Gregory Wheeler

Modeling a complex phenomena such as the mind presents tremendous computational complexity challenges. Modeling field theory (MFT) addresses these challenges in a non-traditional way. The main idea behind MFT is to match levels of uncertainty of the model (also, a problem or some theory) with levels of uncertainty of the evaluation criterion used to identify that model. When a model becomes more certain, then the evaluation criterion is adjusted dynamically to match that change to the model. This process is called the Dynamic Logic of Phenomena (DLP) for model construction and it mimics processes of the mind and natural evolution. This paper provides a formal description of DLP by specifying its syntax, semantics, and reasoning system. We also outline links between DLP and other logical approaches. Computational complexity issues that motivate this work are presented using an example of polynomial models.

C.P. Snow observed that universities are largely made up of two broad types of people, literary intellectuals and scientists, yet a typical individual of each type is barely able, if able at all, to communicate with his counterpart. Snow's observation, popularized in his 1959 lecture Two Cultures and the Scientific Revolution (reissued by Cambridge 1993), goes some way to explaining the two distinct cultures one hears referred to as "the humanities" and "the sciences." Snow's lecture is a study of these two cultures, their rules, hierarchies, and educational traditions, which raises the following question: to what degree are "the humanities" and "the sciences" a consequence of how we organize and fund modern universities? Rather than a happenstance of interests and temperament, perhaps "humanist" and "scientist" are largely bureaucratic categories

Rabern and Rabern (Analysis 68:105–112 2 ) and Uzquiano (Analysis 70:39–44 4 ) have each presented increasingly harder versions of ‘the hardest logic puzzle ever’ (Boolos The Harvard Review of Philosophy 6:62–65 1 ), and each has provided a two-question solution to his predecessor’s puzzle. But Uzquiano’s puzzle is different from the original and different from Rabern and Rabern’s in at least one important respect: it cannot be solved in less than three questions. In this paper we solve Uzquiano’s puzzle in three questions and show why there is no solution in two. Finally, to cement a tradition, we introduce a puzzle of our own

A bounded formula is a pair consisting of a propositional formula φ in the first coordinate and a real number within the unit interval in the second coordinate, interpreted to express the lower-bound probability of φ. Converting conjunctive/disjunctive combinations of bounded formulas to a single bounded formula consisting of the conjunction/disjunction of the propositions occurring in the collection along with a newly calculated lower probability is called absorption. This paper introduces two inference rules for effecting conjunctive and disjunctive absorption and compares the resulting logical system, called System Y, to axiom System P. Finally, we demonstrate how absorption resolves the lottery paradox and the paradox of the preference.

Focused correlation compares the degree of association within an evidence set to the degree of association in that evidence set given that some hypothesis is true. A difference between the confirmation lent to a hypothesis by one evidence set and the confirmation lent to that hypothesis by another evidence set is robustly tracked by a difference in focused correlations of those evidence sets on that hypothesis, provided that all the individual pieces of evidence are equally, positively relevant to that hypothesis. However, that result depends on a very strong equal relevance condition on individual pieces of evidence. In this essay, we prove tracking results for focused correlation analogous to Wheeler and Scheines’s results but for cases involving unequal relevance. Our result is robust as well, and we retain conditions for bidirectional tracking between incremental confirmation measures and focused correlation.

This paper presents statistical default logic, an expansion of classical (i.e., Reiter) default logic that allows us to model common inference patterns found in standard inferential statistics, including hypothesis testing and the estimation of a populations mean, variance and proportions. The logic replaces classical defaults with ordered pairs consisting of a Reiter default in the first coordinate and a real number within the unit interval in the second coordinate. This real number represents an upper-bound limit on the probability of accepting the consequent of an applied default and that consequent being false. A method for constructing extensions is then defined that preserves this upper bound on the probability of error under a (skeptical) non-monotonic consequence relation.

Modelling a complex phenomenon such as the mind presents tremendous computational complexity challenges. Modelling field theory addresses these challenges in a non-traditional way. The main idea behind MFT is to match levels of uncertainty of the model with levels of uncertainty of the evaluation criterion used to identify that model. When a model becomes more certain, then the evaluation criterion is adjusted dynamically to match that change to the model. This process is called the Dynamic Logic of Phenomena for model construction and it mimics processes of the mind and natural evolution. This paper provides a formal description of DLP by specifying its syntax, semantics, and reasoning system. We also outline links between DLP and other logical approaches. Computational complexity issues that motivate this work are presented using an example of polynomial models.

In the age of big data and a machine epistemology that can anticipate, predict, and intervene on events in our lives, the problem once again is that a few individuals possess the knowledge of how to regulate these activities. But the question we face now is not how to share such knowledge more widely, but rather of how to enjoy the public benefits bestowed by this knowledge without freely sharing it. It is not merely personal privacy that is at stake but a range of unsung benefits that come from ignorance and forgetting, traits that are inherently human and integral to the functioning of our society.

Epistemic naturalism holds that the results or methodologies from the cognitive sciences are relevant to epistemology, and some have maintained that scientific methods are more compatible with externalist theories of justification than with internalist theories. But practically all discussions about naturalized epistemology are framed exclusively in terms of cognitive psychology, which is only one of the cognitive sciences. The question addressed in this essay is whether a commitment to naturalism really does favor externalism over internalism, and we offer reasons for thinking that naturalism in epistemology is compatible with both internalist and externalist conceptions of justification. We also argue that there are some distinctively internalist aims that are currently being studied scientifically and these notions, and others, should be studied by scientific methods

Recent advances in philosophy, artificial intelligence, mathematical psychology, and the decision sciences have brought a renewed focus to the role and interpretation of probability in theories of uncertain reasoning. Henry E. Kyburg, Jr. has long resisted the now dominate Bayesian approach to the role of probability in scientific inference and practical decision. The sharp contrasts between the Bayesian approach and Kyburg's program offer a uniquely powerful framework within which to study several issues at the heart of scientific inference, decision, and reasoning under uncertainty. The commissioned essays for this volume take measure of the scope and impact of Kyburg's views on probability and scientific inference, and include several new and important contributions to the field. Contributors: Gert de Cooman, Clark Glymour, William Harper, Isaac Levi, Ron Loui, Enrique Miranda, John Pollock, Teddy Seidenfeld, Choh Man Teng, Mariam Thalos, Gregory Wheeler, Jon Williamson, and Henry E. Kyburg, Jr.

Classical modal logics, based on the neighborhood semantics of Scott and Montague, provide a generalization of the familiar normal systems based on Kripke semantics. This paper defines AGM revision operators on several first-order monotonic modal correspondents, where each first-order correspondence language is defined by Marc Pauly’s version of the van Benthem characterization theorem for monotone modal logic. A revision problem expressed in a monotone modal system is translated into first-order logic, the revision is performed, and the new belief set is translated back to the original modal system. An example is provided for the logic of Risky Knowledge that uses modal AGM contraction to construct counter-factual evidence sets in order to investigate robustness of a probability assignment given some evidence set. A proof of correctness is given

Two compelling principles, the Reasonable Range Principle and the Preservation of Irrelevant Evidence Principle, are necessary conditions that any response to peer disagreements ought to abide by. The Reasonable Range Principle maintains that a resolution to a peer disagreement should not fall outside the range of views expressed by the peers in their dispute, whereas the Preservation of Irrelevant Evidence Principle maintains that a resolution strategy should be able to preserve unanimous judgments of evidential irrelevance among the peers. No standard Bayesian resolution strategy satisfies the PIE Principle, however, and we give a loss aversion argument in support of PIE and against Bayes. The theory of imprecise probability allows one to satisfy both principles, and we introduce the notion of a set-based credal judgment to frame and address a range of subtle issues that arise in peer disagreements.

In his groundbreaking book, Against Coherence (2005), Erik Olsson presents an ingenious impossibility theorem that appears to show that there is no informative relationship between probabilistic measures of coherence and higher likelihood of truth. Although Olsson's result provides an important insight into probabilistic models of epistemological coherence, the scope of his negative result is more limited than generally appreciated. The key issue is the role conditional independence conditions play within the witness testimony model Olsson uses to establish his result. Olsson maintains that his witness model yields charitable ceteris paribus conditions for any theory of probabilistic coherence. Not so. In fact, Olsson's model, like Bayesian witness models in general, selects a peculiar class of models that are in no way representative of the range of options available to coherence theorists. Recent positive results suggest that there is a way to develop a formal theory of coherence after all. Further, although Bayesian witness models are not conducive to the truth, they are conducive to reliability

Henry Kyburg’s lottery paradox (1961, p. 197) arises from considering a fair 1000 ticket lottery that has exactly one winning ticket. If this much is known about the execution of the lottery it is therefore rational to accept that one ticket will win. Suppose that an event is very likely if the probability of its occurring is greater than 0.99. On these grounds it is presumed rational to accept the proposition that ticket 1 of the lottery will not win. Since the lottery is fair, it is rational to accept that ticket 2 won’t win either—indeed, it is rational to accept for any individual ticket i of the lottery that ticket i will not win. However, accepting that ticket 1 won’t win, accepting that ticket 2 won’t win, . . . , and accepting that ticket 1000 won’t win entails that it is rational to accept that no ticket will win, which entails that it is rational to accept the contradictory proposition that one ticket wins and no ticket wins

Jim Joyce argues for two amendments to probabilism. The first is the doctrine that credences are rational, or not, in virtue of their accuracy or “closeness to the truth” (1998). The second is a shift from a numerically precise model of belief to an imprecise model represented by a set of probability functions (2010). We argue that both amendments cannot be satisfied simultaneously. To do so, we employ a (slightly-generalized) impossibility theorem of Seidenfeld, Schervish, and Kadane (2012), who show that there is no strictly proper scoring rule for imprecise probabilities. The question then is what should give way. Joyce, who is well aware of this no-go result, thinks that a quantifiability constraint on epistemic accuracy should be relaxed to accommodate imprecision. We argue instead that another Joycean assumption — called strict immodesty— should be rejected, and we prove a representation theorem that characterizes all “mildly” immodest measures of inaccuracy.

Michael Dummett famously maintained that analytic philosophy was simply philosophy that followed Frege in treating the philosophy of language as the basis for all other philosophy (1978, 441). But one important insight to emerge from computer science is how difficult it is to animate the linguistic artifacts that the analysis of thought produces. Yet, modeling the effects of thought requires a new skill that goes beyond analysis: procedural literacy. Some of the most promising research in philosophy makes use of a variety of modeling techniques that go beyond basic logic and elementary probability theory. What unifies this approach is a focus on what Alan Perlis called procedural literacy. This essay argues that the future spoils in philosophical research will disproportionally go to those who are procedurally literate.

One goal of normative multi-agent system theory is to formulate principles for normative system change that maintain the rule-like structure of norms and preserve links between norms and individual agent obligations. A central question raised by this problem is whether there is a framework for norm change that is at once specific enough to capture this rule-like behavior of norms, yet general enough to support a full battery of norm and obligation change operators. In this paper we propose an answer to this question by developing a bimodal logic for norms and obligations called NO. A key to our approach is that norms are treated as propositional formulas, and we provide some independent reasons for adopting this stance. Then we define norm change operations for a wide class of modal systems, including the class of NO systems, by constructing a class of modal revision operators that satisfy all the AGM postulates for revision, and constructing a class of modal contraction operators that satisfy all the AGM postulates for contraction. More generally, our approach yields an easily extendable framework within which to work out principles for a theory of normative system change

Dilation occurs when an interval probability estimate of some event E is properly included in the interval probability estimate of E conditional on every event F of some partition, which means that one’s initial estimate of E becomes less precise no matter how an experiment turns out. Critics maintain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. However, two points are often overlooked: (1) knowing that E is stochastically independent of F (for all F in a partition of the underlying state space) is sufficient to avoid dilation, but (2) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence between dilator and dilatee, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models

In this essay we advance the view that analytical epistemology and artificial intelligence are complementary disciplines. Both fields study epistemic relations, but whereas artificial intelligence approaches this subject from the perspective of understanding formal and computational properties of frameworks purporting to model some epistemic relation or other, traditional epistemology approaches the subject from the perspective of understanding the properties of epistemic relations in terms of their conceptual properties. We argue that these two practices should not be conducted in isolation. We illustrate this point by discussing how to represent a class of inference forms found in standard inferential statistics. This class of inference forms is interesting because its members share two properties that are common to epistemic relations, namely defeasibility and paraconsistency. Our modeling of standard inferential statistical arguments exploits results from both logical artificial intelligence and analytical epistemology. We remark how our approach to this modeling problem may be generalized to an interdisciplinary approach to the study of epistemic relation.

Statistical Default Logic (SDL) is an expansion of classical (i.e., Reiter) default logic that allows us to model common inference patterns found in standard inferential statistics, e.g., hypothesis testing and the estimation of a population‘s mean, variance and proportions. This paper presents an embedding of an important subset of SDL theories, called literal statistical default theories, into stable model semantics. The embedding is designed to compute the signature set of literals that uniquely distinguishes each extension on a statistical default theory at a pre-assigned error-bound probability.

Yet, in broader terms, formal epistemology is not merely a methodological tool for epistemologists, but a discipline in its own right. On this programmatic view, formal epistemology is an interdisciplinary research program that covers work by philosophers, mathematicians, computer scientists, statisticians, psychologists, operations researchers, and economists who aim to give mathematical and sometimes computational representations of, along with sound strategies for reasoning about, knowledge, belief, judgment and decision making