It is not my claim that the ‘L¨ owenheim-Skolem paradox’ is an antinomy in formal logic; but I shall argue that it is an antinomy, or something close to it, in philosophy of language. Moreover, I shall argue that the resolution of the antinomy—the only resolution that I myself can see as making sense—has profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language.

This paper concerns Tarski's use of the term "model" in his 1936 paper "On the Concept of Logical Consequence." Against several of Tarski's recent defenders, I argue that Tarski employed a non-standard conception of models in that paper. Against Tarski's detractors, I argue that this non-standard conception is more philosophically plausible than it may appear. Finally, I make a few comments concerning the traditionally puzzling case of Tarski's $\omega$-rule example.

In an earlier paper, I claimed that one version of Putnam's model-theoretic argument against realism turned on a subtle, but philosophically significant, mathematical mistake. Recently, Luca Bellotti has criticized my argument for this claim. This paper responds to Bellotti's criticisms.

Skolem's Paradox involves a seeming conflict between two theorems from classical logic. The Löwenheim Skolem theorem says that if a first order theory has infinite models, then it has models whose domains are only countable. Cantor's theorem says that some sets are uncountable. Skolem's Paradox arises when we notice that the basic principles of Cantorian set theory—i.e., the very principles used to prove Cantor's theorem on the existence of uncountable sets—can themselves be formulated as a collection of first order sentences. (...) How can the very principles which prove the existence of uncountable sets be satisfied by a model which is itself only countable? How can a countable model satisfy the first order sentence which says that there are uncountably many mathematical objects—e.g., uncountably many real numbers? (shrink)

This paper concerns Tarski’s use of the term “model” in his 1936 paper “On the Concept of Logical Consequence.” Against several of Tarski’s recent defenders, I argue that Tarski employed a non-standard conception of models in that paper. Against Tarski’s detractors, I argue that this non-standard conception is more philosophically plausible than it may appear. Finally, I make a few comments concerning the traditionally puzzling case of Tarski’s ω-rule example.

This paper discusses two combinatorial problems in stability theory. First we prove a partition result for subsets of stable models: for any A and B, we can partition A into |B|$^{ |B|, then we try to find A' $\subset$ A and B' $\subset$ B such that |A'| is as large as possible, |B'| is as small as possible, and A' $\&2ADD;$ $\underset{B'}$ B. We prove some positive results in this direction, and we discuss the optimality of these results under ZFC (...) + GCH. (shrink)

The Lowenheim-Skolem theorems say that if a first-order theory has infinite models, then it has models which are only countably infinite. Cantor's theorem says that some sets are uncountable. Together, these theorems induce a puzzle known as Skolem's Paradox: the very axioms of set theory which prove the existence of uncountable sets can be satisfied by a merely countable model. ;This dissertation examines Skolem's Paradox from three perspectives. After a brief introduction, chapters two and three examine several formulations of (...) Skolem's Paradox in order to disentangle the roles which set theory, model theory, and philosophy play in these formulations. In these chapters, I accomplish three things. First, I clear up some of the mathematical ambiguities which have all too often infected discussions of Skolem's Paradox. Second, I isolate a key assumption upon which Skolem's Paradox rests, and I show why this assumption has to be false. Finally, I argue that there is no single explanation as to how a countable model can satisfy the axioms of set theory ;In chapter four, I turn to a second puzzle. Why, even though philosophers have known since the early 1920's that Skolem's Paradox has a relatively simple technical solution, have they continued to find this paradox so troubling? I argue that philosophers' attitudes towards Skolem's Paradox have been shaped by the acceptance of certain, fairly specific, claims in the philosophy of language. I then tackle these philosophical claims head on. In some cases, I argue that the claims depend on an incoherent account of mathematical language. In other cases, I argue that the claims are so powerful that they render Skolem's Paradox trivial. In either case, though, examination of the philosophical underpinnings of Skolem's Paradox renders that paradox decidedly unparadoxical. ;Finally, in chapter five, I turn away from "generic" formulations of Skolem's Paradox to examine Hilary Putnam's "model-theoretic argument against realism." I show that Putnam's argument involves mistakes of both the mathematical and the philosophical variety, and that these two types of mistake are closely related. Along the way, I clear up some of the mutual charges of question begging which have characterized discussions between Putnam and his critics. (shrink)

In this dissertation we study two-cardinal phenomena—both of the admitting cardinals variety and of the Chang’s Conjecture variety—under the assumption that all our models have stable theories. All our results involve two, relatively widely accepted generalizations of the traditional definitions in this area. First, we allow the relevant subsets of our models to be picked out by (perhaps infinitary) partial types; second we consider δ-cardinal problems as well as two-cardinal problems.

I present two generalizations of Putnam's model-theoretic argument against realism. The first replaces Putnam's model theory with some new, and substantially simpler, model theory, while the second replaces Putnam's model theory with some more accessible results from astronomy. By design, both of these new arguments fail. But the similarities between these new arguments and Putnam's original arguments illuminate the latter's overall structure, and the flaws in these new arguments highlight the corresponding flaws in Putnam's arguments.

In his new paper, “Eligibility and Inscrutability,” J. R. G. Williams presents a surprising new challenge to David Lewis’ theory of interpretation. Although Williams frames this challenge primarily as a response to Lewis’ criticisms of Putnam’s model-theoretic argument, the challenge itself goes to the heart of Lewis’ own account of interpretation. Further, and leaving Lewis’ project aside for a moment, Williams’ argument highlights some important—and some fairly general—points concerning the relationship between model theory and semantic determinacy.

This paper discusses two combinatorial problems in stability theory. First we prove a partition result for subsets of stable models: for any A and B, we can partition A into |B |<κ(T ) pieces, Ai | i < |B |<κ(T ) , such that for each Ai there is a Bi ⊆ B where |Bi| < κ(T ) and Ai..

According to a growing trend in theoretical neuroscience, the human perceptual system is akin to a Bayesian machine. The aim of this article is to clearly articulate the claims that perception can be considered Bayesian inference and that the brain can be considered a Bayesian machine, some of the epistemological challenges to these claims; and some of the implications of these claims. We address two questions: (i) How are Bayesian models used in theoretical neuroscience? (ii) From the use of Bayesian (...) models in theoretical neuroscience, have we learned or can we hope to learn that perception is Bayesian inference or that the brain is a Bayesian machine? From actual practice in theoretical neuroscience, we argue for three claims. First, currently Bayesian models do not provide mechanistic explanations; instead they are useful devices for predicting and systematizing observational statements about people's performances in a variety of perceptual tasks. That is, currently we should have an instrumentalist attitude towards Bayesian models in neuroscience. Second, the inference typically drawn from Bayesian behavioural performance in a variety of perceptual tasks to underlying Bayesian mechanisms should be understood within the three-level framework laid out by David Marr ( [1982] ). Third, we can hope to learn that perception is Bayesian inference or that the brain is a Bayesian machine to the extent that Bayesian models will prove successful in yielding secure and informative predictions of both subjects' perceptual performance and features of the underlying neural mechanisms. (shrink)

These are chapters from a book forthcoming from MIT Press. Comments to the author at [email protected] would be most welcome. Still time for changes.

The focus of this commentary is what Andy Clark takes to be the most groundbreaking of the philosophical import of the ‘bidirectional hierarchical model of brain functions’, namely, the claim that perceptual representations represent probabilities. This is what makes his account Bayesian and this is a philosophical or theoretical conclusion that neuroscientists and psychologists are also quick and happy to draw. My claim is that nothing in the ‘bidirectional hierarchical models of brain functions’ implies that perceptual representations are probabilistic, (...) or that they represent or ‘encode probability density distributions’. There is a much more parsimonious way of describing the representations in the bidirectional hierarchical model of brain functions: they attribute properties to objects (or to the perceived scene) that are not fully determinate. (shrink)

Causal Bayes nets (CBNs) can be used to model causal relationships up to whole mechanisms. Though modelling mechanisms with CBNs comes with many advantages, CBNs might fail to adequately represent some biological mechanisms because—as Kaiser (2016) pointed out—they have problems with capturing relevant spatial and structural information. In this paper we propose a hybrid approach for modelling mechanisms that combines CBNs and cellular automata. Our approach can incorporate spatial and structural information while, at the same time, it comes with (...) all the merits of a CBN representation of mechanisms. (shrink)

According to [Bayesian] models” in cognitive neuroscience, says a recent textbook, “the human mind behaves like a capable data scientist”. Do they? That is to say, do such model show we are rational? I argue that Bayesian models of cognition, perhaps surprisingly, do not and indeed cannot, show that we are Bayesian-rational. The key reason is that such models appeal to approximations, a fact that carries significant implications. After outlining the argument, I critique two responses, seen in recent cognitive (...) neuroscience. One says that the mind can be seen as approximately Bayes-rational, while the other reconceives norms of rationality. (shrink)

Causal Bayes nets have been developed in philosophy, statistics, and computer sciences to provide a formalism to represent causal structures, to induce causal structure from data and to derive predictions. Causal Bayes nets have been used as psychological theories in at least two ways. They were used as rational, computational models of causal reasoning and they were used as formal models of mental causal models. A crucial assumption made by them is the Markov condition, which informally states that variables are (...) independent of other variables that are not their direct or indirect effects conditional on their immediate causes. Whether people’s inferences conform to the causal Markov and the faithfulness condition has recently been investigated empirically. A review of respective research indicates that inferences frequently violate these conditions. This finding challenges some uses of causal Bayes nets in psychology. They entail that causal Bayes nets may not be appropriate to derive predictions for causal model theories of causal reasoning. They also question whether causal Bayes nets as a rational model are empirically descriptive. They do not challenge, however, causal Bayes nets as normative models and their usage as formal models of causal reasoning. (shrink)

Many of our decisions pertain to causal systems. Nevertheless, only recently has it been claimed that people use causal models when making judgments, decisions and predictions, and that causal Bayes nets allow us to formally describe these inferences. Experimental research has been limited to simple, artificial problems, which are unrepresentative of the complex dynamic systems we successfully deal with in everyday life. For instance, in social interactions, we can explain the actions of other's on the fly and we can generalize (...) from limited observations to predict future actions and their consequences. Our main argument is that none of these inferences (i.e., induction, generalization, explanation, and prediction) can be achieved without causal reasoning. As a case in point we use the popular television series desperate housewives and show how causal Bayes nets are able to explain the inferences made in social contexts. Crucially, causal Bayes nets also allow us to understand why we can infer so much from so little when making sense of a protagonist's behavior. (shrink)

Bayes' Theorem is a simple mathematical formula used for calculating conditional probabilities. It figures prominently in subjectivist or Bayesian approaches to epistemology, statistics, and inductive logic. Subjectivists, who maintain that rational belief is governed by the laws of probability, lean heavily on conditional probabilities in their theories of evidence and their models of empirical learning. Bayes' Theorem is central to these enterprises both because it simplifies the calculation of conditional probabilities and because it clarifies significant features of subjectivist position. Indeed, (...) the Theorem's central insight — that a hypothesis is confirmed by any body of data that its truth renders probable — is the cornerstone of all subjectivist methodology. (shrink)

Causal Bayes nets (CBNs) provide one of the most powerful tools for modelling coarse-grained type-level causal structure. As in other fields (e.g., thermodynamics) the question arises how such coarse-grained characterisations are related to the characterisation of their underlying structure (in this case: token-level causal relations). Answering this question meets what is called a “coherence-requirement” in the reduction debate: How are different accounts of one and the same system (or kind of system) related to each other. We argue that CBNs as (...) tools for type-level causal inference are abstract enough to roughly fit any current token-level theory of causation as long as certain modelling assumptions are satisfied, but accounts of actual causation, i.e. accounts that attempt to infer token-causation based on CBNs, for the very same reason, face certain limitations. (shrink)

A study of stage structured model of fish population is presented. This model focuse on the anchovy population in the Bay of Biscay (Engraulis encrasicolus L.) is presented. The method of study is based on an intermediate complexity mathematical model, taking into account the spatialisation, the environmental conditions and the stage-structure of the fishes. First, to test the model, we show mathematical properties, such as unicity of the solution of structural stability. Then we provide numerical simulations, (...) to validate the model and to test the dynamics according to the variations of the parameters. (shrink)

While Bayesian models have been applied to an impressive range of cognitive phenomena, methodological challenges have been leveled concerning their role in the program of rational analysis. The focus of the current article is on computational impediments to probabilistic inference and related puzzles about empirical confirmation of these models. The proposal is to rethink the role of Bayesian methods in rational analysis, to adopt an independently motivated notion of rationality appropriate for computationally bounded agents, and to explore broad conditions under (...) which Bayesian agents would be rational. The proposal is illustrated with a characterization of costs inspired by thermodynamics. (shrink)

People are adept at inferring novel causal relations, even from only a few observations. Prior knowledge about the probability of encountering causal relations of various types and the nature of the mechanisms relating causes and effects plays a crucial role in these inferences. We test a formal account of how this knowledge can be used and acquired, based on analyzing causal induction as Bayesian inference. Five studies explored the predictions of this account with adults and 4-year-olds, using tasks in which (...) participants learned about the causal properties of a set of objects. The studies varied the two factors that our Bayesian approach predicted should be relevant to causal induction: the prior probability with which causal relations exist, and the assumption of a deterministic or a probabilistic relation between cause and effect. Adults’ judgments (Experiments 1, 2, and 4) were in close correspondence with the quantitative predictions of the model, and children’s judgments (Experiments 3 and 5) agreed qualitatively with this account. (shrink)

On a widely held view, episodes of inner speech provide at least one way in which we become conscious of our thoughts. However, it can be argued, on the one hand, that consciousness of thoughts in virtue of inner speech presupposes interpretation of the simulated speech. On the other hand, the need for such self-interpretation seems to clash with distinctive first-personal characteristics that we would normally ascribe to consciousness of one’s own thoughts: a special reliability; a lack of conscious ambiguity (...) and incomprehensibility; and a sense of causal agency. I try to resolve this puzzle by proposing an account for the requisite self-interpretation of inner speech in terms of Bayesian probabilistic inference. Drawing on “perceptual loop” accounts of speech control, I argue that such interpretive probabilistic inferences are used for the control of inner speech, and that as a consequence of this function, they are biased toward the correct interpretations. I conclude by showing how this model can explain the first-personal characteristics of consciousness of one’s own thoughts. In the case of the sense of causal agency, the resulting explanation yields novel accounts for “audible thoughts” and thought insertion. (shrink)

Some naturalistic philosophers of mind subscribing to the predictive processing theory of mind have adopted a realist attitude towards the results of Bayesian cognitive science. In this paper, we argue that this realist attitude is unwarranted. The Bayesian research program in cognitive science does not possess special epistemic virtues over alternative approaches for explaining mental phenomena involving uncertainty. In particular, the Bayesian approach is not simpler, more unifying, or more rational than alternatives. It is also contentious that the Bayesian approach (...) is overall better supported by the empirical evidence. So, to develop philosophical theories of mind on the basis of a realist interpretation of results from Bayesian cognitive science is unwarranted. Naturalistic philosophers of mind should instead adopt an anti-realist attitude towards these results and remain agnostic as to whether Bayesian models are true. For continuing on with an exclusive focus and praise of Bayes within debates about the predictive processing theory will impede progress in philosophical understanding of scientific practice in computational cognitive science as well as of the architecture of the mind. (shrink)

A fundamental problem in science is how to make logical inferences from scientific data. Mere data does not suffice since additional information is necessary to select a domain of models or hypotheses and thus determine the likelihood of each model or hypothesis. Thomas Bayes’ Theorem relates the data and prior information to posterior probabilities associated with differing models or hypotheses and thus is useful in identifying the roles played by the known data and the assumed prior information when making (...) inferences. Scientists, philosophers, and theologians accumulate knowledge when analyzing different aspects of reality and search for particular hypotheses or models to fit their respective subject matters. Of course, a main goal is then to integrate all kinds of knowledge into an all-encompassing worldview that would describe the whole of reality. A generous description of the whole of reality would span, in the order of complexity, from the purely physical to the supernatural. These two extreme aspects of reality are bridged by a nonphysical realm, which would include elements of life, man, consciousness, rationality, mental and mathematical abstractions, etc. An urgent problem in the theory of knowledge is what science is and what it is not. Albert Einstein’s notion of science in terms of sense perception is refined by defining operationally the data that makes up the subject matter of science. It is shown, for instance, that theological considerations included in the prior information assumed by Isaac Newton is irrelevant in relating the data logically to the model or hypothesis. In addition, the concepts of naturalism, intelligent design, and evolutionary theory are critically analyzed. Finally, Eugene P. Wigner’s suggestions concerning the nature of human consciousness, life, and the success of mathematics in the natural sciences is considered in the context of the creative power endowed in humans by God. (shrink)

The advent of formal definitions of the simplicity of a theory has important implications for model selection. But what is the best way to define simplicity? Forster and Sober ([1994]) advocate the use of Akaike's Information Criterion (AIC), a non-Bayesian formalisation of the notion of simplicity. This forms an important part of their wider attack on Bayesianism in the philosophy of science. We defend a Bayesian alternative: the simplicity of a theory is to be characterised in terms of Wallace's (...) Minimum Message Length (MML). We show that AIC is inadequate for many statistical problems where MML performs well. Whereas MML is always defined, AIC can be undefined. Whereas MML is not known ever to be statistically inconsistent, AIC can be. Even when defined and consistent, AIC performs worse than MML on small sample sizes. MML is statistically invariant under 1-to-1 re-parametrisation, thus avoiding a common criticism of Bayesian approaches. We also show that MML provides answers to many of Forster's objections to Bayesianism. Hence an important part of the attack on Bayesianism fails. (shrink)

Despite their popularity, relatively scant attention has been paid to the upshot of Bayesian and predictive processing models of cognition for views of overall cognitive architecture. Many of these models are hierarchical ; they posit generative models at multiple distinct "levels," whose job is to predict the consequences of sensory input at lower levels. I articulate one possible position that could be implied by these models, namely, that there is a continuous hierarchy of perception, cognition, and action control comprising levels (...) of generative models. I argue that this view is not entailed by a general Bayesian/predictive processing outlook. Bayesian approaches are compatible with distinct formats of mental representation. Focusing on Bayesian approaches to motor control, I argue that the junctures between different types of mental representation are places where the transitivity of hierarchical prediction may be broken, and I consider the upshot of this conclusion for broader discussions of cognitive architecture. (shrink)

Bayesian methods are ubiquitous in contemporary observational cosmology. They enter into three main tasks: cross-checking datasets for consistency; fixing constraints on cosmological parameters; and model selection. This article explores some epistemic limits of using Bayesian methods. The first limit concerns the degree of informativeness of the Bayesian priors and an ensuing methodological tension between task and task. The second limit concerns the choice of wide flat priors and related tension between parameter estimation and model selection. The Dark Energy (...) Survey and its recent Year 1 results illustrate both these limits concerning the use of Bayesianism. (shrink)

On a widely held view, episodes of inner speech provide at least one way in which we become conscious of our thoughts. However, it can be argued, on the one hand, that consciousness of thoughts in virtue of inner speech presupposes interpretation of the simulated speech. On the other hand, the need for such self-interpretation seems to clash with distinctive first-personal characteristics that we would normally ascribe to consciousness of one’s own thoughts: a special reliability; a lack of conscious ambiguity (...) and incomprehensibility; and a sense of causal agency. I try to resolve this puzzle by proposing an account for the requisite self-interpretation of inner speech in terms of Bayesian probabilistic inference. Drawing on “perceptual loop” accounts of speech control, I argue that such interpretive probabilistic inferences are used for the control of inner speech, and that as a consequence of this function, they are biased toward the correct interpretations. I conclude by showing how this model can explain the first-personal characteristics of consciousness of one’s own thoughts. In the case of the sense of causal agency, the resulting explanation yields novel accounts for “audible thoughts” and thought insertion. (shrink)

Jeffrey conditioning is said to provide a more general method of assimilating uncertain evidence than Bayesian conditioning. We show that Jeffrey learning is merely a particular type of Bayesian learning if we accept either of the following two observations: – Learning comprises both probability kinematics and proposition kinematics. – What can be updated is not the same as what can do the updating; the set of the latter is richer than the set of the former. We address the problem of (...) commutativity and isolate commutativity from invariance upon conditioning on conjunctions. We also present a disjunctive model of Bayesian learning which suggests that Jeffrey conditioning is better understood as providing a method for incorporating unspecified but certain evidence rather than providing a method for incorporating specific but uncertain evidence. The results also generalize over many other subjective probability update rules, such as those proposed by Field and Gallow. (shrink)

Oaksford & Chater's (O&C's) account of deductive reasoning is parsimonious at a local level (because a rational model is used to explain a wide range of behavior) and at a global level (because their Bayesian approach connects to other areas of research). Their emphasis on environmental structure is especially important, and the power of their approach is seen at both the computational and algorithmic levels.

Bayesian learning theory and evolutionary theory both formalize adaptive competition dynamics in possibly high‐dimensional, varying, and noisy environments. What do they have in common and how do they differ? In this paper, we discuss structural and dynamical analogies and their limits, both at a computational and an algorithmic‐mechanical level. We point out mathematical equivalences between their basic dynamical equations, generalizing the isomorphism between Bayesian update and replicator dynamics. We discuss how these mechanisms provide analogous answers to the challenge of adapting (...) to stochastically changing environments at multiple timescales. We elucidate an algorithmic equivalence between a sampling approximation, particle filters, and the Wright‐Fisher model of population genetics. These equivalences suggest that the frequency distribution of types in replicator populations optimally encodes regularities of a stochastic environment to predict future environments, without invoking the known mechanisms of multilevel selection and evolvability. A unified view of the theories of learning and evolution comes in sight. (shrink)

It is “well known” that in linear models: (1) testable constraints on the marginal distribution of observed variables distinguish certain cases in which an unobserved cause jointly influences several observed variables; (2) the technique of “instrumental variables” sometimes permits an estimation of the influence of one variable on another even when the association between the variables may be confounded by unobserved common causes; (3) the association (or conditional probability distribution of one variable given another) of two variables connected by a (...) path or pair of paths with a single common vertex (a trek) can be computed directly from the parameter values associated with each edge in the trek; (4) the association of two variables produced by multiple treks can be computed from the parameters associated with each trek; and (5) the independence of two variables conditional on a third implies the corresponding independence of the sums of the variables over all units conditional on the sums over all units of each of the original conditioning variables. (shrink)

one takes to be the most salient, any pair could be judged more similar to each other than to the third. Goodman uses this second problem to showthat there can be no context-free similarity metric, either in the trivial case or in a scientifically ...

Bayesian models of legal arguments generally aim to produce a single integrated model, combining each of the legal arguments under consideration. This combined approach implicitly assumes that variables and their relationships can be represented without any contradiction or misalignment, and in a way that makes sense with respect to the competing argument narratives. This paper describes a novel approach to compare and ‘average’ Bayesian models of legal arguments that have been built independently and with no attempt to make them (...) consistent in terms of variables, causal assumptions or parameterization. The approach involves assessing whether competing models of legal arguments are explained or predict facts uncovered before or during the trial process. Those models that are more heavily disconfirmed by the facts are given lower weight, as model plausibility measures, in the Bayesian model comparison and averaging framework adopted. In this way a plurality of arguments is allowed yet a single judgement based on all arguments is possible and rational. (shrink)

We propose that children employ specialized cognitive systems that allow them to recover an accurate “causal map” of the world: an abstract, coherent, learned representation of the causal relations among events. This kind of knowledge can be perspicuously understood in terms of the formalism of directed graphical causal models, or “Bayes nets”. Children’s causal learning and inference may involve computations similar to those for learning causal Bayes nets and for predicting with them. Experimental results suggest that 2- to 4-year-old children (...) construct new causal maps and that their learning is consistent with the Bayes net formalism. (shrink)

Contemporary literature in philosophy of science has begun to emphasize the practice of modeling, which differs in important respects from other forms of representation and analysis central to standard philosophical accounts. This literature has stressed the constructed nature of models, their autonomy, and the utility of their high degrees of idealization. What this new literature about modeling lacks, however, is a comprehensive account of the models that figure in to the practice of modeling. This paper offers a new account of (...) both concrete and mathematical models, with special emphasis on the intentions of theorists, which are necessary for evaluating the model-world relationship during the practice of modeling. Although mathematical models form the basis of most of contemporary modeling, my discussion begins with more traditional, concrete models such as the San Francisco Bay model. (shrink)