## 1. Introduction: the many roles of analogy



Analogies are widely recognized as playing an important
*heuristic* role, as aids to discovery. They have been
employed, 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 and
electricity,



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



Priestley may be over-stating the case, but there is no doubt that
analogies have suggested fruitful lines of inquiry in many fields.
Because of their heuristic value, analogies and analogical reasoning
have been a particular focus of AI research. Hajek (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 an
analogical argument is explicitly offered in support of some
conclusion. The intended degree of support for the conclusion can vary
considerably. At one extreme, these arguments can be strongly
predictive. For example:




***Example 1*.**
*Hydrodynamic analogies* exploit mathematical similarities
between the equations governing ideal fluid flow and torsional
problems. To predict stresses in a planned structure, one can
construct a fluid model, i.e., a system of pipes through which water
passes (Timoshenko and Goodier 1970). Within the limits of
idealization, such analogies allow us to make demonstrative
inferences, for example, from a measured quantity in the fluid model
to the analogous value in the torsional problem. In practice, there
are numerous complications (Sterrett 2006).



At the other extreme, an analogical argument may provide very weak
support for its conclusion, establishing no more than minimal
plausibility. Consider:



***Example 2*.**
 Thomas Reid's (1785) argument for the existence of life on
other planets (Stebbing 1933; Mill 1843/1930; Robinson 1930; Copi
1961). Reid notes a number of similarities between Earth and the other
planets in our solar system: all orbit and are illuminated by the sun;
several have moons; all revolve on an axis. In consequence, he
concludes, it is "not unreasonable to think, that those planets
may, like our earth, be the habitation of various orders of living
creatures" (1785: 24).



Such modesty is not uncommon. Often the point of an analogical
argument is just to persuade people to take an idea seriously. For
instance:




***Example 3*.**
 Darwin takes himself to be using an analogy between artificial and
natural selection to argue for the plausibility of the latter:



> 
> 
> Why may I not invent the hypothesis of Natural Selection (which from
> the analogy of domestic productions, and from what we know of the
> struggle of existence and of the variability of organic beings, is, in
> some very slight degree, in itself probable) and try whether this
> hypothesis of Natural Selection does not explain (as I think it does)
> a large number of facts.... (*Letter to Henslow*, May 1860
> in Darwin 1903)
> 
> 
> 



Here it appears, by Darwin's own admission, that his analogy is
employed to show that the hypothesis is probable to some "slight
degree" and thus merits further investigation. Some, however,
reject this characterization of Darwin's reasoning (Richards
1997; Gildenhuys 2004).




Sometimes analogical reasoning is the only available form of
justification for a hypothesis. The method of *ethnographic
analogy* is used to interpret



> 
> 
> the nonobservable behaviour of the ancient inhabitants of an
> archaeological site (or ancient culture) based on the similarity of
> their artifacts to those used by living peoples. (Hunter and Whitten
> 1976: 147)
> 
> 
> 



For example:



***Example 4*.**
 Shelley (1999, 2003) describes how ethnographic analogy was used to
determine the probable significance of odd markings on the necks of
Moche clay pots found in the Peruvian Andes. Contemporary potters in
Peru use these marks (called *signales*) to indicate
ownership; the marks enable them to reclaim their work when several
potters share a kiln or storage facility. Analogical reasoning may be
the only avenue of inference to the past in such cases, though this
point is subject to dispute (Gould and Watson 1982; Wylie 1982, 1985).
Analogical reasoning may have similar significance for cosmological
phenomena that are inaccessible due to limits on observation
(Dardashti et al. 2017). See
 SS5.1
 for further discussion.



As philosophers and historians such as Kuhn (1996) have repeatedly
pointed out, there is not always a clear separation between the two
roles that we have identified, discovery and justification. Indeed,
the two functions are blended in what we might call the
*programmatic* (or *paradigmatic*) role of analogy: over
a period of time, an analogy can shape the development of a program of
research. For example:




***Example 5*.**
 An 'acoustical analogy' was employed for many years by
certain nineteenth-century physicists investigating spectral lines.
Discrete spectra were thought to be



> 
> 
> completely analogous to the acoustical situation, with atoms (and/or
> molecules) serving as oscillators originating or absorbing the
> vibrations in the manner of resonant tuning forks. (Maier 1981:
> 51)
> 
> 
> 



Guided by this analogy, physicists looked for groups of spectral lines
that exhibited frequency patterns characteristic of a harmonic
oscillator. This analogy served not only to underwrite the
plausibility of conjectures, but also to guide *and limit*
discovery by pointing scientists in certain directions.




More generally, analogies can play an important programmatic role by
guiding conceptual development (see
 SS5.2).
 In some cases, a programmatic analogy culminates in the theoretical
unification of two different areas of inquiry.



***Example 6*.**
 Descartes's (1637/1954) correlation between geometry and
algebra provided methods for systematically handling geometrical
problems that had long been recognized as analogous. A very different
relationship between analogy and discovery exists when a programmatic
analogy breaks down, as was the ultimate fate of the acoustical
analogy. That atomic spectra have an entirely different explanation
became clear with the advent of quantum theory. In this case, novel
discoveries emerged *against* background expectations shaped by
the guiding analogy. There is a third possibility: an unproductive or
misleading programmatic analogy may simply become entrenched and
self-perpetuating as it leads us to "construct... data that
conform to it" (Stepan 1996: 133). Arguably, the danger of this
third possibility provides strong motivation for developing a critical
account of analogical reasoning and analogical arguments.



*Analogical cognition*, which embraces all cognitive processes
involved in discovering, constructing and using analogies, is broader
than analogical reasoning (Hofstadter 2001; Hofstadter and Sander
2013). Understanding these processes is an important objective of
current cognitive science research, and an objective that generates
many questions. How do humans identify analogies? Do nonhuman animals
use analogies in ways similar to humans? How do analogies and
metaphors influence concept formation?



This entry, however, concentrates specifically on analogical
arguments. Specifically, it focuses on three central epistemological
questions:


1. What criteria should we use to evaluate analogical arguments?
2. What philosophical justification can be provided for analogical
inferences?
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 of
analogical arguments, the entry reviews selected attempts to provide
answers to these three questions. To find such answers would
constitute an important first step towards understanding the nature of
analogical reasoning. To isolate these questions, however, is to make
the non-trivial assumption that there can be a *theory of*
*analogical arguments*--an assumption which, as we shall
see, is attacked in different ways by both philosophers and cognitive
scientists.


## 2. Analogical arguments


### 2.1 Examples



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



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



Two examples from the history of science:



***Example 8***
 (Morphine and meperidine). In 1934, the pharmacologist Schaumann was
testing synthetic compounds for their anti-spasmodic effect. These
drugs had a chemical structure similar to morphine. He observed that
one of the compounds--*meperidine*, also known as
*Demerol*--had a physical effect on mice that was
previously observed only with morphine: it induced an S-shaped tail
curvature. By analogy, he conjectured that the drug might also share
morphine's narcotic effects. Testing on rats, rabbits, dogs and
eventually humans showed that meperidine, like morphine, was an
effective pain-killer (Lembeck 1989: 11; Reynolds and Randall 1975:
273).



***Example 9***
 (Priestley on electrostatic force). In 1769, Priestley suggested that
the absence of electrical influence inside a hollow charged spherical
shell was evidence that charges attract and repel with an inverse
square force. He supported his hypothesis by appealing to the
analogous situation of zero gravitational force inside a hollow shell
of uniform density.



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 found
the manufacturer of a bottle of ginger beer liable for damages to a
consumer who became ill as a result of a dead snail in the bottle. The
court argued that the manufacturer had a duty to take
"reasonable care" in creating a product that could
foreseeably result in harm to the consumer in the absence of such
care, and where the consumer had no possibility of intermediate
examination. The principle articulated in this famous case was
extended, by analogy, to allow recovery for harm against an
engineering firm whose negligent repair work caused the collapse of a
lift (*Haseldine v. CA Daw & Son Ltd.* 1941 2 KB 343). By
contrast, the principle was not applicable to a case where a workman
was injured by a defective crane, since the workman had opportunity to
examine the crane and was even aware of the defects (*Farr v.
Butters Brothers & Co.* 1932 2 KB 606).


### 2.2 Characterization



What, if anything, do all of these examples have in common? We begin
with a simple, quasi-formal characterization. Similar formulations are
found in elementary critical thinking texts (e.g., Copi and Cohen
2005) and in the literature on argumentation theory (e.g., Govier
1999, Guarini 2004, Walton and Hyra 2018). An analogical argument has
the 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. The
argument form is *ampliative*; the conclusion is not guaranteed
to follow from the premises.



\(S\) and \(T\) are referred to as the *source domain* and
*target domain*, respectively. A *domain* is a set of
objects, properties, relations and functions, together with a set of
accepted statements about those objects, properties, relations and
functions. More formally, a domain consists of a set of objects and an
interpreted set of statements about them. The statements need not
belong to a first-order language, but to keep things simple, any
formalizations employed here will be first-order. We use unstarred
symbols \((a, P, R, f)\) to refer to items in the source domain and
starred symbols \((a^\*, P^\*, R^\*, f^\*)\) to refer to corresponding
items in the target domain. In
 *Example 9*,
 the source domain items pertain to gravitation; the target items
pertain to electrostatic attraction.



Formally, an *analogy* between \(S\) and \(T\) is a one-to-one
mapping between objects, properties, relations and functions in \(S\)
and those in \(T\). Not all of the items in \(S\) and \(T\) need to be
placed in correspondence. Commonly, the analogy only identifies
correspondences between a select set of items. In practice, we specify
an analogy simply by indicating the most significant similarities (and
sometimes differences).



We can improve on this preliminary characterization of the argument
from analogy by introducing the *tabular representation* found
in Hesse (1966). We place corresponding objects, properties, relations
and propositions side-by-side in a table of two columns, one for each
domain. For instance, Reid's argument
 (*Example 2*)
 can be represented as follows (using \(\Rightarrow\) for the
analogical inference):





|  | Earth \((S)\) |  | Mars \((T)\) |
| --- | --- | --- | --- |
| **\(\leftarrow\)vertical\(
\rightarrow\)** | Known similarities: |
| orbits the sun | \(\leftarrow\)horizontal\(
\rightarrow\) | orbits the sun |
| has a moon | has moons |
| revolves on axis | revolves on axis |
| subject to gravity | subject to gravity |
| Inferred similarity: |
|  | supports life | \(\Rightarrow\) | *may* support life |



Figure 1.




Hesse introduced useful terminology based on this tabular
representation. The *horizontal relations* in an analogy are
the relations of similarity (and difference) in the mapping between
domains, while the *vertical relations* are those between the
objects, relations and properties within each domain. The
correspondence (similarity) between earth's having a moon and
Mars' having moons is a horizontal relation; the causal relation
between having a moon and supporting life is a vertical relation
within the source domain (with the possibility of a distinct such
relation existing in the target as well).



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




***Positive analogy*.** Let \(P\) stand for a list
of accepted propositions \(P\_1 , \ldots ,P\_n\) about the source domain
\(S\). Suppose that the corresponding propositions \(P^\*\_1 , \ldots
,P^\*\_n\), abbreviated as \(P^\*\), are all accepted as holding for the
target domain \(T\), so that \(P\) and \(P^\*\) represent accepted (or
known) similarities. Then we refer to \(P\) as the *positive
analogy*.



***Negative analogy*.** Let \(A\) stand for a list
of propositions \(A\_1 , \ldots ,A\_r\) accepted as holding in \(S\),
and \(B^\*\) for a list \(B\_1^\*, \ldots ,B\_s^\*\) of propositions
holding in \(T\). Suppose that the analogous propositions \(A^\* =
A\_1^\*, \ldots ,A\_r^\*\) fail to hold in \(T\), and similarly the
propositions \(B = B\_1 , \ldots ,B\_s\) fail to hold in \(S\), so that
\(A, {\sim}A^\*\) and \({\sim}B, B^\*\) represent accepted (or known)
differences. Then we refer to \(A\) and \(B\) as the *negative
analogy*.



***Neutral analogy*.** The *neutral*
analogy consists of accepted propositions about \(S\) for which it is
not known whether an analogue holds in \(T\).




Finally we have:




***Hypothetical analogy*.** The
*hypothetical* analogy is simply the proposition \(Q\) in the
neutral analogy that is the focus of our attention.




These concepts allow us to provide a characterization for an
individual analogical argument that is somewhat richer than the
original one.

 

\[\tag{4} \text{Augmented tabular representation} \\
\begin{array}{ccc}
 \text{Source } (S) & \text{Target } (T) & \\
 P & P^\* & \text{[positive analogy]} \\
 A & {\sim}A^\* & \text{[negative analogy]} \\
{\sim}B & B^\* & \\
 Q & & \\
\hline
 & Q^\* & \text{(plausibly)}
\end{array}\]


An analogical argument may thus be summarized:



It is plausible that \(Q^\*\) holds in the target, *because* of
certain 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 say
something about the meaning of 'plausibly.' To ensure
broad applicability over analogical arguments that vary greatly in
strength, we interpret plausibility rather liberally as meaning
'with some degree of support'. In general, judgments of
plausibility are made after a claim has been formulated, but prior to
rigorous testing or proof. The next sub-section provides further
discussion.



Note that this characterization is incomplete in a number of ways. The
manner in which we list similarities and differences, the nature of
the correspondences between domains: these things are left
unspecified. Nor does this characterization accommodate reasoning with
multiple analogies (i.e., multiple source domains), which is
ubiquitous in legal reasoning and common elsewhere. To characterize
the argument form more fully, however, is not possible without either
taking a step towards a substantive theory of analogical reasoning or
restricting attention to certain classes of analogical arguments.



Arguments by analogy are extensively discussed within argumentation
theory. There is considerable debate about whether they constitute a
species of deductive inference (Govier 1999; Waller 2001; Guarini
2004; Kraus 2015). Argumentation theorists also make use of tools such
as speech act theory (Bermejo-Luque 2012), argumentation schemes and
dialogue types (Macagno et al. 2017; Walton and Hyra 2018) to
distinguish different types of analogical argument.



Arguments by analogy are also discussed in the vast literature on
scientific models and model-based reasoning, following the lead of
Hesse (1966). Bailer-Jones (2002) draws a helpful distinction between
analogies and models. While "many models have their roots in an
analogy" (2002: 113) and analogy "can act as a catalyst to
aid modeling," Bailer-Jones observes that "the aim of
modeling has nothing intrinsically to do with analogy." In
brief, models are tools for prediction and explanation, whereas
analogical arguments aim at establishing plausibility. An analogy is
evaluated in terms of source-target similarity, while a model is
evaluated on how successfully it "provides access to a
phenomenon in that it interprets the available empirical data about
the phenomenon." If we broaden our perspective beyond analogical
*arguments*, however, the connection between models and
analogies is restored. Nersessian (2009), for instance, stresses the
role of analog models in concept-formation and other cognitive
processes.


### 2.3 Plausibility



To say that a hypothesis is plausible is to convey that it has
epistemic support: we have some reason to believe it, even prior to
testing. An assertion of plausibility within the context of an inquiry
typically has pragmatic connotations as well: to say that a hypothesis
is plausible suggests that we have some reason to investigate it
further. For example, a mathematician working on a proof regards a
conjecture as plausible if it "has some chances of
success" (Polya 1954 (v. 2): 148). On both points, there is
ambiguity as to whether an assertion of plausibility is categorical or
a matter of degree. These observations point to the existence of two
distinct conceptions of plausibility, *probabilistic* and
*modal*, either of which may reflect the intended conclusion of
an analogical argument.



On the *probabilistic* conception, plausibility is naturally
identified with rational credence (rational subjective degree of
belief) and is typically represented as a probability. A classic
expression may be found in Mill's analysis of the argument from
analogy in *A System of Logic*:



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



In the terminology introduced in
 SS2.2,
 Mill's idea is that each element of the positive analogy boosts
the probability of the conclusion. Contemporary
'structure-mapping' theories
 (SS3.4)
 employ a restricted version: each *structural* similarity
between two domains contributes to the overall measure of similarity,
and hence to the strength of the analogical argument.



On the alternative *modal* conception, 'it is plausible
that \(p\)' is not a matter of degree. The meaning, roughly
speaking, is that there are sufficient initial grounds for taking
\(p\) seriously, i.e., for further investigation (subject to
feasibility and interest). Informally: \(p\) passes an initial
screening procedure. There is no assertion of degree. Instead,
'It is plausible that' may be regarded as an epistemic
modal operator that aims to capture a notion, *prima facie*
plausibility, that is somewhat stronger than ordinary epistemic
possibility. The intent is to single out \(p\) from an
undifferentiated mass of ideas that remain bare epistemic
possibilities. To illustrate: in 1769, Priestley's argument
 (*Example 9*),
 if successful, would establish the *prima facie* plausibility
of an inverse square law for electrostatic attraction. The set of
*epistemic* possibilities--hypotheses about electrostatic
attraction compatible with knowledge of the day--was much larger.
Individual analogical arguments in mathematics (such as
 *Example 7*)
 are almost invariably directed towards *prima facie*
plausibility.



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



> 
> 
> But in order that a theory may be valuable it must ... display an
> analogy. The propositions of the hypothesis must be analogous to some
> known laws.... (1957: 129)
> 
> 
> 



Commenting on the role of analogy in Fourier's theory of heat
conduction, Campbell writes:



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



The interesting notion here is that of a "valuable"
theory. We may not agree with Campbell that the existence of analogy
is "essential" for a novel theory to be
"valuable." But consider the weaker thesis that an
acceptable analogy is *sufficient* to establish that a theory
is "valuable", or (to qualify still further) that an
acceptable analogy provides defeasible grounds for taking the theory
seriously. (Possible defeaters might include internal inconsistency,
inconsistency with accepted theory, or the existence of a (clearly
superior) rival analogical argument.) The point is that Campbell,
following the lead of 19th century philosopher-scientists
such as Herschel and Whewell, thinks that analogies can establish this
sort of *prima facie* plausibility. Snyder (2006) provides a
detailed discussion of the latter two thinkers and their ideas about
the role of analogies in science.



In general, analogical arguments may be directed at establishing
either sort of plausibility for their conclusions; they can have a
probabilistic use or a modal use.
 *Examples 7*
 through
 9
 are best interpreted as supporting modal conclusions. In those
arguments, an analogy is used to show that a conjecture is worth
taking seriously. To insist on putting the conclusion in probabilistic
terms distracts attention from the point of the argument. The
conclusion might be modeled (by a Bayesian) as having a certain
probability value *because* it is deemed *prima facie*
plausible, but not vice versa.
 *Example 2*,
 perhaps, might be regarded as directed primarily towards a
probabilistic conclusion.



There should be connections between the two conceptions. Indeed, we
might think that the same analogical argument can establish both
*prima facie* plausibility and a degree of probability for a
hypothesis. But it is difficult to translate between epistemic modal
concepts and probabilities (Cohen 1980; Douven and Williamson 2006;
Huber 2009; Spohn 2009, 2012). We cannot simply take the probabilistic
notion as the primitive one. It seems wise to keep the two conceptions
of plausibility separate.


### 2.4 Analogical inference rules?



Schema (4)
 is a template that represents *all* analogical arguments, good
and bad. It is not an inference rule. Despite the confidence with
which particular analogical arguments are advanced, nobody has ever
formulated an acceptable rule, or set of rules, for valid analogical
inferences. There is not even a plausible candidate. This situation is
in marked contrast not only with deductive reasoning, but also with
elementary forms of inductive reasoning, such as induction by
enumeration.



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



(5) 
 Suppose \(S\) and \(T\) are the source and target domains.
Suppose \(P\_1 , \ldots ,P\_n\) (with \(n \ge 1)\) represents the
positive 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 reasons
for thinking otherwise, infer that \(Q^\*\) holds in the target domain
with degree of support \(p \gt 0\), where \(p\) is an increasing
function of \(n\) and a decreasing function of \(r\) and \(s\). 


Rule (5) is modeled on the
 straight rule for enumerative induction
 and inspired by Mill's view of analogical inference, as
described in
 SS2.3.
 We use the generic phrase 'degree of support' in place of
probability, since other factors besides the analogical argument may
influence our probability assignment for \(Q^\*\).



It is pretty clear that (5) is a non-starter. The main problem is that
the rule justifies too much. The only substantive requirement
introduced by (5) is that there be a nonempty positive analogy.
Plainly, there are analogical arguments that satisfy this condition
but establish no *prima facie* plausibility and no measure of
support for their conclusions.



Here is a simple illustration. Achinstein (1964: 328) observes that
there is a formal analogy between swans and line segments if we take
the 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 positive
support from this analogy for the idea that we are likely to find
congruent lines clustered in groups of two or more, just because swans
of the same color are commonly found in groups. The positive analogy
is antecedently known to be irrelevant to the hypothetical analogy. In
such a case, the analogical inference should be utterly rejected. Yet
rule (5) would wrongly assign non-zero degree of support.



To generalize the difficulty: not every similarity increases the
probability of the conclusion and not every difference decreases it.
Some similarities and differences are known to be (or accepted as
being) utterly irrelevant and should have no influence whatsoever on
our probability judgments. To be viable, rule (5) would need to be
supplemented with considerations of *relevance*, which depend
upon the subject matter, historical context and logical details
particular to each analogical argument. To search for a simple rule of
analogical inference thus appears futile.



Carnap and his followers (Carnap 1980; Kuipers 1988; Niiniluoto 1988;
Maher 2000; Romeijn 2006) have formulated principles of analogy for
inductive 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's
concept of analogy and analogical prediction. His approach is a hybrid
of Carnap-style inductive rules and a Bayesian model. Such an approach
would need to be generalized to handle the kinds of arguments
described in
 SS2.1.
 It remains unclear that the Carnapian approach can provide a general
rule for analogical inference.



Norton (2010, and 2018--see Other Internet Resources) has argued
that the project of formalizing inductive reasoning in terms of one or
more simple formal schemata is doomed. His criticisms seem especially
apt when applied to analogical reasoning. He writes:



> 
> 
> If analogical reasoning is required to conform only to a simple formal
> schema, the restriction is too permissive. Inferences are authorized
> that clearly should not pass muster... The natural response has
> been to develop more elaborate formal templates... The familiar
> difficulty is that these embellished schema never seem to be quite
> embellished enough; there always seems to be some part of the analysis
> that must be handled intuitively without guidance from strict formal
> rules. (2018: 1)
> 
> 
> 



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



To embrace a purely formal approach to analogy and to abjure
formalization entirely are two extremes in a spectrum of strategies.
There are intermediate positions. Most recent analyses (both
philosophical and computational) have been directed towards
elucidating criteria and procedures, rather than formal rules, for
reasoning by analogy. So long as these are not intended to provide a
universal 'logic' of analogy, there is room for such
criteria even if one accepts Norton's basic point. The next
section 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 evaluating
analogical 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 important
ones:



(G1) 
 The more similarities (between two domains), the stronger the
analogy. 
(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 than
those not involving causal relations. 
(G6) 
 Structural analogies are stronger than those based on superficial
similarities. 
(G7) 
 The relevance of the similarities and differences to the
conclusion (i.e., to the hypothetical analogy) must be taken into
account. 
(G8) 
 Multiple analogies supporting the same conclusion make the
argument stronger. 


These principles can be helpful, but are frequently too vague to
provide much insight. How do we count similarities and differences in
applying (G1) and (G2)? Why are the structural and causal analogies
mentioned in (G5) and (G6) especially important, and which structural
and causal features merit attention? More generally, in connection
with the all-important (G7): how do we determine which similarities
and differences are relevant to the conclusion? Furthermore, what are
we to say about similarities and differences that have been omitted
from an analogical argument but might still be relevant?



An additional problem is that the criteria can pull in different
directions. To illustrate, consider Reid's argument for life on
other 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 of
our vast ignorance of properties that might be relevant (G3).



There is a further problem that relates to the distinction just made
(in
 SS2.3)
 between two kinds of plausibility. Each of the above criteria apart
from (G7) is expressed in terms of the strength of the argument, i.e.,
the degree of support for the conclusion. The criteria thus appear to
presuppose the *probabilistic* interpretation of plausibility.
The problem is that a great many analogical arguments aim to establish
*prima facie* plausibility rather than any degree of
probability. Most of the guidelines are not directly applicable to
such arguments.


### 3.2 Aristotle's theory



Aristotle sets the stage for all later theories of analogical
reasoning. In his theoretical reflections on analogy and in his most
judicious examples, we find a sober account that lays the foundation
both for the commonsense guidelines noted above and for more
sophisticated analyses.



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



The *argument from example* (*paradeigma*) is described
in the *Rhetoric* and the *Prior Analytics*:



> 
> 
> Enthymemes based upon example are those which proceed from one or more
> similar cases, arrive at a general proposition, and then argue
> deductively to a particular inference. (*Rhetoric* 1402b15)
> 
> 
> 
> Let \(A\) be evil, \(B\) making war against neighbours, \(C\)
> Athenians against Thebans, \(D\) Thebans against Phocians. If then we
> wish to prove that to fight with the Thebans is an evil, we must
> assume that to fight against neighbours is an evil. Conviction of this
> is obtained from similar cases, e.g., that the war against the
> Phocians was an evil to the Thebans. Since then to fight against
> neighbours is an evil, and to fight against the Thebans is to fight
> against neighbours, it is clear that to fight against the Thebans is
> an evil. (*Pr. An.* 69a1)
> 
> 
> 



Aristotle notes two differences between this argument form and
induction (69a15ff.): it "does not draw its proof from all the
particular 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 to
single-case induction followed by deductive inference. It has the
following structure (using \(\supset\) for the conditional):



![[a tree diagram where S is source domain and T is target domain. First node is P(S)&Q(S) in the lower left corner. It is connected by a dashed arrow to (x)(P(x) superset Q(x)) in the top middle which in turn connects by a solid arrow to P(T) and on the next line P(T) superset Q(T) in the lower right. It in turn is connected by a solid arrow to Q(T) below it.]](example.png)

Figure 2.




In the terminology of
 SS2.2,
 \(P\) is the positive analogy and \(Q\) is the hypothetical analogy.
In Aristotle's example, \(S\) (the source) is war between
Phocians and Thebans, \(T\) (the target) is war between Athenians and
Thebans, \(P\) is war between neighbours, and \(Q\) is evil. The first
inference (dashed arrow) is inductive; the second and third (solid
arrows) are deductively valid.



The *paradeigma* has an interesting feature: it is amenable to
an alternative analysis as a purely *deductive* argument form.
Let us concentrate on Aristotle's assertion, "we must
assume that to fight against neighbours is an evil," represented
as \(\forall x(P(x) \supset Q(x))\). Instead of regarding this
intermediate step as something reached by induction from a single
case, we might instead regard it as a hidden presupposition. This
transforms the *paradeigma* into a syllogistic argument with a
missing or *enthymematic* premise, and our attention shifts to
possible means for establishing that premise (with single-case
induction as one such means). Construed in this way, Aristotle's
*paradeigma* argument foreshadows deductive analyses of
analogical reasoning (see
 SS4.1).



The *argument from likeness* (*homoiotes*) seems to be
closer than the *paradeigma* to our contemporary understanding
of analogical arguments. This argument form receives considerable
attention in *Topics* I, 17 and 18 and again in VIII, 1. The
most important passage is the following.



> 
> 
> Try to secure admissions by means of likeness; for such admissions are
> plausible, and the universal involved is less patent; e.g. that as
> knowledge and ignorance of contraries is the same, so too perception
> of contraries is the same; or vice versa, that since the perception is
> the same, so is the knowledge also. This argument resembles induction,
> but is not the same thing; for in induction it is the universal whose
> admission is secured from the particulars, whereas in arguments from
> likeness, what is secured is not the universal under which all the
> like cases fall. (*Topics* 156b10-17)
> 
> 
> 



This passage occurs in a work that offers advice for framing
dialectical arguments when confronting a somewhat skeptical
interlocutor. In such situations, it is best not to make one's
argument depend upon securing agreement about any universal
proposition. The argument from likeness is thus clearly distinct from
the *paradeigma*, where the universal proposition plays an
essential role as an intermediate step in the argument. The argument
from likeness, though logically less straightforward than the
*paradeigma*, is exactly the sort of analogical reasoning we
want when we are unsure about underlying generalizations.



In *Topics* I 17, Aristotle states that any shared attribute
contributes some degree of likeness. It is natural to ask when the
degree of likeness between two things is sufficiently great to warrant
inferring a further likeness. In other words, when does the argument
from likeness succeed? Aristotle does not answer explicitly, but a
clue is provided by the way he justifies particular arguments from
likeness. As Lloyd (1966) has observed, Aristotle typically justifies
such arguments by articulating a (sometimes vague) causal principle
which governs the two phenomena being compared. For example, Aristotle
explains the saltiness of the sea, by analogy with the saltiness of
sweat, as a kind of residual earthy stuff exuded in natural processes
such as heating. The common principle is this:



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



From this method of justification, we might conjecture that Aristotle
believes that the important similarities are those that enter into
such general causal principles.



Summarizing, Aristotle's theory provides us with four important
and influential criteria for the evaluation of analogical
arguments:


* *The strength of an analogy depends upon the number of
similarities.*
* *Similarity reduces to identical properties and
relations.*
* *Good analogies derive from underlying common causes or general
laws.*
* *A good analogical argument need not pre-suppose acquaintance
with the underlying universal (generalization).*



These four principles form the core of a *common-sense model*
for evaluating analogical arguments (which is not to say that they are
correct; indeed, the first three will shortly be called into
question). The first, as we have seen, appears regularly in textbook
discussions of analogy. The second is largely taken for granted, with
important exceptions in computational models of analogy
 (SS3.4).
 Versions of the third are found in most sophisticated theories. The
final point, which distinguishes the argument from likeness and the
argument from example, is endorsed in many discussions of analogy
(e.g., Quine and Ullian 1970).



A slight generalization of Aristotle's first principle helps to
prepare the way for discussion of later developments. As that
principle suggests, Aristotle, in common with just about everyone else
who has written about analogical reasoning, organizes his analysis of
the argument form around overall similarity. In the terminology of
 section 2.2,
*horizontal relationships* drive the reasoning: *the
greater the overall similarity of the two domains, the stronger the
analogical argument*. Hume makes the same point, though stated
negatively, in his *Dialogues 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 to
> a very weak analogy, which is confessedly liable to error and
> uncertainty. (1779/1947: 144)
> 
> 
> 



Most theories of analogy agree with Aristotle and Hume on this general
point. Disagreement relates to the appropriate way of measuring
overall similarity. Some theories assign greatest weight to
*material analogy*, which refers to shared, and typically
observable, features. Others give prominence to *formal
analogy*, emphasizing high-level structural correspondence. The
next two sub-sections discuss representative accounts that illustrate
these 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. She
formulates three requirements that an analogical argument must satisfy
in order to be acceptable:


1. *Requirement of material analogy*. The horizontal relations
must include similarities between observable properties.
2. *Causal condition*. The vertical relations must be causal
relations "in some acceptable scientific sense" (1966:
87).
3. *No-essential-difference condition*. The essential
properties and causal relations of the source domain must not have
been 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 she
refers to as *material analogy*. Material analogy is contrasted
with *formal analogy*. Two domains are formally analogous if
both are "interpretations of the same formal theory"
(1966: 68). *Nomic isomorphism* (Hempel 1965) is a special case
in which the physical laws governing two systems have identical
mathematical form. Heat and fluid flow exhibit nomic isomorphism. A
second example is the analogy between the flow of electric current in
a wire and fluid in a pipe. Ohm's law

 

\[\tag{6}
\Delta v = iR
\]


states that voltage difference along a wire equals current times a
constant resistance. This has the same mathematical form as
Poiseuille's law (for ideal fluids):

 

\[\tag{7}
\Delta p = \dot{V}k
\]


which states that the pressure difference along a pipe equals the
volumetric flow rate times a constant. Both of these systems can be
represented by a common equation. While formal analogy is linked to
common mathematical structure, it should not be limited to nomic
isomorphism (Bartha 2010: 209). The idea of formal analogy generalizes
to cases where there is a common mathematical structure between
*models* for two systems. Bartha offers an even more liberal
definition (2010: 195): "Two features are formally similar if
they occupy corresponding positions in formally analogous theories.
For example, pitch in the theory of sound corresponds to color in the
theory of light."



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



There are good reasons not to accept Hesse's requirement of
material analogy, construed in this narrow way. First, it is apparent
that formal analogies *are* the starting point in many
important inferences. That is certainly the case in mathematics, a
field in which material analogy, in Hesse's sense, plays no role
at all. Analogical arguments based on formal analogy have also been
extremely influential in physics (Steiner 1989, 1998).



In Norton's broad sense, however, 'material analogy'
simply refers to similarities rooted in factual knowledge of the
source 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 transferred
to the target domain, be *causally related* to the positive
analogy. In her words, the essential requirement for a good argument
from analogy is "a tendency to co-occurrence", i.e., a
causal relationship. She states the requirement as follows:



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



The causal condition rules out analogical arguments where there is no
causal knowledge of the source domain. It derives support from the
observation that many analogies do appear to involve a transfer of
causal knowledge.



The causal condition is on the right track, but is arguably too
restrictive. For example, it rules out analogical arguments in
mathematics. Even if we limit attention to the empirical sciences,
persuasive analogical arguments may be founded upon strong statistical
correlation in the absence of any *known* causal connection.
Consider
 (*Example 11*)
 Benjamin Franklin's prediction, in 1749, that pointed metal
rods would attract lightning, by analogy with the way they attracted
the "electrical fluid" in the laboratory:



> 
> 
> Electrical fluid agrees with lightning in these particulars: 1. Giving
> light. 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 inflammable
> substances. 12. Sulphureous smell.--The electrical fluid is
> attracted by points.--We do not know whether this property is in
> lightning.--But since they agree in all the particulars wherein
> we can already compare them, is it not probable they agree likewise in
> this? Let the experiment be made. (*Benjamin Franklin's
> Experiments*, 334)
> 
> 
> 



Franklin's hypothesis was based on a long list of properties
common to the target (lightning) and source (electrical fluid in the
laboratory). There was no known causal connection between the twelve
"particulars" and the thirteenth property, but there was a
strong correlation. Analogical arguments may be plausible even where
there are no known causal relations.


#### 3.3.3 No-essential-difference condition



Hesse's final requirement is that the "essential
properties and causal relations of the [source] have not been shown to
be part of the negative analogy" (1966: 91). Hesse does not
provide a definition of "essential," but suggests that a
property or relation is essential if it is "causally closely
related to the known positive analogy." For instance, an analogy
with fluid flow was extremely influential in developing the theory of
heat conduction. Once it was discovered that heat was not conserved,
however, the analogy became unacceptable (according to Hesse) because
conservation was so central to the theory of fluid flow.



This requirement, though once again on the right track, seems too
restrictive. It can lead to the rejection of a good analogical
argument. Consider the analogy between a two-dimensional rectangle and
a three-dimensional box
 (*Example 7*).
 Broadening Hesse's notion, it seems that there are many
'essential' differences between rectangles and boxes. This
does not mean that we should reject every analogy between rectangles
and boxes out of hand. The problem derives from the fact that
Hesse's condition is applied to the analogy *relation*
independently of the use to which that relation is put. What counts as
essential should vary with the analogical argument. Absent an
inferential 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 of
analogical reasoning. The causal condition and the
no-essential-difference condition incorporate local factors, as urged
by Norton, into the assessment of analogical arguments. These
conditions, singly or taken together, imply that an analogical
argument can fail to generate any support for its conclusion, even
when there is a non-empty positive analogy. Hesse offers no theory
about the 'degree' of analogical support. That makes her
account one of the few that is oriented towards the modal, rather than
probabilistic, use of analogical arguments
 (SS2.3).


### 3.4 Formal criteria: the structure-mapping theory



Many people take the concept of
 model-theoretic isomorphism
 to set the standard for thinking about similarity and its role in
analogical reasoning. They propose *formal criteria* for
evaluating analogies, based on overall structural or syntactical
similarity. Let us refer to theories oriented around such criteria as
*structuralist*.



A number of leading computational models of analogy are structuralist.
They are implemented in computer programs that begin with (or
sometimes build) representations of the source and target domains, and
then construct possible analogy mappings. Analogical inferences emerge
as a consequence of identifying the 'best mapping.' In
terms of criteria for analogical reasoning, there are two main ideas.
First, the goodness of an *analogical argument* is based on the
goodness of the associated *analogy mapping*. Second, the
goodness of the analogy mapping is given by a metric that indicates
how closely it approximates isomorphism.



The most influential structuralist theory has been Gentner's
*structure-mapping* theory, implemented in a program called the
*structure-mapping engine* (SME). In its original form (Gentner
1983), the theory assesses analogies on purely structural grounds.
Gentner asserts:



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



In order to clarify this thesis, Gentner introduces a distinction
between *properties*, or monadic predicates, and
*relations*, which have multiple arguments. She further
distinguishes among different *orders* of relations and
functions, defined inductively (in terms of the order of the relata or
arguments). The best mapping is determined by *systematicity*:
the extent to which it places higher-order relations, and items that
are nested in higher-order relations, in correspondence.
Gentner's *Systematicity Principle* states:



> 
> 
> A predicate that belongs to a mappable system of mutually
> interconnecting relationships is more likely to be imported into the
> target than is an isolated predicate. (1983: 163)
> 
> 
> 



A systematic analogy (one that places high-order relations and their
components in correspondence) is better than a less systematic
analogy. Hence, an analogical inference has a degree of plausibility
that increases monotonically with the degree of systematicity of the
associated analogy mapping. Gentner's fundamental criterion for
evaluating candidate analogies (and analogical inferences) thus
depends solely upon the syntax of the given representations and not at
all 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 earliest
version of the theory is vulnerable to worries about hand-coded
representations of source and target domains. Gentner and her
colleagues have attempted to solve this problem in later work that
generates LISP representations from natural language text (see Turney
2008 for a different approach).



The most important challenges for the structure-mapping approach
relate to the *Systematicity Principle* itself. Does the value
of an analogy derive entirely, or even chiefly, from systematicity?
There appear to be two main difficulties with this view. First: it is
not always appropriate to give priority to systematic, high-level
relational matches. Material criteria, and notably what Gentner refers
to as "superficial feature matches," can be extremely
important in some types of analogical reasoning, such as ethnographic
analogies which are based, to a considerable degree, on surface
resemblances between artifacts. Second and more significantly:
systematicity seems to be at best a fallible *marker* for good
analogies rather than the essence of good analogical reasoning.



Greater systematicity is neither necessary nor sufficient for a more
plausible analogical inference. It is obvious that increased
systematicity is not *sufficient* for increased plausibility.
An implausible analogy can be represented in a form that exhibits a
high degree of structural parallelism. High-order relations can come
cheap, as we saw with Achinstein's "swan" example
 (SS2.4).



More pointedly, increased systematicity is not *necessary* for
greater plausibility. Indeed, in causal analogies, it may even weaken
the inference. That is because *systematicity* takes no account
of the *type* of causal relevance, positive or negative. (McKay
1993) notes that microbes have been found in frozen lakes in
Antarctica; by analogy, simple life forms might exist on Mars.
Freezing temperatures are *preventive* or counteracting causes;
they are *negatively relevant* to the existence of life. The
climate of Mars was probably *more favorable* to life 3.5
billion years ago than it is today, because temperatures were warmer.
Yet the analogy between Antarctica and present-day Mars is *more
systematic* than the analogy between Antarctica and ancient Mars.
According to the *Systematicity Principle*, the analogy with
Antarctica provides stronger support for life on Mars today than it
does for life on ancient Mars.



The point of this example is that increased systematicity does not
always increase plausibility, and reduced systematicity does not
always decrease it (see Lee and Holyoak 2008). The more general point
is that systematicity can be misleading, unless we take into account
the *nature* of the relationships between various factors and
the hypothetical analogy. Systematicity does not magically produce or
explain the plausibility of an analogical argument. When we reason by
analogy, we must determine which features of both domains are relevant
and *how* they relate to the analogical conclusion. There is no
short-cut via syntax.



Schlimm (2008) offers an entirely different critique of the
structure-mapping theory from the perspective of analogical reasoning
in mathematics--a domain where one might expect a formal approach
such as *structure mapping* to perform well. Schlimm introduces
a simple distinction: a domain is *object-rich* if the number
of objects is greater than the number of relations (and properties),
and *relation-rich* otherwise. Proponents of the
structure-mapping theory typically focus on relation-rich examples
(such as the analogy between the solar system and the atom). By
contrast, analogies in mathematics typically involve domains with an
enormous number of objects (like the real numbers), but relatively few
relations and functions (addition, multiplication, less-than).



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


### 3.5 Other theories


#### 3.5.1 Connectionist models



During the past thirty-five years, cognitive scientists have conducted
extensive research on analogy. Gentner's SME is just one of many
computational theories, implemented in programs that construct and use
analogies. Three helpful anthologies that span this period are Helman
1988; Gentner, Holyoak, and Kokinov 2001; and Kokinov, Holyoak, and
Gentner 2009.



One predominant objective of this research has been to model the
cognitive processes involved in using analogies. Early models tended
to be oriented towards "understanding the basic constraints that
govern human analogical thinking" (Hummel and Holyoak 1997:
458). Recent connectionist models have been directed towards
uncovering the psychological mechanisms that come into play when we
use analogies: *retrieval* of a relevant source domain,
*analogical mapping* across domains, and *transfer* of
information and *learning* of new categories or schemas.



In some cases, such as the structure-mapping theory (SS3.4), this
research overlaps directly with the normative questions that are the
focus of this entry; indeed, Gentner's *Systematicity
Principle* may be interpreted normatively. In other cases, we
might view the projects as *displacing* those traditional
normative questions with up-to-date, computational forms of
 naturalized epistemology.
 Two approaches are singled out here because both raise important
challenges to the very idea of finding sharp answers to those
questions, and both suggest that connectionist models offer a more
fruitful approach to understanding analogical reasoning.



The first is the ***constraint-satisfaction
model*** (also known as the ***multiconstraint
theory***), developed by Holyoak and Thagard (1989, 1995).
Like Gentner, Holyoak and Thagard regard the heart of analogical
reasoning as *analogy mapping*, and they stress the importance
of systematicity, which they refer to as a *structural*
constraint. Unlike Gentner, they acknowledge two additional types of
constraints. *Pragmatic* constraints take into account the
goals and purposes of the agent, recognizing that "the purpose
will guide selection" of relevant similarities.
*Semantic* constraints represent estimates of the degree to
which people regard source and target items as being alike, rather
like Hesse's "pre-theoretic" similarities.



The novelty of the multiconstraint theory is that these
*structural*, *semantic* and *pragmatic*
constraints are implemented not as rigid rules, but rather as
'pressures' supporting or inhibiting potential pairwise
correspondences. The theory is implemented in a connectionist program
called **ACME** (Analogical Constraint Mapping Engine),
which assigns an initial activation value to each possible pairing
between elements in the source and target domains (based on semantic
and pragmatic constraints), and then runs through cycles that update
the activation values based on overall coherence (structural
constraints). The best global analogy mapping emerges under the
pressure of these constraints. Subsequent connectionist models, such
as Hummel and Holyoak's LISA program (1997, 2003), have made
significant advances and hold promise for offering a more complete
theory of analogical reasoning.



The second example is Hofstadter and Mitchell's
***Copycat*** program (Hofstadter 1995; Mitchell
1993). The program is "designed to discover insightful
analogies, and to do so in a psychologically realistic way"
(Hofstadter 1995: 205). Copycat operates in the domain of
letter-strings. The program handles the following type of problem:



Suppose the letter-string *abc* were changed to *abd*;
how would you change the letter-string *ijk* in "the same
way"?



Most people would answer *ijl*, since it is natural to think
that *abc* was changed to *abd* by the
"transformation rule": replace the rightmost letter with
its successor. Alternative answers are possible, but do not agree with
most people's sense of what counts as the natural analogy.



Hofstadter and Mitchell believe that analogy-making is in large part
about the *perception* of novel patterns, and that such
perception requires concepts with "fluid" boundaries.
Genuine analogy-making involves "slippage" of concepts.
The Copycat program combines a set of core concepts pertaining to
letter-sequences (*successor*, *leftmost* and so forth)
with probabilistic "halos" that link distinct concepts
dynamically. Orderly structures emerge out of random low-level
processes and the program produces plausible solutions. Copycat thus
shows that analogy-making can be modeled as a process akin to
perception, even if the program employs mechanisms distinct from those
in human perception.



The multiconstraint theory and Copycat share the idea that analogical
cognition involves cognitive processes that operate below the level of
abstract reasoning. Both computational models--to the extent that
they are capable of performing successful analogical
reasoning--challenge the idea that a successful model of
analogical reasoning must take the form of a set of quasi-logical
criteria. Efforts to develop a quasi-logical *theory* of
analogical reasoning, it might be argued, have failed. In place of
faulty inference schemes such as those described earlier
 (SS2.2,
 SS2.4), computational models substitute *procedures*
that can be judged on their performance rather than on traditional
philosophical standards.



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


#### 3.5.2 Articulation model



Most prominent theories of analogy, philosophical and computational,
are based on overall similarity between source and target
domains--defined in terms of some favoured subset of
Hesse's *horizontal relations* (see
 SS2.2).
 Aristotle and Mill, whose approach is echoed in textbook discussions,
suggest counting similarities. Hesse's theory
 (SS3.3)
 favours "pre-theoretic" correspondences. The
structure-mapping theory and its successors
 (SS3.4)
 look to systematicity, i.e., to correspondences involving complex,
high-level networks of relations. In each of these approaches, the
problem is twofold: overall similarity is not a reliable guide to
plausibility, and it fails to explain the plausibility of any
analogical argument.



Bartha's ***articulation model*** (2010)
proposes a different approach, beginning not with horizontal
relations, but rather with a classification of analogical arguments on
the basis of the *vertical relations* within each domain. The
fundamental idea is that a good analogical argument must satisfy two
conditions:



*Prior Association*. There must be a clear connection, in the
source domain, between the known similarities (the positive analogy)
and the further similarity that is projected to hold in the target
domain (the hypothetical analogy). This relationship determines which
features of the source are *critical* to the analogical
inference.



*Potential for Generalization*. There must be reason to think
that the same kind of connection could obtain in the target domain.
More pointedly: there must be no *critical* disanalogy between
the domains.



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



To see how this works, consider
 *Example 7*
 (Rectangles and boxes). In this analogical argument, the source
domain is two-dimensional geometry: we know that of all rectangles
with a fixed perimeter, the square has maximum area. The target domain
is three-dimensional geometry: by analogy, we conjecture that of all
boxes with a fixed surface area, the cube has maximum volume. This
argument should be evaluated *not* by counting similarities,
looking to pre-theoretic resemblances between rectangles and boxes, or
constructing connectionist representations of the domains and
computing a systematicity score for possible mappings. Instead, we
should begin with a precise articulation of the prior association in
the source domain, which amounts to a specific proof for the result
about rectangles. We should then identify, relative to that proof, the
critical features of the source domain: namely, the concepts and
assumptions used in the proof. Finally, we should assess the potential
for generalization: whether, in the three-dimensional setting, those
critical features are known to lack analogues in the target domain.
The articulation model is meant to reflect the conversations that can
and do take place between an *advocate* and a *critic*
of an analogical argument.


### 3.6 Practice-based approaches

 Studies of analogical reasoning based on scientific
practice provide valuable perspectives on criteria for evaluating
analogical arguments.

#### 3.6.1 Norton's material theory of analogy



As noted in
 SS2.4,
 Norton rejects analogical inference rules. But even if we agree with
Norton on this point, we might still be interested in having an
account 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 local
facts that must be investigated and justified empirically. First,
there is "the fact of the analogy": in practice, a
low-level uniformity that embraces both the source and target systems.
Second, there are additional factual properties of the target system
which, when taken together with the uniformity, warrant the analogical
inference. Consider Galileo's famous inference
 (***Example 12***)
 that there are mountains on the moon (Galileo 1610). Through his
newly invented telescope, Galileo observed points of light on the moon
ahead of the advancing edge of sunlight. Noting that the same thing
happens on earth when sunlight strikes the mountains, he concluded
that there must be mountains on the moon and even provided a
reasonable estimate of their height. In this example, Norton tells us,
the *the fact of the analogy* is that shadows and other optical
phenomena are generated in the same way on the earth and on the moon;
the additional fact about the target is the existence of points of
light ahead of the advancing edge of sunlight on the moon.



What are the implications of Norton's material theory when it
comes to evaluating analogical arguments? *The fact of the
analogy* is a local uniformity that powers the inference.
Norton's theory works well when such a uniformity is patent or
naturally inferred. It doesn't work well when the uniformity is
itself the *target* (rather than the *driver*) of the
inference. That happens with explanatory analogies such as
 *Example 5*
 (the *Acoustical Analogy*), and mathematical analogies such as
 *Example 7*
 (*Rectangles and Boxes*). Similarly, the theory doesn't
work well when the underlying uniformity is unclear, as in
 *Example 2*
 (*Life on other Planets*),
 *Example 4*
 (*Clay Pots*), and many other cases. In short, if
Norton's theory is accepted, then for most analogical arguments
there are no useful evaluation criteria.


#### 3.6.2 Field-specific criteria



For those who sympathize with Norton's skepticism about
universal inductive schemes and theories of analogical reasoning, yet
recognize that his approach may be too local, an appealing strategy is
to move up one level. We can aim for field-specific "working
logics" (Toulmin 1958; Wylie and Chapman 2016; Reiss 2015). This
approach has been adopted by philosophers of archaeology, evolutionary
biology and other historical sciences (Wylie and Chapman 2016; Currie
2013; Currie 2016; Currie 2018). In place of schemas, we find
'toolkits', i.e., lists of criteria for evaluating
analogical reasoning.



For example, Currie (2016) explores in detail the use of ethnographic
analogy
 (***Example 13***)
 between shamanistic motifs used by the contemporary San people and
similar motifs in ancient rock art, found both among ancestors of the
San (direct historical analogy) and in European rock art (indirect
historical analogy). Analogical arguments support the hypothesis that
in each of these cultures, rock art symbolizes hallucinogenic
experiences. Currie examines criteria that focus on assumptions about
stability of cultural traits and environment-culture relationships.
Currie (2016, 2018) and Wylie (Wylie and Chapman 2016) also stress the
importance of robustness reasoning that combines analogical arguments
of moderate strength with other forms of evidence to yield strong
conclusions.



Practice-based approaches can thus yield specific guidelines unlikely
to be matched by any general theory of analogical reasoning. One
caveat is worth mentioning. Field-specific criteria for ethnographic
analogy are elicited against a background of decades of methodological
controversy (Wylie and Chapman 2016). Critics and defenders of
ethnographic analogy have appealed to general models of scientific
method (e.g., hypothetico-deductive method or Bayesian confirmation).
To advance the methodological debate, practice-based approaches must
either make connections to these general models or explain why the
lack of any such connection is unproblematic.


#### 3.6.3 Formal analogies in physics



Close attention to analogical arguments in practice can also provide
valuable challenges to general ideas about analogical inference. In an
interesting discussion, Steiner (1989, 1998) suggests that many of the
analogies that played a major role in early twentieth-century physics
count as "Pythagorean." The term is meant to connote
mathematical mysticism: a "Pythagorean" analogy is a
purely formal analogy, one founded on mathematical similarities that
have no known physical basis at the time it is proposed. One example
is Schrodinger's use of analogy
 (***Example 14***)
 to "guess" the form of the relativistic wave equation. In
Steiner's view, Schrodinger's reasoning relies upon
manipulations and substitutions based on purely mathematical
analogies. 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 that
a purely formal analogy, with no physical significance, can support a
plausible analogical inference in physics. Thus, Steiner's
arguments provide a serious challenge.



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


## 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 analogical
arguments deliver plausible conclusions? There have been several ideas
for answering this question. One natural strategy assimilates
analogical reasoning to some other well-understood argument pattern, a
form of deductive or inductive reasoning
 (SS4.1,
 SS4.2). A few philosophers have explored the possibility of
*a priori* justification
 (SS4.3).
 A pragmatic justification may be available for practical applications
of analogy, notably in legal reasoning
 (SS4.4).



Any attempt to provide a general justification for analogical
reasoning faces a basic dilemma. The demands of generality require a
high-level formulation of the problem and hence an abstract
characterization of analogical arguments, such as schema (4). On the
other hand, as noted previously, many analogical arguments that
conform to schema (4) are bad arguments. So a general justification of
analogical reasoning cannot provide support for all arguments that
conform to (4), on pain of proving too much. Instead, it must first
specify a subset of putatively 'good' analogical
arguments, and link the general justification to this specified
subset. The *problem of justification* is linked to the
*problem of characterizing good analogical arguments*. This
difficulty afflicts some of the strategies described in this
section.


### 4.1 Deductive justification



Analogical reasoning may be cast in a *deductive* mold. If
successful, 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 by
Aristotle's treatment of the argument from example
 (SS3.2),
 the *paradeigma*. On this analysis, an analogical argument
between source domain \(S\) and target \(T\) begins with the
assumption of positive analogy \(P(S)\) and \(P(T)\), as well as the
additional information \(Q(S)\). It proceeds via the generalization
\(\forall x(P(x) \supset Q(x))\) to the conclusion: \(Q(T)\). Provided
we can treat that intermediate generalization as an independent
premise, we have a deductively valid argument. Notice, though, that
the existence of the generalization renders the analogy irrelevant. We
can derive \(Q(T)\) from the generalization and \(P(T)\), without any
knowledge of the source domain. The literature on analogy in
argumentation theory
 (SS2.2)
 offers further perspectives on this type of analysis, and on the
question of whether analogical arguments are properly characterized as
deductive.



Some recent analyses follow Aristotle in treating analogical arguments
as reliant upon extra (sometimes tacit) premises, typically drawn from
background knowledge, that convert the inference into a deductively
valid argument--but without making the source domain
irrelevant. Davies and Russell introduce a version that relies upon
what they call *determination rules* (Russell 1986; Davies and
Russell 1987; Davies 1988). Suppose that \(Q\) and \(P\_1 , \ldots
,P\_m\) are variables, and we have background knowledge that the value
of \(Q\) is determined by the values of \(P\_1 , \ldots ,P\_m\). In the
simplest case, where \(m = 1\) and both \(P\) and \(Q\) are binary
Boolean variables, this reduces to

 

\[\tag{8}
\forall x(P(x) \supset Q(x)) \vee \forall x(P(x) \supset{\sim}Q(x)), 
\]


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



\[\tag{9}
Q = F(P\_1 , \ldots ,P\_m), 
\]


i.e., \(Q\) is a function of \(P\_1,\ldots\), \(P\_m\). If we assume
such a rule as part of our background knowledge, then an analogical
argument with conclusion \(Q(T)\) is deductively valid. More
precisely, and allowing for the case where \(Q\) is not a binary
variable: if we have such a rule, and also premises stating that the
source \(S\) 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 and
simple justification for analogical reasoning. Note that, in contrast
to the Aristotelian analysis via the generalization \(\forall x(P(x)
\supset Q(x))\), a determination rule does not trivialize the
analogical argument. Only by combining the rule with information about
the source domain can we derive the value of \(Q(T)\). To illustrate
by adapting one of the examples given by Russell and Davies
 (***Example 16***),
 let's suppose that the value \((Q)\) of a used car (relative to
a particular buyer) is determined by its year, make, mileage,
condition, color and accident history (the variables \(P\_i)\). It
doesn't matter if one or more of these factors are redundant or
irrelevant. Provided two cars are indistinguishable on each of these
points, they will have the same value. Knowledge of the source domain
is necessary; we can't derive the value of the second car from
the determination rule alone. Weitzenfeld (1984) proposes a variant of
this approach, advancing the slightly more general thesis that
analogical 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 providing
a justification for analogical arguments? In general: no. Analogies
are commonly applied to problems such as
 *Example 8*
 (*morphine and meperidine*), where we are not even aware of
all relevant factors, let alone in possession of a determination rule.
Medical researchers conduct drug tests on animals without knowing all
attributes that might be relevant to the effects of the drug. Indeed,
one of the main objectives of such testing is to guard against
reactions unanticipated by theory. On the "determination
rule" analysis, we must either limit the scope of such arguments
to cases where we have a well-supported determination rule, or focus
attention on formulating and justifying an appropriate determination
rule. For cases such as animal testing, neither option seems
realistic.



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


### 4.2 Inductive justification



Some philosophers have attempted to portray, and justify, analogical
reasoning 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 single
case. The second treats it as a kind of sampling argument. The third
recognizes the argument from analogy as a distinctive form, but treats
past successes as evidence for future success.


#### 4.2.1 Single-case induction



Let's reconsider Aristotle's argument from example or
*paradeigma*
 (SS3.2),
 but this time regard the generalization as justified via induction
from a single case (the source domain). Can such a simple analysis of
analogical arguments succeed? In general: no.



A single instance can sometimes lead to a justified generalization.
Cartwright (1992) argues that we can sometimes generalize from a
single careful experiment, "where we have sufficient control of
the materials and our knowledge of the requisite background
assumptions is secure" (51). Cartwright thinks that we can do
this, 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 with
natural kinds.



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



Interpreting the argument from analogy as single-case induction is
also counter-productive in another way. The simplistic analysis does
nothing to advance the search for criteria that help us to distinguish
between relevant and irrelevant similarities, and hence between good
and bad analogical arguments.


#### 4.2.2 Sampling arguments



On the sampling conception of analogical arguments, acknowledged
similarities between two domains are treated as statistically relevant
evidence for further similarities. The simplest version of the
sampling argument is due to Mill (1843/1930). An argument from
analogy, he writes, is "a competition between the known points
of agreement and the known points of difference." Agreement of
\(A\) and \(B\) in 9 out of 10 properties implies a probability of 0.9
that \(B\) will possess any other property of \(A\): "we can
reasonably expect resemblance in the same proportion" (367). His
only restriction has to do with sample size: we must be relatively
knowledgeable about both \(A\) and \(B\). Mill saw no difficulty in
using analogical reasoning to infer characteristics of newly
discovered species of plants or animals, given our extensive knowledge
of botany and zoology. But if the extent of unascertained properties
of \(A\) and \(B\) is large, similarity in a small sample would not be
a reliable guide; hence, Mill's dismissal of Reid's
argument about life on other planets
 (*Example 2*).



The sampling argument is presented in more explicit mathematical form
by 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 the
attribute of also belonging to \(T\) (the target domain). If the
majority of known properties that belong to \(S\) also belong to
\(T\), then we should expect most other properties of \(S\) to belong
to \(T\), for it is unlikely that we would have come to know just the
common properties. In effect, Harrod proposes a binomial distribution,
modeling 'random selection' of properties on random
selection of balls from an urn.



There are grave difficulties with Harrod's and Mill's
analyses. One obvious difficulty is the *counting*
*problem*: the 'population' of properties is poorly
defined. How are we to count similarities and differences? The ratio
of shared to total known properties varies dramatically according to
how we do this. A second serious difficulty is the *problem of
bias*: we cannot justify the assumption that the sample of known
features is random. In the case of the urn, the selection process is
arranged so that the result of each choice is not influenced by the
agent's intentions or purposes, or by prior choices. By
contrast, the presentation of an analogical argument is always
partisan. Bias enters into the initial representation of similarities
and differences: an advocate of the argument will highlight
similarities, while a critic will play up differences. The paradigm of
repeated selection from an urn seems totally inappropriate. Additional
variations of the sampling approach have been developed (e.g., Russell
1988), but ultimately these versions also fail to solve either the
counting problem or the problem of bias.


#### 4.2.3 Argument from past success



Section
 3.6
 discussed Steiner's view that appeal to
'Pythagorean' analogies in physics "evokes, or
should evoke, puzzlement" (1989: 454). Liston (2000) offers a
possible response: physicists are entitled to use Pythagorean
analogies 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 strategies
> have worked well in the past is already reason enough to continue
> pursuing them hoping for success in the present instance. (200)
> 
> 
> 



Setting aside familiar worries about arguments from success, the real
problem here is to determine what counts as a similar strategy. In
essence, that amounts to isolating the features of successful
Pythagorean analogies. As we have seen (SS2.4), nobody has yet
provided a satisfactory scheme that characterizes successful
analogical arguments, let alone successful Pythagorean analogical
arguments.


### 4.3 *A priori* justification



An *a priori* approach traces the validity of a pattern of
analogical reasoning, or of a particular analogical argument, to some
broad and fundamental principle. Three such approaches will be
outlined here.



The first is due to Keynes (1921). Keynes appeals to his famous
Principle of the Limitation of Independent Variety, which he
articulates as follows:



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


Armed with this Principle and some additional assumptions, Keynes is
able to show that in cases where there is *no negative
analogy*, knowledge of the positive analogy increases the
(logical) probability of the conclusion. If there is a non-trivial
negative analogy, however, then the probability of the conclusion
remains unchanged, as was pointed out by Hesse (1966). Those familiar
with Carnap's theory of logical probability will recognize that
in setting up his framework, Keynes settled on a measure that permits
no learning from experience.



Hesse offers a refinement of Keynes's strategy, once again along
Carnapian lines. In her (1974), she proposes what she calls the
*Clustering Postulate*: the assumption that our epistemic
probability function has a built-in bias towards generalization. The
objections to such postulates of uniformity are well-known (see Salmon
1967), but even if we waive them, her argument fails. The main
objection here--which also applies to Keynes--is that a
purely syntactic axiom such as the *Clustering Postulate* fails
to discriminate between analogical arguments that are good and those
that are clearly without value (according to Hesse's own
material criteria, for example).



A different *a priori* strategy, proposed by Bartha (2010),
limits the scope of justification to analogical arguments that satisfy
tentative criteria for 'good' analogical reasoning. The
criteria are those specified by the articulation model
 (SS3.5).
 In simplified form, they require the existence of non-trivial
positive analogy and no known critical disanalogy. The scope of
Bartha's argument is also limited to analogical arguments
directed at establishing *prima facie* plausibility, rather
than degree of probability.



Bartha's argument rests on a principle of symmetry reasoning
articulated by van Fraassen (1989: 236): "problems which are
essentially the same must receive essentially the same
solution." A modal extension of this principle runs roughly as
follows: if problems *might be* essentially the same, then they
*might have* essentially the same solution. There are two
modalities here. Bartha argues that satisfaction of the criteria of
the articulation model is sufficient to establish the modality in the
antecedent, i.e., that the source and target domains 'might be
essentially the same' in relevant respects. He further suggests
that *prima facie* plausibility provides a reasonable reading
of the modality in the consequent, i.e., that the problems in the two
domains 'might have essentially the same solution.' To
call a hypothesis *prima facie* plausible is to elevate it to
the point where it merits investigation, since it might be
correct.



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


### 4.4 Pragmatic justification



Arguably, the most promising available defense of analogical reasoning
may be found in its application to case law (see
 Precedent and Analogy in Legal Reasoning).
 Judicial decisions are based on the verdicts and reasoning that have
governed relevantly similar cases, according to the doctrine of
*stare decisis* (Levi 1949; Llewellyn 1960; Cross and Harris
1991; Sunstein 1993). Individual decisions by a court are
*binding* on that court and lower courts; judges are obligated
to decide future cases 'in the same way.' That is, the
reasoning applied in an individual decision, referred to as the
*ratio decidendi*, must be applied to similar future cases (see
 *Example 10*).
 In practice, of course, the situation is extremely complex. No two
cases are identical. The *ratio* must be understood in the
context of the facts of the original case, and there is considerable
room for debate about its generality and its applicability to future
cases. If a consensus emerges that a past case was wrongly decided,
later judgments will *distinguish* it from new cases,
effectively restricting the scope of the *ratio* to the
original case.



The practice of following precedent can be justified by two main
practical considerations. First, and above all, the practice is
*conservative*: it provides a relatively stable basis for
replicable decisions. People need to be able to predict the actions of
the courts and formulate plans accordingly. *Stare decisis*
serves as a check against arbitrary judicial decisions. Second, the
practice is still reasonably *progressive*: it allows for the
gradual evolution of the law. Careful judges distinguish bad
decisions; new values and a new consensus can emerge in a series of
decisions over time.



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



Bartha (2010) offers a preliminary attempt to provide such a
justification by shifting from social values to epistemic values. The
general idea is that reasoning by analogy is especially well suited to
the attainment of a common set of epistemic goals or values. In simple
terms, analogical reasoning--when it conforms to certain
criteria--achieves an excellent (perhaps optimal) balance between
the competing demands of stability and innovation. It supports both
conservative epistemic values, such as simplicity and coherence with
existing belief, and progressive epistemic values, such as
fruitfulness and theoretical unification (McMullin (1993) provides a
classic list).


## 5. Beyond analogical arguments



As emphasized earlier, *analogical reasoning* takes in a great
deal more than analogical arguments. In this section, we examine two
broad contexts in which analogical reasoning is important.



The first, still closely linked to analogical arguments, is the
confirmation of scientific hypotheses. Confirmation is the process by
which a scientific hypothesis receives inductive support on the basis
of evidence (see
 evidence,
 confirmation, and
 Bayes' Theorem).
 Confirmation may also signify the *logical relationship* of
inductive support that obtains between a hypothesis \(H\) and a
proposition \(E\) that expresses the relevant evidence. Can analogical
arguments play a role, either in the process or in the logical
relationship? Arguably yes (to both), but this role has to be
delineated carefully, and several obstacles remain in the way of a
clear account.



The second context is *conceptual and theoretical development*
in cutting-edge scientific research. Analogies are used to suggest
possible extensions of theoretical concepts and ideas. The reasoning
is linked to considerations of plausibility, but there is no
straightforward analysis in terms of analogical arguments.


### 5.1 Analogy and confirmation



How is analogical reasoning related to the confirmation of scientific
hypotheses? The examples and philosophical discussion from earlier
sections suggest that a good analogical argument can indeed provide
support for a hypothesis. But there are good reasons to doubt the
claim 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 between
propositions. Christensen (1999: 441) offers a helpful general
characterization:



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



In the Bayesian model, 'confidence' is represented in
terms of subjective probability. A Bayesian agent starts with an
assignment of subjective probabilities to a class of propositions.
Confirmation is understood as a three-place relation:



(11) 
 *Bayesian confirmation*
  

\(E\) confirms \(H\) relative to \(K \leftrightarrow Pr(H \mid E \cdot
K) \gt Pr(H \mid K)\). 


\(E\) represents a proposition about accepted evidence, \(H\) stands
for a hypothesis, \(K\) for background knowledge and \(Pr\) for the
agent's subjective probability function. To confirm \(H\) is to
raise its conditional probability, relative to \(K\). The shift from
*prior probability* \(Pr(H \mid K)\) to *posterior
probability* \(Pr(H \mid E \cdot K)\) is referred to as
*conditionalization* on \(E\). The relation between these two
probabilities is typically given by Bayes' Theorem (setting
aside more complex forms of conditionalization):

 

\[\tag{12}
Pr(H \mid E \cdot K) = \frac{Pr(H \mid K) Pr(E \mid H \cdot K)}{Pr(E \mid K)}
\]


For Bayesians, here is the logical difficulty: it seems that an
analogical argument cannot provide confirmation. In the first place,
it is not clear that we can encapsulate the information contained in
an analogical argument in a single proposition, \(E\). Second, even if
we can formulate a proposition \(E\) that expresses that information,
it is typically not appropriate to treat it as evidence because the
information contained in \(E\) is *already* part of the
background, \(K\). This means that \(E \cdot K\) is equivalent to
\(K\), and hence \(Pr(H \mid E \cdot K) = Pr(H \mid K)\). According to
the Bayesian definition, we don't have confirmation. (This is a
version of the problem of old evidence; see
 confirmation.)
 Third, and perhaps most important, analogical arguments are often
applied to novel hypotheses \(H\) for which the prior probability
\(Pr(H \mid K)\) is not even defined. Again, the definition of
confirmation in terms of Bayesian conditionalization seems
inapplicable.



If analogies don't provide inductive support via ordinary
conditionalization, is there an alternative? Here we face a second
difficulty, once again most easily stated within a Bayesian framework.
Van Fraassen (1989) has a well-known objection to any belief-updating
rule other than conditionalization. This objection applies to any rule
that allows us to boost credences when there is no new evidence. The
criticism, made vivid by the tale of Bayesian Peter, is that these
'ampliative' rules are vulnerable to a
 Dutch Book.
 Adopting any such rule would lead us to acknowledge as fair a system
of bets that foreseeably leads to certain loss. Any rule of this type
for analogical reasoning appears to be vulnerable to van
Fraassen's objection.



There appear to be at least three routes to avoiding these
difficulties and finding a role for analogical arguments within
Bayesian epistemology. First, there is what we might call *minimal
Bayesianism*. Within the Bayesian framework, some writers
(Jeffreys 1973; Salmon 1967, 1990; Shimony 1970) have argued that a
'seriously proposed' hypothesis must have a sufficiently
high prior probability to allow it to become preferred as the result
of observation. Salmon has suggested that analogical reasoning is one
of the most important means of showing that a hypothesis is
'serious' in this sense. If analogical reasoning is
directed primarily towards *prior* probability assignments, it
can provide inductive support while remaining formally distinct from
confirmation, avoiding the logical difficulties noted above. This
approach is minimally Bayesian because it provides nothing more than
an entry point into the Bayesian apparatus, and it only applies to
novel hypotheses. An orthodox Bayesian, such as de Finetti (de Finetti
and Savage 1972, de Finetti 1974), might have no problem in allowing
that analogies play this role.



The second approach is *liberal Bayesianism*: we can change our
prior probabilities in a *non-rule-based fashion*. Something
along these lines is needed if analogical arguments are supposed to
shift opinion about an already existing hypothesis *without*
any new evidence. This is common in fields such as archaeology, as
part of a strategy that Wylie refers to as "mobilizing old data
as new evidence" (Wylie and Chapman 2016: 95). As Hawthorne
(2012) notes, some Bayesians simply accept that both initial
assignments and *ongoing revision* of prior probabilities
(based on plausibility arguments) can be rational, but



the *logic* of Bayesian induction (as described here) has
nothing to say about what values the prior plausibility assessments
for hypotheses should have; and it places no restrictions on how they
might change.



In other words, by not stating any rules for this type of probability
revision, we avoid the difficulties noted by van Fraassen. This
approach admits analogical reasoning into the Bayesian tent, but
acknowledges a dark corner of the tent in which rationality operates
without any clear rules.



Recently, a third approach has attracted interest: *analogue
confirmation* or *confirmation via analogue simulation*. As
described in (Dardashti et al. 2017), the idea is as follows:



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



Dardashti and his co-authors concentrate on a particular example
 (***Example 17***):
 'dumb holes' and other analogues to gravitational black
holes (Unruh 1981; Unruh 2008). Unlike real black holes, some of these
analogues can be (and indeed have been) implemented and studied in the
lab. Given the exact formal analogy between our models for these
systems and our models of black holes, and certain important
additional assumptions, Dardashti et al. make the controversial claim
that observations made about the analogues provide evidence about
actual black holes. For instance, the observation of phenomena
analogous to Hawking radiation in the analogue systems would provide
confirmation for the existence of Hawking radiation in black holes. In
a second paper (Dardashti et al. 2018, Other Internet Resources), the
case for confirmation is developed within a Bayesian framework.



The appeal of a clearly articulated mechanism for analogue
confirmation is obvious. It would provide a tool for exploring
confirmation of inaccessible phenomena not just in cosmology, but also
in historical sciences such as archaeology and evolutionary biology,
and in areas of medical science where ethical constraints rule out
experiments on human subjects. Furthermore, as noted by Dardashti et
al., analogue confirmation relies on *new evidence* obtained
from the analogue system, and is therefore not vulnerable to the
logical difficulties noted above.



Although the concept of analogue confirmation is not entirely new
(think of animal testing, as in
 *Example 8*),
 the claims of (Dardashti et al. 2017, 2018 [Other Internet
Resources]) require evaluation. One immediate difficulty for the black
hole example: if we think in terms of ordinary analogical arguments,
there is *no positive analogy* because, to put it simply, we
have 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 framework
used in the derivation of Hawking radiation *actually describes
black holes in the first place.*" This may not concern
Dardashti et al., since they claim that analogue confirmation is
distinct from ordinary analogical arguments. It may turn out that
analogue confirmation is different for cases such as animal testing,
where we have a basis of known similarities, and for cases where our
only access to the target domain is via a theoretical model.


### 5.2 Conceptual change and theory development



In
 SS3.6,
 we saw that practice-based studies of analogy provide insight into
the criteria for evaluating analogical arguments. Such studies also
point to *dynamical* or *programmatic* roles for
analogies, which appear to require evaluative frameworks that go
beyond those developed for analogical arguments.



Knuttila and Loettgers (2014) examine the role of analogical reasoning
in synthetic biology, an interdisciplinary field that draws on
physics, chemistry, biology, engineering and computational science.
The main role for analogies in this field is not the construction of
individual analogical arguments but rather the development of concepts
such as "noise" and "feedback loops". Such
concepts undergo constant refinement, guided by both positive and
negative analogies to their analogues in engineered and physical
systems. Analogical reasoning here is "transient, heterogeneous,
and programmatic" (87). Negative analogies, seen as problematic
obstacles for individual analogical arguments, take on a prominent and
constructive role when the focus is theoretical construction and
concept refinement.



Similar observations apply to analogical reasoning in its application
to another cutting-edge field: emergent gravity. In this area of
physics, distinct theoretical approaches portray gravity as emerging
from different microstructures (Linneman and Visser 2018).
"Novel and robust" features not present at the micro-level
emerge in the gravitational theory. Analogies with other emergent
phenomena, such as hydrodynamics and thermodynamics, are exploited to
shape these proposals. As with synthetic biology, analogical reasoning
is not directed primarily towards the formulation and assessment of
individual arguments. Rather, its role is to develop different
theoretical models of gravity.



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



> 
> 
> We have taken up the challenge of making explicit arguments in favour
> of an emergent gravity paradigm... That arguments can only be
> plausibility arguments at the heuristic level does not mean that they
> are immune to scrutiny and critical assessment tout court. The
> philosopher of physics' job in the process of discovery of
> quantum gravity... should amount to providing exactly this kind
> of assessments. (Linneman and Visser 2018: 12)
> 
> 
> 



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