholly relevant to the truth of the sentences they verify. Truth-maker semantics generalizes collective, fusion-based theories of conjunction; alternative-based theories of disjunction; and nonstandard negation semantics. This article provides a gentle introduction to truthmaker semantics aimed at linguists; describes applications to various natural language phenomena such as imperatives, ignorance implicatures, and negative events; and discusses its similarities and differences to related frameworks such as event semantics, situation semantics, alternative semantics, and inquisitive semantics.
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3373703.881109
In Part 2, I explained some stuff you can do with graphs whose edges are labeled by elements of a rig. Remember, a rig is like a ring, but it might not have negatives. A great example is the boolean rig, whose elements are truth values:
The addition in this rig is ‘or’ and the multiplication is ‘and’. …
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3421110.881122
In the previous post, I showed that Goodman and Quine’s counting method fails for objects that have too much overlap. I think (though the technical parts here are more difficult) that the same is true for their definition of the ancestral or transitive closure of a relation. …
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3760348.881137
Generative artificial intelligence (AI) applications based on large language models have not enjoyed much success in symbolic processing and reasoning tasks, thus making them of little use in mathematical research. However, recently DeepMind’s AlphaProof and AlphaGeometry 2 applications have recently been reported to perform well in mathematical problem solving. These applications are hybrid systems combining large language models with rule-based systems, an approach sometimes called neuro-symbolic AI. In this paper, I present a scenario in which such systems are used in research mathematics, more precisely in theorem proving. In the most extreme case, such a system could be an autonomous automated theorem prover (AATP), with the potential of proving new humanly interesting theorems and even presenting team in research papers. The use of such AI applications would be transformative to mathematical practice and demand clear ethical guidelines. In addition to that scenario, I identify other, less radical, uses of generative AI in mathematical research. I analyse how guidelines set for ethical AI use in scientific research can be applied in the case of mathematics, arguing that while there are many similarities, there is also a need for mathematics-specific guidelines.
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3818060.881153
The universal conception of necessity says that necessary truth is truth in all possible worlds. This idea is well studied in the context of classical possible worlds models, and there its logic is S5. The universal conception of necessity is less well studied in models for non-classical logics. We will present some preliminary results on universal necessity on models for intuitionistic logic, first-degree entailment, and relevant logics. We will close by discussing a way in which universal necessity is a very classical concept.
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3818088.881166
A challenge for relevant logicians is to delimit their area of study. I propose and explore the definition of a relevant logic as a logic satisfying a variable-sharing property and closed under detachment and adjunction. This definition is, I argue, a good definition that captures many familiar logics and raises interesting new questions concerning relevant logics. As is familiar to readers of Entailment or Relevant Logics and Their Rivals, the motivations for relevant logics have a strong intuitive pull. The philosophical picture put forward by Anderson and Belnap (1975), for example, is compelling and has led to many fruitful developments. With some practice, one can develop a feel for what sorts of axioms or rules lead to violations of relevance in standard relevant logics. These sorts of intuitions only go so far, as some principles that lead to violations of relevance in stronger logics are compatible with it in weaker logics. There is a large number of relevant logics, but there is not much discussion of precise characterizations of the class of relevant logics.
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3933616.881178
The idea that life is to be understood in terms of information has strongly taken hold in recent decades. I discuss two attempts to carry this through mathematically. G. J. Chaitin, co-founder of algorithmic information theory, proposes an information-theoretic definition of life in terms of organized complexity (Chaitin 1990a and 1990b). More recently, William Dembski, Winston Ewart, and Robert Marks have attempted to formulate in information-theoretic terms Dembski’s concept of specified complexity, using a mathematically hybrid entity they term “algorithmic specified complexity” (Ewart, Dembski, and Marks 2013a, 2014, 2015a, 2015b), and Dembski and Ewart have reformulated this concept in their newly revised edition (Dembski and Ewart 2023) of Dembski’s The Design Inference (Dembski 1998). The aim in both cases is to mathematically distinguish informational properties of biological complexity, in contrast to simple order, on the one hand, and mere randomness on the other. Moreover, the respective mathematical strategies are the same: To take an informational measure and subtract out its randomness, leaving a remainder of organization (Chaitin) or specified complexity (Dembski et al.).
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3976251.881192
Linsky & Zalta (1994) argued that simplest quantified modal logic (SQML), with its fixed domain, can be given an actualist interpretation if the Barcan formula is interpreted to conditionally assert the existence of contingently nonconcrete objects. But SQML itself doesn’t require the existence of such objects; in interpretations of SQML in which there is only one possible world, there are no contingent objects, nonconcrete or otherwise. I defend an axiom for SQML that will provably (a) force the domain to have the relevant objects and thereby (b) force the existence of more than one possible world, thereby forestalling modal collapse. I show that the new axiom can be justified by describing the theorems that can be proved when it is added to SQML. I further justify the axiom by the reviewing the theorems the axiom allows us to prove when we assume object theory (‘OT’), in its latest incarnation, as a background framework. Finally, I consider the conclusions one can draw when we consider the new axiom in connection with actualism, as this view has been (re-)characterized in recent work.
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4003898.881214
I’m talking about ‘causal loop diagrams’, which are graph with edges labeled by ‘polarities’. Often the polarities are simply and signs, like here:
But polarities can be elements of any monoid, and last time I argued that things work even better if they’re elements of a rig, so you can not only multiply them but also add them. …
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4101703.8813
In Part 1 I explained ‘causal loop diagrams’, which are graphs with edges labeled by polarities. These are a way to express qualitatively, rather than quantitatively, how entities affect one another. For example, here’s how causal loop diagrams us say that alcoholism ‘tends to increase’ domestic violence:
We don’t need to specify any numbers, or even need to say what we mean by ‘tends to increase’, though that leads to the danger of using the term in a very loose way. …
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4173972.881322
This is a progress report on some joint work with Xiaoyan Li, Nathaniel Osgood and Evan Patterson. Together with collaborators we have been developing software for ‘system dynamics’ modelling, and applying it to epidemiology—though it has many other uses. …
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4534791.881344
I have elsewhere shown the consistency of the theory commonly called New Foundations or NF, originally proposed by W. v. O. Quine in his paper “New foundations for mathematical logic”. In this note, I review that original paper and may eventually review some other sources one might consult for information about this theory . Quine himself made some errors in this paper and later in his discussion of NF, and there are other characteristic difficulties that people have with this system which such a review might allow us to discuss.