This paper argues that we should solve paradoxes for propositions (such as the Russell–Myhill paradox) in essentially the same way that we solve Russellian paradoxes for sets. That is, the standard, iterative approach to sets is extended to include properties, and then the resulting hierarchy of sets and properties is used to construct propositions. Propositions on this account are structured in the sense of mirroring the sentences that express them, and they would seem to serve the needs of philosophers of (...) language and metaphysicians who rely on such propositions. The resulting account has limitations, comparable to those faced by the iterative approach to sets, but it is argued to be in important ways preferable to others that have been proposed. (shrink)

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). A (...) modal stage theory, MST, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that MST interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory. (shrink)

We describe a model of iterated belief revision that extends the AGM theory of revision to account for the effect of a revision on the conditional beliefs of an agent. In particular, this model ensures that an agent makes as few changes as possible to the conditional component of its belief set. Adopting the Ramsey test, minimal conditional revision provides acceptance conditions for arbitrary right-nested conditionals. We show that problem of determining acceptance of any such nested conditional can be reduced (...) to acceptance tests for unnested conditionals. Thus, iterated revision can be accomplished in a "virtual" manner, using uniterated revision. (shrink)

In this paper it is argued that, in order to solve the problem of iterated belief change, both the belief state and its input should be represented as epistemic entrenchment (EE) relations. A belief revision operation is constructed that updates a given EE relation to a new one in light of an evidential EE relation. It is shown that the operation in question satisfies generalized versions of the Gärdenfors revision postulates. The account offered is motivated by Spohn's ordinal conditionalization functions, (...) and can be seen as the Jeffrization of a proposal considered by Rott. (shrink)

Epistemic iteration principles are principles according to which some or another epistemic operator automatically iterates---e.g., if it is known that P, then it is known that P, or there is evidence that P, then there is evidence that there is evidence that P. This article provides a survey of various arguments for and against epistemic iteration principles, with a focus on arguments relevant to a wide range of such principles.

Relative to a hyperstrong cardinal, it is consistent that measure one covering fails relative to HOD. In fact it is consistent that there is a superstrong cardinal and for every regular cardinal κ, κ + is greater than κ + of HOD. The proof uses a very general lemma showing that homogeneity is preserved through certain reverse Easton iterations.

Using matrix iterations of ccc posets, we prove the consistency with ZFC of some cases where the cardinals on the right hand side of Cichon’s diagram take two or three arbitrary values (two regular values, the third one with uncountable cofinality). Also, mixing this with the techniques in J Symb Log 56(3):795–810, 1991, we can prove that it is consistent with ZFC to assign, at the same time, several arbitrary regular values on the left hand side of Cichon’s diagram.

Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead. Contrasts are drawn to accounts of induction by Poincaré, Heyting, and Kreisel. On the phenomenological side, the analysis provides an (...) alternative to that in Tieszen’s Mathematical Intuition, and confirms a view of Gödel on his Dialectica Interpretation. (shrink)

The semantic automata framework, developed originally in the 1980s, provides computational interpretations of generalized quantifiers. While recent experimental results have associated structural features of these automata with neuroanatomical demands in processing sentences with quantifiers, the theoretical framework has remained largely unexplored. In this paper, after presenting some classic results on semantic automata in a modern style, we present the first application of semantic automata to polyadic quantification, exhibiting automata for iterated quantifiers. We also discuss the role of semantic automata in (...) linguistic theory and offer new empirical predictions for sentence processing with embedded quantifiers. (shrink)

Two of the major problems in AGM-style belief revision, namely the difficulties in accounting for iterated change and for Ramsey test conditionals, have satisfactory solutions in descriptor revision. In descriptor revision, the input is a metalinguistic sentence specifying the success condition of the operation. The choice mechanism selects one of the potential outcomes in which the success condition is satisfied. Iteration of this operation is unproblematic. Ramsey test conditionals can be introduced without giving rise to the paradoxical results that they (...) generate in other systems. In addition to standard Ramsey test conditionals, a more general variant of epistemic conditionals is defined, representing statements of the form ”if the belief state is changed to satisfy condition A then it will satisfy condition B”. An axiomatic characterization of such descriptor conditionals is presented. It is related in intricate ways to the KLM postulates for cumulative reasoning. (shrink)

This is a discussion of the problem of extending the basic AGM belief revision theory to iterated belief revision: the problem of formulating rules, not only for revising a basic belief state in response to potential new information, but also for revising one’s revision rules in response to potential new information. The emphasis in the paper is on foundational questions about the nature of and motivation for various constraints, and about the methodology of the evaluation of putative counterexamples to proposed (...) constraints. Some specific constraints that have been proposed are criticized. The paper emphasizes the importance of meta-information—information about one’s sources of information—and argues that little of substance can be said about constraints on iterated belief revision at a level of abstraction that lacks the resources for explicit representation of meta-information. (shrink)

We present a novel, perspicuous framework for building iterated ultrapowers. Furthermore, our framework naturally lends itself to the construction of a certain type of order indiscernibles, here dubbed tight indiscernibles, which are shown to provide smooth proofs of several results in general model theory.

If $U$ is a normal ultrafilter on a measurable cardinal $\kappa$, then the intersection of the $\omega$ first iterated ultrapowers of the universe by $U$ is a Prikry generic extension of the $\omega$th iterated ultrapower.

The axiom of recovery, while capturing a central intuition regarding belief change, has been the source of much controversy. We argue briefly against putative counterexamples to the axiom—while agreeing that some of their insight deserves to be preserved—and present additional recovery-like axioms in a framework that uses epistemic states, which encode preferences, as the object of revisions. This makes iterated revision possible and renders explicit the connection between iterated belief change and the axiom of recovery. We provide a representation theorem (...) that connects the semantic conditions we impose on iterated revision and our additional syntactical properties. We show interesting similarities between our framework and that of Darwiche–Pearl (Artificial Intelligence 89:1–29 1997). In particular, we show that intuitions underlying the controversial (C2) postulate are captured by the recovery axiom and our recovery-like postulates (the latter can be seen as weakenings of (C2)). We present postulates for contraction, in the same spirit as the Darwiche–Pearl postulates for revision, and provide a theorem that connects our syntactic postulates with a set of semantic conditions. Lastly, we show a connection between the contraction postulates and a generalisation of the recovery axiom. (shrink)

A neologicist set theory based on an abstraction principle (NewerV) codifying the iterative conception of set is investigated, and its strength is compared to Boolos's NewV. The new principle, unlike NewV, fails to imply the axiom of replacement, but does secure powerset. Like NewV, however, it also fails to entail the axiom of infinity. A set theory based on the conjunction of these two principles is then examined. It turns out that this set theory, supplemented by a principle stating that (...) there are infinitely many nonsets, captures all (or enough) of standard second-order ZFC. Issues pertaining to the axiom of foundation are also investigated, and I conclude by arguing that this treatment provides the neologicist with the most viable reconstruction of set theory he is likely to obtain. (shrink)

Hilary Putnam once suggested that “the actual existence of sets as ‘intangible objects’ suffers… from a generalization of a problem first pointed out by Paul Benacerraf… are sets a kind of function or are functions a sort of set?” Sadly, he did not elaborate; my aim, here, is to do so on his behalf. There are well-known methods for treating sets as functions and functions as sets. But these do not raise any obvious philosophical or foundational puzzles. For that, we (...) first need to provide a full-fledged function theory. I supply such a theory: it axiomatizes the iterative notion of function in exactly the same sense that ZF axiomatizes the iterative notion of set. Indeed, this function theory is synonymous with ZF. It might seem that set theory and function theory present us with rival foundations for mathematics, since they postulate different ontologies. But appearances are deceptive. Set theory and function theory provide the very same judicial foundation for mathematics. They do not supply rival metaphysical foundations; indeed, if they supply metaphysical foundations at all, then they supply the very same metaphysical foundations. (shrink)

This paper considers the consensus control problem of nonlinear spatial-temporal hyperbolic partial difference multiagent systems and parabolic partial difference multiagent systems with time delay. Based on the system’s own fixed topology and the method of generating the desired trajectory by introducing virtual leader, using the consensus tracking error between the agent and the virtual leader agent and neighbor agents in the last iteration, an iterative learning algorithm is proposed. The sufficient condition for the system consensus error to converge along the (...) iterative axis is given. When the iterative learning number k approaches infinity, the consensus error in the sense of the L 2 norm between all agents in the system will converge to zero. Furthermore, simulation results illustrate the effectiveness of the algorithm. (shrink)

The phrase ‘The iterative conception of sets’ conjures up a picture of a particular settheoretic universe – the cumulative hierarchy – and the constant conjunction of phrasewith-picture is so reliable that people tend to think that the cumulative hierarchy is all there is to the iterative conception of sets: if you conceive sets iteratively, then the result is the cumulative hierarchy. In this paper, I shall be arguing that this is a mistake: the iterative conception of set is a good (...) one, for all the usual reasons. However, the cumulative hierarchy is merely one way among many of working out this conception, and arguments in favour of an iterative conception have been mistaken for arguments in favour of this one special instance of it. (This may be the point to get out of the way the observation that although philosophers of mathematics write of the iterative conception of set, what they really mean – in the terminology of modern computer science at least – is the recursive conception of sets. Nevertheless, having got that quibble off my chest, I shall continue to write of the iterative conception like everyone else.). (shrink)

Iterative characterizations of computable unary functions are useful patterns for the definition of programming languages based on iterative constructs. The features of such a characterization depend on the pairing producing it: this paper offers an infinite class of pairings involving very nice features.

Pristine experience is inner experience that is directly ongoing before it is disturbed by any attempt at apprehension; we live our lives immersed in our pristine experiences. I argue that an iterative method -- one that successively approximates the desired result -- facilitates the faithful apprehension of pristine experience. There are four main aspects of an iterative method: the refreshment by new experience; the improvement of the observations; the multiple perspectives on experience; and the open- beginningedness of the process. Because (...) an iterative exploration of experience is open-beginninged, first interviews occupy a unique position in an iterative method. I comment on the transcript of a first interview, showing why and how an iterative procedure is desirable, if not necessary. (shrink)

Let N be a transitive model of ZFC such that ωN ⊂ N and P(R) ⊂ N. Assume that both V and N satisfy "the core model K exists." Then KN is an iterate of K. i.e., there exists an iteration tree J on K such that J has successor length and $\mathit{M}_{\infty}^{\mathit{J}}=K^{N}$. Moreover, if there exists an elementary embedding π: V → N then the iteration map associated to the main branch of J equals π ↾ K. (This answers (...) a question of W. H. Woodin, M. Gitik, and others.) The hypothesis that P(R) ⊂ N is not needed if there does not exist a transitive model of ZFC with infinitely many Woodin cardinals. (shrink)

Close connections between probability theory and the theory of belief change emerge if the codomain of probability functions is extended from the real-valued interval [0, 1] to a hyperreal interval with the same limits. Full beliefs are identified as propositions with a probability at most infinitesimally smaller than 1. Full beliefs can then be given up, and changes in the set of full beliefs follow a pattern very close to that of AGM revision. In this contribution, iterated revision is investigated. (...) The iterated changes in the set of full beliefs generated by repeated revisions of a hyperreal probability function can, semantically, be modelled with the same basic structure as the sphere models of belief change theory. The changes on the set of full beliefs induced by probability revision satisfy the Darwiche–Pearl postulates for iterated belief change. (shrink)

We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let Ω be a regular uncountable cardinal. Let m < ω and M be an m-sound premouse and Σ be...

The prequel to this paper introduced the topic of iteration principles in epistemology and surveyed some arguments in support of them. In this sequel, I'll consider two influential families of objection to iteration principles. The first turns on the idea that they lead to some variety of skepticism, and the second turns on ‘margin for error’ considerations adduced by Timothy Williamson.

Why are natural theories pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. This approach was inspired by Martin's Conjecture, one of the most prominent conjectures in recursion theory. Fixing a reasonable subsystem $T$ of arithmetic, the goal was to classify the recursive functions that are monotone with respect to the Lindenbaum algebra of $T$. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate $\mathsf{Con}_T^\alpha$ of the consistency operator ``in (...) the limit'' within the ultrafilter of sentences that are true in the standard model. In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as $\mathsf{Con}_T$ in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as $\mathsf{Con}_T$ in the limit nor as strong as $\mathsf{Con}_T^2$ in the limit. In fact, for every $\alpha$, we produce a function that is cofinally equivalent to $\mathsf{Con}^\alpha_T$ yet cofinally equivalent to $\mathsf{Con}_T$. (shrink)

The set of unary functions of complexity classes defined by using bounded primitive recursion is inductively characterized by means of bounded iteration. Elementary unary functions, linear space computable unary functions and polynomial space computable unary functions are then inductively characterized using only composition and bounded iteration.

This paper contrasts two influential theoretical accounts of language change and evolution – Iterated Learning and Social Coordination. The contrast is based on an experiment that compares drawings produced with Garrod et al’s ‘pictionary’ task with those produced in an Iterated Learning version of the same task. The main finding is that Iterated Learning does not lead to the systematic simplification and increased symbolicity of graphical signs produced in the standard interactive version of the task. A second finding is that (...) Iterated Learning leads to less conceptual and structural alignment between participants than observed for those in the interactive condition. The paper concludes with a comparison of the two accounts in relation to how each promotes signs that are effi cient, systematic and learnable. (shrink)

This paper contrasts two influential theoretical accounts of language change and evolution – Iterated Learning and Social Coordination. The contrast is based on an experiment that compares drawings produced with Garrod et al’s ‘pictionary’ task with those produced in an Iterated Learning version of the same task. The main finding is that Iterated Learning does not lead to the systematic simplification and increased symbolicity of graphical signs produced in the standard interactive version of the task. A second finding is that (...) Iterated Learning leads to less conceptual and structural alignment between participants than observed for those in the interactive condition. The paper concludes with a comparison of the two accounts in relation to how each promotes signs that are effi cient, systematic and learnable. (shrink)

Discusses the metaphysics of the iterative conception of set.

The Question of Iterated Grounding (QIG) asks what grounds the grounding facts. Although the question received a lot of attention in the past few years, it is usually discussed independently of another important issue: the connection between metaphysical explanation and the relation or relations that supposedly “back” it. I will show that once we get clear on the distinction between metaphysical explanation and the relation(s) backing it, we can distinguish no fewer than four questions lumped under QIG. I will also (...) argue that given some plausible assumptions about what it would take for a relation to back metaphysical explanation, many salient views about grounding allow us to give “easy” answers to these questions—easy in the sense that we can straightforwardly derive them from the respective conception of grounding without getting into the sorts of complexities that typically inform answers to QIG. The paper's main upshot is that we cannot expect to make much progress on QIG without first addressing the difficult issue of how exactly grounding is related to metaphysical explanation. (shrink)

The semantic automata framework, developed originally in the 1980s, provides computational interpretations of generalized quantifiers. While recent experimental results have associated structural features of these automata with neuroanatomical demands in processing sentences with quantifiers, the theoretical framework has remained largely unexplored. In this paper, after presenting some classic results on semantic automata in a modern style, we present the first application of semantic automata to polyadic quantification, exhibiting automata for iterated quantifiers. We also discuss the role of semantic automata in (...) linguistic theory and offer new empirical predictions for sentence processing with embedded quantifiers. (shrink)

We show that \, the basic theory of bounded arithmetic corresponding to the complexity class \, proves the \ axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the \ iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, \ can also prove the integer division axiom, and the \-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories \ and \. As a side (...) result, we also prove that there is a well-behaved \ definition of modular powering in \\). (shrink)

For “natural enough” systems of ordinal notation we show that α times iterated local reflection schema over a sufficiently strong arithmetic T proves the same Π 1 0 -sentences as ω α times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at ε-numbers. We also derive the following more general “mixed” formulas estimating the consistency strength of iterated local reflection: for all ordinals α ⩾ 1 and all β, β ≡ Π 1 0 (...) T ω α · , α ≡ Π 1 0 T β + ω α . Here T α stands for α times iterated local reflection over T , T β stands for β times iterated consistency, and ≡ Π 1 0 denotes mutual Π 1 0 -conservativity. In an appendix to this paper we develop our notion of “natural enough” system of ordinal notation and show that such systems do exist for every recursive ordinal. (shrink)

We define infinitary extensions to classical epistemic logic systems, and add also a common belief modality, axiomatized in a finitary, fixed-point manner. In the infinitary K system, common belief turns to be provably equivalent to the conjunction of all the finite levels of mutual belief. In contrast, in the infinitary monotonic system, common belief implies every transfinite level of mutual belief but is never implied by it. We conclude that the fixed-point notion of common belief is more powerful than the (...) iterative notion of common belief. (shrink)

The numerous systems designed to facilitate animals’ use of computers often are specific to the animals involved, their unique context, and the applications – enrichment among them. Hence, several development methods have arisen in parallel, largely transposed from the human-computer interaction (HCI) domain. In light of that prior work, the paper presents a step-by-step guide for iteratively designing and constructing interactive computers for animals, informed by the rich history of HCI yet applying animal-centred principles, to enrich animal-computer interaction. For each (...) stage in the iterative design (requirements, ideation, prototyping, and testing), the author reflects on real-world experience of building interactive devices for various animals. The paper concludes with overarching considerations vital for future practice of developing interactive computers for animals. Thus, it serves as a valuable reference and information source for researchers designing novel computer systems for animals. (shrink)

Recent work has considered the problem of extending to the case of iterated belief change the so-called `Harper Identity' (HI), which defines single-shot contraction in terms of single-shot revision. The present paper considers the prospects of providing a similar extension of the Levi Identity (LI), in which the direction of definition runs the other way. We restrict our attention here to the three classic iterated revision operators--natural, restrained and lexicographic, for which we provide here the first collective characterisation in the (...) literature, under the appellation of `elementary' operators. We consider two prima facie plausible ways of extending (LI). The first proposal involves the use of the rational closure operator to offer a `reductive' account of iterated revision in terms of iterated contraction. The second, which doesn't commit to reductionism, was put forward some years ago by Nayak et al. We establish that, for elementary revision operators and under mild assumptions regarding contraction, Nayak's proposal is equivalent to a new set of postulates formalising the claim that contraction by ¬A should be considered to be a kind of `mild' revision by A. We then show that these, in turn, under slightly weaker assumptions, jointly amount to the conjunction of a pair of constraints on the extension of (HI) that were recently proposed in the literature. Finally, we consider the consequences of endorsing both suggestions and show that this would yield an identification of rational revision with natural revision. We close the paper by discussing the general prospects for defining iterated revision in terms of iterated contraction. (shrink)

Thomas More’s 281 epigrams form a diverse and seemingly haphazard collection of occasional and programmatic pieces written in a variety of meters on diverse topics. Since most of More’s papers disappeared in the years immediately following his death, it is difficult and perhaps impossible to reconstruct on the basis of external evidence the rationale behind the selection and distribution of his epigrams. Despite this challenge, internal evidence provides some clues. Nearly half of the epigrams are translations of Greek originals. Some (...) of these Greek originals serve as the basis for sequences of epigrams, one of which is the “Lame Beggars Sequence.” Through a process of iterative amplificatio, More progressively cultivates eloquence as he extends the meaning of his original to encompass moral and political themes. In turn, he develops the theme into original compositions before producing a final translation that encompasses both the literal and moral senses of the original Greek. (shrink)

We introduce a class of set-theoretic operators on a tolerance space that models the process of minimal belief contraction, and therefore a natural process of iterated contraction can be defined. We characterize the class of contraction operators and study the properties of the associated iterated belief contraction.

We consider a classM of Boolean algebras with strictly positive, finitely additive measures. It is shown thatM is closed under iterations with finite support and that the forcing via such an algebra does not destroy the Lebesgue measure structure from the ground model. Also, we deduce a simple characterization of Martin's Axiom reduced to the classM.

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty (...) set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded. (shrink)

The standard rule of single privative modification replaces privative modifiers by Boolean negation. This rule is valid, for sure, but also simplistic. If an individual a instantiates the privatively modified property (MF) then it is true that a instantiates the property of not being an F, but the rule fails to express the fact that the properties (MF) and F have something in common. We replace Boolean negation by property negation, enabling us to operate on contrary rather than contradictory properties. (...) To this end, we apply our theory of intensional essentialism, which operates on properties (intensions) rather than their extensions. We argue that each property F is necessarily associated with an essence, which is the set of the so-called requisites of F that jointly define F. Privation deprives F of some but not all of its requisites, replacing them by their contradictories. We show that properties formed from iterated privatives, such as being an imaginary fake banknote, give rise to a trifurcation of cases between returning to the original root property or to a property contrary to it or being semantically undecidable for want of further information. In order to determine which of the three forks the bearers of particular instances of multiply modified properties land upon we must examine the requisites, both of unmodified and modified properties. Requisites underpin our presuppositional theory of positive predication. Whereas privation is about being deprived of certain properties, the assignment of requisites to properties makes positive predication possible, which is the predication of properties the bearers must have because they have a certain property formed by means of privation. (shrink)

Following the pioneer work of Bruno De Finetti [12], conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable as Popper functions, and (...) that every Popper function is representable in terms of the standard real values of some infinitesimal measure. Our main goal in this article is to study the constraints on (qualitative and probabilistic) change imposed by an extended version of McGee's result. We focus on an extension capable of allowing for iterated changes of view. Such extension, we argue, seems to be needed in almost all considered applications. Since most of the available axiomatizations stipulate (definitionally) important constraints on iterated change, we propose a non-questionbegging framework, Iterative Probability Systems (IPS) and we show that every Popper function can be regarded as a Bayesian IPS. A generalized version of McGee's result is then proved and several of its consequences considered. In particular we note that our proof requires the imposition of Cumulativity, i.e. the principle that a proposition that is accepted at any stage of an iterative process of acceptance will continue to be accepted at any later stage. The plausibility and range of applicability of Cumulativity is then studied. In particular we appeal to a method for defining belief from conditional probability (first proposed in [42] and then slightly modified in [6] and [3]) in order to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic supposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30]. (shrink)

Languages are transmitted from person to person and generation to generation via a process of iterated learning: people learn a language from other people who once learned that language themselves. We analyze the consequences of iterated learning for learning algorithms based on the principles of Bayesian inference, assuming that learners compute a posterior distribution over languages by combining a prior (representing their inductive biases) with the evidence provided by linguistic data. We show that when learners sample languages from this posterior (...) distribution, iterated learning converges to a distribution over languages that is determined entirely by the prior. Under these conditions, iterated learning is a form of Gibbs sampling, a widely-used Markov chain Monte Carlo algorithm. The consequences of iterated learning are more complicated when learners choose the language with maximum posterior probability, being affected by both the prior of the learners and the amount of information transmitted between generations. We show that in this case, iterated learning corresponds to another statistical inference algorithm, a variant of the expectation-maximization (EM) algorithm. These results clarify the role of iterated learning in explanations of linguistic universals and provide a formal connection between constraints on language acquisition and the languages that come to be spoken, suggesting that information transmitted via iterated learning will ultimately come to mirror the minds of the learners. (shrink)

Machine awareness is a disputed research topic, in some circles considered a crucial step in realising Artificial General Intelligence. Understanding what that is, under which conditions such feature could arise and how it can be controlled is still a matter of speculation. A more concrete object of theoretical analysis is algorithmic iteration for computational intelligence, intended as the theoretical and practical ability of algorithms to design other algorithms for actions aimed at solving well-specified tasks. We know this ability is already (...) shown by current AIs, and understanding its limits is an essential step in qualifying claims about machine awareness and Super-AI. We propose a formal translation of algorithmic iteration in a fragment of modal logic, formulate principles of transparency and faithfulness across human and machine intelligence, and consider the relevance to theoretical research on -AI as well as the practical import of our results. (shrink)