Ananalogy is a comparison between two objects, or systems ofobjects, that highlights respects in which they are thought to besimilar.Analogical reasoning is any type of thinking thatrelies upon an analogy. Ananalogicalargument is anexplicit representation of a form of analogical reasoning that citesaccepted similarities between two systems to support the conclusionthat some further similarity exists. In general (but not always), sucharguments belong in the category of ampliative reasoning, since theirconclusions do not follow with certainty but are only supported withvarying degrees of strength. However, the proper characterization ofanalogical arguments is subject to debate (see§2.2).

Analogical reasoning is fundamental to human thought and, arguably, tosome nonhuman animals as well. Historically, analogical reasoning hasplayed an important, but sometimes mysterious, role in a wide range ofproblem-solving contexts. The explicit use of analogical arguments,since antiquity, has been a distinctive feature of scientific,philosophical and legal reasoning. This article focuses primarily onthe nature, evaluation and justification of analogical arguments.Related topics includemetaphor,models in science, andprecedent and analogy in legal reasoning.

• 1. Introduction: the many roles of analogy
• 2. Analogical arguments
  • 2.1 Examples
  • 2.2 Characterization
  • 2.3 Plausibility
  • 2.4 Analogical inference rules?
• 3. Criteria for evaluating analogical arguments
  • 3.1 Commonsense guidelines
  • 3.2 Aristotle’s theory
  • 3.3 Material criteria: Hesse’s theory
  • 3.4 Formal criteria: the structure-mapping theory
  • 3.5 Other theories
  • 3.6 Practice-based approaches
• 4. Philosophical foundations for analogical reasoning
  • 4.1 Deductive justification
  • 4.2 Inductive justification
  • 4.3A priori justification
  • 4.4 Pragmatic justification
• 5. Beyond analogical arguments
  • 5.1 Analogy and confirmation
  • 5.2 Conceptual change and theory development
• Bibliography
• Academic Tools
• Other Internet Resources
  • Websites
  • Online Manuscript
• Related Entries

1. Introduction: the many roles of analogy

Analogies are widely recognized as playing an importantheuristic role, as aids to discovery. They have beenemployed, in a wide variety of settings and with considerable success,to generate insight and to formulate possible solutions to problems.According to Joseph Priestley, a pioneer in chemistry andelectricity,

analogy is our best guide in all philosophical investigations; and alldiscoveries, which were not made by mere accident, have been made bythe help of it. (1769/1966: 14)

Priestley may be over-stating the case, but there is no doubt thatanalogies have suggested fruitful lines of inquiry in many fields.Because of their heuristic value, analogies and analogical reasoninghave been a particular focus of AI research. Hájek (2018)examines analogy as a heuristic tool in philosophy.

Analogies have a related (and not entirely separable)justificatory role. This role is most obvious where ananalogical argument is explicitly offered in support of someconclusion. The intended degree of support for the conclusion can varyconsiderably. At one extreme, these arguments can be stronglypredictive. For example:

Example 1.Hydrodynamic analogies exploit mathematical similaritiesbetween the equations governing ideal fluid flow and torsionalproblems. To predict stresses in a planned structure, one canconstruct a fluid model, i.e., a system of pipes through which waterpasses (Timoshenko and Goodier 1970). Within the limits ofidealization, such analogies allow us to make demonstrativeinferences, for example, from a measured quantity in the fluid modelto the analogous value in the torsional problem. In practice, thereare numerous complications (Sterrett 2006).

At the other extreme, an analogical argument may provide very weaksupport for its conclusion, establishing no more than minimalplausibility. Consider:

Example 2. Thomas Reid’s (1785) argument for the existence of life onother planets (Stebbing 1933; Mill 1843/1930; Robinson 1930; Copi1961). Reid notes a number of similarities between Earth and the otherplanets in our solar system: all orbit and are illuminated by the sun;several have moons; all revolve on an axis. In consequence, heconcludes, it is “not unreasonable to think, that those planetsmay, like our earth, be the habitation of various orders of livingcreatures” (1785: 24).

Such modesty is not uncommon. Often the point of an analogicalargument is just to persuade people to take an idea seriously. Forinstance:

Sometimes analogical reasoning is the only available form ofjustification for a hypothesis. The method ofethnographicanalogy is used to interpret

the nonobservable behaviour of the ancient inhabitants of anarchaeological site (or ancient culture) based on the similarity oftheir artifacts to those used by living peoples. (Hunter and Whitten1976: 147)

For example:

Example 4. Shelley (1999, 2003) describes how ethnographic analogy was used todetermine the probable significance of odd markings on the necks ofMoche clay pots found in the Peruvian Andes. Contemporary potters inPeru use these marks (calledsígnales) to indicateownership; the marks enable them to reclaim their work when severalpotters share a kiln or storage facility. Analogical reasoning may bethe only avenue of inference to the past in such cases, though thispoint is subject to dispute (Gould and Watson 1982; Wylie 1982, 1985).Analogical reasoning may have similar significance for cosmologicalphenomena that are inaccessible due to limits on observation(Dardashti et al. 2017). See§5.1 for further discussion.

As philosophers and historians such as Kuhn (1996) have repeatedlypointed out, there is not always a clear separation between the tworoles that we have identified, discovery and justification. Indeed,the two functions are blended in what we might call theprogrammatic (orparadigmatic) role of analogy: overa period of time, an analogy can shape the development of a program ofresearch. For example:

More generally, analogies can play an important programmatic role byguiding conceptual development (see§5.2). In some cases, a programmatic analogy culminates in the theoreticalunification of two different areas of inquiry.

Example 6. Descartes’s (1637/1954) correlation between geometry andalgebra provided methods for systematically handling geometricalproblems that had long been recognized as analogous. A very differentrelationship between analogy and discovery exists when a programmaticanalogy breaks down, as was the ultimate fate of the acousticalanalogy. That atomic spectra have an entirely different explanationbecame clear with the advent of quantum theory. In this case, noveldiscoveries emergedagainst background expectations shaped bythe guiding analogy. There is a third possibility: an unproductive ormisleading programmatic analogy may simply become entrenched andself-perpetuating as it leads us to “construct… data thatconform to it” (Stepan 1996: 133). Arguably, the danger of thisthird possibility provides strong motivation for developing a criticalaccount of analogical reasoning and analogical arguments.

Analogical cognition, which embraces all cognitive processesinvolved in discovering, constructing and using analogies, is broaderthan analogical reasoning (Hofstadter 2001; Hofstadter and Sander2013). Understanding these processes is an important objective ofcurrent cognitive science research, and an objective that generatesmany questions. How do humans identify analogies? Do nonhuman animalsuse analogies in ways similar to humans? How do analogies andmetaphors influence concept formation?

This entry, however, concentrates specifically on analogicalarguments. Specifically, it focuses on three central epistemologicalquestions:

1. What criteria should we use to evaluate analogical arguments?
2. What philosophical justification can be provided for analogicalinferences?
3. How do analogical arguments fit into a broader inferential context(i.e., how do we combine them with other forms of inference),especially theoretical confirmation?

Following a preliminary discussion of the basic structure ofanalogical arguments, the entry reviews selected attempts to provideanswers to these three questions. To find such answers wouldconstitute an important first step towards understanding the nature ofanalogical reasoning. To isolate these questions, however, is to makethe non-trivial assumption that there can be atheory ofanalogical arguments—an assumption which, as we shallsee, is attacked in different ways by both philosophers and cognitivescientists.

2. Analogical arguments

2.1 Examples

Analogical arguments vary greatly in subject matter, strength andlogical structure. In order to appreciate this variety, it is helpfulto increase our stock of examples. First, a geometric example:

Example 7 (Rectangles and boxes). Suppose that you have established that of allrectangles with a fixed perimeter, the square has maximum area. Byanalogy, you conjecture that of all boxes with a fixed surface area,the cube has maximum volume.

Two examples from the history of science:

Example 8 (Morphine and meperidine). In 1934, the pharmacologist Schaumann wastesting synthetic compounds for their anti-spasmodic effect. Thesedrugs had a chemical structure similar to morphine. He observed thatone of the compounds—meperidine, also known asDemerol—had a physical effect on mice that waspreviously observed only with morphine: it induced an S-shaped tailcurvature. By analogy, he conjectured that the drug might also sharemorphine’s narcotic effects. Testing on rats, rabbits, dogs andeventually humans showed that meperidine, like morphine, was aneffective pain-killer (Lembeck 1989: 11; Reynolds and Randall 1975:273).

Example 9 (Priestley on electrostatic force). In 1769, Priestley suggested thatthe absence of electrical influence inside a hollow charged sphericalshell was evidence that charges attract and repel with an inversesquare force. He supported his hypothesis by appealing to theanalogous situation of zero gravitational force inside a hollow shellof uniform density.

Finally, an example from legal reasoning:

Example 10 (Duty of reasonable care). In a much-cited case (Donoghue v.Stevenson 1932 AC 562), the United Kingdom House of Lords foundthe manufacturer of a bottle of ginger beer liable for damages to aconsumer who became ill as a result of a dead snail in the bottle. Thecourt argued that the manufacturer had a duty to take“reasonable care” in creating a product that couldforeseeably result in harm to the consumer in the absence of suchcare, and where the consumer had no possibility of intermediateexamination. The principle articulated in this famous case wasextended, by analogy, to allow recovery for harm against anengineering firm whose negligent repair work caused the collapse of alift (Haseldine v. CA Daw & Son Ltd. 1941 2 KB 343). Bycontrast, the principle was not applicable to a case where a workmanwas injured by a defective crane, since the workman had opportunity toexamine the crane and was even aware of the defects (Farr v.Butters Brothers & Co. 1932 2 KB 606).

2.2 Characterization

What, if anything, do all of these examples have in common? We beginwith a simple, quasi-formal characterization. Similar formulations arefound in elementary critical thinking texts (e.g., Copi and Cohen2005) and in the literature on argumentation theory (e.g., Govier1999, Guarini 2004, Walton and Hyra 2018). An analogical argument hasthe following form:

1. \(S\) is similar to \(T\) in certain (known) respects.
2. \(S\) has some further feature \(Q\).
3. Therefore, \(T\) also has the feature \(Q\), or some feature\(Q^*\) similar to \(Q\).

(1) and (2) are premises. (3) is the conclusion of the argument. Theargument form isampliative; the conclusion is not guaranteedto follow from the premises.

\(S\) and \(T\) are referred to as thesource domain andtarget domain, respectively. Adomain is a set ofobjects, properties, relations and functions, together with a set ofaccepted statements about those objects, properties, relations andfunctions. More formally, a domain consists of a set of objects and aninterpreted set of statements about them. The statements need notbelong to a first-order language, but to keep things simple, anyformalizations employed here will be first-order. We use unstarredsymbols \((a, P, R, f)\) to refer to items in the source domain andstarred symbols \((a^*, P^*, R^*, f^*)\) to refer to correspondingitems in the target domain. InExample 9, the source domain items pertain to gravitation; the target itemspertain to electrostatic attraction.

Formally, ananalogy between \(S\) and \(T\) is a one-to-onemapping between objects, properties, relations and functions in \(S\)and those in \(T\). Not all of the items in \(S\) and \(T\) need to beplaced in correspondence. Commonly, the analogy only identifiescorrespondences between a select set of items. In practice, we specifyan analogy simply by indicating the most significant similarities (andsometimes differences).

We can improve on this preliminary characterization of the argumentfrom analogy by introducing thetabular representation foundin Hesse (1966). We place corresponding objects, properties, relationsand propositions side-by-side in a table of two columns, one for eachdomain. For instance, Reid’s argument (Example 2) can be represented as follows (using \(\Rightarrow\) for theanalogical inference):

Hesse introduced useful terminology based on this tabularrepresentation. Thehorizontal relations in an analogy arethe relations of similarity (and difference) in the mapping betweendomains, while thevertical relations are those between theobjects, relations and properties within each domain. Thecorrespondence (similarity) between earth’s having a moon andMars’ having moons is a horizontal relation; the causal relationbetween having a moon and supporting life is a vertical relationwithin the source domain (with the possibility of a distinct suchrelation existing in the target as well).

In an earlier discussion of analogy, Keynes (1921) introduced someterminology that is also helpful.

Finally we have:

These concepts allow us to provide a characterization for anindividual analogical argument that is somewhat richer than theoriginal one.

An analogical argument may thus be summarized:

It is plausible that \(Q^*\) holds in the target,because ofcertain known (or accepted) similarities with the source domain,despite certain known (or accepted) differences.

In order for this characterization to be meaningful, we need to saysomething about the meaning of ‘plausibly.’ To ensurebroad applicability over analogical arguments that vary greatly instrength, we interpret plausibility rather liberally as meaning‘with some degree of support’. In general, judgments ofplausibility are made after a claim has been formulated, but prior torigorous testing or proof. The next sub-section provides furtherdiscussion.

Note that this characterization is incomplete in a number of ways. Themanner in which we list similarities and differences, the nature ofthe correspondences between domains: these things are leftunspecified. Nor does this characterization accommodate reasoning withmultiple analogies (i.e., multiple source domains), which isubiquitous in legal reasoning and common elsewhere. To characterizethe argument form more fully, however, is not possible without eithertaking a step towards a substantive theory of analogical reasoning orrestricting attention to certain classes of analogical arguments.

Arguments by analogy are extensively discussed within argumentationtheory. There is considerable debate about whether they constitute aspecies of deductive inference (Govier 1999; Waller 2001; Guarini2004; Kraus 2015). Argumentation theorists also make use of tools suchas speech act theory (Bermejo-Luque 2012), argumentation schemes anddialogue types (Macagno et al. 2017; Walton and Hyra 2018) todistinguish different types of analogical argument.

Arguments by analogy are also discussed in the vast literature onscientific models and model-based reasoning, following the lead ofHesse (1966). Bailer-Jones (2002) draws a helpful distinction betweenanalogies and models. While “many models have their roots in ananalogy” (2002: 113) and analogy “can act as a catalyst toaid modeling,” Bailer-Jones observes that “the aim ofmodeling has nothing intrinsically to do with analogy.” Inbrief, models are tools for prediction and explanation, whereasanalogical arguments aim at establishing plausibility. An analogy isevaluated in terms of source-target similarity, while a model isevaluated on how successfully it “provides access to aphenomenon in that it interprets the available empirical data aboutthe phenomenon.” If we broaden our perspective beyond analogicalarguments, however, the connection between models andanalogies is restored. Nersessian (2009), for instance, stresses therole of analog models in concept-formation and other cognitiveprocesses.

2.3 Plausibility

To say that a hypothesis is plausible is to convey that it hasepistemic support: we have some reason to believe it, even prior totesting. An assertion of plausibility within the context of an inquirytypically has pragmatic connotations as well: to say that a hypothesisis plausible suggests that we have some reason to investigate itfurther. For example, a mathematician working on a proof regards aconjecture as plausible if it “has some chances ofsuccess” (Polya 1954 (v. 2): 148). On both points, there isambiguity as to whether an assertion of plausibility is categorical ora matter of degree. These observations point to the existence of twodistinct conceptions of plausibility,probabilistic andmodal, either of which may reflect the intended conclusion ofan analogical argument.

On theprobabilistic conception, plausibility is naturallyidentified with rational credence (rational subjective degree ofbelief) and is typically represented as a probability. A classicexpression may be found in Mill’s analysis of the argument fromanalogy inA System of Logic:

There can be no doubt that every resemblance [not known to beirrelevant] affords some degree of probability, beyond what wouldotherwise exist, in favour of the conclusion. (Mill 1843/1930:333)

In the terminology introduced in§2.2, Mill’s idea is that each element of the positive analogy booststhe probability of the conclusion. Contemporary‘structure-mapping’ theories (§3.4) employ a restricted version: eachstructural similaritybetween two domains contributes to the overall measure of similarity,and hence to the strength of the analogical argument.

On the alternativemodal conception, ‘it is plausiblethat \(p\)’ is not a matter of degree. The meaning, roughlyspeaking, is that there are sufficient initial grounds for taking\(p\) seriously, i.e., for further investigation (subject tofeasibility and interest). Informally: \(p\) passes an initialscreening procedure. There is no assertion of degree. Instead,‘It is plausible that’ may be regarded as an epistemicmodal operator that aims to capture a notion,prima facieplausibility, that is somewhat stronger than ordinary epistemicpossibility. The intent is to single out \(p\) from anundifferentiated mass of ideas that remain bare epistemicpossibilities. To illustrate: in 1769, Priestley’s argument (Example 9), if successful, would establish theprima facie plausibilityof an inverse square law for electrostatic attraction. The set ofepistemic possibilities—hypotheses about electrostaticattraction compatible with knowledge of the day—was much larger.Individual analogical arguments in mathematics (such asExample 7) are almost invariably directed towardsprima facieplausibility.

The modal conception figures importantly in some discussions ofanalogical reasoning. The physicist N. R. Campbell (1957) writes:

But in order that a theory may be valuable it must … display ananalogy. The propositions of the hypothesis must be analogous to someknown laws…. (1957: 129)

Commenting on the role of analogy in Fourier’s theory of heatconduction, Campbell writes:

Some analogy is essential to it; for it is only this analogywhich distinguishes the theory from the multitude of others…which might also be proposed to explain the same laws. (1957: 142)

The interesting notion here is that of a “valuable”theory. We may not agree with Campbell that the existence of analogyis “essential” for a novel theory to be“valuable.” But consider the weaker thesis that anacceptable analogy issufficient to establish that a theoryis “valuable”, or (to qualify still further) that anacceptable analogy provides defeasible grounds for taking the theoryseriously. (Possible defeaters might include internal inconsistency,inconsistency with accepted theory, or the existence of a (clearlysuperior) rival analogical argument.) The point is that Campbell,following the lead of 19^th century philosopher-scientistssuch as Herschel and Whewell, thinks that analogies can establish thissort ofprima facie plausibility. Snyder (2006) provides adetailed discussion of the latter two thinkers and their ideas aboutthe role of analogies in science.

In general, analogical arguments may be directed at establishingeither sort of plausibility for their conclusions; they can have aprobabilistic use or a modal use.Examples 7 through9 are best interpreted as supporting modal conclusions. In thosearguments, an analogy is used to show that a conjecture is worthtaking seriously. To insist on putting the conclusion in probabilisticterms distracts attention from the point of the argument. Theconclusion might be modeled (by a Bayesian) as having a certainprobability valuebecause it is deemedprima facieplausible, but not vice versa.Example 2, perhaps, might be regarded as directed primarily towards aprobabilistic conclusion.

There should be connections between the two conceptions. Indeed, wemight think that the same analogical argument can establish bothprima facie plausibility and a degree of probability for ahypothesis. But it is difficult to translate between epistemic modalconcepts and probabilities (Cohen 1980; Douven and Williamson 2006;Huber 2009; Spohn 2009, 2012). We cannot simply take the probabilisticnotion as the primitive one. It seems wise to keep the two conceptionsof plausibility separate.

2.4 Analogical inference rules?

Schema (4) is a template that representsall analogical arguments, goodand bad. It is not an inference rule. Despite the confidence withwhich particular analogical arguments are advanced, nobody has everformulated an acceptable rule, or set of rules, for valid analogicalinferences. There is not even a plausible candidate. This situation isin marked contrast not only with deductive reasoning, but also withelementary forms of inductive reasoning, such as induction byenumeration.

Of course, it is difficult to show that no successful analogicalinference rule will ever be proposed. But consider the followingcandidate, formulated using the concepts of schema (4) and taking usonly a short step beyond that basic characterization.

(5)

Suppose \(S\) and \(T\) are the source and target domains.Suppose \(P_1 , \ldots ,P_n\) (with \(n \ge 1)\) represents thepositive analogy, \(A_1 , \ldots ,A_r\) and \({\sim}B_1 , \ldots,{\sim}B_s\) represent the (possibly vacuous) negative analogy, and\(Q\) represents the hypothetical analogy. In the absence of reasonsfor thinking otherwise, infer that \(Q^*\) holds in the target domainwith degree of support \(p \gt 0\), where \(p\) is an increasingfunction of \(n\) and a decreasing function of \(r\) and \(s\).

Rule (5) is modeled on thestraight rule for enumerative induction and inspired by Mill’s view of analogical inference, asdescribed in§2.3. We use the generic phrase ‘degree of support’ in place ofprobability, since other factors besides the analogical argument mayinfluence our probability assignment for \(Q^*\).

It is pretty clear that (5) is a non-starter. The main problem is thatthe rule justifies too much. The only substantive requirementintroduced by (5) is that there be a nonempty positive analogy.Plainly, there are analogical arguments that satisfy this conditionbut establish noprima facie plausibility and no measure ofsupport for their conclusions.

Here is a simple illustration. Achinstein (1964: 328) observes thatthere is a formal analogy between swans and line segments if we takethe relation ‘has the same color as’ to correspond to‘is congruent with’. Both relations are reflexive,symmetric, and transitive. Yet it would be absurd to find positivesupport from this analogy for the idea that we are likely to findcongruent lines clustered in groups of two or more, just because swansof the same color are commonly found in groups. The positive analogyis antecedently known to be irrelevant to the hypothetical analogy. Insuch a case, the analogical inference should be utterly rejected. Yetrule (5) would wrongly assign non-zero degree of support.

To generalize the difficulty: not every similarity increases theprobability of the conclusion and not every difference decreases it.Some similarities and differences are known to be (or accepted asbeing) utterly irrelevant and should have no influence whatsoever onour probability judgments. To be viable, rule (5) would need to besupplemented with considerations ofrelevance, which dependupon the subject matter, historical context and logical detailsparticular to each analogical argument. To search for a simple rule ofanalogical inference thus appears futile.

Carnap and his followers (Carnap 1980; Kuipers 1988; Niiniluoto 1988;Maher 2000; Romeijn 2006) have formulated principles of analogy forinductive logic, using Carnapian \(\lambda \gamma\) rules. Generally,this body of work relates to “analogy by similarity”,rather than the type of analogical reasoning discussed here. Romeijn(2006) maintains that there is a relation between Carnap’sconcept of analogy and analogical prediction. His approach is a hybridof Carnap-style inductive rules and a Bayesian model. Such an approachwould need to be generalized to handle the kinds of argumentsdescribed in§2.1. It remains unclear that the Carnapian approach can provide a generalrule for analogical inference.

Norton (2010, and 2018—see Other Internet Resources) has arguedthat the project of formalizing inductive reasoning in terms of one ormore simple formal schemata is doomed. His criticisms seem especiallyapt when applied to analogical reasoning. He writes:

If analogical reasoning is required to conform only to a simple formalschema, the restriction is too permissive. Inferences are authorizedthat clearly should not pass muster… The natural response hasbeen to develop more elaborate formal templates… The familiardifficulty is that these embellished schema never seem to be quiteembellished enough; there always seems to be some part of the analysisthat must be handled intuitively without guidance from strict formalrules. (2018: 1)

Norton takes the point one step further, in keeping with his“material theory” of inductive inference. He argues thatthere is no universal logical principle that “powers”analogical inference “by asserting that things that share someproperties must share others.” Rather, each analogical inferenceis warranted by some local constellation of facts about the targetsystem that he terms “the fact of analogy”. These localfacts are to be determined and investigated on a case by casebasis.

To embrace a purely formal approach to analogy and to abjureformalization entirely are two extremes in a spectrum of strategies.There are intermediate positions. Most recent analyses (bothphilosophical and computational) have been directed towardselucidating criteria and procedures, rather than formal rules, forreasoning by analogy. So long as these are not intended to provide auniversal ‘logic’ of analogy, there is room for suchcriteria even if one accepts Norton’s basic point. The nextsection discusses some of these criteria and procedures.

3. Criteria for evaluating analogical arguments

3.1 Commonsense guidelines

Logicians and philosophers of science have identified‘textbook-style’ general guidelines for evaluatinganalogical arguments (Mill 1843/1930; Keynes 1921; Robinson 1930;Stebbing 1933; Copi and Cohen 2005; Moore and Parker 1998; Woods,Irvine, and Walton 2004). Here are some of the most importantones:

(G1)

The more similarities (between two domains), the stronger theanalogy.

(G2)

The more differences, the weaker the analogy.

(G3)

The greater the extent of our ignorance about the two domains,the weaker the analogy.

(G4)

The weaker the conclusion, the more plausible the analogy.

(G5)

Analogies involving causal relations are more plausible thanthose not involving causal relations.

(G6)

Structural analogies are stronger than those based on superficialsimilarities.

(G7)

The relevance of the similarities and differences to theconclusion (i.e., to the hypothetical analogy) must be taken intoaccount.

(G8)

Multiple analogies supporting the same conclusion make theargument stronger.

These principles can be helpful, but are frequently too vague toprovide much insight. How do we count similarities and differences inapplying (G1) and (G2)? Why are the structural and causal analogiesmentioned in (G5) and (G6) especially important, and which structuraland causal features merit attention? More generally, in connectionwith the all-important (G7): how do we determine which similaritiesand differences are relevant to the conclusion? Furthermore, what arewe to say about similarities and differences that have been omittedfrom an analogical argument but might still be relevant?

An additional problem is that the criteria can pull in differentdirections. To illustrate, consider Reid’s argument for life onother planets (Example 2). Stebbing (1933) finds Reid’s argument “suggestive”and “not unplausible” because the conclusion is weak (G4),while Mill (1843/1930) appears to reject the argument on account ofour vast ignorance of properties that might be relevant (G3).

There is a further problem that relates to the distinction just made(in§2.3) between two kinds of plausibility. Each of the above criteria apartfrom (G7) is expressed in terms of the strength of the argument, i.e.,the degree of support for the conclusion. The criteria thus appear topresuppose theprobabilistic interpretation of plausibility.The problem is that a great many analogical arguments aim to establishprima facie plausibility rather than any degree ofprobability. Most of the guidelines are not directly applicable tosuch arguments.

3.2 Aristotle’s theory

Aristotle sets the stage for all later theories of analogicalreasoning. In his theoretical reflections on analogy and in his mostjudicious examples, we find a sober account that lays the foundationboth for the commonsense guidelines noted above and for moresophisticated analyses.

Although Aristotle employs the term analogy (analogia) anddiscussesanalogical predication, he never talks about analogical reasoning or analogical argumentsper se. He does, however, identify two argument forms, theargument from example (paradeigma) and theargument from likeness (homoiotes), both closelyrelated to what would we now recognize as an analogical argument.

Theargument from example (paradeigma) is describedin theRhetoric and thePrior Analytics:

Enthymemes based upon example are those which proceed from one or moresimilar cases, arrive at a general proposition, and then arguedeductively to a particular inference. (Rhetoric 1402b15)

Let \(A\) be evil, \(B\) making war against neighbours, \(C\)Athenians against Thebans, \(D\) Thebans against Phocians. If then wewish to prove that to fight with the Thebans is an evil, we mustassume that to fight against neighbours is an evil. Conviction of thisis obtained from similar cases, e.g., that the war against thePhocians was an evil to the Thebans. Since then to fight againstneighbours is an evil, and to fight against the Thebans is to fightagainst neighbours, it is clear that to fight against the Thebans isan evil. (Pr. An. 69a1)

Aristotle notes two differences between this argument form andinduction (69a15ff.): it “does not draw its proof from all theparticular cases” (i.e., it is not a “complete”induction), and it requires an additional (deductively valid)syllogism as the final step. The argument from example thus amounts tosingle-case induction followed by deductive inference. It has thefollowing structure (using \(\supset\) for the conditional):

Figure 2.

In the terminology of§2.2, \(P\) is the positive analogy and \(Q\) is the hypothetical analogy.In Aristotle’s example, \(S\) (the source) is war betweenPhocians and Thebans, \(T\) (the target) is war between Athenians andThebans, \(P\) is war between neighbours, and \(Q\) is evil. The firstinference (dashed arrow) is inductive; the second and third (solidarrows) are deductively valid.

Theparadeigma has an interesting feature: it is amenable toan alternative analysis as a purelydeductive argument form.Let us concentrate on Aristotle’s assertion, “we mustassume that to fight against neighbours is an evil,” representedas \(\forall x(P(x) \supset Q(x))\). Instead of regarding thisintermediate step as something reached by induction from a singlecase, we might instead regard it as a hidden presupposition. Thistransforms theparadeigma into a syllogistic argument with amissing orenthymematic premise, and our attention shifts topossible means for establishing that premise (with single-caseinduction as one such means). Construed in this way, Aristotle’sparadeigma argument foreshadows deductive analyses ofanalogical reasoning (see§4.1).

Theargument from likeness (homoiotes) seems to becloser than theparadeigma to our contemporary understandingof analogical arguments. This argument form receives considerableattention inTopics I, 17 and 18 and again in VIII, 1. Themost important passage is the following.

Try to secure admissions by means of likeness; for such admissions areplausible, and the universal involved is less patent; e.g. that asknowledge and ignorance of contraries is the same, so too perceptionof contraries is the same; or vice versa, that since the perception isthe same, so is the knowledge also. This argument resembles induction,but is not the same thing; for in induction it is the universal whoseadmission is secured from the particulars, whereas in arguments fromlikeness, what is secured is not the universal under which all thelike cases fall. (Topics 156b10–17)

This passage occurs in a work that offers advice for framingdialectical arguments when confronting a somewhat skepticalinterlocutor. In such situations, it is best not to make one’sargument depend upon securing agreement about any universalproposition. The argument from likeness is thus clearly distinct fromtheparadeigma, where the universal proposition plays anessential role as an intermediate step in the argument. The argumentfrom likeness, though logically less straightforward than theparadeigma, is exactly the sort of analogical reasoning wewant when we are unsure about underlying generalizations.

InTopics I 17, Aristotle states that any shared attributecontributes some degree of likeness. It is natural to ask when thedegree of likeness between two things is sufficiently great to warrantinferring a further likeness. In other words, when does the argumentfrom likeness succeed? Aristotle does not answer explicitly, but aclue is provided by the way he justifies particular arguments fromlikeness. As Lloyd (1966) has observed, Aristotle typically justifiessuch arguments by articulating a (sometimes vague) causal principlewhich governs the two phenomena being compared. For example, Aristotleexplains the saltiness of the sea, by analogy with the saltiness ofsweat, as a kind of residual earthy stuff exuded in natural processessuch as heating. The common principle is this:

Everything that grows and is naturally generated always leaves aresidue, like that of things burnt, consisting in this sort of earth.(Mete 358a17)

From this method of justification, we might conjecture that Aristotlebelieves that the important similarities are those that enter intosuch general causal principles.

Summarizing, Aristotle’s theory provides us with four importantand influential criteria for the evaluation of analogicalarguments:

• The strength of an analogy depends upon the number ofsimilarities.
• Similarity reduces to identical properties andrelations.
• Good analogies derive from underlying common causes or generallaws.
• A good analogical argument need not pre-suppose acquaintancewith the underlying universal (generalization).

These four principles form the core of acommon-sense modelfor evaluating analogical arguments (which is not to say that they arecorrect; indeed, the first three will shortly be called intoquestion). The first, as we have seen, appears regularly in textbookdiscussions of analogy. The second is largely taken for granted, withimportant exceptions in computational models of analogy (§3.4). Versions of the third are found in most sophisticated theories. Thefinal point, which distinguishes the argument from likeness and theargument from example, is endorsed in many discussions of analogy(e.g., Quine and Ullian 1970).

A slight generalization of Aristotle’s first principle helps toprepare the way for discussion of later developments. As thatprinciple suggests, Aristotle, in common with just about everyone elsewho has written about analogical reasoning, organizes his analysis ofthe argument form around overall similarity. In the terminology ofsection 2.2,horizontal relationships drive the reasoning:thegreater the overall similarity of the two domains, the stronger theanalogical argument. Hume makes the same point, though statednegatively, in hisDialogues Concerning Natural Religion:

Wherever you depart, in the least, from the similarity of the cases,you diminish proportionably the evidence; and may at last bring it toa very weak analogy, which is confessedly liable to error anduncertainty. (1779/1947: 144)

Most theories of analogy agree with Aristotle and Hume on this generalpoint. Disagreement relates to the appropriate way of measuringoverall similarity. Some theories assign greatest weight tomaterial analogy, which refers to shared, and typicallyobservable, features. Others give prominence toformalanalogy, emphasizing high-level structural correspondence. Thenext two sub-sections discuss representative accounts that illustratethese two approaches.

3.3 Material criteria: Hesse’s theory

Hesse (1966) offers a sharpened version of Aristotle’s theory,specifically focused on analogical arguments in the sciences. Sheformulates three requirements that an analogical argument must satisfyin order to be acceptable:

1. Requirement of material analogy. The horizontal relationsmust include similarities between observable properties.
2. Causal condition. The vertical relations must be causalrelations “in some acceptable scientific sense” (1966:87).
3. No-essential-difference condition. The essentialproperties and causal relations of the source domain must not havebeen shown to be part of the negative analogy.

3.3.1 Requirement of material analogy

For Hesse, an acceptable analogical argument must include“observable similarities” between domains, which sherefers to asmaterial analogy. Material analogy is contrastedwithformal analogy. Two domains are formally analogous ifboth are “interpretations of the same formal theory”(1966: 68).Nomic isomorphism (Hempel 1965) is a special casein which the physical laws governing two systems have identicalmathematical form. Heat and fluid flow exhibit nomic isomorphism. Asecond example is the analogy between the flow of electric current ina wire and fluid in a pipe. Ohm’s law

states that voltage difference along a wire equals current times aconstant resistance. This has the same mathematical form asPoiseuille’s law (for ideal fluids):

which states that the pressure difference along a pipe equals thevolumetric flow rate times a constant. Both of these systems can berepresented by a common equation. While formal analogy is linked tocommon mathematical structure, it should not be limited to nomicisomorphism (Bartha 2010: 209). The idea of formal analogy generalizesto cases where there is a common mathematical structure betweenmodels for two systems. Bartha offers an even more liberaldefinition (2010: 195): “Two features are formally similar ifthey occupy corresponding positions in formally analogous theories.For example, pitch in the theory of sound corresponds to color in thetheory of light.”

By contrast, material analogy consists of what Hesse calls“observable” or “pre-theoretic” similarities.These are horizontal relationships of similarity between properties ofobjects in the source and the target. Similarities between echoes(sound) and reflection (light), for instance, were recognized longbefore we had any detailed theories about these phenomena. Hesse(1966, 1988) regards such similarities as metaphorical relationshipsbetween the two domains and labels them “pre-theoretic”because they draw on personal and cultural experience. We have bothmaterial and formal analogies between sound and light, and it issignificant for Hesse that the former are independent of thelatter.

There are good reasons not to accept Hesse’s requirement ofmaterial analogy, construed in this narrow way. First, it is apparentthat formal analogiesare the starting point in manyimportant inferences. That is certainly the case in mathematics, afield in which material analogy, in Hesse’s sense, plays no roleat all. Analogical arguments based on formal analogy have also beenextremely influential in physics (Steiner 1989, 1998).

In Norton’s broad sense, however, ‘material analogy’simply refers to similarities rooted in factual knowledge of thesource and target domains. With reference to this broader meaning,Hesse proposes two additional material criteria.

3.3.2 Causal condition

Hesse requires that the hypothetical analogy, the feature transferredto the target domain, becausally related to the positiveanalogy. In her words, the essential requirement for a good argumentfrom analogy is “a tendency to co-occurrence”, i.e., acausal relationship. She states the requirement as follows:

The vertical relations in the model [source] are causal relations insome acceptable scientific sense, where there are no compelling apriori reasons for denying that causal relations of the same kind mayhold between terms of the explanandum [target]. (1966: 87)

The causal condition rules out analogical arguments where there is nocausal knowledge of the source domain. It derives support from theobservation that many analogies do appear to involve a transfer ofcausal knowledge.

The causal condition is on the right track, but is arguably toorestrictive. For example, it rules out analogical arguments inmathematics. Even if we limit attention to the empirical sciences,persuasive analogical arguments may be founded upon strong statisticalcorrelation in the absence of anyknown causal connection.Consider (Example 11) Benjamin Franklin’s prediction, in 1749, that pointed metalrods would attract lightning, by analogy with the way they attractedthe “electrical fluid” in the laboratory:

Electrical fluid agrees with lightning in these particulars: 1. Givinglight. 2. Colour of the light. 3. Crooked direction. 4. Swift motion.5. Being conducted by metals. 6. Crack or noise in exploding. 7.Subsisting in water or ice. 8. Rending bodies it passes through. 9.Destroying animals. 10. Melting metals. 11. Firing inflammablesubstances. 12. Sulphureous smell.—The electrical fluid isattracted by points.—We do not know whether this property is inlightning.—But since they agree in all the particulars whereinwe can already compare them, is it not probable they agree likewise inthis? Let the experiment be made. (Benjamin Franklin’sExperiments, 334)

Franklin’s hypothesis was based on a long list of propertiescommon to the target (lightning) and source (electrical fluid in thelaboratory). There was no known causal connection between the twelve“particulars” and the thirteenth property, but there was astrong correlation. Analogical arguments may be plausible even wherethere are no known causal relations.

3.3.3 No-essential-difference condition

Hesse’s final requirement is that the “essentialproperties and causal relations of the [source] have not been shown tobe part of the negative analogy” (1966: 91). Hesse does notprovide a definition of “essential,” but suggests that aproperty or relation is essential if it is “causally closelyrelated to the known positive analogy.” For instance, an analogywith fluid flow was extremely influential in developing the theory ofheat conduction. Once it was discovered that heat was not conserved,however, the analogy became unacceptable (according to Hesse) becauseconservation was so central to the theory of fluid flow.

This requirement, though once again on the right track, seems toorestrictive. It can lead to the rejection of a good analogicalargument. Consider the analogy between a two-dimensional rectangle anda three-dimensional box (Example 7). Broadening Hesse’s notion, it seems that there are many‘essential’ differences between rectangles and boxes. Thisdoes not mean that we should reject every analogy between rectanglesand boxes out of hand. The problem derives from the fact thatHesse’s condition is applied to the analogyrelationindependently of the use to which that relation is put. What counts asessential should vary with the analogical argument. Absent aninferential context, it is impossible to evaluate the importance or‘essentiality’ of similarities and differences.

Despite these weaknesses, Hesse’s ‘material’criteria constitute a significant advance in our understanding ofanalogical reasoning. The causal condition and theno-essential-difference condition incorporate local factors, as urgedby Norton, into the assessment of analogical arguments. Theseconditions, singly or taken together, imply that an analogicalargument can fail to generate any support for its conclusion, evenwhen there is a non-empty positive analogy. Hesse offers no theoryabout the ‘degree’ of analogical support. That makes heraccount one of the few that is oriented towards the modal, rather thanprobabilistic, use of analogical arguments (§2.3).

3.4 Formal criteria: the structure-mapping theory

Many people take the concept ofmodel-theoretic isomorphism to set the standard for thinking about similarity and its role inanalogical reasoning. They proposeformal criteria forevaluating analogies, based on overall structural or syntacticalsimilarity. Let us refer to theories oriented around such criteria asstructuralist.

A number of leading computational models of analogy are structuralist.They are implemented in computer programs that begin with (orsometimes build) representations of the source and target domains, andthen construct possible analogy mappings. Analogical inferences emergeas a consequence of identifying the ‘best mapping.’ Interms of criteria for analogical reasoning, there are two main ideas.First, the goodness of ananalogical argument is based on thegoodness of the associatedanalogy mapping. Second, thegoodness of the analogy mapping is given by a metric that indicateshow closely it approximates isomorphism.

The most influential structuralist theory has been Gentner’sstructure-mapping theory, implemented in a program called thestructure-mapping engine (SME). In its original form (Gentner1983), the theory assesses analogies on purely structural grounds.Gentner asserts:

Analogies are about relations, rather than simple features. No matterwhat kind of knowledge (causal models, plans, stories, etc.), it isthe structural properties (i.e., the interrelationships between thefacts) that determine the content of an analogy. (Falkenhainer,Forbus, and Gentner 1989/90: 3)

In order to clarify this thesis, Gentner introduces a distinctionbetweenproperties, or monadic predicates, andrelations, which have multiple arguments. She furtherdistinguishes among differentorders of relations andfunctions, defined inductively (in terms of the order of the relata orarguments). The best mapping is determined bysystematicity:the extent to which it places higher-order relations, and items thatare nested in higher-order relations, in correspondence.Gentner’sSystematicity Principle states:

A predicate that belongs to a mappable system of mutuallyinterconnecting relationships is more likely to be imported into thetarget than is an isolated predicate. (1983: 163)

A systematic analogy (one that places high-order relations and theircomponents in correspondence) is better than a less systematicanalogy. Hence, an analogical inference has a degree of plausibilitythat increases monotonically with the degree of systematicity of theassociated analogy mapping. Gentner’s fundamental criterion forevaluating candidate analogies (and analogical inferences) thusdepends solely upon the syntax of the given representations and not atall upon their content.

Later versions of the structure-mapping theory incorporate refinements(Forbus, Ferguson, and Gentner 1994; Forbus 2001; Forbus et al. 2007;Forbus et al. 2008; Forbus et al 2017). For example, the earliestversion of the theory is vulnerable to worries about hand-codedrepresentations of source and target domains. Gentner and hercolleagues have attempted to solve this problem in later work thatgenerates LISP representations from natural language text (see Turney2008 for a different approach).

The most important challenges for the structure-mapping approachrelate to theSystematicity Principle itself. Does the valueof an analogy derive entirely, or even chiefly, from systematicity?There appear to be two main difficulties with this view. First: it isnot always appropriate to give priority to systematic, high-levelrelational matches. Material criteria, and notably what Gentner refersto as “superficial feature matches,” can be extremelyimportant in some types of analogical reasoning, such as ethnographicanalogies which are based, to a considerable degree, on surfaceresemblances between artifacts. Second and more significantly:systematicity seems to be at best a falliblemarker for goodanalogies rather than the essence of good analogical reasoning.

Greater systematicity is neither necessary nor sufficient for a moreplausible analogical inference. It is obvious that increasedsystematicity is notsufficient for increased plausibility.An implausible analogy can be represented in a form that exhibits ahigh degree of structural parallelism. High-order relations can comecheap, as we saw with Achinstein’s “swan” example (§2.4).

More pointedly, increased systematicity is notnecessary forgreater plausibility. Indeed, in causal analogies, it may even weakenthe inference. That is becausesystematicity takes no accountof thetype of causal relevance, positive or negative. (McKay1993) notes that microbes have been found in frozen lakes inAntarctica; by analogy, simple life forms might exist on Mars.Freezing temperatures arepreventive or counteracting causes;they arenegatively relevant to the existence of life. Theclimate of Mars was probablymore favorable to life 3.5billion years ago than it is today, because temperatures were warmer.Yet the analogy between Antarctica and present-day Mars ismoresystematic than the analogy between Antarctica and ancient Mars.According to theSystematicity Principle, the analogy withAntarctica provides stronger support for life on Mars today than itdoes for life on ancient Mars.

The point of this example is that increased systematicity does notalways increase plausibility, and reduced systematicity does notalways decrease it (see Lee and Holyoak 2008). The more general pointis that systematicity can be misleading, unless we take into accountthenature of the relationships between various factors andthe hypothetical analogy. Systematicity does not magically produce orexplain the plausibility of an analogical argument. When we reason byanalogy, we must determine which features of both domains are relevantandhow they relate to the analogical conclusion. There is noshort-cut via syntax.

Schlimm (2008) offers an entirely different critique of thestructure-mapping theory from the perspective of analogical reasoningin mathematics—a domain where one might expect a formal approachsuch asstructure mapping to perform well. Schlimm introducesa simple distinction: a domain isobject-rich if the numberof objects is greater than the number of relations (and properties),andrelation-rich otherwise. Proponents of thestructure-mapping theory typically focus on relation-rich examples(such as the analogy between the solar system and the atom). Bycontrast, analogies in mathematics typically involve domains with anenormous number of objects (like the real numbers), but relatively fewrelations and functions (addition, multiplication, less-than).

Schlimm provides an example of an analogical reasoning problem ingroup theory that involves a single relation in each domain. In thiscase, attaining maximal systematicity is trivial. The difficulty isthat, compatible with maximal systematicity, there are different waysin which the objects might be placed in correspondence. Thestructure-mapping theory appears to yield the wrong inference. Wemight put the general point as follows: in object-rich domains,systematicity ceases to be a reliable guide to plausible analogicalinference.

3.5 Other theories

3.5.1 Connectionist models

During the past thirty-five years, cognitive scientists have conductedextensive research on analogy. Gentner’s SME is just one of manycomputational theories, implemented in programs that construct and useanalogies. Three helpful anthologies that span this period are Helman1988; Gentner, Holyoak, and Kokinov 2001; and Kokinov, Holyoak, andGentner 2009.

One predominant objective of this research has been to model thecognitive processes involved in using analogies. Early models tendedto be oriented towards “understanding the basic constraints thatgovern human analogical thinking” (Hummel and Holyoak 1997:458). Recent connectionist models have been directed towardsuncovering the psychological mechanisms that come into play when weuse analogies:retrieval of a relevant source domain,analogical mapping across domains, andtransfer ofinformation andlearning of new categories or schemas.

In some cases, such as the structure-mapping theory (§3.4), thisresearch overlaps directly with the normative questions that are thefocus of this entry; indeed, Gentner’sSystematicityPrinciple may be interpreted normatively. In other cases, wemight view the projects asdisplacing those traditionalnormative questions with up-to-date, computational forms ofnaturalized epistemology. Two approaches are singled out here because both raise importantchallenges to the very idea of finding sharp answers to thosequestions, and both suggest that connectionist models offer a morefruitful approach to understanding analogical reasoning.

The first is theconstraint-satisfactionmodel (also known as themulticonstrainttheory), developed by Holyoak and Thagard (1989, 1995).Like Gentner, Holyoak and Thagard regard the heart of analogicalreasoning asanalogy mapping, and they stress the importanceof systematicity, which they refer to as astructuralconstraint. Unlike Gentner, they acknowledge two additional types ofconstraints.Pragmatic constraints take into account thegoals and purposes of the agent, recognizing that “the purposewill guide selection” of relevant similarities.Semantic constraints represent estimates of the degree towhich people regard source and target items as being alike, ratherlike Hesse’s “pre-theoretic” similarities.

The novelty of the multiconstraint theory is that thesestructural,semantic andpragmaticconstraints are implemented not as rigid rules, but rather as‘pressures’ supporting or inhibiting potential pairwisecorrespondences. The theory is implemented in a connectionist programcalledACME (Analogical Constraint Mapping Engine),which assigns an initial activation value to each possible pairingbetween elements in the source and target domains (based on semanticand pragmatic constraints), and then runs through cycles that updatethe activation values based on overall coherence (structuralconstraints). The best global analogy mapping emerges under thepressure of these constraints. Subsequent connectionist models, suchas Hummel and Holyoak’s LISA program (1997, 2003), have madesignificant advances and hold promise for offering a more completetheory of analogical reasoning.

The second example is Hofstadter and Mitchell’sCopycat program (Hofstadter 1995; Mitchell1993). The program is “designed to discover insightfulanalogies, and to do so in a psychologically realistic way”(Hofstadter 1995: 205). Copycat operates in the domain ofletter-strings. The program handles the following type of problem:

Suppose the letter-stringabc were changed toabd;how would you change the letter-stringijk in “the sameway”?

Most people would answerijl, since it is natural to thinkthatabc was changed toabd by the“transformation rule”: replace the rightmost letter withits successor. Alternative answers are possible, but do not agree withmost people’s sense of what counts as the natural analogy.

Hofstadter and Mitchell believe that analogy-making is in large partabout theperception of novel patterns, and that suchperception requires concepts with “fluid” boundaries.Genuine analogy-making involves “slippage” of concepts.The Copycat program combines a set of core concepts pertaining toletter-sequences (successor,leftmost and so forth)with probabilistic “halos” that link distinct conceptsdynamically. Orderly structures emerge out of random low-levelprocesses and the program produces plausible solutions. Copycat thusshows that analogy-making can be modeled as a process akin toperception, even if the program employs mechanisms distinct from thosein human perception.

The multiconstraint theory and Copycat share the idea that analogicalcognition involves cognitive processes that operate below the level ofabstract reasoning. Both computational models—to the extent thatthey are capable of performing successful analogicalreasoning—challenge the idea that a successful model ofanalogical reasoning must take the form of a set of quasi-logicalcriteria. Efforts to develop a quasi-logicaltheory ofanalogical reasoning, it might be argued, have failed. In place offaulty inference schemes such as those described earlier (§2.2,§2.4), computational models substituteproceduresthat can be judged on their performance rather than on traditionalphilosophical standards.

In response to this argument, we should recognize the value of theconnectionist models while acknowledging that we still need a theorythat offers normative principles for evaluating analogical arguments.In the first place, even if the construction and recognition ofanalogies are largely a matter of perception, this does not eliminatethe need for subsequent critical evaluation of analogical inferences.Second and more importantly, we need to look not just at theconstruction of analogy mappings but at the ways in which individualanalogical arguments are debated in fields such as mathematics,physics, philosophy and the law. These high-level debates requirereasoning that bears little resemblance to the computational processesof ACME or Copycat. (Ashley’s HYPO (Ashley 1990) is one exampleof a non-connectionist program that focuses on this aspect ofanalogical reasoning.) There is, accordingly, room for bothcomputational and traditional philosophical models of analogicalreasoning.

3.5.2 Articulation model

Most prominent theories of analogy, philosophical and computational,are based on overall similarity between source and targetdomains—defined in terms of some favoured subset ofHesse’shorizontal relations (see§2.2). Aristotle and Mill, whose approach is echoed in textbook discussions,suggest counting similarities. Hesse’s theory (§3.3) favours “pre-theoretic” correspondences. Thestructure-mapping theory and its successors (§3.4) look to systematicity, i.e., to correspondences involving complex,high-level networks of relations. In each of these approaches, theproblem is twofold: overall similarity is not a reliable guide toplausibility, and it fails to explain the plausibility of anyanalogical argument.

Bartha’sarticulation model (2010)proposes a different approach, beginning not with horizontalrelations, but rather with a classification of analogical arguments onthe basis of thevertical relations within each domain. Thefundamental idea is that a good analogical argument must satisfy twoconditions:

Prior Association. There must be a clear connection, in thesource domain, between the known similarities (the positive analogy)and the further similarity that is projected to hold in the targetdomain (the hypothetical analogy). This relationship determines whichfeatures of the source arecritical to the analogicalinference.

Potential for Generalization. There must be reason to thinkthat the same kind of connection could obtain in the target domain.More pointedly: there must be nocritical disanalogy betweenthe domains.

The first order of business is to make the prior association explicit.The standards of explicitness vary depending on the nature of thisassociation (causal relation, mathematical proof, functionalrelationship, and so forth). The two general principles are fleshedout via a set of subordinate models that allow us to identify criticalfeatures and hence critical disanalogies.

To see how this works, considerExample 7 (Rectangles and boxes). In this analogical argument, the sourcedomain is two-dimensional geometry: we know that of all rectangleswith a fixed perimeter, the square has maximum area. The target domainis three-dimensional geometry: by analogy, we conjecture that of allboxes with a fixed surface area, the cube has maximum volume. Thisargument should be evaluatednot by counting similarities,looking to pre-theoretic resemblances between rectangles and boxes, orconstructing connectionist representations of the domains andcomputing a systematicity score for possible mappings. Instead, weshould begin with a precise articulation of the prior association inthe source domain, which amounts to a specific proof for the resultabout rectangles. We should then identify, relative to that proof, thecritical features of the source domain: namely, the concepts andassumptions used in the proof. Finally, we should assess the potentialfor generalization: whether, in the three-dimensional setting, thosecritical features are known to lack analogues in the target domain.The articulation model is meant to reflect the conversations that canand do take place between anadvocate and acriticof an analogical argument.

3.6 Practice-based approaches

3.6.1 Norton’s material theory of analogy

As noted in§2.4, Norton rejects analogical inference rules. But even if we agree withNorton on this point, we might still be interested in having anaccount that gives us guidelines for evaluating analogical arguments.How does Norton’s approach fare on this score?

According to Norton, each analogical argument is warranted by localfacts that must be investigated and justified empirically. First,there is “the fact of the analogy”: in practice, alow-level uniformity that embraces both the source and target systems.Second, there are additional factual properties of the target systemwhich, when taken together with the uniformity, warrant the analogicalinference. Consider Galileo’s famous inference (Example 12) that there are mountains on the moon (Galileo 1610). Through hisnewly invented telescope, Galileo observed points of light on the moonahead of the advancing edge of sunlight. Noting that the same thinghappens on earth when sunlight strikes the mountains, he concludedthat there must be mountains on the moon and even provided areasonable estimate of their height. In this example, Norton tells us,the fact of the analogy is that shadows and other opticalphenomena are generated in the same way on the earth and on the moon;the additional fact about the target is the existence of points oflight ahead of the advancing edge of sunlight on the moon.

What are the implications of Norton’s material theory when itcomes to evaluating analogical arguments?The fact of theanalogy is a local uniformity that powers the inference.Norton’s theory works well when such a uniformity is patent ornaturally inferred. It doesn’t work well when the uniformity isitself thetarget (rather than thedriver) of theinference. That happens with explanatory analogies such asExample 5 (theAcoustical Analogy), and mathematical analogies such asExample 7 (Rectangles and Boxes). Similarly, the theory doesn’twork well when the underlying uniformity is unclear, as inExample 2 (Life on other Planets),Example 4 (Clay Pots), and many other cases. In short, ifNorton’s theory is accepted, then for most analogical argumentsthere are no useful evaluation criteria.

3.6.2 Field-specific criteria

For those who sympathize with Norton’s skepticism aboutuniversal inductive schemes and theories of analogical reasoning, yetrecognize that his approach may be too local, an appealing strategy isto move up one level. We can aim for field-specific “workinglogics” (Toulmin 1958; Wylie and Chapman 2016; Reiss 2015). Thisapproach has been adopted by philosophers of archaeology, evolutionarybiology and other historical sciences (Wylie and Chapman 2016; Currie2013; Currie 2016; Currie 2018). In place of schemas, we find‘toolkits’, i.e., lists of criteria for evaluatinganalogical reasoning.

For example, Currie (2016) explores in detail the use of ethnographicanalogy (Example 13) between shamanistic motifs used by the contemporary San people andsimilar motifs in ancient rock art, found both among ancestors of theSan (direct historical analogy) and in European rock art (indirecthistorical analogy). Analogical arguments support the hypothesis thatin each of these cultures, rock art symbolizes hallucinogenicexperiences. Currie examines criteria that focus on assumptions aboutstability of cultural traits and environment-culture relationships.Currie (2016, 2018) and Wylie (Wylie and Chapman 2016) also stress theimportance of robustness reasoning that combines analogical argumentsof moderate strength with other forms of evidence to yield strongconclusions.

Practice-based approaches can thus yield specific guidelines unlikelyto be matched by any general theory of analogical reasoning. Onecaveat is worth mentioning. Field-specific criteria for ethnographicanalogy are elicited against a background of decades of methodologicalcontroversy (Wylie and Chapman 2016). Critics and defenders ofethnographic analogy have appealed to general models of scientificmethod (e.g., hypothetico-deductive method or Bayesian confirmation).To advance the methodological debate, practice-based approaches musteither make connections to these general models or explain why thelack of any such connection is unproblematic.

3.6.3 Formal analogies in physics

Close attention to analogical arguments in practice can also providevaluable challenges to general ideas about analogical inference. In aninteresting discussion, Steiner (1989, 1998) suggests that many of theanalogies that played a major role in early twentieth-century physicscount as “Pythagorean.” The term is meant to connotemathematical mysticism: a “Pythagorean” analogy is apurely formal analogy, one founded on mathematical similarities thathave no known physical basis at the time it is proposed. One exampleis Schrödinger’s use of analogy (Example 14) to “guess” the form of the relativistic wave equation. InSteiner’s view, Schrödinger’s reasoning relies uponmanipulations and substitutions based on purely mathematicalanalogies. Steiner argues that the success, and even the plausibility,of such analogies “evokes, or should evoke, puzzlement”(1989: 454). Both Hesse (1966) and Bartha (2010) reject the idea thata purely formal analogy, with no physical significance, can support aplausible analogical inference in physics. Thus, Steiner’sarguments provide a serious challenge.

Bartha (2010) suggests a response: we can decompose Steiner’sexamples into two or more steps, and then establish that at least onestep does, in fact, have a physical basis. Fraser (forthcoming),however, offers a counterexample that supports Steiner’sposition. Complex analogies between classical statistical mechanics(CSM) and quantum field theory (QFT) have played a crucial role in thedevelopment and application of renormalization group (RG) methods inboth theories (Example 15). Fraser notes substantial physical disanalogies between CSM and QFT,and concludes that the reasoning is based entirely on formalanalogies.

4. Philosophical foundations for analogical reasoning

What philosophical basis can be provided for reasoning by analogy?What justification can be given for the claim that analogicalarguments deliver plausible conclusions? There have been several ideasfor answering this question. One natural strategy assimilatesanalogical reasoning to some other well-understood argument pattern, aform of deductive or inductive reasoning (§4.1,§4.2). A few philosophers have explored the possibility ofa priori justification (§4.3). A pragmatic justification may be available for practical applicationsof analogy, notably in legal reasoning (§4.4).

Any attempt to provide a general justification for analogicalreasoning faces a basic dilemma. The demands of generality require ahigh-level formulation of the problem and hence an abstractcharacterization of analogical arguments, such as schema (4). On theother hand, as noted previously, many analogical arguments thatconform to schema (4) are bad arguments. So a general justification ofanalogical reasoning cannot provide support for all arguments thatconform to (4), on pain of proving too much. Instead, it must firstspecify a subset of putatively ‘good’ analogicalarguments, and link the general justification to this specifiedsubset. Theproblem of justification is linked to theproblem of characterizing good analogical arguments. Thisdifficulty afflicts some of the strategies described in thissection.

4.1 Deductive justification

Analogical reasoning may be cast in adeductive mold. Ifsuccessful, this strategy neatly solves the problem of justification.A valid deductive argument is as good as it gets.

An early version of the deductivist approach is exemplified byAristotle’s treatment of the argument from example (§3.2), theparadeigma. On this analysis, an analogical argumentbetween source domain \(S\) and target \(T\) begins with theassumption of positive analogy \(P(S)\) and \(P(T)\), as well as theadditional information \(Q(S)\). It proceeds via the generalization\(\forall x(P(x) \supset Q(x))\) to the conclusion: \(Q(T)\). Providedwe can treat that intermediate generalization as an independentpremise, we have a deductively valid argument. Notice, though, thatthe existence of the generalization renders the analogy irrelevant. Wecan derive \(Q(T)\) from the generalization and \(P(T)\), without anyknowledge of the source domain. The literature on analogy inargumentation theory (§2.2) offers further perspectives on this type of analysis, and on thequestion of whether analogical arguments are properly characterized asdeductive.

Some recent analyses follow Aristotle in treating analogical argumentsas reliant upon extra (sometimes tacit) premises, typically drawn frombackground knowledge, that convert the inference into a deductivelyvalid argument––but without making the source domainirrelevant. Davies and Russell introduce a version that relies uponwhat they calldetermination rules (Russell 1986; Davies andRussell 1987; Davies 1988). Suppose that \(Q\) and \(P_1 , \ldots,P_m\) are variables, and we have background knowledge that the valueof \(Q\) is determined by the values of \(P_1 , \ldots ,P_m\). In thesimplest case, where \(m = 1\) and both \(P\) and \(Q\) are binaryBoolean variables, this reduces to

i.e., whether or not \(P\) holds determines whether or not \(Q\)holds. More generally, the form of a determination rule is

i.e., \(Q\) is a function of \(P_1,\ldots\), \(P_m\). If we assumesuch a rule as part of our background knowledge, then an analogicalargument with conclusion \(Q(T)\) is deductively valid. Moreprecisely, and allowing for the case where \(Q\) is not a binaryvariable: if we have such a rule, and also premises stating that thesource \(S\) agrees with the target \(T\) on all of the values\(P_i\), then we may validly infer that \(Q(T) = Q(S)\).

The “determination rule” analysis provides a clear andsimple justification for analogical reasoning. Note that, in contrastto the Aristotelian analysis via the generalization \(\forall x(P(x)\supset Q(x))\), a determination rule does not trivialize theanalogical argument. Only by combining the rule with information aboutthe source domain can we derive the value of \(Q(T)\). To illustrateby adapting one of the examples given by Russell and Davies (Example 16), let’s suppose that the value \((Q)\) of a used car (relative toa particular buyer) is determined by its year, make, mileage,condition, color and accident history (the variables \(P_i)\). Itdoesn’t matter if one or more of these factors are redundant orirrelevant. Provided two cars are indistinguishable on each of thesepoints, they will have the same value. Knowledge of the source domainis necessary; we can’t derive the value of the second car fromthe determination rule alone. Weitzenfeld (1984) proposes a variant ofthis approach, advancing the slightly more general thesis thatanalogical arguments are deductive arguments with a missing(enthymematic) premise that amounts to a determination rule.

Do determination rules give us a solution to the problem of providinga justification for analogical arguments? In general: no. Analogiesare commonly applied to problems such asExample 8 (morphine and meperidine), where we are not even aware ofall relevant factors, let alone in possession of a determination rule.Medical researchers conduct drug tests on animals without knowing allattributes that might be relevant to the effects of the drug. Indeed,one of the main objectives of such testing is to guard againstreactions unanticipated by theory. On the “determinationrule” analysis, we must either limit the scope of such argumentsto cases where we have a well-supported determination rule, or focusattention on formulating and justifying an appropriate determinationrule. For cases such as animal testing, neither option seemsrealistic.

Recasting analogy as a deductive argument may help to bring outbackground assumptions, but it makes little headway with the problemof justification. That problem re-appears as the need to state andestablish the plausibility of a determination rule, and that is atleast as difficult as justifying the original analogical argument.

4.2 Inductive justification

Some philosophers have attempted to portray, and justify, analogicalreasoning in terms of some well-understood inductive argument pattern.There have been three moderately popular versions of this strategy.The first treats analogical reasoning as generalization from a singlecase. The second treats it as a kind of sampling argument. The thirdrecognizes the argument from analogy as a distinctive form, but treatspast successes as evidence for future success.

4.2.1 Single-case induction

Let’s reconsider Aristotle’s argument from example orparadeigma (§3.2), but this time regard the generalization as justified via inductionfrom a single case (the source domain). Can such a simple analysis ofanalogical arguments succeed? In general: no.

A single instance can sometimes lead to a justified generalization.Cartwright (1992) argues that we can sometimes generalize from asingle careful experiment, “where we have sufficient control ofthe materials and our knowledge of the requisite backgroundassumptions is secure” (51). Cartwright thinks that we can dothis, for example, in experiments with compounds that have stable“Aristotelian natures.” In a similar spirit, Quine (1969)maintains that we can have instantial confirmation when dealing withnatural kinds.

Even if we accept that there are such cases, the objection tounderstanding all analogical arguments as single-case induction isobvious: the view is simply too restrictive. Most analogical argumentswill not meet the requisite conditions. We may not know that we aredealing with a natural kind or Aristotelian nature when we make theanalogical argument. We may not know which properties are essential.An insistence on the ‘single-case induction’ analysis ofanalogical reasoning is likely to lead to skepticism (Agassi 1964,1988).

Interpreting the argument from analogy as single-case induction isalso counter-productive in another way. The simplistic analysis doesnothing to advance the search for criteria that help us to distinguishbetween relevant and irrelevant similarities, and hence between goodand bad analogical arguments.

4.2.2 Sampling arguments

On the sampling conception of analogical arguments, acknowledgedsimilarities between two domains are treated as statistically relevantevidence for further similarities. The simplest version of thesampling argument is due to Mill (1843/1930). An argument fromanalogy, he writes, is “a competition between the known pointsof agreement and the known points of difference.” Agreement of\(A\) and \(B\) in 9 out of 10 properties implies a probability of 0.9that \(B\) will possess any other property of \(A\): “we canreasonably expect resemblance in the same proportion” (367). Hisonly restriction has to do with sample size: we must be relativelyknowledgeable about both \(A\) and \(B\). Mill saw no difficulty inusing analogical reasoning to infer characteristics of newlydiscovered species of plants or animals, given our extensive knowledgeof botany and zoology. But if the extent of unascertained propertiesof \(A\) and \(B\) is large, similarity in a small sample would not bea reliable guide; hence, Mill’s dismissal of Reid’sargument about life on other planets (Example 2).

The sampling argument is presented in more explicit mathematical formby Harrod (1956). The key idea is that the known properties of \(S\)(the source domain) may be considered a random sample of all\(S\)’s properties—random, that is, with respect to theattribute of also belonging to \(T\) (the target domain). If themajority of known properties that belong to \(S\) also belong to\(T\), then we should expect most other properties of \(S\) to belongto \(T\), for it is unlikely that we would have come to know just thecommon properties. In effect, Harrod proposes a binomial distribution,modeling ‘random selection’ of properties on randomselection of balls from an urn.

There are grave difficulties with Harrod’s and Mill’sanalyses. One obvious difficulty is thecountingproblem: the ‘population’ of properties is poorlydefined. How are we to count similarities and differences? The ratioof shared to total known properties varies dramatically according tohow we do this. A second serious difficulty is theproblem ofbias: we cannot justify the assumption that the sample of knownfeatures is random. In the case of the urn, the selection process isarranged so that the result of each choice is not influenced by theagent’s intentions or purposes, or by prior choices. Bycontrast, the presentation of an analogical argument is alwayspartisan. Bias enters into the initial representation of similaritiesand differences: an advocate of the argument will highlightsimilarities, while a critic will play up differences. The paradigm ofrepeated selection from an urn seems totally inappropriate. Additionalvariations of the sampling approach have been developed (e.g., Russell1988), but ultimately these versions also fail to solve either thecounting problem or the problem of bias.

4.2.3 Argument from past success

Section3.6 discussed Steiner’s view that appeal to‘Pythagorean’ analogies in physics “evokes, orshould evoke, puzzlement” (1989: 454). Liston (2000) offers apossible response: physicists are entitled to use Pythagoreananalogies on the basis of induction from their past success:

[The scientist] can admit that no one knows how [Pythagorean]reasoning works and argue that the very fact that similar strategieshave worked well in the past is already reason enough to continuepursuing them hoping for success in the present instance. (200)

Setting aside familiar worries about arguments from success, the realproblem here is to determine what counts as a similar strategy. Inessence, that amounts to isolating the features of successfulPythagorean analogies. As we have seen (§2.4), nobody has yetprovided a satisfactory scheme that characterizes successfulanalogical arguments, let alone successful Pythagorean analogicalarguments.

4.3A priori justification

Ana priori approach traces the validity of a pattern ofanalogical reasoning, or of a particular analogical argument, to somebroad and fundamental principle. Three such approaches will beoutlined here.

The first is due to Keynes (1921). Keynes appeals to his famousPrinciple of the Limitation of Independent Variety, which hearticulates as follows:

(LIV)

The amount of variety in the universe is limited in such a waythat there is no one object so complex that its qualities fall into aninfinite number of independent groups (i.e., groups which might existindependently as well as in conjunction) (1921: 258).

Armed with this Principle and some additional assumptions, Keynes isable to show that in cases where there isno negativeanalogy, knowledge of the positive analogy increases the(logical) probability of the conclusion. If there is a non-trivialnegative analogy, however, then the probability of the conclusionremains unchanged, as was pointed out by Hesse (1966). Those familiarwith Carnap’s theory of logical probability will recognize thatin setting up his framework, Keynes settled on a measure that permitsno learning from experience.

Hesse offers a refinement of Keynes’s strategy, once again alongCarnapian lines. In her (1974), she proposes what she calls theClustering Postulate: the assumption that our epistemicprobability function has a built-in bias towards generalization. Theobjections to such postulates of uniformity are well-known (see Salmon1967), but even if we waive them, her argument fails. The mainobjection here—which also applies to Keynes—is that apurely syntactic axiom such as theClustering Postulate failsto discriminate between analogical arguments that are good and thosethat are clearly without value (according to Hesse’s ownmaterial criteria, for example).

A differenta priori strategy, proposed by Bartha (2010),limits the scope of justification to analogical arguments that satisfytentative criteria for ‘good’ analogical reasoning. Thecriteria are those specified by the articulation model (§3.5). In simplified form, they require the existence of non-trivialpositive analogy and no known critical disanalogy. The scope ofBartha’s argument is also limited to analogical argumentsdirected at establishingprima facie plausibility, ratherthan degree of probability.

Bartha’s argument rests on a principle of symmetry reasoningarticulated by van Fraassen (1989: 236): “problems which areessentially the same must receive essentially the samesolution.” A modal extension of this principle runs roughly asfollows: if problemsmight be essentially the same, then theymight have essentially the same solution. There are twomodalities here. Bartha argues that satisfaction of the criteria ofthe articulation model is sufficient to establish the modality in theantecedent, i.e., that the source and target domains ‘might beessentially the same’ in relevant respects. He further suggeststhatprima facie plausibility provides a reasonable readingof the modality in the consequent, i.e., that the problems in the twodomains ‘might have essentially the same solution.’ Tocall a hypothesisprima facie plausible is to elevate it tothe point where it merits investigation, since it might becorrect.

The argument is vulnerable to two sorts of concerns. First, there arequestions about the interpretation of the symmetry principle. Second,there is a residual worry that this justification, like all theothers, proves too much. The articulation model may be too vague ortoo permissive.

4.4 Pragmatic justification

Arguably, the most promising available defense of analogical reasoningmay be found in its application to case law (seePrecedent and Analogy in Legal Reasoning). Judicial decisions are based on the verdicts and reasoning that havegoverned relevantly similar cases, according to the doctrine ofstare decisis (Levi 1949; Llewellyn 1960; Cross and Harris1991; Sunstein 1993). Individual decisions by a court arebinding on that court and lower courts; judges are obligatedto decide future cases ‘in the same way.’ That is, thereasoning applied in an individual decision, referred to as theratio decidendi, must be applied to similar future cases (seeExample 10). In practice, of course, the situation is extremely complex. No twocases are identical. Theratio must be understood in thecontext of the facts of the original case, and there is considerableroom for debate about its generality and its applicability to futurecases. If a consensus emerges that a past case was wrongly decided,later judgments willdistinguish it from new cases,effectively restricting the scope of theratio to theoriginal case.

The practice of following precedent can be justified by two mainpractical considerations. First, and above all, the practice isconservative: it provides a relatively stable basis forreplicable decisions. People need to be able to predict the actions ofthe courts and formulate plans accordingly.Stare decisisserves as a check against arbitrary judicial decisions. Second, thepractice is still reasonablyprogressive: it allows for thegradual evolution of the law. Careful judges distinguish baddecisions; new values and a new consensus can emerge in a series ofdecisions over time.

In theory, then,stare decisis strikes a healthy balancebetween conservative and progressive social values. This justificationis pragmatic. It pre-supposes a common set of social values, and linksthe use of analogical reasoning to optimal promotion of those values.Notice also that justification occurs at the level of the practice ingeneral; individual analogical arguments sometimes go astray. A fullexamination of the nature and foundations forstare decisisis beyond the scope of this entry, but it is worth asking thequestion: might it be possible to generalize the justification forstare decisis? Is a parallel pragmatic justificationavailable for analogical arguments in general?

Bartha (2010) offers a preliminary attempt to provide such ajustification by shifting from social values to epistemic values. Thegeneral idea is that reasoning by analogy is especially well suited tothe attainment of a common set of epistemic goals or values. In simpleterms, analogical reasoning—when it conforms to certaincriteria—achieves an excellent (perhaps optimal) balance betweenthe competing demands of stability and innovation. It supports bothconservative epistemic values, such as simplicity and coherence withexisting belief, and progressive epistemic values, such asfruitfulness and theoretical unification (McMullin (1993) provides aclassic list).

5. Beyond analogical arguments

As emphasized earlier,analogical reasoning takes in a greatdeal more than analogical arguments. In this section, we examine twobroad contexts in which analogical reasoning is important.

The first, still closely linked to analogical arguments, is theconfirmation of scientific hypotheses. Confirmation is the process bywhich a scientific hypothesis receives inductive support on the basisof evidence (seeevidence,confirmation, andBayes’ Theorem). Confirmation may also signify thelogical relationship ofinductive support that obtains between a hypothesis \(H\) and aproposition \(E\) that expresses the relevant evidence. Can analogicalarguments play a role, either in the process or in the logicalrelationship? Arguably yes (to both), but this role has to bedelineated carefully, and several obstacles remain in the way of aclear account.

The second context isconceptual and theoretical developmentin cutting-edge scientific research. Analogies are used to suggestpossible extensions of theoretical concepts and ideas. The reasoningis linked to considerations of plausibility, but there is nostraightforward analysis in terms of analogical arguments.

5.1 Analogy and confirmation

How is analogical reasoning related to the confirmation of scientifichypotheses? The examples and philosophical discussion from earliersections suggest that a good analogical argument can indeed providesupport for a hypothesis. But there are good reasons to doubt theclaim that analogies provide actual confirmation.

In the first place, there is a logical difficulty. To appreciate this,let us concentrate on confirmation as a relationship betweenpropositions. Christensen (1999: 441) offers a helpful generalcharacterization:

Some propositions seem to help make it rational to believe otherpropositions. When our current confidence in \(E\) helps make rationalour current confidence in \(H\), we say that \(E\) confirms \(H\).

In the Bayesian model, ‘confidence’ is represented interms of subjective probability. A Bayesian agent starts with anassignment of subjective probabilities to a class of propositions.Confirmation is understood as a three-place relation:

(11)

Bayesian confirmation
\(E\) confirms \(H\) relative to \(K \leftrightarrow Pr(H \mid E \cdotK) \gt Pr(H \mid K)\).

\(E\) represents a proposition about accepted evidence, \(H\) standsfor a hypothesis, \(K\) for background knowledge and \(Pr\) for theagent’s subjective probability function. To confirm \(H\) is toraise its conditional probability, relative to \(K\). The shift fromprior probability \(Pr(H \mid K)\) toposteriorprobability \(Pr(H \mid E \cdot K)\) is referred to asconditionalization on \(E\). The relation between these twoprobabilities is typically given by Bayes’ Theorem (settingaside more complex forms of conditionalization):

For Bayesians, here is the logical difficulty: it seems that ananalogical argument cannot provide confirmation. In the first place,it is not clear that we can encapsulate the information contained inan analogical argument in a single proposition, \(E\). Second, even ifwe can formulate a proposition \(E\) that expresses that information,it is typically not appropriate to treat it as evidence because theinformation contained in \(E\) isalready part of thebackground, \(K\). This means that \(E \cdot K\) is equivalent to\(K\), and hence \(Pr(H \mid E \cdot K) = Pr(H \mid K)\). According tothe Bayesian definition, we don’t have confirmation. (This is aversion of the problem of old evidence; seeconfirmation.) Third, and perhaps most important, analogical arguments are oftenapplied to novel hypotheses \(H\) for which the prior probability\(Pr(H \mid K)\) is not even defined. Again, the definition ofconfirmation in terms of Bayesian conditionalization seemsinapplicable.

If analogies don’t provide inductive support via ordinaryconditionalization, is there an alternative? Here we face a seconddifficulty, once again most easily stated within a Bayesian framework.Van Fraassen (1989) has a well-known objection to any belief-updatingrule other than conditionalization. This objection applies to any rulethat allows us to boost credences when there is no new evidence. Thecriticism, made vivid by the tale of Bayesian Peter, is that these‘ampliative’ rules are vulnerable to aDutch Book. Adopting any such rule would lead us to acknowledge as fair a systemof bets that foreseeably leads to certain loss. Any rule of this typefor analogical reasoning appears to be vulnerable to vanFraassen’s objection.

There appear to be at least three routes to avoiding thesedifficulties and finding a role for analogical arguments withinBayesian epistemology. First, there is what we might callminimalBayesianism. Within the Bayesian framework, some writers(Jeffreys 1973; Salmon 1967, 1990; Shimony 1970) have argued that a‘seriously proposed’ hypothesis must have a sufficientlyhigh prior probability to allow it to become preferred as the resultof observation. Salmon has suggested that analogical reasoning is oneof the most important means of showing that a hypothesis is‘serious’ in this sense. If analogical reasoning isdirected primarily towardsprior probability assignments, itcan provide inductive support while remaining formally distinct fromconfirmation, avoiding the logical difficulties noted above. Thisapproach is minimally Bayesian because it provides nothing more thanan entry point into the Bayesian apparatus, and it only applies tonovel hypotheses. An orthodox Bayesian, such as de Finetti (de Finettiand Savage 1972, de Finetti 1974), might have no problem in allowingthat analogies play this role.

The second approach isliberal Bayesianism: we can change ourprior probabilities in anon-rule-based fashion. Somethingalong these lines is needed if analogical arguments are supposed toshift opinion about an already existing hypothesiswithoutany new evidence. This is common in fields such as archaeology, aspart of a strategy that Wylie refers to as “mobilizing old dataas new evidence” (Wylie and Chapman 2016: 95). As Hawthorne(2012) notes, some Bayesians simply accept that both initialassignments andongoing revision of prior probabilities(based on plausibility arguments) can be rational, but

thelogic of Bayesian induction (as described here) hasnothing to say about what values the prior plausibility assessmentsfor hypotheses should have; and it places no restrictions on how theymight change.

In other words, by not stating any rules for this type of probabilityrevision, we avoid the difficulties noted by van Fraassen. Thisapproach admits analogical reasoning into the Bayesian tent, butacknowledges a dark corner of the tent in which rationality operateswithout any clear rules.

Recently, a third approach has attracted interest:analogueconfirmation orconfirmation via analogue simulation. Asdescribed in (Dardashti et al. 2017), the idea is as follows:

Our key idea is that, in certain circumstances, predictions concerninginaccessible phenomena can be confirmed via an analogue simulation ina different system. (57)

Dardashti and his co-authors concentrate on a particular example (Example 17): ‘dumb holes’ and other analogues to gravitational blackholes (Unruh 1981; Unruh 2008). Unlike real black holes, some of theseanalogues can be (and indeed have been) implemented and studied in thelab. Given the exact formal analogy between our models for thesesystems and our models of black holes, and certain importantadditional assumptions, Dardashti et al. make the controversial claimthat observations made about the analogues provide evidence aboutactual black holes. For instance, the observation of phenomenaanalogous to Hawking radiation in the analogue systems would provideconfirmation for the existence of Hawking radiation in black holes. Ina second paper (Dardashti et al. 2018, Other Internet Resources), thecase for confirmation is developed within a Bayesian framework.

The appeal of a clearly articulated mechanism for analogueconfirmation is obvious. It would provide a tool for exploringconfirmation of inaccessible phenomena not just in cosmology, but alsoin historical sciences such as archaeology and evolutionary biology,and in areas of medical science where ethical constraints rule outexperiments on human subjects. Furthermore, as noted by Dardashti etal., analogue confirmation relies onnew evidence obtainedfrom the analogue system, and is therefore not vulnerable to thelogical difficulties noted above.

Although the concept of analogue confirmation is not entirely new(think of animal testing, as inExample 8), the claims of (Dardashti et al. 2017, 2018 [Other InternetResources]) require evaluation. One immediate difficulty for the blackhole example: if we think in terms of ordinary analogical arguments,there isno positive analogy because, to put it simply, wehave no basis of known similarities between a ‘dumb hole’and a black hole. As Crowther et al. (2018, Other Internet Resources)argue, “it is not known if the particular modelling frameworkused in the derivation of Hawking radiationactually describesblack holes in the first place.” This may not concernDardashti et al., since they claim that analogue confirmation isdistinct from ordinary analogical arguments. It may turn out thatanalogue confirmation is different for cases such as animal testing,where we have a basis of known similarities, and for cases where ouronly access to the target domain is via a theoretical model.

5.2 Conceptual change and theory development

In§3.6, we saw that practice-based studies of analogy provide insight intothe criteria for evaluating analogical arguments. Such studies alsopoint todynamical orprogrammatic roles foranalogies, which appear to require evaluative frameworks that gobeyond those developed for analogical arguments.

Knuttila and Loettgers (2014) examine the role of analogical reasoningin synthetic biology, an interdisciplinary field that draws onphysics, chemistry, biology, engineering and computational science.The main role for analogies in this field is not the construction ofindividual analogical arguments but rather the development of conceptssuch as “noise” and “feedback loops”. Suchconcepts undergo constant refinement, guided by both positive andnegative analogies to their analogues in engineered and physicalsystems. Analogical reasoning here is “transient, heterogeneous,and programmatic” (87). Negative analogies, seen as problematicobstacles for individual analogical arguments, take on a prominent andconstructive role when the focus is theoretical construction andconcept refinement.

Similar observations apply to analogical reasoning in its applicationto another cutting-edge field: emergent gravity. In this area ofphysics, distinct theoretical approaches portray gravity as emergingfrom different microstructures (Linneman and Visser 2018).“Novel and robust” features not present at the micro-levelemerge in the gravitational theory. Analogies with other emergentphenomena, such as hydrodynamics and thermodynamics, are exploited toshape these proposals. As with synthetic biology, analogical reasoningis not directed primarily towards the formulation and assessment ofindividual arguments. Rather, its role is to develop differenttheoretical models of gravity.

These studies explore fluid and creative applications of analogy toshape concepts on the front lines of scientific research. An adequateanalysis would certainly take us beyond the analysis of individualanalogical arguments, which have been the focus of our attention.Knuttila and Loettgers (2014) are led to reject the idea that theindividual analogical argument is the “primary unit” inanalogical reasoning, but this is a debatable conclusion. Linneman andVisser (2018), for instance, explicitly affirm the importance ofassessing the case for different gravitational models through“exemplary analogical arguments”:

We have taken up the challenge of making explicit arguments in favourof an emergent gravity paradigm… That arguments can only beplausibility arguments at the heuristic level does not mean that theyare immune to scrutiny and critical assessment tout court. Thephilosopher of physics’ job in the process of discovery ofquantum gravity… should amount to providing exactly this kindof assessments. (Linneman and Visser 2018: 12)

Accordingly, Linneman and Visser formulate explicit analogicalarguments for each model of emergent gravity, and assess them usingfamiliar criteria for evaluating individual analogical arguments.Arguably, even the most ambitious heuristic objectives still dependupon considerations of plausibility that benefit by being expressed,and examined, in terms of analogical arguments.

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Online Manuscript

• Crowther, K., N. Linnemann, and C. Wüthrich, 2018, “What we cannot learn from analogue experiments,” online at arXiv.org.
• Dardashti, R., S. Hartmann, K. Thébault, and E. Winsberg,2018, “Hawking Radiation and Analogue Experiments: A Bayesian Analysis,” online at PhilSci Archive.
• Norton, J., 2018. “Analogy”, unpublished draft, University of Pittsburgh.

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• Resources for Research on Analogy: a Multi-Disciplinary Guide (University of Windsor)
• UCLA Reasoning Lab (UCLA)
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