Using hitherto unpublished manuscripts from the Zermelo Nachlass, I describe the development of the notion of definiteness and the discussion about it, giving a conclusive picture of Zermelo's thoughts up to the late thirties. As it turns out, Zermelo's considerations about definiteness are intimately related to his concept of a Cantorian universe of categorically definable sets that may be considered an inner model of set theory in an ideationally given universe of classes.

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz” in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world of objects that could only be grasped adequately (...) by infinitary means. So the refutation might serve as a final clue to his epistemological credo.Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic. However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”. (shrink)

This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most (...) striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic. (shrink)

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz”(“Relativism in Set Theory and the So-Called Theorem of Skolem”) in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a (...) world of objects that could only be grasped adequately by infinitary means. So the refutation might serve as a final clue to his epistemological credo.Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic (cf. [8]). However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, (cf. [1]) he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”. (shrink)

I defend the thesis that Kantian analytic judgments are about objects (as opposed to concepts) against two challenges raised by recent scholars. First, can it accommodate cases like “A two-sided polygon is two-sided”, where no object really falls under the subject-concept as Kant sees it? Second, is it compatible with Kant’s view that analytic judgments make no claims about objects in the world and that we can know them to be true without going beyond the given concepts? I address these (...) challenges in two steps. First, given Kant’s distinction between an object in general = x from an object of sensible intuition, I argue that analytic judgments are about objects in the former sense, no matter whether the purported objects can be given in our intuition. Second, using Kant’s method of representing certain logical relations of concepts with such figures as circles, I construct a model to show that analytic truths are truths about objects in general = x and yet can be determined solely by the intensional relation between the given concepts plus certain Kantian-logical laws. Analytic truths are thus shown as formal in the Kantian sense that they do not presuppose the purported objects as givable in our intuition. This account of the formality of analytic truths captures Kant’s diagnosis of the Leibnizian illusion that we can make material claims about the world by analytically true judgments. (shrink)

The experience of embodiment may be studied using the rubber hand illusion. Little is known about the cognitive mechanism that elicits the feeling of embodiment. In previous models of the rubber hand illusion, bodily signals are processed sequentially. Such models cannot explain some more recent findings. Carruthers (2013) proposed a multidimensional model of embodiment, in which the processing of embodiment is understood in terms of conceptual hand space. Visual features of hands are represented along several dimensions. The rubber hand (...) illusion is then explained as the erroneous matching of the online representation of the artificial hand to the stored prototype in a space defined by those dimensions. We conducted the first experimental tests to investigate the multidimensional conceptual space account. First, participants performed a series of odd-one-out judgments for triads of hand images (including their own hand) and we then employed multidimensional scaling analyses. We found that a multidimensional model of perceived hand similarity could be fitted to our data. Second, we tested if a multisensory bodily signal manipulation (the rubber hand illusion) influences the position of the viewed artificial hand in hand space. We employed synchronous and asynchronous stroking and found that in the synchronous condition, which elicits the rubber hand illusion, compared to the asynchronous condition, which does not, the artificial hand was closer to the center of hand space; that is, to a prototype hand. We discuss these findings in the context of the Carruthers (2013) conceptual space model as well as other rubber hand illusion models. (shrink)

One central argument in favor of perceptual accounts of emotions concerns recalcitrant emotions: emotions that persist in the face of repudiating judgments. For, it is argued, to understand how the conflict between recalcitrant emotions and judgment falls short of incoherence in judgment, we need to understand recalcitrant emotions to be something like perceptual illusions of value, so that in normal, non-recalcitrant cases emotions are non-illusory perceptions of value. I argue that these arguments fail and that a closer examination of recalcitrant (...) emotions reveals important disanalogies with perception that undermine the perceptual model of emotions. (shrink)

Investigating the influence of perception on the control of visually guided action typically involves controlled experimentation within the laboratory setting. When appropriate, however, behavioral research of this nature may benefit from the use of methods that allow for remote data collection outside of the lab. This study tested the feasibility of using remote data collection methods to explore the influence of perceived target size on visually guided cursor movements using the Ebbinghaus illusion. Participants completed the experiment remotely, using the (...) trackpad of their personal laptop computers. The task required participants to click on a single circular target presented at either the left or right side of their screen as quickly and accurately as possible, or to emphasize speed or accuracy. On each trial the target was either surrounded by small or large context circles, or no context circles. Participants’ judgments of the targets’ perceived size were influenced by the illusion, however, the illusion failed to produce differences in click-point accuracy or movement time. Interestingly, the illusion appeared to affect participants’ movement of the cursor toward the target; more directional changes were made when clicking the Perceived Large version of the illusion compared to the Perceived Small version. These results suggest the planning of the cursor movement may have been influenced by the illusion, while later stages of the movement were not, and cursor movements directed toward targets perceived as smaller required less correction compared to targets perceived as larger. (shrink)

Scientific modelling is a value-laden process: the decisions involved can seldom be made using ‘scientific’ criteria alone, but rather draw on social and ethical values. In this paper, we draw on a body of philosophical literature to analyze a COVID-19 vaccination model, presenting a case study of social and ethical value judgments in health-oriented modelling. This case study urges us to make value judgments in health-oriented models explicit and interpretable by non-experts and to invite public involvement in making them.

The question of whether time is its own best representation is explored. Though there is theoretical debate between proponents of internal models and embedded cognition proponents (e.g. Brooks R 1991 Artificial Intelligence 47 139–59) concerning whether the world is its own best model, proponents of internal models are often content to let time be its own best representation. This happens via the time update of the model that simply allows the model’s state to evolve along with the (...) state of the modeled domain. I argue that this is neither necessary nor advisable. I show that this is not necessary by describing how internal modeling approaches can be generalized to schemes that explicitly represent time by maintaining trajectory estimates rather than state estimates. Though there are a variety of ways this could be done, I illustrate the proposal with a scheme that combines filtering, smoothing and prediction to maintain an estimate of the modeled domain’s trajectory over time. I show that letting time be its own representation is not advisable by showing how trajectory estimation schemes can provide accounts of temporal illusions, such as apparent motion, that pose serious difficulties for any scheme that lets time be its own representation. (shrink)

Scientific modelling is a value-laden process: the decisions involved can seldom be made using 'scientific' criteria alone, but rather draw on social and ethical values. In this paper, we draw on a body of philosophical literature to analyze a COVID-19 vaccination model, presenting a case study of social and ethical value judgments in health-oriented modelling. This case study urges us to make value judgments in health-oriented models explicit and interpretable by non-experts and to invite public involvement in making them.

Dual-systems frameworks propose that moral judgments are derived from both an immediate emotional response, and controlled/rational cognition. Recently Cushman (2013) proposed a new dual-system theory based on model-free and model-based reinforcement learning. Model-free learning attaches values to actions based on their history of reward and punishment, and explains some deontological, non-utilitarian judgments. Model-based learning involves the construction of a causal model of the world and allows for far-sighted planning; this form of learning fits well with (...) utilitarian considerations that seek to maximize certain kinds of outcomes. I present three concerns regarding the use of model-free reinforcement learning to explain deontological moral judgment. First, many actions that humans find aversive from model-free learning are not judged to be morally wrong. Moral judgment must require something in addition to model-free learning. Second, there is a dearth of evidence for central predictions of the reinforcement account—e.g., that people with different reinforcement histories will, all else equal, make different moral judgments. Finally, to account for the effect of intention within the framework requires certain assumptions which lack support. These challenges are reasonable foci for future empirical/theoretical work on the model-free/model-based framework. (shrink)

Problems of definition and identification in the integrated evolutionary model of sociopathy are suggested by Schoenfeld's (1974) criticism of the field of race differences in intelligence. Moral judgments by those labeled primary and secondary sociopaths may offer a way to validate the assumptions of the model.

Recent studies by experimental philosophers demonstrate puzzling asymmetries in people’s judgments about intentional action, leading many philosophers to propose that normative factors are inappropriately influencing intentionality judgments. In this paper, I present and defend the Deep Self Model of judgments about intentional action that provides a quite different explanation for these judgment asymmetries. The Deep Self Model is based on the idea that people make an intuitive distinction between two parts of an agent’s psychology, an Acting Self that (...) contains the desires, means-end beliefs, and intentions that are the immediate causal source of an agent’s actions, and a Deep Self, which contains an agent’s stable and central psychological attitudes, including the agent’s values, principles, life goals, and other more fundamental attitudes. The Deep Self Model proposes that when people are asked to make judgments about whether an agent brought about an outcome intentionally, in addition to standard criteria proposed in traditional models, people also assess an additional ‘Concordance Criterion’: Does the outcome concord with the psychological attitudes of the agent’s Deep Self? I show that the Deep Self Model can explain a very complex pattern of judgment asymmetries documented in the experimental philosophy literature, and does so in a way that has significant advantages over competing models. (shrink)

Can we ever experimentally disentangle phenomenal consciousness from the cognitive accessibility inherent to conscious reports? In this commentary, we suggest that (1) Block's notion of phenomenal consciousness remains intractably entangled with the need to obtain subjective reports about it; and (2) many experimental paradigms suggest that the intuitive notion of a rich but non-reportable phenomenal world is, to a large extent illusory – in a sense that requires clarification.

Rational choice models are characterized by the image of the self-interested Homo economicus. The role of moral concerns, which may involve a concern for others' welfare in people's judgments and choices, questions the descriptive validity of such models. Increasing evidence of a role for perceived moral obligation within the expectancy-value-based theory of reasoned action and the theory of planned behavior indicates the importance of moral-normative influences in social behavior. In 2 studies, the influence of moral judgments on attitudes toward food (...) produced with the use of genetic engineering techniques and toward meat consumption is addressed. The reasons participants provide for their moral judgments indicate some foci of their moral concerns. The results of both studies corroborate earlier findings that perceived moral obligation has independent effects on behavioral intentions; they also provide evidence that such judgments may affect attitudes themselves. The results are discussed in relation to the need for attitude-behavior models to reflect the role of moral evaluations in judgment and choice. (shrink)

The authors challenge the reigning “causal power framework” as an explanation for whether a particular outcome was actually caused by a specific potential cause. They test a new measure of causal attribution in two experiments by embedding the measure within the Structure Induction model of Singular Causation (SISC, Stephan & Waldmann, 2016).

The hermeneutical model of illusion, just as that of projection, has always been part of the psychoanalytic views of religion. The author presents a brief critical summary on this subject, and underlines that in relational psychoanalysis, the concept of illusion refers not to religion as such, but to the subjective experiences of desire and relatedness, that is, the source of the desire for God in man. Because of personal conflicts and their outcome, besides illusions one encounters also in such (...) experiences, disillusion, disappointment, and even delusion. The author, while challenging the views of many scholars taking part in this debate: a) maintains that psychoanalytic interpretation is not concerned with the question of religious truth but with the formation and transformation of the process of believing; b) calls for special attention to the fact when speaking of religious representation, the focus is on the process rather than on the objects represented; c) raises his criticism at the often used expression, "unconscious representation of God", because according to him, religion gives a name to the object of desire only when placed at the conscious and cultural level. Coherent with his basic distinction between conscious religious behaviour and the deep psychological pre-conditions, the author underlines the differences between the unconscious processes of desire and the religious concepts, particularly, between the representation of God and the concept of God; between the parental imagos and the transcendent God; and between the capacity to "believe in anything at all" and religious faith. The author concludes that in the illusion model, it is suggested that as part of religious maturity, one could construct the representation of God as something that is deeply connected with primary objects, while at the same time, assuming the capacity to take the necessary distance from personal desires and projections. (shrink)

In this article, we try to clarify some of the issues raised by S. C. Draine, A. G. Greenwald, and M. R. Banaji concerning our investigation into the gender bias in fame judgments . First, we did not test the general hypothesis and did not draw the general conclusion that Drain et al. suggest we did. Second, we did not reject M. R. Banaji and A. G. Greenwald's assumptions about the familiarity of male and female names in the fame judgment (...) task, but we showed how one could have come to reject it using a widespread measurement model for the process dissociation procedure. Third, we argue that the processes which Draine et al. suggest should also be included in the measurement model we used are probably negligible, and if they are not, then the validity of the results of a number of fame judgment experiments must be called into question. In general, however, we agree with what seems to be the main message ofM. R. Banaji and A. G. Greenwald's research, namely, that social categories have to be considered whenever priming is investigated within a social domain. (shrink)

When attempting to predict future events, people commonly rely on historical data. One psychological characteristic of judgmental forecasting of time series, established by research, is that when people make forecasts from series, they tend to underestimate future values for upward trends and overestimate them for downward ones, so-called trend-damping. Events in a time series can be experienced sequentially, or they can also be retrospectively viewed simultaneously, not experienced individually in real time. In one experiment, we studied the influence of (...) presentation mode on two sorts of judgment: predictions of the next event and estimation of the average value of all the events in the presented series. Participants' responses in dynamic mode were anchored on more recent events than in static mode for all types of judgment but with different consequences; hence, dynamic presentation improved prediction accuracy, but not estimation. These results are not anticipated by existing theoretical accounts; we develop and present an agent-based model—the adaptive anchoring model —to account for the difference between processing sequences of dynamically and statically presented stimuli. ADAM captures how variation in presentation mode produces variation in responses in both forecasting and judgment tasks. ADAM's model predictions for the forecasting and judgment tasks fit better with the response data than a linear-regression time series model. Moreover, ADAM outperformed autoregressive-integrated-moving-average and exponential-smoothing models, while neither of these models accounts for people's responses on the average estimation task. (shrink)