In this three-part paper, my concern is to expound and defend a conception of science, close to Einstein's, which I call aim-oriented empiricism. I argue that aim-oriented empiricsim has the following virtues. (i) It solve the problem of induction; (ii) it provides decisive reasons for rejecting van Fraassen's brilliantly defended but intuitively implausible constructive empiricism; (iii) it solves the problem of verisimilitude, the problem of explicating what it can mean to speak of scientific progress given that science advances from (...) one false theory to another; (iv) it enables us to hold that appropriate scientific theories, even though false, can nevertheless legitimately be interpreted realistically, as providing us with genuine , even if only approximate, knowledge of unobservable physical entities; (v) it provies science with a rational, even though fallible and non-mechanical, method for the discovery of fundamental new theories in physics. In the third part of the paper I show that Einstein made essential use of aim-oriented empiricism in scientific practice in developing special and general relativity. I conclude by considering to what extent Einstein came explicitly to advocate aim-oriented empiricism in his later years. (shrink)
The pessimistic induction holds that successful past scientific theories are completely false, so successful current ones are completely false too. I object that past science did not perform as poorly as the pessimistic induction depicts. A close study of the history of science entitles us to construct an optimistic induction that would neutralize the pessimistic induction. Also, even if past theories were completely false, it does not even inductively follow that the current theories will also turn (...) out to be completely false because the current theories are more successful and have better birth qualities than the past theories. Finally, the extra success and better birth qualities justify an anti-induction in favor of the present theories. (shrink)
In this paper I adduce a new argument in support of the claim that IBE is an autonomous (indispensable) form of inference, based on a familiar, yet surprisingly, under-discussed, problem for Hume’s theory of induction. I then use some insights thereby gleaned to argue for the (reductionist) claim that induction is really IBE, and draw some normative conclusions.
A Mug's Game? Solving the Problem of Induction with Metaphysical Presuppositions Nicholas Maxwell Emeritus Reader in Philosophy of Science at University College London Email: email@example.com Website: www.nick-maxwell.demon.co.uk Abstract This paper argues that a view of science, expounded and defended elsewhere, solves the problem of induction. The view holds that we need to see science as accepting a hierarchy of metaphysical theses concerning the comprehensibility and knowability of the universe, these theses asserting less and less as we go up (...) the hierarchy. It may seem that this view must suffer from vicious circularity, in so far as accepting physical theories is justified by an appeal to metaphysical theses in turn justified by the success of science. But this is rebutted. A thesis high up in the hierarchy asserts that the universe is such that the element of circularity, just indicated, is legitimate and justified, and not vicious. Acceptance of the thesis is in turn justified without appeal to the success of science. It may seem that the practical problem of induction can only be solved along these lines if there is a justification of the truth of the metaphysical theses in question. It is argued that this demand must be rejected as it stems from an irrational conception of science. (shrink)
Necessity holds that, if a proposition A supports another B, then it must support B. John Greco contends that one can resolve Hume's Problem of Induction only if she rejects Necessity in favor of reliabilism. If Greco's contention is correct, we would have good reason to reject Necessity and endorse reliabilism about inferential justification. Unfortunately, Greco's contention is mistaken. I argue that there is a plausible reply to Hume's Problem that both endorses Necessity and is at least as good (...) as Greco's alternative. Hence, Greco provides a good reason for neither rejecting Necessity nor endorsing inferential reliabilism. (shrink)
In this paper, I consider the pessimistic induction construed as a deductive argument (specifically, reductio ad absurdum) and as an inductive argument (specifically, inductive generalization). I argue that both formulations of the pessimistic induction are fallacious. I also consider another possible interpretation of the pessimistic induction, namely, as pointing to counterexamples to the scientific realist’s thesis that success is a reliable mark of (approximate) truth. I argue that this interpretation of the pessimistic induction fails, too. If (...) this is correct, then the pessimistic induction is an utter failure that should be abandoned by scientific anti-realists. (shrink)
Israel 2004 claims that numerous philosophers have misinterpreted Goodman’s original ‘New Riddle of Induction’, and weakened it in the process, because they do not define ‘grue’ as referring to past observations. Both claims are false: Goodman clearly took the riddle to concern the maximally general problem of “projecting” any type of characteristic from a given realm of objects into another, and since this problem subsumes Israel’s, Goodman formulated a stronger philosophical challenge than the latter surmises.
In 1955, Goodman set out to 'dissolve' the problem of induction, that is, to argue that the old problem of induction is a mere pseudoproblem not worthy of serious philosophical attention. I will argue that, under naturalistic views of the reflective equilibrium method, it cannot provide a basis for a dissolution of the problem of induction. This is because naturalized reflective equilibrium is -- in a way to be explained -- itself an inductive method, and thus renders (...) Goodman's dissolution viciously circular. This paper, then, examines how the old problem of induction crept back in while nobody was looking. (shrink)
In the mid-eighteenth century David Hume argued that successful prediction tells us nothing about the truth of the predicting theory. But physical theory routinely predicts the values of observable magnitudes within very small ranges of error. The chance of this sort of predictive success without a true theory suggests that Hume's argument is flawed. However, Colin Howson argues that there is no flaw and examines the implications of this disturbing conclusion; he also offers a solution to one of the central (...) problems of Western philosophy, the problem of induction. (shrink)
We formalise a notion of dynamic rationality in terms of a logic of conditional beliefs on (doxastic) plausibility models. Similarly to other epistemic statements (e.g. negations of Moore sentences and of Muddy Children announcements), dynamic rationality changes its meaning after every act of learning, and it may become true after players learn it is false. Applying this to extensive games, we “simulate” the play of a game as a succession of dynamic updates of the original plausibility model: the epistemic situation (...) when a given node is reached can be thought of as the result of a joint act of learning (via public announcements) that the node is reached. We then use the notion of “stable belief”, i.e. belief that is preserved during the play of the game, in order to give an epistemic condition for backward induction: rationality and common knowledge of stable belief in rationality. This condition is weaker than Aumann’s and compatible with the implicit assumptions (the “epistemic openness of the future”) underlying Stalnaker’s criticism of Aumann’s proof. The “dynamic” nature of our concept of rationality explains why our condition avoids the apparent circularity of the “backward induction paradox”: it is consistent to (continue to) believe in a player’s rationality after updating with his irrationality. (shrink)
Alvin plantinga and michael slote, Following ayer, Have attempted to formulate the argument from analogy for the existence of other minds as an enumerative induction. Their way of avoiding the 'generalizing from a single case' objection is shown to be fallacious.
Writing on the justification of certain inductive inferences, the author proposes that sometimes induction is justified and that arguments to prove otherwise are not cogent. In the first part he examines the problem of justifying induction, looks at some attempts to prove that it is justified, and responds to criticisms of these proofs. In the second part he deals with such topics as formal logic, deductive logic, the theory of logical probability, and probability and truth.
In a formal theory of induction, inductive inferences are licensed by universal schemas. In a material theory of induction, inductive inferences are licensed by facts. With this change in the conception of the nature of induction, I argue that the celebrated “problem of induction” can no longer be set up and is thereby dissolved. Attempts to recreate the problem in the material theory of induction fail. They require relations of inductive support to conform to an (...) unsustainable, hierarchical empiricism. (shrink)
In this paper we deal with two types of reasoning: induction, and deduction First, we present a unified computational model of deductive reasoning through models, where deduction occurs in five phases: Construction, Integration, Conclusion, Falsification, and Response. Second, we make an attempt, to analyze induction through the same phases. Our aim is an explorative evaluation of the mental processes possibly shared by deductive and inductive reasoning.
Nelson Goodman’s new riddle of induction forcefully illustrates a challenge that must be confronted by any adequate theory of inductive inference: provide some basis for choosing among alternative hypotheses that fit past data but make divergent predictions. One response to this challenge is to distinguish among alternatives by means of some epistemically significant characteristic beyond fit with the data. Statistical learning theory takes this approach by showing how a concept similar to Popper’s notion of degrees of testability is linked (...) to minimizing expected predictive error. In contrast, formal learning theory appeals to Ockham’s razor, which it justifies by reference to the goal of enhancing efficient convergence to the truth. In this essay, I show that, despite their differences, statistical and formal learning theory yield precisely the same result for a class of inductive problems that I call strongly VC ordered , of which Goodman’s riddle is just one example. (shrink)
John Foster presents a clear and powerful discussion of a range of topics relating to our understanding of the universe: induction, laws of nature, and the existence of God. He begins by developing a solution to the problem of induction - a solution whose key idea is that the regularities in the workings of nature that have held in our experience hitherto are to be explained by appeal to the controlling influence of laws, as forms of natural necessity. (...) His second line of argument focuses on the issue of what we should take such necessitational laws to be, and whether we can even make sense of them at all. Having considered and rejected various alternatives, Foster puts forward his own proposal: the obtaining of a law consists in the causal imposing of a regularity on the universe as a regularity. With this causal account of laws in place, he is now equipped to offer an argument for theism. His claim is that natural regularities call for explanation, and that, whatever explanatory role we may initially assign to laws, the only plausible ultimate explanation is in terms of the agency of God. Finally, he argues that, once we accept the existence of God, we need to think of him as creating the universe by a method which imposes regularities on it in the relevant law-yielding way. In this new perspective, the original nomological-explanatory solution to the problem of induction becomes a theological-explanatory solution. The Divine Lawmaker is bold and original in its approach, and rich in argument. The issues on which it focuses are among the most important in the whole epistemological and metaphysical spectrum. (shrink)
Watkins proposes a neo-Popperian solution to the pragmatic problem of induction. He asserts that evidence can be used non-Inductively to prefer the principle that corroboration is more successful over all human history than that, Say, Counter-Corroboration is more successful either over this same period or in the future. Watkins's argument for rejecting the first counter-Corroborationist alternative is beside the point, However, As whatever is the best strategy over all human history is irrelevant to the pragmatic problem of induction (...) since we are not required to act in the past, And his argument for rejecting the second presupposes induction. (shrink)
Philosophers of mathematics commonly distinguish between explanatory and non-explanatory proofs. An important subclass of mathematical proofs are proofs by induction. Are they explanatory? This paper addresses the question, based on general principles about explanation. First, a recent argument for a negative answer is discussed and rebutted. Second, a case is made for a qualified positive take on the issue.
Future Logic is an original and wide-ranging treatise of formal logic. It deals with deduction and induction, of categorical and conditional propositions, involving the natural, temporal, extensional, and logical modalities. This is the first work ever to strictly formalize the inductive processes of generalization and particularization, through the novel methods of factorial analysis, factor selection and formula revision. This is the first work ever to develop a formal logic of the natural, temporal and extensional types of conditioning (as distinct (...) from logical conditioning), including their production from modal categorical premises. (shrink)
The limited aim here is to explain what John Dewey might say about the formulation of the grue example. Nelson Goodman’s problem of distinguishing good and bad inductive inferences is an important one, but the grue example misconstrues this complex problem for certain technical reasons, due to ambiguities that contemporary logical theory has not yet come to terms with. Goodman’s problem is a problem for the theory of induction and thus for logical theory in general. Behind the whole discussion (...) of these issues over the last several decades is a certain view of logic hammered out by Russell, Carnap, Tarski, Quine, and many others. Goodman’s nominalism hinges in essential ways on a certain view of formal logic with an extensional quantification theory at its core. This raises many issues, but the one issue most germane here is the conception of predicates ensconced in this view of logic. (shrink)
My aim is to evaluate a new realist strategy for addressing the pessimistic induction, Ludwig Fahrbach’s (Synthese 180:139–155, 2011) appeal to the exponential growth of science. Fahrbach aims to show that, given the exponential growth of science, the history of science supports realism. I argue that Fahrbach is mistaken. I aim to show that earlier generations of scientists could construct a similar argument, but one that aims to show that the theories that they accepted are likely true. The problem (...) with this is that from our perspective on the history of science we know their argument is flawed. Consequently, we should not be impressed or persuaded by Fahrbach’s argument. Fahrbach has failed to identify a difference that matters between today’s theories and past theories. But realists need to find such a difference if they are to undermine the pessimistic induction. (shrink)
The standard backward-induction reasoning in a game like the centipede assumes that the players maintain a common belief in rationality throughout the game. But that is a dubious assumption. Suppose the first player X didn't terminate the game in the first round; what would the second player Y think then? Since the backwards-induction argument says X should terminate the game, and it is supposed to be a sound argument, Y might be entitled to doubt X's rationality. Alternatively, Y (...) might doubt that X believes Y is rational, or that X believes Y believes X is rational, or Y might have some higher-order doubt. X’s deviant first move might cause a breakdown in common belief in rationality, therefore. Once that goes, the entire argument fails. The argument also assumes that the players act rationally at each stage of the game, even if this stage could not be reached by rational play. But it is also dubious to assume that past irrationality never exerts a corrupting influence on present play. However, the backwards-induction argument can be reconstructed for the centipede game on a more secure basis.1 It may be implausible to assume a common belief in rationality throughout the game, however the game might go, but the argument requires less than this. The standard idealisations in game theory certainly allow us to assume a common belief in rationality at the beginning of the game. They also allow us to assume this common belief persists so long as no one makes an irrational move. That is enough for the argument to go through. (shrink)
The paper sketches an ontological solution to an epistemological problem in the philosophy of science. Taking the work of Hilary Kornblith and Brian Ellis as a point of departure, it presents a realist solution to the Humean problem of induction, which is based on a scientific essentialist interpretation of the principle of the uniformity of nature. More specifically, it is argued that use of inductive inference in science is rationally justified because of the existence of real, natural kinds of (...) things, which are characterized as such by the essential properties which all members of a kind necessarily possess in common. The proposed response to inductive scepticism combines the insights of epistemic naturalism with a metaphysical outlook that is due to scientific realism. (shrink)
The model function for induction of Goodmans's composite predicate "Grue" was examined by analysis. Two subpredicates were found, each containing two further predicates which are mutually exclusive (green and blue, observed before and after t). The rules for the inductive processing of composite predicates were studied with the more familiar predicate "blellow" (blue and yellow) for violets and primroses. The following rules for induction were violated by processing "grue": From two subpredicates only one (blue after t) appears in (...) the conclusion. As a statement about a future and unobserved condition this subpredicate, however, is not projectible for induction whereas the only suitable predicate (green before t) does not show up in the conclusion. In a disjunction "a v b" where "a" is true and "b" false the disjunction is true. When, however, the only true component is dropped, what remains is necessarily false. An analogous mistake may be observed in the processing of "grue", where the only true component (green) was dropped in the conclusion. - As a potent criterion for correct inductions a check of the necessity of the conclusions is recommended. (shrink)
I first examine John Duns Scotus’ view of contingency, pure possibility, and created possibilities, and his version of the celebrated distinction between ordained and absolute power. Scotus’ views on ethical natural law and his account of induction are characterised, and their dependence on the preceding doctrines detailed. I argue that there is an inconsistency in his treatments of the problem of induction and ethical natural law. Both proceed with God’s contingently willed creation of a given order of laws, (...) which can be revoked and replaced with a new order of laws. In the case of ethical natural law God promulgated the Decalogue, for example; in the case of nature, there are physical laws that can be known by induction. Scotus exalts the freedom of God and the mutability of ethical natural law in order to explain exceptions to it disclosed by revelation (for example, the Old Testament command to Abraham to kill Isaac). Yet he treats ethical natural laws as (mostly) not universal and immutable. In contrast, he holds that we can arrive at knowledge of the universal and immutable laws of nature, except for those regularities that result from free will. Finally, I present several ways of characterising this tension between Scotus’ doctrines. (shrink)
On the basis of the distinction between logical and factual probability, epistemic justification is distinguished from logical justification of induction. It is argued that, contrary to the accepted interpretation of Hume, Hume believes that inductive inferences are epistemically legitimate and justifiable. Hence the beliefs arrived at via (correct) inductive inferences are rational beliefs. According to this interpretation, Hume is not a radical skeptic about induction.
The problem of induction : East and West -- The later Nyaya solution -- The method of generalization : Vyaptigrahopayah -- Counterfactual reasoning : Tarkah -- Universal based extraordinary perception : Samanyalaksanapratyaksa -- Earlier views of adjuncts : Upadhivadah -- The accepted view of adjuncts : Upadhivadasiddhantah -- Classification of adjuncts : Upadhivibhagah -- Sriharsa's Khandanakhandakhadyam on pervasion -- Selected passages from Prabhacandra's Prameyakamalamartanda on critique of pervasion and inference -- Selections from Dharmakirti's Nyayabindu on non-perception as a probans.
We conceive of a player in dynamic games as a set of agents, which are assigned the distinct tasks of reasoning and node-specific choices. The notion of agent connectedness measuring the sequential stability of a player over time is then modeled in an extended type-based epistemic framework. Moreover, we provide an epistemic foundation for backward induction in terms of agent connectedness. Besides, it is argued that the epistemic independence assumption underlying backward induction is stronger than usually presumed.
We present a framework that provides a logic for science by generalizing the notion of logical (Tarskian) consequence. This framework will introduce hierarchies of logical consequences, the first level of each of which is identified with deduction. We argue for identification of the second level of the hierarchies with inductive inference. The notion of induction presented here has some resonance with Popper's notion of scientific discovery by refutation. Our framework rests on the assumption of a restricted class of structures (...) in contrast to the permissibility of classical first-order logic. We make a distinction between deductive and inductive inference via the notions of compactness and weak compactness. Connections with the arithmetical hierarchy and formal learning theory are explored. For the latter, we argue against the identification of inductive inference with the notion of learnable in the limit. Several results highlighting desirable properties of these hierarchies of generalized logical consequence are also presented. (shrink)
The Pessimistic Induction (PI) states: most past scientific theories were radically mistaken; therefore, current theories are probably similarly mistaken. But mistaken in what way? On the usual understanding, such past theories are false. However, on widely held views about reference and presupposition, many theoretical claims of previous scientific theories are neither true nor false. And if substantial portions of past theories are truth-valueless, then the PI leads to semantic antirealism. But most current philosophers of science reject semantic antirealism. So (...) PI proponents face a difficult choice: accept either semantic antirealism or an unorthodox position on reference and presupposition. (shrink)
According to the so-called “Folk Theorem” for repeated games, stable cooperative relations can be sustained in a Prisoner’s Dilemma if the game is repeated an indefinite number of times. This result depends on the possibility of applying strategies that are based on reciprocity, i.e., strategies that reward cooperation with subsequent cooperation and punish defectionwith subsequent defection. If future interactions are sufficiently important, i.e., if the discount rate is relatively small, each agent may be motivated to cooperate by fear of retaliation (...) in the future. For finite games, however, where the number of plays is known beforehand, there is a backward induction argument showing that rational agents will not be able to achieve cooperation. On behalf of the Hobbesian “Foole”, who cannot see any advantage in cooperation, Gregory Kavka (1983, 1986) has presented an argument that significantly extends the range of the backward induction argument. He shows that, for the backward induction argument to be effective, it is not necessary that the precise number of future interactions be known. It is sufficient that there is a known definite upper bound on the number of interactions. A similar argument is developed by John W. Carroll (1987). We will here question the assumption of a known upper bound. When the assumption is made precise in the way needed for the argument to go through, its apparent plausibility evaporates. We then offer a reformulation of the argument, based on weaker, and more plausible, assumptions. (shrink)
The use of the material theory of induction to vindicate a scientist's claims of evidential warrant is illustrated with the cases of Einstein's thermodynamic argument for light quanta of 1905 and his recovery of the anomalous motion of Mercury from general relativity in 1915. In a survey of other accounts of inductive inference applied to these examples, I show that, if it is to succeed, each account must presume the same material facts as the material theory and, in addition, (...) some general principle of inductive inference not invoked by the material theory. Hence these principles are superfluous and the material theory superior in being more parsimonious. (shrink)
We consider the desirability, or otherwise, of various forms of induction in the light of certain principles and inductive methods within predicate uncertain reasoning. Our general conclusion is that there remain conflicts within the area whose resolution will require a deeper understanding of the fundamental relationship between individuals and properties.
How does one become an effective teacher? What can be done to stem high attrition rates among beginning teachers? While many teachers are left to ?sink or swim? in their first year?learning by trial and error, there remain a number of outstanding examples of collaboration and collegiality in teacher induction programs. Analysis of the most exemplary teacher induction programs from Australia, Britain, Canada, France, Germany, Japan, New Zealand and the United States revealed common attributes and exceptional features. The (...) most successful teacher induction programs reported here include opportunities for experts and neophytes to learn together in a supportive environment promoting time for collaboration, reflection and acculturation into the profession of teaching. Furthermore, several practices unique to specific regions were highlighted. These included extended internship programs, specially trained mentors, comprehensive inservice training and reduced teaching assignments for beginning teachers with an emphasis on assistance rather than assessment. (shrink)
Discussion on whether Hume's treatment of induction is descriptive or normative has usually centred on Hume's negative argument, somewhat neglecting the positive argument. In this paper, I will buck this trend, focusing on the positive argument. First, I argue that Hume's positive and negative arguments should be read as addressing the same issues (whether normative or descriptive). I then argue that Hume's positive argument in the Enquiry is normative in nature; drawing on his discussion of scepticism in Section 12 (...) of the Enquiry, I explain a framework by which he provides what I call consequent justification for our inductive practices in his positive argument. Based on this, I argue that his negative argument in the Enquiry should similarly be read as normative in nature. (shrink)
According to a standard objection to the use of backward induction in extensive-form games with perfect information, backward induction (BI) can only work if the players are confident that each player is resiliently rational - disposed to act rationally at each possible node that the game can reach, even at the nodes that will certainly never be reached in actual play - and also confident that these beliefs in the players’ future resilient rationality are robust, i.e. that they (...) would be kept come what may, whatever evidence of irrationality would by then transpire concerning past performance of the players. Since both resiliency and robustness assumptions are extremely strong and their appropriateness as idealizations is quite problematic, it has been argued (by Binmore, Reny, Bicchieri, Pettit and Sugden, among others) that BI is an indefensible procedure. Therefore, we need not be worried that BI can be used to justify seemingly counter-intuitive game solutions. I show, however, that there is a restricted class of extensive-form games in which BI solutions can be defended without assuming resiliency or robustness. For these ”BI-terminating games” (= games in which BI moves always terminate the play, at each choice node), to defend BI solutions, it is enough to make confidence-in-rationality assumptions concerning actual play; stipulations about various counterfactual developments are unnecessary. For this class of games, then, the standard objection to BI is inapplicable. At the same time, however, it will transpire that the class in question contains some well-known games, such as the Centipede in its different versions, in which BI recommends a seemingly unreasonable behaviour. (shrink)
In this paper we argue that an existing theory of concepts called dynamic frame theory, although not developed with that purpose in mind, allows for the precise formulation of a number of problems associated with induction from a single instance. A key role is played by the distinction we introduce between complete and incomplete dynamic frames, for incomplete frames seem to be very elegant candidates for the format of the background knowledge used in induction from a single instance. (...) Furthermore, we show how dynamic frame theory provides the terminology to discuss the justification and the fallibility of incomplete frames. In the Appendix, we give a formal account of incomplete frames and the way these lead to induction from a single instance. (shrink)
If one believes, as Hume did, that all events are loose and separate, then the problem of induction is probably insoluble. Anything could happen. But if one thinks, as scientific essentialists do, that the laws of nature are immanent in the world, and depend on the essential natures of things, then there are strong constraints on what could possibly happen. Given these constraints, the problem of induction may be soluble. For these constraints greatly strengthen the case for conceptual (...) and theoretical conservatism, and rule out Goodmanesque inferences based on alternative descriptions of the world. This may not, in itself, solve the problem, but it significantly changes its nature. (shrink)
The backward induction argument purports to show that rational and suitably informed players will defect throughout a finite sequence of prisoner's dilemmas. It is supposed to be a useful argument for predicting how rational players will behave in a variety of interesting decision situations. Here, I lay out a set of assumptions defining a class of finite sequences of prisoner's dilemmas. Given these assumptions, I suggest how it might appear that backward induction succeeds and why it is actually (...) fallacious. Then, I go on to consider the consequences of adopting a stronger set of assumptions. Focusing my attention on stronger sets that, like the original, obey the informedness condition, I show that any supplementation of the original set that preserves informedness does so at the expense of forcing rational participants in prisoner's dilemma situations to have unexpected beliefs, ones that threaten the usefulness of backward induction. (shrink)
The problem of finding sufficient doxastic conditions for backward induction in games of perfect information is analyzed in a syntactic framework with subjunctive conditionals. This allows to describe the structure of the game by a logical formula and consequently to treat beliefs about this structure in the same way as beliefs about rationality. A backward induction and a non-Nash equilibrium result based on higher level belief in rationality and the structure of the game are derived.
The purpose of this note is to present a simplification of the system of arithmetical axioms given in previous work; specifically, it is shown how the induction principle can in fact be obtained from the remaining axioms, without the need of explicit postulation. The argument might be of more general interest, beyond the specifics of the proposed axiomatization, as it highlights the interaction of the notion of Dedekind-finiteness and the induction principle.
A cornerstone of game theory is backward induction, whereby players reason backward from the end of a game in extensive form to the beginning in order to determine what choices are rational at each stage of play. Truels, or three-person duels, are used to illustrate how the outcome can depend on (1) the evenness/oddness of the number of rounds (the parity problem) and (2) uncertainty about the endpoint of the game (the uncertainty problem). Since there is no known endpoint (...) in the latter case, an extension of the idea of backward induction is used to determine the possible outcomes. The parity problem highlights the lack of robustness of backward induction, but it poses no conflict between foundational principles. On the other hand, two conflicting views of the future underlie the uncertainty problem, depending on whether the number of rounds is bounded (the players invariably shoot from the start) or unbounded (they may all cooperate and never shoot, despite the fact that the truel will end with certainty and therefore be effectively bounded). Some real-life examples, in which destructive behavior sometimes occurred and sometimes did not, are used to illustrate these differences, and some ethical implications of the analysis are discussed. (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)
The present studies tested the hypothesis that strong assumptions about within-category homogeneity impede children’s recognition of the inductive value of diverse samples of evidence. In Study 1a, children (7-year-olds) and adults were randomly assigned to receive a prime emphasizing within-category variability, a prime emphasizing within-category similarities, or to not receive a prime. Only following the variability prime, children demonstrated a reliable preference for evaluating diverse over nondiverse samples to determine whether there is support for a category-wide generalization. Adults demonstrated a (...) robust preference for diverse samples in all conditions. These effects extended beyond the specific categories included in the prime, as well as to multiple types of test questions. Study 1b demonstrated that priming variability leads children to select diverse samples only when doing so is informative for induction. Implications for conceptual development are discussed. (shrink)
It is common to assume that the problem of induction arises only because of small sample sizes or unreliable data. In this paper, I argue that the piecemeal collection of data can also lead to underdetermination of theories by evidence, even if arbitrarily large amounts of completely reliable experimental and observational data are collected. Specifically, I focus on the construction of causal theories from the results of many studies (perhaps hundreds), including randomized controlled trials and observational studies, where the (...) studies focus on overlapping, but not identical, sets of variables. Two theorems reveal that, for any collection of variables V, there exist fundamentally different causal theories over V that cannot be distinguished unless all variables are simultaneously measured. Underdetermination can result from piecemeal measurement, regardless of the quantity and quality of the data. Moreover, I generalize these results to show that, a priori, it is impossible to choose a series of small (in terms of number of variables) observational studies that will be most informative with respect to the causal theory describing the variables under investigation. This final result suggests that scientific institutions may need to play a larger role in coordinating differing research programs during inquiry. (shrink)
The traditional form of the backward induction argument, which concludes that two initially rational agents would always defect, relies on the assumption that they believe they will be rational in later rounds. Philip Pettit and Robert Sugden have argued, however, that this assumption is unjustified. The purpose of this paper is to reconstruct the argument without using this assumption. The formulation offered concludes that two initially rational agents would decide to always defect, and relies only on the weaker assumption (...) that they do not believe they will not be rational in later rounds. The argument employs the idea that decisions justify revocable presumptions about behaviour. (shrink)
Two justifications of backward induction (BI) in generic perfect information games are formulated using Bonanno's (1992; Theory and Decision 33, 153) belief systems. The first justification concerns the BI strategy profile and is based on selecting a set of rational belief systems from which players have to choose their belief functions. The second justification concerns the BI path of play and is based on a sequential deletion of nodes that are inconsistent with the choice of rational belief functions.
The nature of concepts -- Generalizations as hierarchical -- Perceiving first-level causal connections -- Conceptualizing first-level causal connections -- The structure of inductive reasoning -- Galileo's kinematics -- Newton's optics -- The methods of difference and agreement -- Induction as inherent in conceptualization -- The birth of celestial physics -- Mathematics and causality -- The power of mathematics -- Proof of Kepler's theory -- The development of dynamics -- The discovery of universal gravitation -- Discovery is proof -- Chemical (...) elements and atoms -- The kinetic theory of gases -- The unification of chemistry -- The method of proof -- Misapplying the inductive method -- Abandoning the inductive method -- Physics as inherently mathematical -- The science of philosophy -- An end-and a new beginning. (shrink)
We provide eductive foundations for the concept of forward induction, in the class of games with an outside option. The formulation presented tries to capture in a static notion the rest point of an introspective process, achievable from some restricted preliminary beliefs. The former requisite is met by requiring the rest point to be a Nash equilibrium that yields a higher payoff than the outside option. With respect to the beliefs, we propose the Incentive Dominance Criterion. Players should consider (...) one action more likely than another whenever the former is better than getting the outside option for more conjectures over his rival's actions. We apply this model to the case where the subgame is a coordination game with a conflict between payoff dominance and risk dominance. Our results provide support for dominance solvability, but not for Van Damme's notion of forward induction. We show how the forward induction logic helps to select the Pareto dominant equilibrium. This is the case whenever player 1's act of giving up the outside option reverses the incentive dominance relations among 1's pure actions in the subgame. (shrink)