This analogy served not only to underwrite the plausibility of conjectures, but also to guide and limit discovery by pointing scientists in certain directions. In some cases, a programmatic analogy culminates in the theoretical unification of two different areas of inquiry.
A very different relationship between analogy and discovery exists when a programmatic analogy breaks down, as was the ultimate fate of the acoustical analogy. That atomic spectra have an entirely different explanation became clear with the advent of quantum theory.
In this case, novel discoveries emerged against background expectations shaped by the guiding analogy. Arguably, the danger of this third possibility provides strong motivation for developing a critical account of analogical reasoning and analogical arguments.
Analogical cognition , which embraces all cognitive processes involved in discovering, constructing and using analogies, is broader than analogical reasoning Hofstadter ; Hofstadter and Sander Understanding these processes is an important objective of current cognitive science research, and an objective that generates many questions. How do humans identify analogies? Do nonhuman animals use analogies in ways similar to humans? How do analogies and metaphors influence concept formation?
This entry, however, concentrates specifically on analogical arguments. Specifically, it focuses on three central epistemological questions:. Following a preliminary discussion of the basic structure of analogical arguments, the entry reviews selected attempts to provide answers to these three questions. To find such answers would constitute an important first step towards understanding the nature of analogical reasoning.
To isolate these questions, however, is to make the non-trivial assumption that there can be a theory of analogical arguments —an assumption which, as we shall see, is attacked in different ways by both philosophers and cognitive scientists. Analogical arguments vary greatly in subject matter, strength and logical structure. In order to appreciate this variety, it is helpful to increase our stock of examples.
First, a geometric example:. Example 7 Rectangles and boxes. Suppose that you have established that of all rectangles with a fixed perimeter, the square has maximum area. By analogy, you conjecture that of all boxes with a fixed surface area, the cube has maximum volume.
Example 8 Morphine and meperidine. In , the pharmacologist Schaumann was testing synthetic compounds for their anti-spasmodic effect. These drugs had a chemical structure similar to morphine. He observed that one of the compounds— meperidine , also known as Demerol —had a physical effect on mice that was previously observed only with morphine: it induced an S-shaped tail curvature.
Testing on rats, rabbits, dogs and eventually humans showed that meperidine, like morphine, was an effective pain-killer Lembeck 11; Reynolds and Randall Example 9 Priestley on electrostatic force. In , Priestley suggested that the absence of electrical influence inside a hollow charged spherical shell was evidence that charges attract and repel with an inverse square force.
He supported his hypothesis by appealing to the analogous situation of zero gravitational force inside a hollow shell of uniform density. Example 10 Duty of reasonable care.
In a much-cited case Donoghue v. Stevenson AC , the United Kingdom House of Lords found the manufacturer of a bottle of ginger beer liable for damages to a consumer who became ill as a result of a dead snail in the bottle. The principle articulated in this famous case was extended, by analogy, to allow recovery for harm against an engineering firm whose negligent repair work caused the collapse of a lift Haseldine v. By contrast, the principle was not applicable to a case where a workman was injured by a defective crane, since the workman had opportunity to examine the crane and was even aware of the defects Farr v.
What, if anything, do all of these examples have in common? We begin with a simple, quasi-formal characterization. Similar formulations are found in elementary critical thinking texts e. An analogical argument has the following form:. The argument form is ampliative ; the conclusion is not guaranteed to follow from the premises.
S and T are referred to as the source domain and target domain , respectively. A domain is a set of objects, properties, relations and functions, together with a set of accepted statements about those objects, properties, relations and functions. More formally, a domain consists of a set of objects and an interpreted set of statements about them.
The statements need not belong to a first-order language, but to keep things simple, any formalizations employed here will be first-order. In Example 9 , the source domain items pertain to gravitation; the target items pertain to electrostatic attraction. Formally, an analogy between S and T is a one-to-one mapping between objects, properties, relations and functions in S and those in T. Not all of the items in S and T need to be placed in correspondence.
Commonly, the analogy only identifies correspondences between a select set of items. In practice, we specify an analogy simply by indicating the most significant similarities and sometimes differences.
We can improve on this preliminary characterization of the argument from analogy by introducing the tabular representation found in Hesse We place corresponding objects, properties, relations and propositions side-by-side in a table of two columns, one for each domain.
Hesse introduced useful terminology based on this tabular representation. The horizontal relations in an analogy are the relations of similarity and difference in the mapping between domains, while the vertical relations are those between the objects, relations and properties within each domain. In an earlier discussion of analogy, Keynes introduced some terminology that is also helpful. Positive analogy. Let P stand for a list of accepted propositions P 1 , …, P n about the source domain S.
Then we refer to P as the positive analogy. Negative analogy. Then we refer to A and B as the negative analogy. Neutral analogy. The neutral analogy consists of accepted propositions about S for which it is not known whether an analogue holds in T. Hypothetical analogy. The hypothetical analogy is simply the proposition Q in the neutral analogy that is the focus of our attention.
These concepts allow us to provide a characterization for an individual analogical argument that is somewhat richer than the original one. In general, judgments of plausibility are made after a claim has been formulated, but prior to rigorous testing or proof.
The next sub-section provides further discussion. Note that this characterization is incomplete in a number of ways. The manner in which we list similarities and differences, the nature of the correspondences between domains: these things are left unspecified. Nor does this characterization accommodate reasoning with multiple analogies i. To characterize the argument form more fully, however, is not possible without either taking a step towards a substantive theory of analogical reasoning or restricting attention to certain classes of analogical arguments.
Arguments by analogy are extensively discussed within argumentation theory. There is considerable debate about whether they constitute a species of deductive inference Govier ; Waller ; Guarini ; Kraus Argumentation theorists also make use of tools such as speech act theory Bermejo-Luque , argumentation schemes and dialogue types Macagno et al. Arguments by analogy are also discussed in the vast literature on scientific models and model-based reasoning, following the lead of Hesse Bailer-Jones draws a helpful distinction between analogies and models.
Nersessian , for instance, stresses the role of analog models in concept-formation and other cognitive processes. To say that a hypothesis is plausible is to convey that it has epistemic support: we have some reason to believe it, even prior to testing.
An assertion of plausibility within the context of an inquiry typically has pragmatic connotations as well: to say that a hypothesis is plausible suggests that we have some reason to investigate it further.
On both points, there is ambiguity as to whether an assertion of plausibility is categorical or a matter of degree. These observations point to the existence of two distinct conceptions of plausibility, probabilistic and modal , either of which may reflect the intended conclusion of an analogical argument.
On the probabilistic conception, plausibility is naturally identified with rational credence rational subjective degree of belief and is typically represented as a probability. There can be no doubt that every resemblance [not known to be irrelevant] affords some degree of probability, beyond what would otherwise exist, in favour of the conclusion. The meaning, roughly speaking, is that there are sufficient initial grounds for taking p seriously, i.
Informally: p passes an initial screening procedure. There is no assertion of degree. The intent is to single out p from an undifferentiated mass of ideas that remain bare epistemic possibilities.
The set of epistemic possibilities—hypotheses about electrostatic attraction compatible with knowledge of the day—was much larger. Individual analogical arguments in mathematics such as Example 7 are almost invariably directed towards prima facie plausibility.
The modal conception figures importantly in some discussions of analogical reasoning. The physicist N. Campbell writes:. But in order that a theory may be valuable it must … display an analogy.
The propositions of the hypothesis must be analogous to some known laws…. Some analogy is essential to it; for it is only this analogy which distinguishes the theory from the multitude of others… which might also be proposed to explain the same laws. Possible defeaters might include internal inconsistency, inconsistency with accepted theory, or the existence of a clearly superior rival analogical argument.
The point is that Campbell, following the lead of 19 th century philosopher-scientists such as Herschel and Whewell, thinks that analogies can establish this sort of prima facie plausibility.
Snyder provides a detailed discussion of the latter two thinkers and their ideas about the role of analogies in science. In general, analogical arguments may be directed at establishing either sort of plausibility for their conclusions; they can have a probabilistic use or a modal use. Examples 7 through 9 are best interpreted as supporting modal conclusions. In those arguments, an analogy is used to show that a conjecture is worth taking seriously.
To insist on putting the conclusion in probabilistic terms distracts attention from the point of the argument. The conclusion might be modeled by a Bayesian as having a certain probability value because it is deemed prima facie plausible, but not vice versa.
Example 2 , perhaps, might be regarded as directed primarily towards a probabilistic conclusion. There should be connections between the two conceptions. Indeed, we might think that the same analogical argument can establish both prima facie plausibility and a degree of probability for a hypothesis. But it is difficult to translate between epistemic modal concepts and probabilities Cohen ; Douven and Williamson ; Huber ; Spohn , We cannot simply take the probabilistic notion as the primitive one.
It seems wise to keep the two conceptions of plausibility separate. Schema 4 is a template that represents all analogical arguments, good and bad. It is not an inference rule. Despite the confidence with which particular analogical arguments are advanced, nobody has ever formulated an acceptable rule, or set of rules, for valid analogical inferences.
There is not even a plausible candidate. This situation is in marked contrast not only with deductive reasoning, but also with elementary forms of inductive reasoning, such as induction by enumeration. Of course, it is difficult to show that no successful analogical inference rule will ever be proposed. But consider the following candidate, formulated using the concepts of schema 4 and taking us only a short step beyond that basic characterization.
It is pretty clear that 5 is a non-starter. The main problem is that the rule justifies too much. The only substantive requirement introduced by 5 is that there be a nonempty positive analogy.
Plainly, there are analogical arguments that satisfy this condition but establish no prima facie plausibility and no measure of support for their conclusions.
Here is a simple illustration. Both relations are reflexive, symmetric, and transitive. Yet it would be absurd to find positive support from this analogy for the idea that we are likely to find congruent lines clustered in groups of two or more, just because swans of the same color are commonly found in groups.
The positive analogy is antecedently known to be irrelevant to the hypothetical analogy. In such a case, the analogical inference should be utterly rejected. Yet rule 5 would wrongly assign non-zero degree of support.
To generalize the difficulty: not every similarity increases the probability of the conclusion and not every difference decreases it. Some similarities and differences are known to be or accepted as being utterly irrelevant and should have no influence whatsoever on our probability judgments.
To be viable, rule 5 would need to be supplemented with considerations of relevance , which depend upon the subject matter, historical context and logical details particular to each analogical argument. To search for a simple rule of analogical inference thus appears futile. His approach is a hybrid of Carnap-style inductive rules and a Bayesian model. It remains unclear that the Carnapian approach can provide a general rule for analogical inference.
Norton , and —see Other Internet Resources has argued that the project of formalizing inductive reasoning in terms of one or more simple formal schemata is doomed. His criticisms seem especially apt when applied to analogical reasoning. He writes:. If analogical reasoning is required to conform only to a simple formal schema, the restriction is too permissive. Inferences are authorized that clearly should not pass muster… The natural response has been to develop more elaborate formal templates… The familiar difficulty is that these embellished schema never seem to be quite embellished enough; there always seems to be some part of the analysis that must be handled intuitively without guidance from strict formal rules.
These local facts are to be determined and investigated on a case by case basis. To embrace a purely formal approach to analogy and to abjure formalization entirely are two extremes in a spectrum of strategies. There are intermediate positions. Most recent analyses both philosophical and computational have been directed towards elucidating criteria and procedures, rather than formal rules, for reasoning by analogy.
The next section discusses some of these criteria and procedures. Here are some of the most important ones:. These principles can be helpful, but are frequently too vague to provide much insight.
How do we count similarities and differences in applying G1 and G2? Why are the structural and causal analogies mentioned in G5 and G6 especially important, and which structural and causal features merit attention? More generally, in connection with the all-important G7 : how do we determine which similarities and differences are relevant to the conclusion?
Furthermore, what are we to say about similarities and differences that have been omitted from an analogical argument but might still be relevant? An additional problem is that the criteria can pull in different directions. Each of the above criteria apart from G7 is expressed in terms of the strength of the argument, i. The criteria thus appear to presuppose the probabilistic interpretation of plausibility.
The problem is that a great many analogical arguments aim to establish prima facie plausibility rather than any degree of probability. Most of the guidelines are not directly applicable to such arguments. Aristotle sets the stage for all later theories of analogical reasoning.
In his theoretical reflections on analogy and in his most judicious examples, we find a sober account that lays the foundation both for the commonsense guidelines noted above and for more sophisticated analyses.
Although Aristotle employs the term analogy analogia and discusses analogical predication , he never talks about analogical reasoning or analogical arguments per se. He does, however, identify two argument forms, the argument from example paradeigma and the argument from likeness homoiotes , both closely related to what would we now recognize as an analogical argument.
The argument from example paradeigma is described in the Rhetoric and the Prior Analytics :. Enthymemes based upon example are those which proceed from one or more similar cases, arrive at a general proposition, and then argue deductively to a particular inference. Rhetoric b If then we wish to prove that to fight with the Thebans is an evil, we must assume that to fight against neighbours is an evil. Conviction of this is obtained from similar cases, e.
Since then to fight against neighbours is an evil, and to fight against the Thebans is to fight against neighbours, it is clear that to fight against the Thebans is an evil.
Aristotle notes two differences between this argument form and induction 69a15ff. The argument from example thus amounts to single-case induction followed by deductive inference.
The first inference dashed arrow is inductive; the second and third solid arrows are deductively valid. The paradeigma has an interesting feature: it is amenable to an alternative analysis as a purely deductive argument form. Instead of regarding this intermediate step as something reached by induction from a single case, we might instead regard it as a hidden presupposition.
This transforms the paradeigma into a syllogistic argument with a missing or enthymematic premise, and our attention shifts to possible means for establishing that premise with single-case induction as one such means.
The argument from likeness homoiotes seems to be closer than the paradeigma to our contemporary understanding of analogical arguments. The most important passage is the following. Try to secure admissions by means of likeness; for such admissions are plausible, and the universal involved is less patent; e.
This argument resembles induction, but is not the same thing; for in induction it is the universal whose admission is secured from the particulars, whereas in arguments from likeness, what is secured is not the universal under which all the like cases fall.
Topics b10— This passage occurs in a work that offers advice for framing dialectical arguments when confronting a somewhat skeptical interlocutor. The argument from likeness is thus clearly distinct from the paradeigma , where the universal proposition plays an essential role as an intermediate step in the argument.
The argument from likeness, though logically less straightforward than the paradeigma , is exactly the sort of analogical reasoning we want when we are unsure about underlying generalizations. In Topics I 17, Aristotle states that any shared attribute contributes some degree of likeness. It is natural to ask when the degree of likeness between two things is sufficiently great to warrant inferring a further likeness.
In other words, when does the argument from likeness succeed? Aristotle does not answer explicitly, but a clue is provided by the way he justifies particular arguments from likeness.
As Lloyd has observed, Aristotle typically justifies such arguments by articulating a sometimes vague causal principle which governs the two phenomena being compared. For example, Aristotle explains the saltiness of the sea, by analogy with the saltiness of sweat, as a kind of residual earthy stuff exuded in natural processes such as heating.
The common principle is this:. Everything that grows and is naturally generated always leaves a residue, like that of things burnt, consisting in this sort of earth. Mete a From this method of justification, we might conjecture that Aristotle believes that the important similarities are those that enter into such general causal principles.
These four principles form the core of a common-sense model for evaluating analogical arguments which is not to say that they are correct; indeed, the first three will shortly be called into question. The first, as we have seen, appears regularly in textbook discussions of analogy.
Versions of the third are found in most sophisticated theories. The final point, which distinguishes the argument from likeness and the argument from example, is endorsed in many discussions of analogy e.
As that principle suggests, Aristotle, in common with just about everyone else who has written about analogical reasoning, organizes his analysis of the argument form around overall similarity. In the terminology of section 2. Narens L. Prelec D. Quiggin J. Journal of Economic Behavior and Organization — Kluwer Academic Publishers, Boston.
Ramsey F. See Ch. Reprinted in H. Smokler Eds. Studies in Subjective Probability pp. New York: Wiley. Savage L. Schmeidler D. Schneider B. Starmer C. Steingrimsson R. Psychology Ph. Dissertation, U. Behavioral properties of summations and productions.
Behavioral properties linking summations and productions. Form for the Psychophysical Function and Perceptual Filtering. Forms for the Weighting Function. In preparation. Stevens S. Tversky A. Wakker P. The algebraic approach. Zimmer K. Abstracts for Fechner Day pp. Download references. You can also search for this author in PubMed Google Scholar.
Correspondence to R. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors s and do not necessarily reflect the views of the National Science Foundation. I appreciate the detailed and useful comments of my collaborators A. Marley and Ragnar Steingrimsson on earlier versions of the manuscript, as well as those of three referees, one of whom, Ehtibar Dzhafarov, provided very useful comments and criticisms.
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