We introduce the appropriate iterated version of the system of ordinal notations from [G1] whose order type is the familiar Howard ordinal. As in [G1], our ordinal notations are partly inspired by the ideas from [P] where certain crucial properties of the traditional Munich' ordinal notations are isolated and used in the cut-elimination proofs. As compared to the corresponding “impredicative” Munich' ordinal notations (see e.g. [B1, B2, J, Sch1, Sch2, BSch]), our ordinal notations arearbitrary terms in the appropriate simple (...) term algebra based on the notion of collapsing functions (which we would rather identify as projective functions). In Sect. 1 below we define the systems of ordinal notationsPRJ( ), for any primitive recursive limit wellordering . In Sect. 2 we prove the crucial well-foundness property by using the appropriate well-quasi-ordering property of the corresponding binary labeled trees [G3]. In Sect. 3 we interprete inPRJ( ) the familiar Veblen-Bachmann hierarchy of ordinal functions, and in Sect. 4 we show that the corresponding Buchholz's system of ordinal notationsOT( ) is a proper subsystem ofPRJ( ), although it has the same order type according to [G3] together with the interpretation from Sect. 2 in the terms of labeled trees. In Sect. 5 we use Friedman's approach in order to obtain an appropriate purely combinatorial statement which is not provable in the theory of iterated inductive definitions ID< λ, for arbitrarily large limit ordinalλ. Formal theories, axioms, etc. used below are familiar in the proof theory of subsystems of analysis (see [BFPS, T, BSch]). (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)

At first pass, Evidential Decision Theory (EDT) recommends one-boxing in Newcomb’s Problem and Causal Decision Theory (CDT) recommends two-boxing. However, it has been acknowledged that concrete instances of the problem have messy features complicating their analyses. Recently, a third competitor, Functional Decision Theory (FDT) has emerged recommending one-boxing in some versions and two-boxing in others. This paper explores the verdicts of these competing theories in a few variations of the problem involving iteration and habit. We argue that this motivates the (...) re-evaluation of case intuitions in decision theory and greater consideration of formal and structural relationships between CDT and FDT. (shrink)

This article assesses Richard Vernon's attempted reconciliation of compatriot preference with global justice by analyzing the iteration proviso (IP), which says that a group of people can legitimately set out to confer special advantages upon each other if others, outside that group, are free to do the same in their own case. Part I outlines how duties to outsiders are typically characterized in two leading accounts of global justice — moral universalism and associativism. The IP is motivated by Vernon's desire (...) to transcend the binary opposition between, and the limitations of, these two views. Part II sketches the version of contractualism that Vernon deploys to surmount these limitations and explains the role of the IP therein. Part III elucidates two different interpretations of the IP and shows that neither seems plausible. (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)

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)

We show that $$ VTC ^0$$, the basic theory of bounded arithmetic corresponding to the complexity class $$\mathrm {TC}^0$$, proves the $$ IMUL $$ axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the $$\mathrm {TC}^0$$ iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, $$ VTC ^0$$ can also prove the integer division axiom, and (by our previous results) the $$ RSUV $$ -translation of induction and minimization for (...) sharply bounded formulas. Similar consequences hold for the related theories $$\Delta ^b_1\text{- } CR $$ and $$C^0_2$$. As a side result, we also prove that there is a well-behaved $$\Delta _0$$ definition of modular powering in $$I\Delta _0+ WPHP (\Delta _0)$$. (shrink)

The iterative conception of set commonly is regarded as supporting the axioms of Zermelo-Fraenkel set theory (ZF). This paper presents a modified version of the iterative conception of set and explores the consequences of that modified version for set theory. The modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets. It is suggested that this modified iterative conception of set supports the axioms of Quine's set theory NF.

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)

Progressions of iterated reflection principles can be used as a tool for the ordinal analysis of formal systems. We discuss various notions of proof-theoretic ordinals and compare the information obtained by means of the reflection principles with the results obtained by the more usual proof-theoretic techniques. In some cases we obtain sharper results, e.g., we define proof-theoretic ordinals relevant to logical complexity Π1 0 and, similarly, for any class Π n 0. We provide a more general version of the (...) fine structure relationships for iterated reflection principles (due to U. Schmerl [25]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including IΣ n, IΣ n −, IΠ n − and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform Σ1-reflection principle for T is Σ2-conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl's theorem. (shrink)

We show in this paper that the AGM postulates are too weak to ensure the rational preservation of conditional beliefs during belief revision, thus permitting improper responses to sequences of observations. We remedy this weakness by proposing four additional postulates, which are sound relative to a qualitative version of probabilistic conditioning. Contrary to the AGM framework, the proposed postulates characterize belief revision as a process which may depend on elements of an epistemic state that are not necessarily captured by (...) a belief set. We also show that a simple modification to the AGM framework can allow belief revision to be a function of epistemic states. We establish a model-based representation theorem which characterizes the proposed postulates and constrains, in turn, the way in which entrenchment orderings may be transformed under iterated belief revision. (shrink)

Darwiche and Pearl’s seminal 1997 article outlined a number of baseline principles for a logic of iterated belief revision. These principles, the DP postulates, have been supplemented in a number of alternative ways. Most suggestions have resulted in a form of ‘reductionism’ that identifies belief states with orderings of worlds. However, this position has recently been criticised as being unacceptably strong. Other proposals, such as the popular principle (P), aka ‘Independence’, characteristic of ‘admissible’ operators, remain commendably more modest. In this (...) paper, we supplement the DP postulates and (P) with a number of novel conditions. While the DP postulates constrain the relation between a prior and a posterior conditional belief set, our new principles notably govern the relation between two posterior conditional belief sets obtained from a common prior by different revisions. We show that operators from the resulting family, which subsumes both lexicographic and restrained revision, can be represented as relating belief states associated with a ‘proper ordinal interval’ (POI) assignment, a structure more fine-grained than a simple ordering of worlds. We close the paper by noting that these operators satisfy iterated versions of many AGM era postulates, including Superexpansion, that are not sound for admissible operators in general. (shrink)

This paper is about what I call the Question of Iterated Causation (QIC): for any instance of causation in which c1…ck cause effect e, what are the causes of c1…ck’s causing of e? In short: what causes instances of causation or, as I will refer to these instances, the “causal goings‐on”? A natural response (which I call “dismissivism”) is that this is a bad question because causal goings‐on aren’t apt to be caused. After rebutting several versions of dismissivism, I consider (...) the view that QIC, though not illegitimate, is easy to answer: the causal goings‐on are apt to be caused but are plainly uncaused (“brutism”). However, I will argue that brutism too has a serious problem: namely, it leads to a highly implausible kind of armchair indeterminism. Next I consider some substantive candidate answers to QIC, none of which, I argue, is particularly promising. The paper’s final conclusion is twofold: QIC is at least as difficult as the more well‐known Question of Iterated Grounding; moreover, the largely overlooked regress problem that it raises gives us at least some defeasible reason to avoid causation in theory‐building. (shrink)

The conclusion of this chapter is that higher-order vagueness is universal: no sentence whatsoever is definitely true, definitely definitely true, definitely definitely definitely true, and so on ad infinitum. The argument, of which there are several versions, turns on the existence of Sorites sequences of possible worlds connecting the actual world to possible worlds where a given sentence is used in such a way that its meaning is very different. The chapter attempts to be neutral between competing accounts of the (...) nature of vagueness and definiteness. (shrink)

The KK principle states that knowing entails knowing that one knows. This historically popular principle has fallen out of favour among many contemporary philosophers in light of putative counterexamples. Recently, some have defended more palatable versions of KK by weakening the principle. These revisions remain faithful to their predecessor in spirit while escaping crucial objections. This paper examines the prospects of such a strategy. It is argued that revisions of the original principle can be captured by a generalized knowledge iteration (...) principle, Weak-KK, which states that knowing entails the possibility of knowing that one knows. But Weak-KK is vulnerable to an unknowability result and therefore must be rejected. The arguments here suggest that retreating to weaker iteration principles is not an option for the KK enthusiast. (shrink)

In order to clarify the problems of iterated (global) belief change it is useful to study simple cases, in particular consecutive contractions by sentences that are both logically and epistemically independent. Models in which the selection mechanism is kept constant are much more plausible in this case than what they are in general. One such model, namely uniform specified meet contraction, has the advantage of being closely connected with the AGM model. Its properties seem fairly adequate for the intended type (...) of contraction. However, the revision operator based on it via the Levi identity collapses into an implausible operation that loses all old information when revising by new information. A weaker version, semi-uniform specified meet contraction, avoids the collapse but has the disadvantage of a remarkably weak logic. It is left as an open issue whether there is an intermediate class of contraction operators that yields a more satisfactory logic. (shrink)

We discuss an abstract notion of a logical operation and corresponding logics. It is shown that if all the logical operations considered are implicitely definable in a logic ℒ*, then the same holds also for the logic obtained from these operations. As an application we show that certain iterated forms of infinitely deep languages are implicitely definable in game quantifier languages. We consider also relations between structures and show that Karttunen's characterization of elementary equivalence for the ordinary infinitely deep languages (...) can be generalized to hold for the iterated infinitely deep languages. An early version of this work was presented in the Abstracts Section of ICM '78. (shrink)

The idea that the dynamical properties of quantum systems are invariably relative to other systems has recently regained currency. Using Relational Quantum Mechanics (RQM) for a case study, this paper calls attention to a question that has been underappreciated in the debate about quantum relativism: the question of whether relativity iterates. Are there absolute facts about the properties one system possesses relative to a specified reference, or is this again a relative matter, and so on? It is argued that RQM (...) (in its best-known form) is committed to what I call the Unrestricted Iteration Principle (UIP), and thus to an infinite regress of relativisations. This principle plays a crucial role in ensuring the communicability and coherence of interaction outcomes across observers. It is, however, shown to be incompatible with the widespread, conservative reading of RQM in terms of relations, instead necessitating the adoption of the more unorthodox notion of perspectival facts. I conclude with some reflections on the current state of play in perspectivist versions of RQM and quantum relativism more generally, underscoring both the need for further conceptual development and the importance of the iteration principle for an accurate cost-benefit analysis of such interpretations. (shrink)

The paper is essentially a short version Spohn "Strategic Rationality" which emphasizes in particular how the ideas developed there may be used to shed new light on the iterated prisoner's dilemma (and on iterated Newcomb's problem).

The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be denied. Solving such equations requires numerical methods having higher-order convergence with cost-effectiveness, for the equations do not have exact solutions. In the pursuit of efficient numerical methods, an attempt is made to devise a modified strategy for approximating the solution of nonlinear models in either scalar or vector versions. Two numerical methods of second-and sixth-order convergence are carefully merged to obtain a hybrid multi-step numerical method with twelfth-order (...) convergence while using seven function evaluations per iteration, resulting in the efficiency index of about 1.4262. The convergence is also ascertained theoretically, and the asymptotic error constant is computed. Furthermore, various numerical methods of varying orders are used to compare the numerical results obtained with the proposed hybrid method in approximate solution, number of iterations, absolute error, absolute functional value, and the machine time measured in seconds. The obtained results outperformed the chosen methods when applied models arising from physical and natural fields were solved. Finally, to observe the convergence graphically, some complex polynomials are plotted as polynomiographs, wherein the rapid convergence of the proposed method is guaranteed. (shrink)

Non-primary qualities are ubiquitous and humanly quite important. This paper briefly argues that they can be best understood on a dispositional model (or, that they are what is nowadays called in literature “responsedependent”), and offers a particular version of the model, the response-intentionalist one. It then discusses combinations of non-primary qualities and argues that there is a layered structure of iterated response-dependence, underlying aesthetic and other interesting properties such as meaningfulness.

In this paper we distinguish between various kinds of doxastic theories. One distinction is between informal and formal doxastic theories. AGM-type theories of belief change are of the former kind, while Hintikka’s logic of knowledge and belief is of the latter. Then we distinguish between static theories that study the unchanging beliefs of a certain agent and dynamic theories that investigate not only the constraints that can reasonably be imposed on the doxastic states of a rational agent but also rationality (...) constraints on the changes of doxastic state that may occur in such agents. An additional distinction is that between non-introspective theories and introspective ones. Non-introspective theories investigate agents that have opinions about the external world but no higher-order opinions about their own doxasticnstates. Standard AGM-type theories as well as the currently existing versions of Segerberg’s dynamic doxastic logic (DDL) are non-introspective. Hintikka-style doxastic logic is of course introspective but it is a static theory. Thus, the challenge remains to devise doxastic theories that are both dynamic and introspective. We outline the semantics for truly introspective dynamic doxastic logic, i.e., a dynamic doxastic logic that allows us to describe agents who have both the ability to form higher-order beliefs and to reflect upon and change their minds about their own (higher-order) beliefs. This extension of DDL demands that we give up the Preservation condition on revision. We make some suggestions as to how such a non-preservative revision operation can be constructed. We also consider extending DDL with conditionals satisfying the Ramsey test and show that Gärdenfors’ well-known impossibility result applies to such a framework. Also in this case, Preservation has to be given up. (shrink)

Two Hilbert calculi in the use-restricted relevance tradition of Anderson and Belnap are presented. One yields Anderson and Belnap's conjunctive version of the so-called ``entailment theorem,'' whereas the other yields the implication-iterated version. In contrast to previous such use-restricted calculi, those presented in this paper apply to arbitrary strengthenings of the weak contractionless logic B without need for modifying the use-conditions. As such they help to identify the core idea of the ``use-account'' of relevance.

We show that many principles of first-order arithmetic, previously only known to lie strictly between [Formula: see text]-induction and [Formula: see text]-induction, are equivalent to the well-foundedness of [Formula: see text]. Among these principles are the iteration of partial functions of Hájek and Paris, the bounded monotone enumerations principle by Chong, Slaman, and Yang, the relativized Paris–Harrington principle for pairs, and the totality of the relativized Ackermann–Péter function. With this we show that the well-foundedness of [Formula: see text] is a (...) far more widespread than usually suspected. Further, we investigate the [Formula: see text]-iterated version of the bounded monotone iterations principle, and show that it is equivalent to the well-foundedness of the -height [Formula: see text]-tower [Formula: see text]. (shrink)

We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martin's axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the (...) club principle for every stationary subset of limits of ω1 together with ¬CH and Martin's axiom for countable p.o.-sets. (shrink)

Bernard Williams (1973) famously argued that if given the choice to relinquish our mortality we should refuse. We should not choose to always live. His piece provoked an entire literature on the desirability of immortality. Intending to contradict Williams, Thomas Nagel claimed that if given the choice between living for a week and dying in five minutes he would always choose to live. I argue that (1) Nagel’s iterating scenario is closer to the original Makropulos case (Čapek’s) that inspired Williams’s (...) piece; (2) iterating versions of the choice given in the Makropulos case might well be less desirable than a one-time choice; and (3) Nagel’s mathematical induction premise is implausible. I discuss some useful implications of (1)-(3) for the broader discussion of Williams’s arguments and, more generally, for our understanding of the value of mortality and the possibility of mutually consistent but necessarily incompatible wants in ordinary human psychology. (shrink)

We consider the connections between belief revision, conditional logic and nonmonotonic reasoning, using as a foundation the approach to theory change developed by Alchourrón, Gärdenfors and Makinson (the AGM approach). This is first generalized to allow the iteration of theory change operations to capture the dynamics of epistemic states according to a principle of minimal change of entrenchment. The iterative operations of expansion, contraction and revision are characterized both by a set of postulates and by Grove's construction based on total (...) pre-orders on the set of complete theories of the belief logic. We present a sound and complete conditional logic whose semantics is based on our iterative revision operation, but which avoids Gärdenfors's triviality result because of a severely restricted language of beliefs and hence the weakened scope of our extended postulates. In the second part of the paper, we develop a computational approach to theory dynamics using Rott's E-bases as a representation for epistemic states. Under this approach, a ranked E-base is interpreted as standing for the most conservative entrenchment compatible with the base, reflecting a kind of foundationalism in the acceptance of evidence for a belief. Algorithms for the computation of our iterative versions of expansion, contraction and revision are presented. Finally, we consider the relationship between nonmonotonic reasoning and both conditional logic and belief revision. Adapting the approach of Delgrande, we show that the unique extension of a default theory expressed in our conditional logic of belief revision corresponds to the most conservative belief state which respects the theory: however, this correspondence is limited to propositional default theories. Considering first order default theories, we present a belief revision algorithm which incorporates the assumption of independence of default instances and propose the use of a base logic for default reasoning which incorporates uniqueness of names. We conclude with an examination of the behavior of an implemented system on some of Lifschitz's benchmark problems in nonmonotonic reasoning. (shrink)

This paper develops a version of the self-refutation argument against relativism in the teeth of the prevailing response by relativists: that this argument begs the question against them. It is maintained that although weaker varieties of relativism are not self-refuting, strong varieties are faced by this argument with a choice between making themselves absolute (one thing is absolutely true - relativism); or reflexive (relativism is 'true for' the relativist). These positions are in direct conflict. The commonest response, Reflexive Relativism, (...) is shown to be vulnerable to an iterated version of the self-refutation argument. As a result, Reflexive Relativism possesses only the appearance of content, being either incoherent, or a regressively disguised version of Absolute Relativism. Concluding remarks on Absolute Relativism acknowledge this to be a bare, formal possibility, but claim that in fact it must represent one of a range of weaker varieties of relativism that alone remain tenable. (shrink)

Building on previous work by Mints, Buchholz and Schwichtenberg, a simplified version of continuous normalization for the untyped λ-calculus and Gödel’s is presented and analysed in the coalgebraic framework of non-wellfounded terms with so-called repetition constructors.The primitive recursive normalization function is uniformly continuous w.r.t. the natural metric on non-wellfounded terms. Furthermore, the number of necessary repetition constructors is locally related to the number of reduction steps needed to reach the normal form and its size.It is also shown how continuous (...) normal forms relate to derivations of strong normalizability in the typed λ-calculus and how this leads to new bounds for the sum of the height of the reduction tree and the size of the normal form.Finally, the methods are extended to an infinitary λ-calculus with ω-rule and permutative conversions and this is used to derive a strong form of normalization for an iterative version of Gödel’s system, leading to a value table semantics for number-theoretic functions. (shrink)

The simple model of conflict-prone situations called Prisoner’s Dilemma is discussed. Whereas the best strategy for the model is to defect, in the case of its iterated version (Iterated Prisoner’s Dilemma—IPD) it is possible and more profitable to cooperate with the opponent.The simple strategy called Tit for Tat (TFT) which is easy to recognize, never defects first, punishes every defection but is also forgiving is presented. The TFT strategy is very successful being able to establish the cooperation with its (...) opponents. The possible outcome of the competition between the TFT strategy and non-cooperative strategies is considered. It turns out that even small number of TFT strategies can overcome the greater number of more “egoistic” strategies.The described model can give the better understanding of some psychological, sociological and biological processes as well as help to propagate the altruistic attitudes in the society. (shrink)

The paper argues against a commitment to metaphysical necessity, semantic modalities are enough. The best approaches to elucidate the semantic modalities are (still) versions of lingustic ersatzism and fictionalism, even if only developed in parts. Within these necessary properties and the difference between natural and semantic laws can be accounted for. The proper background theory for this is an updated version of Logical Empiricism, which is congenial to recent trends in Structural Realism. The anti-metaphysical attitude of Logical Empiricism deserves (...) revitalization. Another target besides metaphysical necessity are substantial forms of iterated modalities, as used, for instance, in the philosophy of religion. -/- An extended version appeared as book in 2025 (Berlin: Logos). (shrink)

We can now survey the ruins of a Babelian tower of discourse about cosmopolitanism. We speak of “elite travel lounge,” “Davos,” “banal” as well as of “reflexive,” “really existing,” “patriotic,” and “horizontal” cosmopolitanisms. Here, an attempt is made to extract what is normative and ideal in the concept of cosmopolitanism by foregrounding the epistemic and moral dimensions of this attitude towards the world and other cultures. Kant, in a rather unexpected way, is profiled as the exemplification of what is here (...) called “imperial” cosmopolitanism, which is both blind and dismissive of its own material conditions of possibility. Then, through a discussion of the works of Nussbaum, Appiah, Mignolo, Butler, Benhabib and Beck the author elaborates a version of cosmopolitanism that is grounded, enlightened, and reflexive, which corrects and supersedes Kant’s own Eurocentric cosmopolitanism. We do not live in an age of cosmopolitanism, but in an age of cosmopolitization. Democratic iterations that are jurisgenerative are matched at the global level by cosmopolitan iterations that are both jurisgenerative and affect generating. (shrink)

We present probabilistic approaches to check the validity of selected connexive principles within the setting of coherence. Connexive logics emerged from the intuition that conditionals of the form If ∼A, then A, should not hold, since the conditional’s antecedent ∼A contradicts its consequent A. Our approach covers this intuition by observing that for an event A the only coherent probability assessment on the conditional event A|~A is p(A|~A)=0 . Moreover, connexive logics aim to capture the intuition that conditionals should express (...) some “connection” between the antecedent and the consequent or, in terms of inferences, validity should require some connection between the premise set and the conclusion. This intuition is covered by a number of principles, a selection of which we analyze in our contribution. We present two approaches to connexivity within coherence-based probability logic. Specifically, we analyze connections between antecedents and consequents firstly, in terms of probabilistic constraints on conditional events (in the sense of defaults, or negated defaults) and secondly, in terms of constraints on compounds of conditionals and iterated conditionals. After developing different notions of negations and notions of validity, we analyze the following connexive principles within both approaches: Aristotle’s Theses, Aristotle’s Second Thesis, Abelard’s First Principle and selected versions of Boethius’ Theses. We conclude by remarking that coherence-based probability logic offers a rich language to investigate the validity of various connexive principles. (shrink)

In the Divide-the-Dollar (DD) game, two players simultaneously make demands to divide a dollar. Each player receives his demand if the sum of the demands does not exceed one, a payoff of zero otherwise. Note that, in the latter case, both parties are punished severely. A major setback of DD is that each division of the dollar is a Nash equilibrium outcome. Observe that, when the sum of the two demands x and y exceeds one, it is as if Player (...) 1's demand x (or his offer (1−x) to Player 2) suggests that Player 2 agrees to λx < 1 times his demand y so that Player 1's demand and Player 2's modified demand add up to exactly one; similarly, Player 2's demand y (or his offer (1−y) to Player 1) suggests that Player 1 agrees to λyx so that λyx+y = 1. Considering this fact, we change DD's payoff assignment rule when the sum of the demands exceeds one; here in this case, each player's payoff becomes his demand times his λ; i.e., each player has to make the sacrifice that he asks his opponent to make. We show that this modified version of DD has an iterated strict dominant strategy equilibrium in which each player makes the egalitarian demand 1/2. We also provide a natural N-person generalization of this procedure. (shrink)

Time-motion studies revolutionized the design and efficiency of repetitive work last century. Wouldtime-ideastudies revolutionize the rules of intellectual/creative work this century? Collaborating with seven professional dancers, we set out to discover if there were any significant temporal patterns to be found in a timeline coded to show when dancers come up with ideas and when they modify or reject them. On each of 3 days, the dancers were given a choreographic problem (or task) to help them generate a novel, high (...) quality contemporary dance phrase. They were videoed as they worked on this task for sessions of 15, 30, and 45 min. At the end of each 15 min interval during each session, we had them perform the phrase they were creating. They recorded and then coded the video of themselves dancing during these sessions by using a coding language we developed with them to identify when ideas are introduced, modified, and rejected. We found that most ideas are created early and that though these early ideas are aggressively pruned early on, many still make it into the final product. The two competing accounts of creativity in design research make predictions for the temporal structure of creativity. Our results support neither account, rather showing a more blended version of the two. The iterative design view, arguably the dominant view, is that good ideas are the product of generating many ideas, choosing one fairly early, committing to it, and iteratively improving it. The “fail fast fail often” view is that good ideas are the product of rapidly generating and discarding ideas and holding back from early commitment to any one in particular. The result of holding back commitment, typically, is not that an idea is taken up later and then incrementally improved at the last minute, as much as that later designs are not completely novel, instead incorporating the best parts of the entire sequence of ideas. In our study, we found no evidence that one account or the other was more predictive for the domain of contemporary dance. The behavior of the dancers that we studied revealed elements of both, calling into question how predictive these theories are. (shrink)

My goal in this paper is to defend the plausibility of a particular version of collectivism – understood as the evolutionary claim that individual-level cognition is systematically biased in favor of aggregate-level regularities – in the domain of language. Chomsky's (1986) methodological promotion of I-language (speaker-internal knowledge) and the corresponding demotion of E-language (aggregate output of a population of speakers) has led mainstream cognitive science to view language essentially as a property of individual minds/brains whose evolution is best explained (...) as a result of the natural selection of an innate language faculty (Pinker 1994). Such a framework is largely oblivious to the linguistic dynamics arising from iterated learning, i.e. the fact that linguistic information must be periodically mapped from its I-linguistic into its E-linguistic medium and back in order to persist over time. Recent expression/induction models of language evolution, which are heavily inspired by the simulation-based methodology of Artificial Life, suggest that linguistic universals, such as compositional and recursive structure, can emerge from cultural selection alone (Brighton et al. 2005). I will conclude that there is both a biological and a cultural route towards collectivism. (shrink)

We investigate both iteration hypotheses and extender models at the level of one supercompact cardinal. The HOD Conjecture is introduced and shown to be a key conjecture both for the Inner Model Program and for understanding the limits of the large cardinal hierarchy. We show that if the HOD Conjecture is true then this provides strong evidence for the existence of an ultimate version of Gödel's constructible universe L. Whether or not this "ultimate" L exists is now arguably the (...) central issue for the Inner Model Program. (shrink)

The first version of this piece was written for the opening panel of the 2017 Conference of the Association for Feminist Ethics and Social Theory in Florida. The panel, “Decolonial Feminism: Theories and Praxis,” offered the opportunity for Black and Latinx feminist philosophers and decolonial scholars to consider their arrival to decolonial feminisms, their various points of emergence, and the utility of decolonial politics for liberation movements and organizing. I was prepared to discuss some genealogies of US Latina decolonial (...) feminisms with a focus on the relationship of decolonial feminisms to other feminist articulations—for example, a consideration of the relation and divergence between decolonial and postcolonial feminism. I was particularly interested in examining some of the “decolonizing constellations of resistance and love” created by Black, Indigenous, Latinx feminisms. I wanted to track the intergenerational labor of relationality as a part of women of color politics and to discuss how these politics unseat coloniality in its variant iterations. (shrink)

Ainslie challenges our interpretation of the properties of hyperbolic discount curves in an iterated prisoners’ dilemma model. In this reply, we discuss the emergence of hyperbolic discount functions in the behavioral economics literature and evaluate their properties. Furthermore, we present a summarized version of our IPD model and evaluate Ainslie’s points of contention.

In “Iteration, Reiteration, Repetition”, Quentin Meillassoux accuses Deleuze of forming a subjectalist philosophical system, that is to say, despite his critiques of subjectivism and representationalism, Deleuze absolutizes the correlation between thought and being, while failing to grasp absolute exteriority. Meillassoux’s main argument in support of this claim is his interpretation of Deleuze’s ideas of “intensity” and “intensive difference” as a “difference of degree” instead of a “difference in nature”. In this paper, I argue against Meillassoux’s reading, and claim that, in (...) his early works, Deleuze constructs a transcendental philosophy very different from Kantian transcendentalism, in which exteriority is graspable through the evental aspects of our experience. Meillassoux accuses Deleuze of being a philosopher of continuity and homogeneity, and believes that the only way to evade subjectalism is by establishing a Cartesian dualism of subjective and objective realms. I claim, however, that Deleuze is the true philosopher of heterogeneity, and that continuity in Deleuze’s thought is simply a means of inducing heterogeneity everywhere. In order to reach the absolute, we do not need to establish a sharp dualism; what we need is a Deleuzian version of transcendental realism, which induces transcendental/empirical distinction everywhere, into both the inorganic and organic realms. (shrink)

There are two versions of the Proper Iteration Lemma. The stronger version can be used to give simpler proofs of iteration theorems . In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ωω-bounding property.

Software development is an iterative process from designing to implementation, and to testing, in which product development staff should be closely integrated with users. Satisfying user needs effectively is often the pain point for developers. In order to alleviate this, this paper manages to establish the quantitative connection between users' online reviews and APP downloads. By analyzing user online comments, companies can dig out user needs and preferences. This could benefit them by making accurate market positioning of their APP products, (...) and therefore iteratively innovating products based on user needs, which hopefully will increase the volume of APP downloads. This paper regards WeChat APP during 47 updates periods as the research object. Based on Grounded Theory, user needs are extracted after data cleaning. Next, by using semantic analysis and word frequency analysis, we are able to obtain the implicit feedbacks such as emotion tendency, satisfaction and requirements lie under online reviews. Then, we construct a quantile regression model to study the impact of users' online reviews on downloads based on the influencing factors we extracted so as to provide a decision basis for enterprises to iteratively update their products. Results show that: Generally speaking, needs of WeChat users mainly focus on performance, reliability, usability, functional deficiency, functional insufficiency, and system adaptability; For those APP versions with relatively fewer downloads, user needs are mostly about functional deficiency, followed by functional insufficiency, performance, usability, and system adaptability. At this stage, it is found out that users' emotion tendency and user satisfaction significantly affect the volume of downloads; When the volume of APP downloads is moderate, the user needs are functional deficiency, functional insufficiency, and system adaptability. While under this circumstances, users' star ratings have a significant impact on downloads; In addition, when the volume of App downloads is high, user needs are performance, usability, and system adaptability. Our methods effectively extract users' requirements from online reviews and then successfully build up the quantitative connection between the implicit feedbacks from those requirements and APP downloads. (shrink)

We present a uniform version of Di Nola Theorem, this enables to embed all MV-algebras of a bounded cardinality in an algebra of functions with values in a single non-standard ultrapower of the real interval [0,1]. This result also implies the existence, for any cardinal α, of a single MV-algebra in which all infinite MV-algebras of cardinality at most α embed. Recasting the above construction with iterated ultrapowers, we show how to construct such an algebra of values in a (...) definable way, thus providing a sort of “canonical” set of values for the functional representation. (shrink)

Many philosophers have thought that Tolerance Puzzles can be easily dissolved by adopting some form of counterpart theory, which is roughly the view that being possibly a certain way is having a counterpart that is that way. This chapter shows how standard versions of counterpart theory involve radical departures from standard modal logic (going far beyond Iteration-denial) which we claim are unacceptable, and argues that once counterpart theory is developed in such a way as to avoid such logical revisionism, it (...) has no special capacity to resolve the puzzles. (shrink)

According to the KK-principle, knowledge iterates freely. It has been argued, notably in Greco, that accounts of knowledge which involve essential appeal to normality are particularly conducive to defence of the KK-principle. The present article evaluates the prospects for employing normality in this role. First, it is argued that the defence of the KK-principle depends upon an implausible assumption about the logical principles governing iterated normality claims. Once this assumption is dropped, counter-instances to the principle can be expected to arise. (...) Second, it is argued that even if the assumption is maintained, there are other logical properties of normality which can be expected to lead to failures of KK. Such failures are noteworthy, since they do not depend on either a margins-for-error principle or safety condition of the kinds Williamson appeals to in motivating rejection KK. “Introduction: KK and Being in a Position to Know” Section formulates two versions of the KK-Principle; “Inexact Knowledge and Margins for Error” Section presents a version of Williamson’s margins-for-error argument against it; “Knowledge and Normality” and “Iterated Normality” Sections discuss the defence of the KK-Principle due to Greco and show that it is dependent upon the implausible assumption that the logic of normality ascriptions is at least as strong as K4; finally, “Knowledge in Abnormal Conditions” and “Higher-Order Ignorance Inside the Margins” Sections argue that a weakened version of Greco’s constraint on knowledge is plausible and demonstrate that this weakened constraint will, given uncontentious assumptions, systematically generate counter-instances to the KK-principle of a novel kind. (shrink)

Presentists believe that only present things exist. But opponents insist this view has unacceptable implications: if only present things exist, we can’t express singular propositions about the past, since the obvious propositional constituents don’t exist, nor can we account for temporal passage, or the openness of the future. According to such opponents, and in spite of the apparent ‘common sense’ status of the view, presentism should be rejected on the basis of these unacceptable implications. In this paper, I present and (...) defend a version of presentism that avoids the unacceptable implications. The basic strategy I employ is familiar—I postulate presently existing entities to serve as surrogates for non-present entities—but some of the details of my proposal are more novel, and their application to these problems is certainly novel. One overarching thesis of this paper is that Thisness Presentism is preferable to other versions of presentism since it solves important problems facing standard iterations of the view. And I assume that this is a good positive reason in favour of the underlying thisness ontology. (shrink)

In standard Bayesian probability revision, the adoption of full beliefs (propositions with probability 1) is irreversible. Once an agent has full belief in a proposition, no subsequent revision can remove that belief. This is an unrealistic feature, and it also makes probability revision incompatible with belief change theory, which focuses on how the set of full beliefs is modified through both additions and retractions. This problem in probability theory can be solved in a model that (i) lets the codomain of (...) the probability function be a hyperreal-valued rather than the real-valued closed interval [0, 1], and (ii) identifies the full beliefs as the propositions whose probability is either 1 or infinitesimally smaller than 1. In this model, changes in the probability function will result in changes in the set of full beliefs (belief set), which constitutes a submodel that can be conceived as the “tip of the iceberg” within the larger model that also contains beliefs on lower levels of probability. The patterns of change in the set of full beliefs in this modified Bayesian model coincides with the corresponding pattern in a slightly modified version of AGM revision, which is commonly conceived as the gold standard of (dichotomous) belief change. The modification only concerns the marginal case of revision by an inconsistent input sentence. These results show that probability revision and dichotomous belief change can be unified in one and the same framework, or – if we so wish – that belief change theory can be subsumed under a modified version of probability revision that allows for iterated change and for the removal of full beliefs. (shrink)

The dynamic abstractionist approach to set theory canvassed in Chapter 3 is properly developed. We begin with a plural version of Frege’s Basic Law V. While this law is inconsistent in the ordinary static setting, it becomes consistent when transposed to a dynamic setting. Thus transposed, the law ensures that any objects whatsoever can be used to define a set. This is an intuitive and highly explanatory principle of set formation, which traces its roots back to Cantor. The resulting (...) dynamic approach to set theory justifies all of ordinary ZF set theory and provides an explication of the celebrated iterative conception of sets. (shrink)

Global challenge-led research seeks to contribute to solution-generation for complex problems. Multicultural, multidisciplinary, and multisectoral teams must be capable of operating in highly demanding contexts. This brings with it a swathe of ethical conflicts that require quick and effective solutions that respect both international conventions and cultural diversity. The objective of this article is to describe the process of creating a toolkit designed to support global researchers in navigating these ethical challenges. The process of creating the toolkit embodied the model (...) of ethical research practice that it advocates. Specifically, at the heart of ethical decision-making is consideration of the following: Place, solutions must be relevant to the context in which they are to be used; People, those impacted by the outcomes must be partners in co-creation; Principles, ethical projects must be guided by clear values; and Precedent, the existing evidence-base should guide the project and, in turn, the project should extend the evidence-base. It is the thesis underlying the toolkit that consideration of these 4Ps provides a strong basis for understanding ethical conflicts and allows for the generation of potential solutions. This toolkit has been designed in two phases of collaborative work. More than 200 researchers participated from more than 30 countries and more than 60 different disciplines. This allowed us to develop a model for contextual, dynamic analysis of ethical conflicts in global research that is complementary to traditional codes of ethics. It emphasizes the need to consider ethical analysis as an iterative, reflective, process relevant at all stages of the research journey, including, ultimately, in evaluating the legacy of a project. The toolkit is presented as an open access website to promote universal access. A downloadable “pocket guide” version is also now available in 11 languages. (shrink)