Stove, and David Armstrong. Williams argued in The Ground of Induction that it is logically true that one form of probabilistic inductive inference is sound and that this is logically demonstrable in the theory of probability. Stove reiterated the argument with a few reformulations and corrections four decades later. Williams held that induction is a reasonable method. By this he intended not only that it is characterized by ordinary sagacity. Williams , The specific form of induction favored by Williams and Stove is now known as inverse inference ; inference to a characteristic of a population based on premises about that population see the taxonomy in section 3.

Williams and Stove focus on inverse inferences about relative frequency. In particular on inferences of the form:. Williams, followed by Stove, sets out to show that it is necessarily true that the inference from i to ii has high probability:. Given a fair sized sample, then, from any [large, finite] population, with no further material information, we know logically that it very probably is one of those which [approximately] match the population, and hence that very probably the population has a composition similar to that which we discern in the sample.

This is the logical justification of induction. Williams and Stove recognize that induction may depend upon context and also upon the nature of the traits and properties to which it is applied. Williams' initial argument was simple and persuasive. It turns out, however, to have subtle and revealing difficulties.

In response to these difficulties, Stove modified and weakened the argument, but this response may not be sufficient. There is in addition the further problem that the sense of necessity that founds the inferences is not made precise and becomes increasingly stressed as the argument plays out. There are two principles on which the i to ii inference depends: First is the proportional syllogism C8 of section 3.

Second is a rule relating the frequency of a trait in a population to its frequency in samples from that population:. The proof of the Frequency Principle depends upon the fact that the relative frequencies of a trait in samples from such a population are normally distributed with mean equal to the frequency of the trait in the population and dispersion standard deviation that diminishes with the sizes of the population and the samples.

Now let P be a symmetrical probability. Williams assumes symmetry without explicit notice. Given a large population X and a k -sample S of appropriate size from X , in which the relative frequency of the trait R is r , the content of the premise i above can be expressed in two premises:.

## The Problem of Induction (Stanford Encyclopedia of Philosophy/Summer Edition)

We might like to reason in this way, and Williams did reason in this way, but as Stove pointed out in The Rationality of Induction , 65 it ignores the failure of monotonicity. Inductive inference in general and inductive conditional probabilities in particular are not monotonic : adding premises may change a good induction to a bad one and adding conditions to a conditional probability may change, and sometimes reduce, its value. Here, 3 depends on Premise B but suppresses mention of it, thus failing to respect the requirement to take account of all available and relevant evidence.

Maher , Thus the addition of premise B to the condition of 3 might decrease the probability that S 0 resembles the population X : 3b might be low while 3a is high. Conditional probability contrasts with the deductive logic of the material conditional in this respect:. Stove's response to this difficulty was to point out that neither he nor Williams had ever claimed that every inductive inference, nor even every instance of the i to ii inference, was necessarily highly probable.

All that was needed on Stove's view to establish Williams's thesis was to provide one case of values for r, X, S and R for which the inference holds necessarily. This would, Stove claimed, show that at least one inductive inference was necessarily rational. Stove , chapter VII provided examples of specific values for the above parameters that he argued do license the inference. Maher pointed out that the argument depends upon tacit assumptions about the prior probabilities of different populations and their constitutions, and that when this is taken account of the conclusions no longer follow deductively.

Scott Campbell continued the discussion. Williams' original argument when expressed in general terms is simple and seductive: It is a combinatorial fact that the relative frequency of a trait in a large population is close to its relative frequency in most large samples from that population. The proportional syllogism is a truth of probability theory: In the symmetrical case relative frequency equals probability.

From these it looks to be a necessary truth that the probability of a trait in a large population is close to its relative frequency in that population. We have seen that and why the consequence does not follow. Various efforts at weakening the original Williams thesis have been more or less successful. It is in any event plausible that there are at least some examples of inductions for which some form of the Williams thesis is true, but the thesis emerges from this dialectic considerably weakened.

About one-third of What is a Law of Nature Armstrong is devoted to stating and supporting three rationalistic criticisms of what Armstrong calls the regularity theory of law. Armstrong argues against all forms of the regularity theory; laws, on his view, are necessary connections of universals that neither depend nor supervene on the course of worldly events but determine, restrict, and govern those events. The law statement, a linguistic assertion, must in his view be distinguished from the law itself.

Armstrong's rationalism does not lead him, as it did Williams and Stove, to see the resolution of the problem of induction as a matter of demonstrating that induction is necessarily a rational procedure:. I add that not merely is it the case that induction is rational, but it is a necessary truth that it is so. Armstrong , Armstrong does not argue for this principle; it is a premise of an argument to the conclusion that regularity views imply the inevitability of inductive skepticism; the view, attributed to Hume, that inferences from the observed to the unobserved are not rational Armstrong , The problem of induction for Armstrong is to explain why the rationality of induction is a necessary truth Armstrong , One of these is that laws of nature are objective natural necessities and, in particular, that they are necessary connections of universals.

IBE, as its name suggests, is an informal and non-metric form of likelihood methods. IBE is clearly more general than simple enumerative induction, can compare and evaluate competing inductions, and can fill in supportive hypotheses not themselves instances of enumerative induction. Armstrong's affinity for IBE should not lead one to think that he shares other parts of Harman's views on induction.

Such instantiations are states of affairs in their own right. As concerns the problem of induction, the need to explain why inductive inferences are necessarily rational, one part of Armstrong's resolution of the problem can be seen as a response to the challenge put sharply by Goodman: Which universal generalizations are supported by their instances?

Armstrong holds that necessary connections of universals, like N F, G , are lawlike, supported by their instances, and, if true, laws of nature. It remains to show how and why we come to believe these laws. The traditional problem of induction as Hume formulated it concerned what we now know as universal inference see the taxonomy in section 3. The premise. The first of these forms, universal inference, can be codified or schematized by means of two definitions and two principles:.

A simple example, due to C. Hempel, shows that all is not as simple as it might at first appear: By Nicod's principle. Thus if the equivalence principle is to obtain, Nicod's principle cannot be a necessary condition of inductive support, though it may be sufficient. The difficulty is endemic; the structure of logical equivalents may differ, but that of instances cannot. And this is, or at least seems, paradoxical; that a non-raven lends support to a hypothesis about the color of ravens is highly implausible.

The paradox resides in the conflict of this counterintuitive result with our strong intuitive attachment to enumerative induction, both in everyday life and in the methodology of science. The initial resolution of this dilemma was proposed by C. Hempel who credits discussion with Nelson Goodman. Assume first that we ignore all the background knowledge we bring to the question, such as that there are very many things that are either ravens or are not black, and that we look strictly at the truth conditions of the premise this is neither a raven nor black and the supported hypothesis all ravens are black.

The hypothesis says is equivalent to. This hypothesis partitions the world into three exclusive and exhaustive classes of things: non-black ravens, black ravens, and non-ravens. Any member of the first class falsifies the hypothesis.

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Each member of the other two classes confirms it. A non-black non-raven is a member of the third class and is thus a confirming instance. If this seems implausible it is because we in fact do not, as assumed, ignore the context in which the question is raised. We know before considering the inference that there are some black ravens and that there are many more non-ravens, many of which are not black. Observing, for example, a white shoe thus tells us nothing about the colors of ravens that we don't already know, and since sound induction is ampliative, good inductions should increase our knowledge.

If we didn't know that many non-ravens are not black, the observation of a non-black, non-raven would increase our knowledge. On the other hand, we don't know whether any of the unobserved ravens are not black, i. Taken by itself, the statement that the given object is neither black nor a raven confirms the hypothesis that everything that is not a raven is not black as well as the hypothesis that everything that is not black is not a raven. We tend to ignore the former hypothesis because we know it to be false from abundant other evidence—from all the familiar things that are not ravens but are black.

Goodman , The important lesson of the paradox of the ravens and the Hempel-Goodman resolution of it is that inductive inference is sensitive to background information and context. What looks to be a good induction when considered out of context and in isolation turns out not to be so when the context, including background knowledge, is taken into account. The inductive inference from. Recent discussion of the paradox continues and improves on the Hempel-Goodman account by making explicit, and thus licit, the suppressed evidence. Further development, along generally Bayesian lines, generalizes the earlier approach by defining comparative and quantitative concepts of support capable of differentiating support for the two hypotheses in question.

We return to the matter in discussing objective Bayesian approaches to induction below. Suppose that at time t we have observed many emeralds to be green and no emeralds to be any other color. We thus have evidence statements. Then we have also the evidence statements. Hence the same observations support incompatible hypotheses about emeralds to be observed after t ; that they are green and that they are blue.

That the definition of grue includes a time parameter is sometimes advanced as a criticism of the definition. The question here is whether inductive inference should be relative to the language in which it is formulated. Deductive inference is relative in this way as is Carnapian inductive logic. The questions that are raised by the ravens paradox concern the nature of this support and the role of context; not that the nature of the hypothesis itself can be called into question.

The grue paradox, on the other hand, presents us with a quite different question; here is a generalization of appropriate form that is clearly not, indeed apparently cannot, be supported by its instances. That is Goodman's new riddle of induction. But see Mill's remark cited at the beginning of this article where the riddle is anticipated. The old, or traditional problem of induction was to justify induction; to show that induction, typically universal and singular predictive predictive inference, leads always, or in an important proportion of cases, from true premises to true conclusions.

This problem, says Goodman, is, as Hume demonstrated, insoluble, and efforts to solve it are at best a waste of time. We have been looking at the wrong problem; it is only a careless reading of Hume that prevented us from seeing this. Once we see the difficulty, more homely examples than the grue hypothesis are easy to come by:. It is not the least of Goodman's accomplishments to have shown that three questions all issue from the same new riddle:.

Goodman's own response to the new riddle was that those generalizations that are supported by their instances involve predicates that have a history of use in prediction. Such predicates Goodman called projectible. The project of the Carnapian logic of confirmation was to put inductive reasoning on the sure path of a science; to give a unified and objective account of inductive inference including clear rules of procedure, in close analogy to deductive inference see Carnap LFP, section The Hempel-Goodman resolution of the ravens paradox, according to which reference to context may be essential to induction, threatens to undermine this enterprise before it properly begins: If Hempel and Goodman have it right, a is a black raven may confirm all ravens are black in one context and not in another.

This relativity produces yet another problem of induction: How can the objectivity of inductive inference be assured given that it depends upon context? Context dependence must in this regard be distinguished from non-monotonicity: Monotonicity and its contrary concern relations among inductive arguments or inferences; a sound inductive argument can be converted into an unsound argument by adding premises. Context dependence, on the other hand, means that one and the same argument may be sound in one context and not in another.

Context dependence, but not non-monotonicity, entails relativity and loss of objectivity. It is useful to distinguish discursive context, such as that there are many more black things than ravens, from non-discursive context. Hume gives us a famous and striking example of the latter:. THN Something like this must be the right account.

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As concerns discursive context, objective Bayesianism seeks to eliminate or ameliorate the relativity illustrated and exemplified in the Ravens paradox first by supplementing the Nicod principle with a definition or necessary and sufficient condition of inductive support. This is done in terms of one of several appropriate objective probabilities, governed by normative principles. Carnap's measures discussed in section 4. Given such a probability, P , there are a number of alternative definitions of support.

Support principle.

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Branden Fitelson and James Hawthorne undermine the relativity entailed by context dependence by making of conditional probability a three-place function, in which probability is conditioned on evidence and context, the latter to include also background knowledge. This permits the distinction of support relative to i some background knowledge, ii our actual, present background knowledge, iii tautological or necessary background knowledge and iv any background knowledge whatever.

Rudner's argument was simple:. Rudner , 2.

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Sufficiency in such a decision will and should depend upon the importance of getting it right or wrong. The argument is not restricted to scientific inductions; it shows as well that our everyday inferences depend inevitably upon value judgments; how much evidence one collects depends upon the importance of the consequences of the decision. Isaac Levi, in responding to Rudner's claim, and to later formulations of it, distinguished cognitive values from other sorts of values; moral, aesthetic, and so on.

Levi , 43—46 Of course the scientist qua scientist, that is to say in his scientific activity, makes judgments and commitments of cognitive value, but he need not, and in many instances should not, allow other sorts of values fame, riches to weigh upon his scientific inductions.

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See also Jeffrey, for a related response to Rudner's argument. What is in question is the separation of practical reason from theoretical reason. Rudner denies the distinction; Levi does too, but distinguishes practical reason with cognitive ends from other sorts. Recent pragmatic accounts of inductive reasoning are even more radical. Following Ramsey a and Savage , they subsume inductive reasoning under practical reason; reason that aims at and ends in action.

These and their successors, such as Jeffrey , define partial belief on the basis of preferences; preferences among possible worlds for Ramsey, among acts for Savage, and among propositions for Jeffrey. Preferences are in each case highly structured. In all cases beliefs as such are theoretical entities, implicitly defined by more elaborate versions of the pragmatic principle that agents or reasonable agents act or should act in ways they believe will satisfy their desires: If we observe the actions and know the desires preferences we can then interpolate the beliefs.

In any given case the actions and desires will fit distinct, even radically distinct, beliefs, but knowing more desires and observing more actions should, by clever design, let us narrow the candidates. In all these theories the problem of induction is a problem of decision, in which the question is which action to take, or which wager to accept. The pragmatic principle is given a precise formulation in the injunction to act so as to maximize expected utility, to perform that action, A i , among the possible alternatives, that maximizes.

One significant advantage of treating induction as a matter of utility maximization is that the cost of gathering more information, of adding to the evidence for an inductive inference, can be factored into the decision. Put very roughly, the leading idea is to look at gathering evidence as an action on its own.

Suppose that you are facing a decision among acts A i , and that you are concerned only about the occurrence or non-occurrence of a consequence S. The principle of utility maximization directs you to choose that act A i that maximizes. Suppose further that you have the possibility of investigating to see if evidence E , for or against S , obtains. Assume further that this investigation is cost-free. Then should you investigate and find E to be true, utility maximization would direct you to choose that act A i that maximizes utility when your beliefs are conditioned on E :.

Hence if your prior strength of belief in the evidence E is P E , you should choose to maximize the weighted average. About this, several brief remarks:. Notice that the utility of investigation depends upon your beliefs about your future beliefs and desires, namely that you believe now that following the investigation you will maximize utility and update your beliefs. Investigation in the actual world is normally not cost-free. It may take time, trouble and money, and is sometimes dangerous.

A general theory of epistemic utility should consider these factors. Good proved that in the cost-free case U A i can never exceed U E A i and that when the utilities of outcomes are distinct the latter always exceeds the former Skyrms , chapter 4. The question of bad evidence is critical. The evidence gathered might take you further from the truth. Think of drawing a succession of red balls from an urn containing predominantly blacks.

The author would like to thanks to Patrick Maher for helpful comments, and the editors would like to thank Wolfgang Swarz for his suggestions for improvement. The contemporary notion of induction 2. Hume on induction 2. Probability and induction 3. Bayesianism and subjectivism 4. Paradoxes, the new riddle of induction and objectivity 5. Knowledge, values and evaluation 6. This is not to denigrate the leading authority on English vocabulary—until the middle of the previous century induction was understood to be what we now know as enumerative induction or universal inference ; inference from particular inferences: a 1 , a 2 , …, a n are all F s that are also G , to a general law or principle All F s are G The problem of induction was, until recently, taken to be to justify this form of inference; to show that the truth of the premise supported, if it did not entail, the truth of the conclusion.

A few simple counterexamples to the OED definition may suggest the increased breadth of the contemporary notion: There are good inductions with general premises and particular conclusions: All observed emeralds have been green. Therefore, the next emerald to be observed will be green. There are valid deductions with particular premises and general conclusions: New York is east of the Mississippi.

Delaware is east of the Mississippi. Therefore, everything that is either New York or Delaware is east of the Mississippi. But this cannot be a merely descriptive endeavor; accurate description of these operations entails also a considerable normative component, for, as Hume puts it, [o]ur reason [to be taken here quite generally, to include the imagination] must be consider'd as a kind of cause, of which truth is the natural effect; but such-a-one as by the irruption of other causes, and by the inconstancy of our mental powers, may frequently be prevented.

Hume THN, The account must thus not merely describe what goes on in the mind, it must also do this in such a way as to show that and how these mental activities lead naturally, if with frequent exceptions, to true belief see Loeb for further discussion of these questions. THN, 89 And were this premise to be established by reasoning, that reasoning would be either deductive or probabilistic i. The several definitions offered in Enquiries concerning Human Understanding and concerning the Principles of Morals EHU, 60 make this explicit: [W]e may define a cause to be an object, followed by another, and where all objects similar to the first are followed by objects similar to the second.

Or, in other words, where, if the first object had not been, the second never had existed. Another definition defines a cause to be: an object followed by another, and whose appearance always conveys the thought to that other. It is equally clear that the epistemic force of this inference, what Hume calls the necessary connection between the premises and the conclusion, does not reside in the premises alone: All observed F s have also been G s, and a is an F , do not imply a is a G. Hume's account of causal inference raises the problem of induction in an acute form: One would like to say that good and reliable inductions are those that follow the lines of causal necessity; that when All observed F s have also been G s, is the manifestation in experience of a causal connection between F and G , then the inference All observed F s have also been G s, a is an F , Therefore, a , not yet observed, is also a G , is a good induction.

Ramsey a, A satisfactory resolution of the problem of induction would account for this objectivity in the distinction between good and bad inductions. Enumerative induction does not realistically lead from premises All observed F s have also been G s a is an F , to the simple assertion Therefore, a , not yet observed, is also a G. The appropriate conclusion is It is therefore probable that, a , not yet observed, is also a G Hume's response to this THN, 89 is to insist that probabilistic connections, no less than simple causal connections, depend upon habits of the mind and are not to be found in our experience of the world.

They may be falsified and rejected or tentatively accepted if corroborated in the absence of falsification by the proper kinds of tests: [A] theory of induction is superfluous. It has no function in a logic of science. LSD, Popper gave two formulations of the problem of induction; the first is the establishment of the truth of a theory by empirical evidence; the second, slightly weaker, is the justification of a preference for one theory over another as better supported by empirical evidence.

This is not because he thinks that there is a sharp division between ordinary knowledge and scientific knowledge, but rather because he thinks that to study the growth of knowledge one must study scientific knowledge: [M]ost problems connected with the growth of our knowledge must necessarily transcend any study which is confined to common-sense knowledge as opposed to scientific knowledge. For the most important way in which common-sense knowledge grows is, precisely, by turning into scientific knowledge. Popper LSD, 18 3. The laws of probability require that if A is any sentence of the language then P1.

Logically equivalent sentences are always equiprobable. We state without proof the generalization of C5: C6. Given a language L k and a probability P on L k , if any k -sequence of L k is thoroughly independent in P then every k -sequence of L k is thoroughly independent in P. Then if A is thoroughly independent in P , P is symmetrical on L k. The proportional syllogism If P is a symmetrical probability defined on a finite population then the probability that an individual in that population has a trait R is equal to the relative frequency of R in the population.

Direct inference typically infers the relative frequency of a trait in a sample from its relative frequency in the population from which the sample is drawn. Predictive inference is inference from one sample to another sample not overlapping the first. It includes the special case, known as singular predictive inference , in which the second sample consists of just one individual. Inference by analogy is inference from the traits of one individual to those of another on the basis of traits that they share. Inverse inference infers something about a population on the basis of premises about a sample from that population.

Universal inference , mentioned in the opening sentence of this article, is inference from a sample to a hypothesis of universal form. Bayesianism and subjectivism Bayesian induction incorporates a subjectivistic view of probability, according to which probability is identified with strength of belief. Hume, on the other hand, according to Williams held that: [A]lthough our nervous tissue is so composed that when we have encountered a succession on M 's which are P we naturally expect the rest of M 's to be P , and although this expectation has been borne out by the event in the past, the series of observations never provided a jot of logical reason for the expectation, and the fact that the inductive habit succeeded in the past is itself only a gigantic coincidence, giving no reason for supposing it will succeed in the future.

Williams , 97 Williams and Stove recognize that induction may depend upon context and also upon the nature of the traits and properties to which it is applied. Second is a rule relating the frequency of a trait in a population to its frequency in samples from that population: Frequency Principle : It is necessarily true that the relative frequency of a trait in a large finite population is close to that of most large samples from that population.

We say that such samples resemble the population. Given a large population X and a k -sample S of appropriate size from X , in which the relative frequency of the trait R is r , the content of the premise i above can be expressed in two premises: Premise A. S is a k -sample from X. Premise B. The relative frequency of R in S is r , i. Maher , Thus the addition of premise B to the condition of 3 might decrease the probability that S 0 resembles the population X : 3b might be low while 3a is high. Armstrong, like Williams and Stove, is a rationalist about induction.

Armstrong's rationalism does not lead him, as it did Williams and Stove, to see the resolution of the problem of induction as a matter of demonstrating that induction is necessarily a rational procedure: [O]rdinary inductive inference, ordinary inference from the observed to the unobserved, is, although invalid , nevertheless a rational form of inference. Armstrong , 52 Armstrong does not argue for this principle; it is a premise of an argument to the conclusion that regularity views imply the inevitability of inductive skepticism; the view, attributed to Hume, that inferences from the observed to the unobserved are not rational Armstrong , Armstrong , IBE, as its name suggests, is an informal and non-metric form of likelihood methods.

An instantiation of a law is of the form N F , G a 's being F , a 's being G where a is an individual. Armstrong , 5. Paradoxes, the new riddle of induction and objectivity The traditional problem of induction as Hume formulated it concerned what we now know as universal inference see the taxonomy in section 3. Inductively supports All A 's are B 's.

And singular predictive inference : One or several A 's have been observed to be B 's, and no A 's are known to be not- B. Nicod's principle: Universal generalizations are supported or confirmed by their positive instances and falsified by their negative instances Nicod , Equivalence principle : Whatever confirms a generalization confirms as well all its logical equivalents. The hypothesis says is equivalent to Everything is either a black raven or is not a raven. Goodman , 72 The important lesson of the paradox of the ravens and the Hempel-Goodman resolution of it is that inductive inference is sensitive to background information and context.

The inductive inference from a is a white shoe to all ravens are black is not so much unsound as it is uninteresting and uninformative. We thus have evidence statements Emerald a is green, emerald b is green, etc. Then we have also the evidence statements Emerald a is grue, emerald b is grue, etc. A few cautionary remarks about this frequently misunderstood paradox: The grue hypothesis is not well supported by its instances. The paradox makes it clear that there is something wrong with instance confirmation and enumerative induction as initially characterized.

Neither the grue evidence statements nor the grue hypothesis entails that any emeralds change color. This is a common confusion. The grue paradox cannot be resolved, as was the ravens paradox, by looking to background knowledge as would be the case if it entailed color changes. Of course we know that it is extremely unlikely that unobserved emeralds are grue. That just restates the point of the paradox and does nothing to resolve it. Once we see the difficulty, more homely examples than the grue hypothesis are easy to come by: Albert is in this room and safe from freezing.

For the same reasons Everyone in this room is safe from freezing. But Albert is in this room and is a third son. Nor does Everyone in this room is a third son. It is not the least of Goodman's accomplishments to have shown that three questions all issue from the same new riddle: What is the difference between those generalizations that are supported by their instances and those that are not?

Which generalizations support counterfactual conditionals? How are lawlike generalizations to be distinguished from accidental generalizations? Hume gives us a famous and striking example of the latter: [A] man, who being hung out from a high tower in a cage of iron cannot forbear trembling, when he surveys the precipice below him, tho he knows himself to be perfectly safe from falling.

Maher also accounts for our initial and mistaken rejection of the paradoxical conclusion; PC. Given that a is not a raven a is not black supports all ravens are black See also Hempel , They write P K H, E to indicate relativity to background knowledge, including perhaps knowledge of discursive context. Rudner's argument was simple: [S]ince no hypothesis is ever completely verified, in accepting a hypothesis the scientist must make the decision that the evidence is sufficiently strong or that the probability is sufficiently high to warrant the acceptance of the hypothesis.

Rudner , 2 Sufficiency in such a decision will and should depend upon the importance of getting it right or wrong. About this, several brief remarks: Notice that the utility of investigation depends upon your beliefs about your future beliefs and desires, namely that you believe now that following the investigation you will maximize utility and update your beliefs.

Nominalism vs. Realism , Cambridge: Cambridge University Press. Ayer, A. Sylla trans. Brown, M.

First published Dretske, F. Edwards, W. Lindman, and L. Annales de l'Institut Henri Poincare , 7: 1— Fitelson, Branden and J. Eels and J. Fetzer eds. Frege, Gottlob, , Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens , Halle Friedman, Michael and Richard Creath eds.

## Symmetry and its discontents

Giere, Ronald N. Good, I. Goldman, Alvin L.

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