We construct a machine that knows its own code, at the price of not knowing its own factivity.

We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern.

We argue that C. Darwin and more recently W. Hennig worked at times under the simplifying assumption of an eternal biosphere. So motivated, we explicitly consider the consequences which follow mathematically from this assumption, and the infinite graphs it leads to. This assumption admits certain clusters of organisms which have some ideal theoretical properties of species, shining some light onto the species problem. We prove a dualization of a law of T.A. Knight and C. Darwin, and sketch a decomposition result (...) involving the internodons of D. Kornet, J. Metz and H. Schellinx. A further goal of this paper is to respond to B. Sturmfels’ question, “Can biology lead to new theorems?”. (shrink)

Elementary patterns of resemblance notate ordinals up to the ordinal of Pi^1_1-CA_0. We provide ordinal multiplication and exponentiation algorithms using these notations.

Reinhardt’s conjecture, a formalization of the statement that a truthful knowing machine can know its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal.

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

Inspired by recent progress in multi-agent Reinforcement Learning (RL), in this work we examine the collective intelligent behaviour of theoretical universal agents by introducing a weighted mixture operation. Given a weighted set of agents, their weighted mixture is a new agent whose expected total reward in any environment is the corresponding weighted average of the original agents' expected total rewards in that environment. Thus, if RL agent intelligence is quantified in terms of performance across environments, the weighted mixture's intelligence is (...) the weighted average of the original agents' intelligences. This operation enables various interesting new theorems that shed light on the geometry of RL agent intelligence, namely: results about symmetries, convex agent-sets, and local extrema. We also show that any RL agent intelligence measure based on average performance across environments, subject to certain weak technical conditions, is identical (up to a constant factor) to performance within a single environment dependent on said intelligence measure. (shrink)

A fundamental question asked in modal logic is whether a given theory is consistent. But consistent with what? A typical way to address this question identifies a choice of background knowledge axioms (say, S4, D, etc.) and then shows the assumptions codified by the theory in question to be consistent with those background axioms. But determining the specific choice and division of background axioms is, at least sometimes, little more than tradition. This paper introduces generic theories for propositional modal logic (...) to address consistency results in a more robust way. As building blocks for background knowledge, generic theories provide a standard for categorical determinations of consistency. We argue that the results and methods of this paper help to elucidate problems in epistemology and enjoy sufficient scope and power to have purchase on problems bearing on modalities in judgement, inference, and decision making. (shrink)

We present a game mechanic called pseudo-visibility for games inhabited by non-player characters (NPCs) driven by reinforcement learning (RL). NPCs are incentivized to pretend they cannot see pseudo-visible players: the training environment simulates an NPC to determine how the NPC would act if the pseudo-visible player were invisible, and penalizes the NPC for acting differently. NPCs are thereby trained to selectively ignore pseudo-visible players, except when they judge that the reaction penalty is an acceptable tradeoff (e.g., a guard might accept (...) the penalty in order to protect a treasure because losing the treasure would hurt even more). We describe an RL agent transformation which allows RL agents that would not otherwise do so to perform some limited self-reflection to learn the training environments in question. (shrink)

Can an AGI create a more intelligent AGI? Under idealized assumptions, for a certain theoretical type of intelligence, our answer is: “Not without outside help”. This is a paper on the mathematical structure of AGI populations when parent AGIs create child AGIs. We argue that such populations satisfy a certain biological law. Motivated by observations of sexual reproduction in seemingly-asexual species, the Knight-Darwin Law states that it is impossible for one organism to asexually produce another, which asexually produces another, and (...) so on forever: that any sequence of organisms (each one a child of the previous) must contain occasional multi-parent organisms, or must terminate. By proving that a certain measure (arguably an intelligence measure) decreases when an idealized parent AGI single-handedly creates a child AGI, we argue that a similar Law holds for AGIs. (shrink)

Legg and Hutter, as well as subsequent authors, considered intelligent agents through the lens of interaction with reward-giving environments, attempting to assign numeric intelligence measures to such agents, with the guiding principle that a more intelligent agent should gain higher rewards from environments in some aggregate sense. In this paper, we consider a related question: rather than measure numeric intelligence of one Legg- Hutter agent, how can we compare the relative intelligence of two Legg-Hutter agents? We propose an elegant answer (...) based on the following insight: we can view Legg-Hutter agents as candidates in an election, whose voters are environments, letting each environment vote (via its rewards) which agent (if either) is more intelligent. This leads to an abstract family of comparators simple enough that we can prove some structural theorems about them. It is an open question whether these structural theorems apply to more practical intelligence measures. (shrink)

We define a notion of the intelligence level of an idealized mechanical knowing agent. This is motivated by efforts within artificial intelligence research to define real-number intelligence levels of compli- cated intelligent systems. Our agents are more idealized, which allows us to define a much simpler measure of intelligence level for them. In short, we define the intelligence level of a mechanical knowing agent to be the supremum of the computable ordinals that have codes the agent knows to be codes (...) of computable ordinals. We prove that if one agent knows certain things about another agent, then the former necessarily has a higher intelligence level than the latter. This allows our intelligence no- tion to serve as a stepping stone to obtain results which, by themselves, are not stated in terms of our intelligence notion (results of potential in- terest even to readers totally skeptical that our notion correctly captures intelligence). As an application, we argue that these results comprise evidence against the possibility of intelligence explosion (that is, the no- tion that sufficiently intelligent machines will eventually be capable of designing even more intelligent machines, which can then design even more intelligent machines, and so on). (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This is a paper about the general theory of measuring or estimating social intelligence via benchmarks. Hernández-Orallo and Dowe described a problem with certain proposed intelligence measures. The problem suggests that those intelligence measures might not accurately capture social intelligence. We argue that Hernández-Orallo and Dowe's problem is even more general than how they stated it, applying to many subdomains of AGI, not just the one subdomain in which they stated it. We then propose a solution. In our solution, instead (...) of using test-cases within the given AGI subdomain to estimate an AI's intelligence, one would use test-cases in an extended subdomain where test-cases have the ability to simulate the AI being tested. Surprisingly, AIs only designed for the original subdomain can be tested with test-cases in the extended subdomain anyway. By extending the subdomain in this way, we might avoid Hernández-Orallo and Dowe's problem. (shrink)

After generalizing the Archimedean property of real numbers in such a way as to make it adaptable to non-numeric structures, we demonstrate that the real numbers cannot be used to accurately measure non-Archimedean structures. We argue that, since an agent with Artificial General Intelligence (AGI) should have no problem engaging in tasks that inherently involve non-Archimedean rewards, and since traditional reinforcement learning rewards are real numbers, therefore traditional reinforcement learning probably will not lead to AGI. We indicate two possible ways (...) traditional reinforcement learning could be altered to remove this roadblock. (shrink)

A variation of Fitch’s paradox is given, where no special rules of inference are assumed, only axioms. These axioms follow from the familiar assumptions which involve rules of inference. We show (by constructing a model) that by allowing that possibly the knower doesn’t know his own soundness (while still requiring he be sound), Fitch’s paradox is avoided. Provided one is willing to admit that sound knowers may be ignorant of their own soundness, this might offer a way out of the (...) paradox. (shrink)

The Voluntary Simplicity Movement can be understood broadly as a diverse social movement made up of people who are resisting high consumption lifestyles and who are seeking, in various ways, a lower consumption but higher quality of life alternative. The central argument of this paper is that the Voluntary Simplicity Movement or something like it will almost certainly need to expand, organise, radicalise and politicise, if anything resembling a degrowth society is to emerge in law through democratic processes. In a (...) sentence, that is the 'grass-roots' or 'bottom up' theory of legal and political transformation that will be expounded and defended in this paper. The essential reasoning here is that legal, political and economic structures will never reflect a post-growth ethics of macro-economic sufficiency until a post-consumerist ethics of micro-economic sufficiency is embraced and mainstreamed at the cultural level. (shrink)

We provide an intuitive motivation for the hyperreal numbers via electoral axioms. We do so in the form of a Socratic dialogue, in which Protagoras suggests replacing big-oh complexity classes by real numbers, and Socrates asks some troubling questions about what would happen if one tried to do that. The dialogue is followed by an appendix containing additional commentary and a more formal proof.

Recognized as one of the earliest Jewish neo-Platonist writers, Isaac ben Solomon Israeli influenced Muslim, Jewish, and Christian scholars through the Middle Ages. A native of Egypt who wrote in Arabic, Israeli explored definitions of such terms as imagination, sense-perception, desire, love, creation, and “coming-to-be” in his writings. This classic volume contains English translations of Israeli’s philosophical writings, including the _Book of Definitions_, the _Book of Substances,_ and the _Book on Spirit and Soul_. Additionally, _Isaac Israeli_ features a biographical sketch (...) of the philosopher and extensive notes and comments on the texts, as well as a survey and appraisal of his philosophy. Restored to print for the first time in decades, _Isaac Israeli _will be essential reading for students and scholars of medieval philosophy and Jewish studies. (shrink)

Do we live in a computer simulation? I will present an argument that the results of a certain experiment constitute empirical evidence that we do not live in, at least, one type of simulation. The type of simulation ruled out is very specific. Perhaps that is the price one must pay to make any kind of Popperian progress.

One shortcoming of the chain rule is that it does not iterate: it gives the derivative of f(g(x)), but not (directly) the second or higher-order derivatives. We present iterated differentials and a version of the multivariable chain rule which iterates to any desired level of derivative. We first present this material informally, and later discuss how to make it rigorous (a discussion which touches on formal foundations of calculus). We also suggest a finite calculus chain rule (contrary to Graham, Knuth (...) and Patashnik's claim that "there's no corresponding chain rule of finite calculus"). (shrink)

We propose that, for the purpose of studying theoretical properties of the knowledge of an agent with Artificial General Intelligence (that is, the knowledge of an AGI), a pragmatic way to define such an agent’s knowledge (restricted to the language of Epistemic Arithmetic, or EA) is as follows. We declare an AGI to know an EA-statement φ if and only if that AGI would include φ in the resulting enumeration if that AGI were commanded: “Enumerate all the EA-sentences which you (...) know.” This definition is non-circular because an AGI, being capable of practical English communication, is capable of understanding the everyday English word “know” independently of how any philosopher formally defines knowledge; we elaborate further on the non-circularity of this circular-looking definition. This elegantly solves the problem that different AGIs may have different internal knowledge definitions and yet we want to study knowledge of AGIs in general, without having to study different AGIs separately just because they have separate internal knowledge definitions. Finally, we suggest how this definition of AGI knowledge can be used as a bridge which could allow the AGI research community to import certain abstract results about mechanical knowing agents from mathematical logic. (shrink)

I present an argument that for any computer-simulated civilization we design, the mathematical knowledge recorded by that civilization has one of two limitations. It is untrustworthy, or it is weaker than our own mathematical knowledge. This is paradoxical because it seems that nothing prevents us from building in all sorts of advantages for the inhabitants of said simulation.

In his dissertation, Wadge defined a notion of guessability on subsets of the Baire space and gave two characterizations of guessable sets. A set is guessable if and only if it is in the second ambiguous class, if and only if it is eventually annihilated by a certain remainder. We simplify this remainder and give a new proof of the latter equivalence. We then introduce a notion of guessing with an ordinal limit on how often one can change one’s mind. (...) We show that for every ordinal $\alpha$, a guessable set is annihilated by $\alpha$ applications of the simplified remainder if and only if it is guessable with fewer than $\alpha$ mind changes. We use guessability with fewer than $\alpha$ mind changes to give a semi-characterization of the Hausdorff difference hierarchy, and indicate how Wadge’s notion of guessability can be generalized to higher-order guessability, providing characterizations of ${\mathbf{\Delta}}^{0}_{\alpha}$ for all successor ordinals $\alpha\gt 1$. (shrink)

We consider an extended notion of reinforcement learning in which the environment can simulate the agent and base its outputs on the agent's hypothetical behavior. Since good performance usually requires paying attention to whatever things the environment's outputs are based on, we argue that for an agent to achieve on-average good performance across many such extended environments, it is necessary for the agent to self-reflect. Thus weighted-average performance over the space of all suitably well-behaved extended environments could be considered a (...) way of measuring how self-reflective an agent is. We give examples of extended environments and introduce a simple transformation which experimentally seems to increase some standard RL agents' performance in a certain type of extended environment. (shrink)

We extend first-order logic to include variadic function symbols, and prove a substitution lemma. Two applications are given: one to bounded quantifier elimination and one to the definability of certain Borel sets.

In their thought-provoking paper, Legg and Hutter consider a certain abstrac- tion of an intelligent agent, and define a universal intelligence measure, which assigns every such agent a numerical intelligence rating. We will briefly summarize Legg and Hutter’s paper, and then give a tongue-in-cheek argument that if one’s goal is to become more intelligent by cultivating music appreciation, then it is bet- ter to use classical music (such as Bach, Mozart, and Beethoven) than to use more recent pop music. The (...) same argument could be adapted to other media: books beat films, card games beat first-person shooters, parables beat dissertations, etc. We leave it to the reader to decide whether this argument tells us something about classical music, something about Legg-Hutter intelligence, or something about both. (shrink)

Can you find an xy-equation that, when graphed, writes itself on the plane? This idea became internet-famous when a Wikipedia article on Tupper’s self-referential formula went viral in 2012. Under scrutiny, the question has two flaws: it is meaningless (it depends on fonts) and it is trivial. We fix these flaws by formalizing the problem.

A suprise may occur if we use a similar strategy to the Liar's paradox to mathematically formalize "This sentence does not contain the symbol X".

Mathematicians and software developers use the word "function" very differently, and yet, sometimes, things that are in practice implemented using the software developer's "function", are mathematically formalized using the mathematician's "function". This mismatch can lead to inaccurate formalisms. We consider a special case of this meta-problem. Various kinds of agents might, in actual practice, make use of private memory, reading and writing to a memory-bank invisible to the ambient environment. In some sense, we humans do this when we silently subvocalize (...) thoughts about the actions we are taking (at least when the environment we're in is too primitive to probe the contents of our brains). Mathematical function formalizations of agents often ignore this ability. We show that in a general agent-environment framework (of which reinforcement learning is a special case), in a technical sense, such private memories do not enable qualitatively different agent behavior. (shrink)

A biologically unavoidable sequence is an infinite gender sequence which occurs in every gendered, infinite genealogical network satisfying certain tame conditions. We show that every eventually periodic sequence is biologically unavoidable (this generalizes König's Lemma), and we exhibit some biologically avoidable sequences. Finally we give an application of unavoidable sequences to cellular automata.

Excerpt from Spinoza and Time The Arthur Davis Memorial Lecture was founded in 1917, under the auspices of the Jewish Historical Society of England, by his collaborators in the translation of "The Service of the Synagogue," with the object of fostering Hebraic thought and learning in honour of an unworldly scholar. The Lecture is to be given annually in the anniversary week of his death, and the lectureship is to be open to men or women of any race or creed, (...) who are to have absolute liberty in the treatment of their subject. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

In their paper 'Reward is enough', Silver et al conjecture that the creation of sufficiently good reinforcement learning (RL) agents is a path to artificial general intelligence (AGI). We consider one aspect of intelligence Silver et al did not consider in their paper, namely, that aspect of intelligence involved in designing RL agents. If that is within human reach, then it should also be within AGI's reach. This raises the question: is there an RL environment which incentivises RL agents to (...) design RL agents? (shrink)

Internodons are a formalization of Hennig's concept of species. We present an alternative construction of internodons imposing a tree structure on the genealogical network. We prove that the segments (trivial unary trees) from this tree structure are precisely the internodons. We obtain the following spin-offs. First, the generated tree turns out to be an organismal tree of life. Second, this organismal tree is homeomorphic to the phylogenetic Hennigian species tree of life, implying the discovery of a multi-level tree of life: (...) this phylogenetic tree can be obtained by zooming out from the organismal tree, or conversely, the organismal tree of life can be generated by expanding the phylogenetic nodes into unary trees. Finally, the definition of the organismal tree allows an efficient algorithmic transformation of a given genealogical network into its corresponding phylogenetic species tree of life. The latter will be presented in a separate paper. (shrink)

Fitch's Paradox and the Paradox of the Knower both make use of the Factivity Principle. The latter also makes use of a second principle, namely the Knowledge-of-Factivity Principle. Both the principle of factivity and the knowledge thereof have been the subject of various discussions, often in conjunction with a third principle known as Closure. In this paper, we examine the well-known Surprise Examination paradox considering both the principles on which this paradox rests and some formal characterisations of the surprise notion, (...) crucial in this paradox. Standard formalizations of the Surprise Examination paradox in modal logic do not seem, at first glance, to depend on either factivity or knowledge-of-factivity, but we will argue that both factivity and knowledge-of-factivity play a key implicit role in the paradox. Namely, they are implicitly, perhaps unintentionally, used in order to simplify the definition of surprise. We analyze modal logical formalizations of three versions of the paradox concluding that the Surprise Examination paradox is the result of two flaws: the assumption of knowledge-of-factivity, and the over-simplification of the definition of "surprise" accordingly. By fixing these two flaws, the Surprise Examination paradox vanishes. (shrink)

We describe in our own words a visual phenomenon first described by Gallagher and Tsuchiya in 2020. The key to the phenomenon (as we describe it) is to direct one’s left eye at the image of one's left eye, while simultaneously directing one's right eye at the image of one's right eye. We suggest that one would naturally arrive at this phenomenon if one took a sufficiently literal reading of certain words of Socrates preserved in Plato's Alcibiades. We speculate that (...) perhaps Socrates was aware of the visual phenomenon in question. (shrink)

Our beliefs utilize various sources: perception, memory, induction, etc. We trust these sources to provide reliable information about the world around us. My dissertation investigates how this trust could be justified. Chapter one introduces background material. I argue that justification rather than knowledge is of primary epistemological importance, discuss the internalism/externalism debate, and introduce an evidentialist thesis that provides a starting point/framework for epistemological theorizing. Chapter two introduces a puzzle concerning justification. Can a belief source provide justification absent prior justification (...) for believing it's reliable? Any answer appears to either make justifying the reliability of a source intellectually unsatisfying or all together impossible. Chapter three considers and rejects a plethora of proposed solutions to our puzzle. Investigating these solutions illustrates the need to further investigate evidence, evidence possession, and evidential support. Chapter four discusses the metaphysics of evidence. I argue that evidence always consists of a set of facts and that fact-proposition pairs stand in confirmation relations isomorphic to those holding between pairs of propositions. Chapter five argues that justification requires what I call actually connected possession of supporting evidence: a subject must be aware of supporting evidence and of the support relation itself. Chapter six argues that the relation constitutive of a set of facts being justificatory evidence is a sui generis and irreducible relation that is knowable a priori. Chapter seven begins by showing how Richard Fumerton's acquaintance theory meets the constraints on a theory of justification laid down in previous chapters. I modify the theory so as to: make room for fallible foundational justification, and allow inferential justification absent higher-order beliefs about evidential connections. Chapter eight applies the developed theory of justification to our initial puzzle. I show how my modified acquaintance theory is in a unique position to vindicate the idea that necessarily a source provides a person with justification only if she is aware of evidence for the reliability of that source. However, this awareness of evidence for a source's reliability falls short of a justified belief and thereby avoids impalement from our dilemma's skeptical horn. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Despite the growing importance of ‘social value’ as a central feature of research ethics, the term remains both conceptually vague and to a certain extent operationally rigid. And yet, perhaps because the rhetorical appeal of social value appears immediate and self-evident, the concept has not been put to rigorous investigation in terms of its definition, strength, function, and scope. In this article, we discuss how the anthropological concept of liminality can illuminate social value and differentiate and reconfigure its variegated approaches. (...) Employing liminality as a heuristic encourages a reassessment of how we understand the mobilization of ‘social value’ in bioethics. We argue that social value as seen through the lens of liminality can provide greater clarity of its function and scope for health research. Building on calls to understand social value as a dynamic, rather than a static, concept, we emphasize the need to appraise social value iteratively throughout the entire research as something that transforms over multiple times and across multiple spaces occupied by a range of actors. (shrink)

Thomas Samuel Kuhn (1922–1996) is one of the most influential philosophers of science of the twentieth century, perhaps the most influential. His 1962 book The Structure of Scientific Revolutions is one of the most cited academic books of all time. Kuhn’s contribution to the philosophy of science marked not only a break with several key positivist doctrines, but also inaugurated a new style of philosophy of science that brought it closer to the history of science. His account of the (...) development of science held that science enjoys periods of stable growth punctuated by revisionary revolutions. To this thesis, Kuhn added the controversial ‘incommensurability thesis’, that theories from differing periods suffer from certain deep kinds of failure of comparability. (shrink)

This essay examines Gloria Anzaldúa’s critical appropriation of two Mexican philosophers in the writing of Borderlands/La Frontera: Samuel Ramos and Octavio Paz. We argue that although neither of these authors is cited in her seminal work, Anzaldúa had them both in mind through the writing process and that their ideas are present in the text itself. Through a genealogical reading of Borderlands/La Frontera, and aided by archival research, we demonstrate how Anzaldúa’s philosophical vision of the “new mestiza” is a (...) critical continuation of the broader tradition known as la filosofía de lo mexicano, which flourished during a golden age of Mexican philosophy (1910–1960). Our aim is to open new directions in Latinx and Latin American philosophy by presenting Anzaldúa’s Borderlands/La Frontera as a profound scholarly encounter with two classic works of Mexican philosophy, Ramos’ Profile of Man and Culture in Mexico and Paz’s The Labyrinth of Solitude. (shrink)