REASONING BY ANALOGY
Abstract: Analogy plays an important role in science as well as in non-scientific domains such
as taxonomy or learning. We make explicit the difference and complementarity between the
concept of analogical statement, which merely states that two objects have a relevant
similarity, and the concept of analogical inference, which relies on the former in order to draw
a conclusion from some premises. For the first, we show that it is not possible to give an
absolute definition of what it means for two objects to be analogous; a relative definition of
analogy is introduced, only relevant from some point of view. For the second, we argue that
it is necessary to introduce a background over-hypothesis relating two sets of properties; the
belief strength of the conclusion is then directly related to the belief strength of the over-
hypothesis. Moreover, we assert the syntactical identity between analogical inference and
one case induction despite important pragmatic differences.
Keywords: analogy, induction, reasoning mode, similarity, taxonomy
Analogies are factual statements, on which a reasoning procedure may rely in order to draw
a new conclusion. Such reasoning by analogy is a very usual mode of reasoning, explicit or
implicit in epistemic practice. Many examples punctuate the history of science, for instance
the parallelism of structure between hydraulic and electric systems. Some typical applications
are at work in everyday life, for instance when a child learns a new language or when a lawyer
compares different normative situations. Various analogies are used to exemplify some ideas,
for instance when comparing natural and artificial selection. Finally, analogies help to build
taxonomies, classes being created by selecting the most pregnant similarities.
More precisely, three main functions of analogies and analogical reasoning may be considered.
The didactic function aims at providing a simple evocating image, either realist or poetic, of
some complex phenomenon, in order to fulfill a communicative or a pedagogic aim. The
heuristic function consists in suggesting the possible existence of some new property
possessed by an object when it is similar to another in other respects. The argumentative
1EHESS (54 Bd Raspail, Paris, France)
2CMLA (ENS Paris Saclay, 61 avenue du Président Wilson, 94235 Cachan, France) and IHPST (CNRS, ENS Ulm, 13 rue du Four, 75006, Paris,
function intends to sustain more firmly the belief in a new property attributed to some object
on the basis of its similarity to another object. The first function concerns situations where the
focus is on analogical judgments only, the two others concerns analogical reasoning, with
different epistemic status of its conclusion.
Of course, any analogical reasoning can be judged as more or less relevant. Hesse (1966)
asserts that some deep analogical reasoning forms the core of scientific research. Conversely,
Bouveresse (1999) observes that several sociological analogies result from a fanciful mode of
reasoning. A majority of observers think that analogies and analogical reasoning have to be
accepted or discarded through a one by one judgment. More constructively, some
epistemological works try to link and even to reduce analogical reasoning to more classical
ones, while others view it as a specific reasoning mode.
The paper assumes that reasoning by analogy obeys the same syntactical principles whatever
its field of application or its function. These principles will be expressed in a formal way by
avoiding as much as possible some frequent ill-defined or ambiguous concepts (“essential”,
“causal”, “relevant”). Such an underlying structure allows to compare analogical reasoning to
other formalized reasoning modes such as deductive or inductive. It helps moreover to
evaluate the rationality of analogical arguments.
Reasoning by analogy will be decomposed in two successive steps. An “analogical statement”
relies on a certain kind of similarity between two objects based on common properties, in a
relative rather than absolute sense. An “analogical inference” or an “analogical argument”
relies on an analogical statement in order to transfer some additional property from one
object to the other, by taking into account an explicit over-hypothesis. These two steps are
strongly linked since an analogical statement may prepare an analogical argument, contrary
to a mere similarity. The complete procedure allows to evaluate the rational belief in its
conclusion with regard to its assumptions.
A general framework, exclusively composed of objects and properties, is first introduced. In
order to ground the analogical statement, a relative definition of analogy between two objects
is expressed from a given point of view (§2). To specify the analogical inference, a background
over-hypothesis linking two sets of properties is made precise. The degree of belief in the
conclusion of the inference process depends then on the degree of belief in the over-
It is easily shown that analogical reasoning is syntactically identical to one case induction, even
if their more common examples are pragmatically different, and a new insight is proposed on
the deep reason of this difference. The analogical reasoning mode may also be related to case-
based reasoning (§4). A critical overview of the more recent philosophical and logical works
related to analogy is further provided (§5). Some conclusions about the specificity of our
approach are sketched and some insights for future analysis are suggested (§6).
2.1. General framework
We adopt the framework of first order logic. We assume the existence of a universe X of
objects, which are constants of the language, and denoted X, Y,… Objects can be material
(people, cars, trees) as well as symbolic (numbers, propositions, values). Objects can be
specific (John, my car, the Eiffel tower) or generic (a man, a car, a monument). Note that a
specific object is just a given entity while a generic object is already a class of specific ones.
We suppose moreover that a set P of properties, defined on the universe X of objects and
denoted P, Q, … is given. Properties are predicates (which can be one-place or many-place).
They can concern either material aspects (to be red, to be heavy) or symbolic ones (to be
greater than 3, to be nice). For a given object X, a property P is said to be relevant or not
whether it applies or not to that object. For instance, a ball is red or not, but this is irrelevant
for a number: so the property “to be red” is undefined for a number.
An analogy always relates in an oriented way two or more objects belonging respectively to
two different domains, the “source domain” and the “target domain”. Two types of relations
are introduced when linking these domains (Hesse, 1966). The horizontal relations link similar
properties present in the two domains. The vertical relations link different properties present
in the same domain. An analogical statement can be of different types.
The simplest analogical statement is called “notional analogy”. It expresses that “A is like B”,
written “A ~ B”, and just states that there is a specific kind of similarity between two specific
objects (John is like Ophelia) or between two generic ones (an airplane is like a bird). The
second object B belongs to the source domain while the first A belongs to the target domain.
This inversion comes from the fact that, in an analogical inference, it is a property of the source
that is transferred to the target (see §2.2).
A more elaborate form is called “relational analogy”. It expresses that “A is to B what C is to
D”, written “A:B :: C:D”, and actually points to a similarity between two couples of objects. It
can concern specific objects (Dante is to Italy what Shakespeare is to England), generic ones
(a hoof is to a horse what a foot is to a man) or even mixed ones (beer is to Belgium what wine
is to France).
This form is in fact not logically different from the previous one since it can be re-written as a
notional analogy between two couples: “(A,B) ~ (C,D)”. Relational analogy has often been
confused with analogy in general, since it gave its name to the concept in Aristotle’s work,
αναλογία applying to an identity in proportion. That is why it is often called “proportional
analogy” in the literature, though it does not rely only on the numerical concept of proportion.
An analogical statement can be extended with the same method to a t-uple of objects, and
stays always reducible to the simplest notional form. Hence, an analogy may concern two
formal models, where a model represents a multidimensional generic object. It is called
“structural analogy” and is written [A] ~ [B] where [X] is a model. Such a structural analogy can
be defined at two levels. It is “formal” when it concerns only the respective structure of related
variables and equations (the Lotka-Volterra model in biology is like the Goodwin model in
economics); Hempel (1965a) speaks of a nomic isomorphism. It is “substantial” when some
common interpretation is involved (the Fechner law in psychology is like the utility of money
in decision theory; they both concern the marginally decreasing subjective effect of a material
2.2. Absolute analogy
It is generally accepted that analogy, although it logically relies on a similarity ex post in a
symmetrical way is not symmetrical ex ante, contrary to the general concept of similarity.
Analogy is associated with an illocutionary intention that distinguishes a part which inherits a
property from a part for which the property is well known. When one says “your eyes are blue
like the sky”, the well-known property of the intense blue of the sky is attributed to someone
for making her a compliment. However, this remark concerns the intentional aspect contrary
to a purely formal syntactic point of view.
Despite this pragmatic non-symmetrical feature, a notional analogy may be considered
syntactically as an equivalence relation satisfying the following axioms:
- Reflexivity: A ~ A (an airplane is like an airplane)
- Symmetry: if A ~ B, then B ~ A (if an airplane is like a bird, then a bird is like an airplane)
- Transitivity: If A ~ B and B ~ C, then A ~ C (if an airplane is like a bird and a bird is like a
bee, then an airplane is like a bee).
For a relational analogy, these principles become:
- Horizontal Reflexivity: A:B :: A:B
- Horizontal Symmetry: if A:B :: C:D then C:D :: A:B
- Horizontal Transitivity: if A:B :: C:D and C:D :: E:F then A:B :: E:F
Since not every equivalence relation pretends to be an analogy, we need more principles in
order to characterize what it is. Of course, it is necessary that analogy does not reduce to
identity (if any couple is formed of same elements) or to triviality (if any possible couple
satisfies it). It is also required to propose a logical definition which is independent of any
specific field of objects.
However, following Quine (1969), it can be shown that it is not possible to propose a logical
definition of absolute analogy, since it is not even possible to give a logical definition of the
weaker notion of similarity. In any case, analogy is defined according to the properties shared
by two objects.
The simplest intuitive definition is:
(1) A ~ B iff there exists some property P such that P(A) & P(B)
Analogy is defined by the existence of a property commonly shared by both objects. Of course,
it is an equivalence relation but this definition is far too lax and is even trivial: it is always
possible to find a common property between any two objects. For example: a tiger is like a
zebra, they have stripes.
A more restrictive attempt is the following:
(2) A ~ B iff for any property P, P(A) & P(B)
But this covalence of properties leads to an extreme situation which reduces analogy to
Quine (1969) proposes to try an intermediary definition of similarity:
(3) A ~ B iff A and B have “many” common properties
But, as he highlights, this notion is too vague because one cannot tell how many properties
are required. In fact, the question is to determine what counts as a property. If any set of
objects counts as a property, then any two objects will be members of an arbitrary number of
sets and will share “many” properties. If one restricts the type of sets to the properties
collecting similar objects, we are led to a circular definition.
An attempt to escape from the problem faced by those general definitions may be to restrict
the admissible properties to a unique subset of externally defined properties that are relevant
for any object, say W. Hence, amended formulations of (1), (2), (3) may be:
(1a) A ~ B iff there exists a property P belonging to W such that P(A) & P(B)
(2a) A ~ B iff for any property P belonging to W, P(A) & P(B)
(3a) A ~ B iff A and B have “many” common properties P belonging to W
Clearly, (1a) may prevent the triviality of (1), (2a) may prevent the reducibility of (2) to identity,
(3a) may prevent the arbitrary nature of (3). The question is then to be able to give a relevant
definition for a universal W since universality is required for a definition of absolute analogy.
An intuitive way to do this would be to identify W with the set of “natural kinds”, also
considered by Quine (1969), who pointed the intuitive relationship between this concept and
the notion of similarity: a natural kind is a collection of similar objects and reversely similar
objects seem to be those very objects which are instances of the same natural kind. However,
this approach leads to many philosophical issues and is highly controversial, unless one
accepts a very specific essentialist position (see Bird and Tobin, 2015, for a critical presentation
of this position).
Firstly, the properties which define natural kinds are supposed to be the properties which are
“really important” for classifying objects in “genuinely natural ways”. But critics of that
position deny that any of our classifications is natural. Classifications are mere human tools
built within current language and science for practical purposes, not hard categories in the
world as for Plato. Dupré (1993) argues for instance in favour of a "ubiquitous realism,"
stressing that there are always a large number of ways to build taxonomies and kinds,
depending on the theoretical interests pursued. In fact, what is called "natural kinds" does not
correspond to essences or necessities existing in nature but to evolving categories that are
established according to complex pragmatic optimizations.
Secondly, Quine’s argumentation leads to the conclusion that defining natural kinds relies
again on too vague notions or to obvious circularities with the notion of similarity. Hence there
is neither philosophical nor logical way to define “natural kinds”, and to identify which set of
properties could be a proper W.
But the impossibility to find a formal definition of absolute analogy comes moreover from the
debates necessarily open when an analogy is refuted by a counter-analogy. The counter-
analogy suggests a better partner than the one proposed, for the source as well as for the
For instance, for a notional analogy:
- Bruges is the Venice of the North; it is a city built on canals.
- No, Bruges is not the Venice of the North; it is a city that never had any major
-It is Antwerp which is the Venice of the North, since it was a European economic
capital like Venice.
Likewise, for a relational analogy:
- Freud is to psychoanalysis what Piaget is to cognitive developmental psychology, its
most well-known inventor;
- No, Freud is not to psychoanalysis what Piaget is to cognitive developmental
psychology, he was not its first inventor.
- It is Joseph Breuer who is to psychoanalysis what Piaget is to cognitive
developmental psychology, since he is the first inventor of psychoanalysis according to
These debates underline the vacuity of vindicating any absolute analogy: even if A and B were
similar with respect to one point of view, they would usually differ from another point of view.
No pair of non-identical objects are similar from the standpoint of all possible properties, even
if we limit these properties to current categories: is an apple similar to a pear, because it is a
fruit, or to a tennis ball, because it is round ?
2.3. Relative analogy
So we are led to consider that any analogy has always to be expressed with respect to one
property or to a domain of properties. This is obviously an appropriate answer to the debates
on analogical statements: there is no more any assertion stated from a universal standpoint
but only relative points of views on the similarity between two objects.
The simplest way to represent relative analogy for notional analogy is again:
(1b) (A ~P B) iff there is some property P such that P(A) & P(B)3
For instance, an apple is like a pear, relatively to “fruitness”, they are fruits.
For relational analogy, the condition states:
(1b’) A:B ::R C:D iff there is a relation R (which is a two-place predicate), such that R(A,B) &
For instance, Paul is to Ana what Bob is to Julia, relatively to “sonness”, he is her son.
But on second thought, this seems to be a very restrictive way to express things. Indeed, an
analogy is significant because it spotlights that two objects share one particular property
among a list of other possible properties all pertaining to a certain way of describing the
objects. These two cars are analogous relatively to the colour if they are both blue, these two
animals are analogous relatively to the species if they are both dogs. But it would be rather
odd to say that these two cars are analogous relatively to their “blueness” or that these two
animals are analogous relatively to their “dogness”.
What is meant is the fact that these objects are analogous relatively to some “point of view”
which can be expressed by a set of related properties (for example colours or animal species).
They share the same property in this set while they could have two different ones (one car
could be blue and the other one red, one animal could be a dog and the other one a cat). What
is stressed by the analogy is that it is not the case that they have two different properties
inside the set which is considered. So these cars are analogous relatively to their colour, these
animals are analogous relatively to their specie.
Let’s define a domain Z as a set of possible disjoint properties4 (or disjoint values of one
property) which are associated with the same point of view. The relativization of any analogy
to a given domain expresses the speaker's intention to choose a specific point of view and her
intention to speak only with regard to this aspect of the world. A “point of view” is a mental
attitude consisting of applying a filter on the properties of things or events. This notion is
represented by the set Z which is not any set of properties, but a set of disjoint properties
which are then correlated through this disjunction.
Relative notional analogy can then be defined by:
(1c) (A ~Z B) iff there exists a property P within the domain Z, such as P(A) & P(B)
For instance, an apple is like a pear, with regard to vegetal kinds, they are fruits
Likewise, relative relational analogy is defined by:
(1c’) A:B ::Z C:D iff there is a relation R within the domain Z, such as R(A,B) & R(C,D)
3Be careful not to confuse definition (1b) with definition (1) whose form is very near. The latter was intended to
define analogy in an absolute form. The former uses the same condition (the existence of a common property)
but states that the two objects are similar only relatively to this property. That’s why it is denoted ~P.
4 The fact that these properties are disjoint means that no object can satisfy simultaneously two properties.
For instance, Paul is to Ana what Bob is to Julia, with regard to family relationship, he is her
One can check that relative notional analogy expressed by (1c) or by (1c’) satisfies all the
minimal principles required for a relevant definition of analogy. It can be used for any kind of
object and satisfies the usual theoretical properties:
- it is an equivalence relation.
- it is neither reduced to identity nor to triviality.
- it is not circular since Z is chosen by the agent with respect to the point of view she
wants to stress and does not need analogy to be defined.
Observe that reflexivity and symmetry are obvious and transitivity comes from the fact that Z
is defined as a list of disjoint properties. It is noticeable that transitivity is not respected with
the usual definition of absolute analogy since the properties shared by A and B are not
necessary the same than those shared by B and C, which cannot be the case here.
Il It could be possible to argue that the situation can be a little bit more complex if one
considers several common properties or relations involved in the relative analogy. For
- An apple is like a pear, with regard to vegetal kinds and colours, they are yellow fruits.
- Paul is to Ana what Bob is to Julia, with regard to family relationship and social
relationship, he is her son and he doesn’t care about her.
But these situations may be reduced to the simplest one by using a conjunction of predicates.
For instance: to be yellow and to be a fruit for the first example and to be a son and to not
care about his mother, for the second. The unique domain Z is then the set of disjoint
conjunctions formed by using these two predicates or relations.
The situation may be even more complex when the common properties involved in analogical
statements are different, though appearing to “correspond”, as several authors (Hesse (1966),
Juthe (2005), Bartha (2010)) pointed out. The problem is then to define rigorously this intuitive
but vague notion of “correspondence” between properties, and to ask if this should induce a
simple refinement or a radical change in our general definition of analogical statements.
We can start with a simple example. While lungs (A) make it possible to breathe in the air (P1 ),
gills (B) make it possible to breathe in the water (P2). The lungs therefore seem to be analogous
to the gills, although their properties are indeed different: the associated chemical
transformations are not the same. This is because they share a common property, “making
extraction of oxygen possible”, from water or from air. Each of those properties is an
application of this general property to animals that are aerial or aquatic. It is in this precise
meaning that they "correspond".
This idea may be made more precise.
The property to breathe in the air and the property to breathe in the water both imply the
property to breath. Hence lungs and gills share a same general property: to make possible to
breath, which is one of the alternative functions of vital organs (others being for instance
digestion, blood circulation…) whose set is the domain from the point of view of which the
analogy is expressed.
Let’s make it more general:
Let Z be a domain of properties over a domain of objects D, Z1 be the restriction of Z on D1 ⊂ D,
Z2 be the restriction of Z on D2 ⊂ D,
“P1 corresponds to P2 in Z” if there are properties P1 within Z1, P2 within Z2, P within Z such that
∀ X P1(X) → P(X) and ∀ X P2(X) → P(X). Then, we see that:
If [P1(A), P2(B), (P1 corresponds to P2 in Z)], then (A~B)z because P(A) and P(B).
Thus the cases of analogies with corresponding but different properties may be easily casted
in the general definition of analogies.
This reduction to the simpler general case can easily be extended to the correspondence
between properties Q1 and Q2 transferred by analogical inferences.
This form of analogy is frequent for scientific models, for instance when the equations of
different domains express the same mathematical relations between obviously different but
“corresponding” measures, for example those of electric and hydraulic networks. One can say
they share a common point of view, that of “constrained flows”. The electric intensity is like
the hydraulic debit because they imply both a quantity of fluids. The tension is like the
pressure variation because they imply both a potential of movement. Moreover, the law of
nodes (for intensities as well as debits) and the law of loops (for tensions as well as pressure
variations) apply to both of them. Finally, the Ohm law linking linearly intensity to tension is
analogous to the law linking linearly debit to pressure variation.
3. Analogical inference
3.1. Basic concepts
Typically, analogical inference consists in using a similarity between two objects as a premise
for inferring new similarities as conclusions. Then, in a way, it is considered that an analogical
statement is a kind of similarity judgement which justifies its extension to other properties: it
is this point which differentiates “mere similarities” from “analogies” or “relevant similarities”.
As expressed by Bartha (2010): “An analogical argument is an explicit representation of
analogical reasoning that cites accepted similarities between two systems in support of the
conclusion that some further similarities exist”.
An analogical inference uses an analogical statement to transfer the properties of an object to
another object. The analogical statement may be explicit or not, and represented by facts
which imply it, as illustrated for instance by the “violinist argument” (Thomson, 1971). Let Q
be a new one-place predicate in P or S a new two-place predicate. In the two basic forms
(notional analogy and relational analogy), an analogical inference states that:
(4) [A ~Z B, Q(B)]Q(A)
(4’) [A:B ::Z C:D, S(C,D)]S(A,B)
An analogical inference inherits the asymmetric nature of an analogical statement: it is based
on the property of the “source” transferred to the “target”. The symbol denotes a relation
of entailment between premises and conclusion. It has to be further characterized, and may
be assimilated to one of the many relations of entailment that have been described in the
literature. Besides deduction, one can cite non-monotonic logic (Krauss, Lehman and Magidor,
1990), belief revision (Alchourron, Gärdenfors & Makinson, 1985), inductive logic or
probabilistic logic. Our thesis is that should not be assimilated univocally to anyone of these
relations of entailment but instead that it should be interpreted accordingly to the status of a
background over-hypothesis that is necessary for understanding the analogical reasoning.
For the very same reasons that led us to introduce a domain Z listing the considered properties
for defining the relative analogy which is the premise, we introduce a domain Z’ listing the
properties that will be considered for the conclusion. The analogical reasoning (4) can now be
(5) [P ∊ Z, Q ∊ Z’, P(A), P(B), Q(B)]Q(A),
Why should we accept this inference scheme? Our answer does not rely on the construction
of a new consequence relation but on external supplementary hypotheses used by the person
who argues in favour of the conclusion. It is a meta-linguistic analysis (Jackson, 1991).More
precisely, one feels that Z and Z’ must be linked in some way but it is not easy to define
formally what this link is. In fact, it will be shown that this link is achieved by complementary
hypotheses that will play the role of selecting the relevant domain Z’ knowing Z.
For simplicity, these complementary hypotheses will be summarized in an over-hypothesis HE
such that in its presence, the belief in the conclusion is related to the premises in an intuitively
relevant way. The prior role of this assumption is not to transform these arguments into
deduction, but to discard the irrelevant reasoning associated with possible instances of the
purely syntactical criterion.
This external assumption HE will be integrated in the reasoning as follows:
(6) [(A ~Z B), Q(B), HE]Q(A)
and developed into:
(7) [P∊ Z, Q ∊ Z’, P(A), P(B), Q(B), HE]Q(A)
The symbol is used instead of in order to acknowledge the fact that including HE in the
premises gives a better epistemic status to the inference. We will discuss later the precise
status that must be given to . The extension to relational analogies or to analogies between
t-uples is obvious, replacing P and Q by two-place or t-place predicates.
Before being complemented by HE, these reasoning modes look like enthymemes, i.e.
inferences to which a premise lacks (according to the modern meaning of a concept created
by Aristotle with the broader meaning of “deductions from likelihoods and indices”, see Boyer,
1995). Musgrave (1989) was one of the first to suggest the transformation of inductive
inferences into deductive enthymemes. But understanding analogy and induction requires
giving a relevant formal account of these enthymemes, which fulfills several constraints and
does not necessarily lead to deductive conclusions.
3.2. Structure of the over-hypothesis
In order for HE to be relevant for explaining how analogical inference runs, we impose to it
the two following additional principles:
- HE must not lead to “trivialize” the reasoning in a way which would make unnecessary
the consultation of one of the other premises. This is the “non-redundancy” condition.
- the empirical protocol for believing in HE must be coherent, accessible and itself not
exposed to a higher-level redundancy.
We consider now more and more sophisticated expressions of HE. Let’s examine a first
(8) HE1 : ∀X, [P(X)→ Q(X)]
Trivialization is obviously at work in this extreme case. Due to the redundancy of the premises,
the reasoning becomes both trivial and deductive (the conclusion is certain). Analogical
inference becomes a “focusing” operation from a prior generic belief on a specific case:
knowing that P(A), the conclusion Q(A) is acquired without any need to refer to P(B).
Changing HE1 into a probabilistic relationship between Q(X) and P(X) would not change
drastically this result. Indeed, the preceding over-hypothesis HE1 is deterministic and
describes an explanation scheme of Q by P like the “Hempel-Oppenheim explanation”. Then
one may rely on a weaker entailment of Q by P as in Hempel’s (1965a)” inductive-statistical
(9) HE2: Pr(Q(X) / P(X)) = α
Analogical inference is no more deductive since it is now defeasible: a further observation may
question the conclusion. But it is still redundant: it is not necessary to consult B in order to get
a probability degree over Q(A). So it does not do the job.
To go further, Davies & Russell (1987) proposed an interesting candidate for HE, called the
“determination clause”. It is initially written as follows:
(10) HE3: [∀X, (P(X) →Q(X))] or [∀X, (P(X) →-Q(X))]
The conclusion is again obtained deductively, but involves all premises: it is necessary to use
Q(B) to infer Q(A). For instance, if HE3 states that: any golden object is insensible to acid or
any golden object is sensible to acid, the fact that my watch is golden and insensible to acid
implies that yours which is also golden is insensible to acid.
How is it possible to acquire the knowledge of a hypothesis like HE3? If all X such as P(X) have
been observed, the fact that Q(X) or -Q(X) for all these X is already known and there is no need
for the analogical reasoning to know that Q(A). Of course, if -Q(x) is the case, HE3 is irrelevant
for this analogical reasoning which is false. But if the fact that Q(X) or -Q(X) for all these X is
already known, the belief in HE3 is reduced to the belief in one of the two possibilities, either
to HE1 or to its contrary and we are back to redundancy. If only some X such as P(X) have been
observed and if all associated are such that Q(X) or all such that -Q(X), the belief in HE3 may
stem from an inductive process leading possibly to a probabilistic belief but again, the belief
will concern only one of the two possibilities mentioned in HE3. Every empirical observation
which does not refute HE3 will lead to believe either in its first part or in its second part. Then
there is no way to learn a hypothesis such as HE3 as it is.
The problem comes from the fact that H3 is not the right over-hypothesis that one needs in
order to complete the analogical reasoning. This is again a question of the right level of
properties to express things. The domains Z and Z’ whose important role has been noticed
have a role to play in the over-hypothesis. A more relevant over-hypothesis, whose important
difference with HE3 is not mentioned in Davies & Russel (1987), is then the following:
(11) HE4: For any property P in Z, for any property Q in Z’, [∀X, (P(X) →Q(X))] or [∀X,
It is important to stress the fact that due to the quantification over Z and Z’, HE4 is actually a
set of hypotheses for each object X. The meaning of HE4 is worth to be explained. Z is a list of
disjoint properties, Z’ is a list of disjoint properties. Then each object X can satisfy at most one
property in Z and at most one in Z’. HE4 means that all objects satisfying one given property
in Z must satisfy one same other property in Z’. In a way, HE4 links each property in Z with (at
most) one property in Z’.
For instance, consider that this car is like mine, it's a Chevrolet Silverado, and I want to infer
that it costs nearly the same price than mine. Davies & Russel’s HE3 hypothesis would state:
every Chevrolet Silverado costs between 28 000€ and 32 000 € or no Chevrolet Silverado costs
between 28 000 € and 32 000 €; my Chevrolet Silverado costs 30 000 €; hence this other
Chevrolet Silverado should cost between 28 000 € and 32 000 €. The proposed HE4 over-
hypothesis states instead: the cost of any car of a given type is situated in a range of prices;
my Chevrolet Silverado costs between 28000€ and 32000 €; hence this other Chevrolet
Silverado should cost between 28000 € and 32000 €. Ranges of price are defined exogenously
by the brand of cars.
It is exactly the situation which was suggested by Goodman (1947) in his “prospects for a
theory of projection”. Suppose that we are interested in the colors k of the marbles drawn
from a bag h which belongs to a stack of bags. What he calls “over-hypothesis” H of a
hypothesis G such as “all the marbles in the bag B are red” is a hypothesis H such as “every
bagful of the stack is uniform in color”. Goodman considers the situation where many bags of
the stack have been observed (but not the bag B itself) and where this observation leads to
confirm H. Having H in mind, observing a red marble from the bag B will support G. In this
case, Z is the set of predicates “to belong to the bag number h” and Z’ is the set of predicates
“to be of color k”. The over-hypothesis is the fact that each bag is associated with only one
Let’s come back to the example of the car. HE3 states that every Chevrolet Silverado costs
between 28.000€ and 32.000 € or no Chevrolet Silverado costs between 28.000 € and 32.000
€. As noticed, knowing that would mean having examined all (or a very large number) of
Chevrolet Silverado and noted that all cost between 28.000 € and 32.000 € (otherwise this
would be incompatible with the premise that mine costs between 28.000 € and 32.000 €). But
in this case, it is not HE3 that we would believe in but in Every Chevrolet Silverado costs
between 28.000€ and 32.000 €. Imagine now that we have observed a large number of
different models of cars and noted that all cars of the same type cost approximately the same
price. This observation, which is really plausible, would lead to the general hypothesis: For any
car of a given type, the cost is between the same range.
So, if Z is a list of types of cars and Z’ a list of mutually exclusive ranges of price, this would be
expressed in HE4 form as:
[For any car A, for any range of price T, [∀car, (car of model A → price of the car is in range T]
or [∀car, (car of model A → price of the car is not in range T]]
We immediately see that, contrary to HE3, the process allowing to acquire the belief in HE4 is
realist and it does not lead to a mutilation of the hypothesis in only one part of the alternative.
Moreover, it appears that HE3 is very strange. In absence of the premise “my Chevrolet
Silverado costs 30.000 €”, one does not see where the hypothesis “every Chevrolet Silverado
costs between 28.000€ and 32.000 € or no Chevrolet Silverado costs between 28.000 € and
32.000 €” could come from. Why using this particular range? The only answer seems to be
because many Chevrolet Silverado whose price lies in this range have been observed. But in
this case, why stating the whole HE3 and not only “every Chevrolet Silverado costs between
28.000€ and 32.000 €”? On the contrary, HE4 does not mention any particular range. It just
states that, having made a partition of all possible prices in some arbitrary ranges for any car,
each model of car is associated with only one range. This is not only more reasonable but
intuitively, this is exactly the way we think in various subjects.
Indeed, such an over-hypothesis frequently states a regularity between classes of objects
- species determines what animals eat;
- age and skills determines the class followed by pupils;
- nationality determines the mother tongue;
Each one of these examples is an example of HE4: A partition of a set Z of possible properties
is set in a functional relation with a partition of another set Z’ of possible properties.
This type of belief is usually acquired in a very natural way: having seen many objects satisfying
one property of a list of properties of the same type, it appears that each of them also satisfies
one property of another list and that to each one first property, the same second property is
always attached. This process is typically inductive in that we infer a general law from a limited
(but possibly large) number of cases. But the over-hypothesis could also result from an
abductive process since the conclusions may be the generic best explanation of the
One may worry about the fact that this way of conceiving an analogical inference is leading to
an infinite regression, since it relies on an over-hypothesis which has itself to be justified. But
it is not the case since foundationalism is not the goal here. The way to acquire HE4 is not
itself a part of analogical reasoning: what is required is only the fact that it is possible to
attribute to HE4 a degree of belief by a clear empirical protocol, without redundancy with the
analogical reasoning that it supports. But this degree of belief is exogenous to the present
3.3. Uncertainty on the over-hypothesis
Of course, considering the ways they may be acquired (inductively, abductively5), hypotheses
of type HE4 are most of the time not certain. One just has a certain degree of belief in them,
depending on the process by which they were acquired. Sometimes, they can be attached to
a probability, sometimes they can be represented by a non-monotonic inference or by other
types of quantitative measures of uncertainty. The important point is that in general, as they
are not certain, the conclusion of the analogical reasoning inherits the uncertainty that affects
From the premises [P∊ Z, Q ∊ Z’, P(A), P(B), Q(B), HE4] supposed all true, the conclusion Q(A)
is obviously obtained deductively. What makes the analogical reasoning [P∊ Z, Q ∊ Z’, P(A),
P(B), Q(B), HE4]Q(A) not deductive is the fact that the premise HE is not known to be true
but has only a certain degree of belief. So the analogical reasoning is not a new consequence
relation but is a specific “inferential scheme” which may rely on different kinds of beliefs. If an
agent has a belief of type Bel(HE4)) then he should have a similar belief Bel(Q(A)) in Q(A). A
good analogical inference is not an inference relation leading to a plausible conclusion, it is an
inferential scheme that gives a degree of belief in the conclusion which is coherent with the
degree of belief in the over-hypothesis HE. Four typical different situations which are worth
to be analysed can then arise.
If the degree of belief in HE4 is so strong that HE4 is “accepted” (meaning for instance that its
probability is close to 1 as formalized by Adam’s semantics (Pearl,1988), then the analogical
reasoning will get a real strength and the conclusion will be accepted with the same strong
degree of belief: my car is a Twingo like yours. Its price is well below 10000 €. Hence, yours
must also cost less than 10000 € (because the model of a car determines its price). The fact
that the model of a car determines its price is almost certain so I can be almost sure of the
5 See Walliser et al. (2005) for a formalisation of the abductive reasoning.
The degree of belief in HE4 can be simply higher than the prior degree of belief in Q(A). In this
case, the degree of belief in Q(A) increases to the level of the degree of belief in HE4. This
corresponds to a relative confirmation (Zwirn & Zwirn, 1996) in which the belief in the
conclusion in strengthened while being not enough to lead to accept it (absolute confirmation).
The strength of the analogical reasoning in situation 2 is lower than in situation 1: Bjorn and
Anna are Swedish students. Bjorn speaks English fluently. So I can believe that Anna speaks
also English very well (because the nationality and the level of studies determine fairly well the
general linguistic competences). A priori, I cannot know if Anna who is Swedish speaks English
fluently and my degree of belief in it is low. On the other hand, I have a fairly good confidence
in the fact that the nationality and the level of studies determine the linguistic competence.
Then, when I learn that Bjorn, who is a Swedish student like Anna is, speaks English fluently, I
increase my belief in the fact that so does Anna.
HE4 can be a mere possibility, totally uncertain, whose (subjective) probability is unknown.
It’s even possible that no degree of belief is attached to it. In this case, the analogical reasoning
will have no proof value but may have a purely heuristic interest to help noticing a new
possibility worth to be explored: Mars and Earth rotate around the sun not too far from it,
they have similar gravity and surface temperature. There is life on Earth. Perhaps there is life
on Mars (because rotating not too far around the sun, having gravity and a surface
temperature similar to those of the Earth are conditions in which life appears). It is clear that
nobody today knows the real conditions that are necessary and sufficient for life to appear.
So the over-hypothesis here is very risky and even probably false. Nonetheless, it seems
worthier exploring if Mars can harbour life than exploring if Pluto can, since the last is totally
different from Earth.
The belief in HE4 can be very low or HE4 can even be considered as a silly hypothesis. In this
case, the conclusion of the analogical reasoning is not taken seriously and is considered as silly
itself. This corresponds to the cases where the analogical reasoning is considered as a bad
reasoning mode: the tiger has a tail as well as my cat. My cat is kind hence the tiger is kind
(because the tail induces the behaviour).
There may be many intermediate situations but the general principle is the same: the belief
in Q(A) is determined by the belief in HE4. Of course, HE4 is not always made explicit by the
agent in her analogical reasoning. But even if not made explicit, a hypothesis of type HE4 has
always to be in the background belief of the agent when following such a reasoning scheme
with some consistency. It can also happen that an adversary makes this hypothesis explicit
just in order to show the weakness of the reasoning: “to infer that, you need this background
over-hypothesis, yet it is clearly absurd or has a very low probability”.
Of course, because it always relies on relative analogies, reasoning by analogy is open to
revision. Especially, the over-hypothesis can be modified in its structure as well as in its degree
of belief. For instance, a counter-analogy, true for A and B in a domain Z* different from Z,
may relativize the conclusion of a first analogical inference, even if the belief in the over-
hypothesis is high, insofar as this counter-analogy could be associated with another over-
hypothesis which leads to an exception to the first one and lessens the belief in it. This is the
case with the over-hypothesis “all bodies fall on the floor“ which is refuted by balloons. The
reason here is that some relevant causes (“hidden factors”) necessary to explain the fall of
bodies were ignored.
Ideally, a rational agent will consider the total evidence available to her to form her belief,
which will lead directly or by discussion with other rational agents to attribute to the over-
hypothesis a belief that already anticipates the possible counter-analogies.
Comparison with other reasoning modes
The raw form of reasoning by analogy [P(A), P(B), Q(B)]Q(A), is identical from a syntactical
point of view to one case induction. Actually, both modes of reasoning transfer a property
from one object to another, relying on the fact that both objects share already another
property. A property Q observed for an object of “type P” is transferred on another object of
“type P”, like in the standard example: this raven in front of me is black, hence the next raven
I’ll see should be black. That does not mean that reasoning by analogy is reducible to induction
but that the first step of any induction consists in reasoning by analogy. One case induction is
nothing else that a kind of analogical reasoning, but there are different ways to express such
reasoning in the usual language since it can be set in different pragmatic contexts.
In one case induction, objects of the analogical statement are usually directly designed by the
property they share according to this statement, preceded by a demonstrative pronoun of
time, place, ownership etc.:
- This P is Q, then this other P is also Q
- These P are Q, then these other P are also Q
For instance, induction implies that since my canary is yellow, yours should be yellow too.
In other kinds of reasoning by analogy, which are the kinds which are the most popular
because they seem different from one-case induction, objects are primarily designed by their
name, or by any designator independent of the properties that they are supposed to share
according to the analogical statement, which are attributed to them in a sentence:
- A is P, B is P, B is Q, then A is Q
- A’s are P, B’s are P, B’s are Q, then A’s are Q
For instance, a canary has wings and is able to fly, then since an airplane has wings too it
should be able to fly.
In one case induction, the property P shared by two objects is used as the “principal designator”
(to be a canary). In reasoning by analogy, the shared property is used only as a secondary
qualifier; it is explicitly mentioned as the shared property of two objects which are designated
through another qualifier (to have wings for a canary). Common properties used in one case
induction are pre-existing: they correspond often to a class inside a current taxonomy. These
taxonomies and these classes have been built inside the language precisely because they are
well suited for maximising causal effects with other properties, hence to build over-
hypotheses of type HE. Common properties used in other analogical reasoning are usually
selected at the moment when the reasoning is made. They are not used as current principal
designators of any class in a usual taxonomy, and can be fanciful.
This helps understanding why one case induction is often considered more reliable than other
analogical reasoning. The basic idea is that current categories are defined in a way which
exactly takes into account the relation of determination between many properties that define
each category. For example, the fact to be a canary has a lot of other consequences. This is
why this category is useful. Hypotheses of type HE4 linked to this kind of category are enough
to help drawing inferences from the fact of knowing that something is a canary.
In a nutshell, one case induction is nothing else than a reasoning by analogy but in a
normalized context where it relies on over-hypotheses that are linked to the properties which
are used as principal designators for the objects which are considered. These over-hypotheses
are well entrenched because they are linked to categories currently used inside the language.
On the other hand, over-hypotheses used in other cases of analogical reasoning are linked to
properties that are less general and hence less entrenched. Both reasoning modes are
essentially of the same nature but used in different pragmatic contexts.
Of course, that does not mean that one case induction cannot be fanciful too: since this stone
is small, this other stone is small too. The fact that the designator is a current category does
not imply that all over-hypotheses that could be linked to it are relevant. The fact to be a stone
has no effect on its size.
Finally, the role of principal designator can be contextual. Consider, for example, as principal
designator “being a New-Yorker”, to whom we can attribute some secondary properties such
as “wearing purple shorts”. In general, the first will serve as a basis for classical inductive
reasoning (this New-Yorker is a runner, then this other too ...) and interest for the second will
be found only in reasoning by analogy in which one will designate the individuals concerned
by their names (Paul is wearing a purple short). But if in a basketball competition one sees
individuals wearing either purple shorts or white shorts, this property may become
contextually a principal designator: as it is learned that this purple short holder belongs to the
team A, it will be inferred that this other purple short wearer also belongs to team A. These
contextual situations do not contradict the preceding remarks: the particular context adds
data on the situation, which makes properties usually subsidiary become relevant as principal
designators; these properties may then be used in this context as meta-hypotheses of type
HE4 (e.g. the team to which belongs an athlete in a competition determines the color of the
shorts that he is wearing).
4.2. Case-based reasoning
Reasoning by analogy is a mode of reasoning which can be applied in more and more applied
reasoning schemes. The best example is case-based reasoning. In a given situation, an expert
has to make an expectation of some outcome or to form a decision about it. In order to do so,
he gathers a set of past situations which are similar to the situation at hand and observes what
outcome or decision was realized.
A first example concerns the effect of some innovation on a given system. Different systems
are considered, differing by the characteristics of the innovation, the influence mechanisms
of the innovation and the distribution of the population involved, the output is the efficiency
of the innovation measured by an aggregate index. This index is a positive one, the idea being
that similar situations should lead to similar outcomes.
A second example concerns the judgment associated to a judicial trial. Different cases are
examined, differing by various circumstances, the operation mode of the suspect, the
personality of the suspect. The output is the penalty imposed to the suspect by the judge. This
penalty is a normative one, the idea being that in similar situations, the same verdict has to
More generally, the evolution of taxonomies generally proceeds by an analogical reasoning.
Analogical statements allow adding new objects to a given class of objects. Analogical
inferences allow a restructuration of the classes of objects (when all objects of the class satisfy
a new property) or to split already constituted classes in new ones by these new properties
(when only some objects of the class satisfy a new property).
More ambitiously, analogical reasoning is at work when generalizing scientific models in order
to realize an economy of thought. Following Walliser (1994), analogical statements allow to
generalize “by enlargement”, i.e. by widening the domain of applicability of a model, while
analogical inferences allow to generalize a model “by completion”, i.e. by adding new
variables to the model and extending its equations.
Some related works
5.1. Philosophical works
Reasoning by analogy has been the subject of many contributions, dating back to Aristotle.
These works are well documented elsewhere, especially in Bartha (2010; 2013). The present
survey relies on them in order to emphasize the resemblances and differences with our own
work when expressed in our own framework.
The most current intuitive theory of analogical inference states that a good analogical
inference relies simply on a good analogy, associated to the fact that the two objects share
many common properties. The structure of an analogical inference may then be associated to
a general kind of enthymematic reasoning:
- A and B share “many” properties P1, P2, ….Pn
- B has property Q
- H0 : objects sharing many properties generally share other properties
- A has also property Q
But this enthymematic reasoning is very different from ours, and is problematic:
- It relies on an absolute notion of analogy, whose logical limits has been shown.
- Unless HE4, H0 is a “general principle of analogy”, looking like the “principle of
uniformity of nature”, and has the flavor of a metaphysical assumption begging the
- It raises intuitive objections: the properties have at least to be “relevant”, there should
be some kind of “causal” relation between one of the Pi and Q.
Several authors tried to add structure in the analogical inference in order to make more
precise the principle H0 and to list the properties which make it more robust and relevant.
Hesse (1966) and Bartha (2010, 2013) are two major contributions for doing this job.
Mary Hesse (1966) proposes an interesting tabular representation of an analogical argument,
which separates the source domain and the target domain, each domain including a set of
objects, properties and relations. She defines the “vertical relations” as the relations within
each domain and the “horizontal relations” as the relations between the domains. Then, she
formulates several qualitative requirements in order for an analogical argument to be
acceptable. Bartha (2010, 2013) proposes a synthesis and critique of her requirements, which
he shows to be too restrictive in some situations. More specifically, he shows that Hesse’s
conditions do not depend enough of the use that will be made of the analogy in such or such
specific analogical argument. Finally, Hesse’s theory keeps many vague concepts of the
intuitive theory of analogical reasoning, such as “causal relations in an acceptable scientific
sense” or “essential properties”.
Bartha proposes himself a more elaborated theory, called “the articulation model”. His thesis
is that, contrary to other philosophical analysis which concentrated on the horizontal relations
(for instance the number of similarities), one should investigate more the vertical relations.
His analysis is very rich detailed, with many subcases and illustrated by several precise
He starts by listing all « potential relevant factors » of the analogical reasoning, without a
formal definition of these factors. They may be « variables, hypothesis, conditions…. ». These
factors, say Fi, may be present or absent in the source and in the target domains (as such or
through “correspondences” between factors in the source and factors in the target). Q is the
property present in the source domain that the analogical argument proposes to extend to
the target domain (as such or through a “corresponding” property Q*).
The first condition for building a good analogical argument is to state a “Prior association”,
meaning that there is a relation between some of the Fi and Q, in the source domain. Typically,
it may be a “causal relationship” when the analogical argument states an empirical prediction.
Then Bartha proposes several further conditions for an analogical argument to be “plausible”,
distinguishing between a modal concept of “prima facie plausibility” and a stronger
quantitative concept of (strangely called) “qualitative plausibility”.
Bartha’s analysis could in fact be simplified and casted in our own analysis in the following
- There are 2 objects (instead of vague “factors”), let’s say the source A and the target B
(this can be easily generalized to more complex situations involving t-tuples of objects). To
each object one can associate properties, known or unknown, for instance P(A), Q(A),
P*(B), Q*(B). P* and Q* “correspond” to P and Q, in an intuitive way. The analogical
argument relies then on P(A), Q(A), P*(B) and infers Q*(B).
- The “Prior association” condition states that there is a relationship between P and Q in the
source domain, for instance a causal relationship known with certainty [∀X, (P(X) →Q(X)],
with some non-monotonic exceptions or with some probability Pr(Q(X)/ P(X) = x). We will
focus on the last case for simplicity.
- The “Overlap condition” (required for “Prima Facie Plausibility”) states that some of the
common properties have a positive effect on Q, which is the case as soon as we assume
that Pr(Q(X)/ P(X)) > s > ½.
- In some cases of analogical inference, P = P* and Q = Q*. It is for instance the case of the
Earth / Mars / Life example. For those cases, the fact to know that Pr(Q(X)/ P(X)) = > s is
equivalent to know that Pr(Q*(X)/ P*(X)) > s. This means that Bartha’s criterion leads to
- Then the only interesting cases are those where P ≠ P* and Q≠ Q*. In those cases, one
does not see why [Pr(Q(X)/ P(X)) > s] should imply that [Pr(Q*(X)/ P*(X)) > s]? The intuition
relies on the notion of “correspondence” between properties, which is not precisely
defined by Bartha. As it is easy to see from what has been shown in §2.3, this concept can
be explained away by saying that two properties P and P* correspond if there exists one
common property Π and two other one’s Z and Z*such that Π.Z = P and Π.Z*=P*. Z and
Z* may express different properties of A and B (for instance, A is a vital organ of aerial
animals and B is a vital organ of aquatic animals).
- Reasoning by analogy in this context means then to believe that:
Pr(Ξ(X).Z(X) / Π(X).Z(X)) > s Pr(Ξ(X).Z*(X) / Π(X).Z*(X)) > s
Where Ξ(X).Z(X) = Q(X) and Ξ(X).Z*(X) = Q*(X).
Intuitively, Z and Z* should be neutral factors which do not disturb the conditional probability
between Ξ(X) and Π(X). This recalls the Sure Thing Principle in decision theory. Bartha’s
numerous conditions for prima facie or qualitative plausibility of analogical inferences can be
interpreted as intuitive conditions for accepting that neutrality. For instance:
- The “No critical difference” condition (required for “Prima Facie Plausibility”) means
that there does not exist a property within Z which would be important for the
conditional probability Pr(Ξ(X).Z(X) / Π(X).Z(X)) and which is not a property within Z’.
- The “Strength of the prior association” condition states that a stronger prior
association induces a stronger analogical argument. Indeed, the strength of the
conclusion is higher when s is higher, but only if we find good reasons to transfer the
Prior association to the target domain.
- The “Counteracting causes” condition means that there exists a property R within Z
with an independent negative effect on the conditional probability. The fact that
Pr(Ξ(X).Z(X) / Π(X).Z(X)) > s despite of R reinforce the intuitive confidence in the fact
that Pr(Ξ(X).Z*(X) / Π(X).Z*(X)) > s, especially is R is also valid within Z* and could have
minored this confidence.
But, none of those condition can definitively ensure that the conclusion of an analogical
argument is safe, with the same belief degree than the belief degree of the Prior association.
That is why, as noticed by Norton (2018), this way of warranting analogical inferences is
endless. It is always possible to add a new case which would contradict the intuitive inference
and has to be specified by a new condition. The only way out of this endless process is to:
- First reduce analogies between “corresponding properties” to simple analogies
between one common property, by using the method indicated in §2.3.
- Second to consider an over-hypothesis such as HE4, which encompass all the cases in
one single hypothesis, while preventing any redundancy in general.
5.2. Logical works
Davies & Russel (1987) proposed a clear formal analysis of analogical reasoning, which has
already been commented, and developed the key notion of determination rules. The syntactic
expression of determination rules takes in charge many intricate intuitions of other
philosophical works on analogical reasoning, and stresses the important notion of non-
redundancy. However, as has been shown, they miss to analyze precisely why there is an
important difference between an over-hypothesis like HE3 and an over-hypothesis like HE4
for expressing those rules without redundancy and to analyse the empirical process explaining
how these over-hypotheses can been learnt.
Bartha (2010) criticizes the Davies and Russel‘s determination rules (supposed to reduce
analogical reasoning to deduction) on another ground, with the following argument:
“Scientific analogies are commonly applied to problems where we do not possess useful
determination rules. In many (perhaps most) cases, researchers are not aware of all relevant
factors”. But this argument lies on a confusion about the role played by these rules: even when
an agent does not know them, or is uncertain about them, they always play a normative role
for evaluating the strength of the analogical argument. The link between the belief in these
background hypotheses and the strength of the analogical argument is missing in Davies and
Russel’s paper, but this does not lead to conclude that the role of these hypotheses is not
universal on logical grounds.
Miller (1995) proposes a solution close to Davies and Russel’s HE3 one, which can be
translated in the present language by:
∀X, ∀Y, [O(X), P(X), Q(X), -O(Y), P(Y)] → Q(Y)
where O(X) is an extra predicate meaning that X has been observed.
Miller proves that this formula is the weakest universal proposition which entails Q(Y) in the
presence of P(X), Q(X), P(Y).For him, it is the weakest form of an over-hypothesis. But the fact
that it is the weakest one is not necessarily a good or required criterion of relevance, as Miller
seems to believe. Indeed, if one drops O(X) and -O(Y) in Miller’s proposition, it can be checked
that it is logically equivalent to Davies and Russell’s one. Adding the fact that X has been
observed while Y has not makes the proposition logically weaker in an uninteresting way since
the difference concerns only the cases where the premise includes -Q(B). In this case, Davies
and Russel would lead to conclude that –Q(A), while Miller would not conclude. But we are
trying to explain the reasoning precisely in the only cases where Q(B).
In an important number of articles (e.g. Prade & Richard, 2011, 2012a, 2012b, 2014, Amgoud,
Ouannani & Prade, 2014), Prade and al. develop a very different approach, which intends to
provide a logical definition of analogy in a propositional framework. This approach focuses on
the definition of 4-term formulas such as a:b :: c:d (where a, b, c, d are propositions) which
are the basic structure of relational analogies.
On one side, they propose a list of principles that such formulas could satisfy, not limited to
those who constitute an equivalence relation. On the other side, they consider several criteria
of “logical proportions”, whose components are analyzed in terms of "similarity indicators"
(a & b; -a & -b) and "dissimilarity indicators" (a & -b; -a & b). The preferred formula for analogy
is what they called “Analogical Proportion”, which is expressed by:
(AP) a:b :: c:d iff ((a& -b) ≡ c& -d)) & ((-a & b) ≡ (-c &d))
This is supposed to mean that “a is to b what c is to d”, and the authors argue that this
definition satisfies the more relevant principles for representing a relational analogy in this
This conclusion seems to conflict with the thesis of the present paper since it may imply that,
despite our previous arguments, it is possible to give a relevant logical definition of absolute
analogy: no restriction to a domain is mentioned in definition (AP).However, the work of
Prade& al. concerns only Boolean propositions as very specific objects of analogy, and cannot
be considered as a general theory of analogy. It is assumed to be a transposition in a Boolean
framework of the standard “proportional analogy”:
(PA) x:y :: z: t iff x/y = z/t, where x, y, z, t are real numbers.
As relevant as this classical example of analogy may be for numbers, it cannot be taken
seriously as a basis for a definition of the philosophical concept of analogy, since it cannot be
applied as such to any other objects, e.g. ravens, apples or human beings…. The same applies
to (AP) which concerns only propositions. The misleading aspect of Prade & al. suggestion is
that it seems that, contrary to numbers, propositions may express our beliefs “in general”. But
in fact, even if analogical statements are themselves propositions, they are propositions
expressing beliefs which stand between objects, not between propositions. It is not possible
to consider the definition (AP) in a propositional way in order to represent this general relation
between objects. Hence, these attempts to define analogy in a propositional framework are
at best interesting Boolean exercises but they are of no help for the philosophical
understanding neither of the very concept of analogy nor for the one of reasoning by analogy
Polya (1945) proposed another propositional interpretation of analogy in terms of
non-monotonic reasoning. He identified some patterns of plausible reasoning among which
(PR) If a and b are analogous and a is true, then the truth of b is more credible.
In terms of non-monotonic reasoning, the implied definition, suggested in Prade& Richard
(NM) a~b iff |~ a≡ b
It is easy to check that [PR] is satisfied if the inference relation |~ in (NM) is a preferential non
monotonic consequence, as defined in Kraus, Lehmann and Magidor (1990).
However this definition raises three problems which manifest the fact that their analysis is of
little help for a philosophical analysis of analogy:
- It concerns again only propositions and it is unclear how it may apply to the very
general notion of analogy between objects, such as “an apple is like a pear”.
- It introduces an absolute definition of analogy (NM) which is too restrictive: a and b
are analogous if and only if they are true in exactly the same“normal”possible worlds.
This almost reduces analogy to identity, with the only exception of the most
- It does not explain how an object A could be similar to an object B from a point of view
and not from another, since the normal worlds are a fixed subset of the possible worlds.
We develop an analytic scheme of reasoning by analogy which is decomposed into two steps.
The analogical assessment states that some objects are similar as concerns a fixed set of
properties. The analogical inference states that a new property possessed by these similar
objects can be transferred to another object already similar to the considered ones. But our
conceptual scheme differs from the existing works in several respects.
Our analysis considers that an analogical assessment is not true or false, good or bad in an
absolute way, but is relative to a point of view expressed by a domain of properties. If some
debates are raised about it and make it defeasible, they concern rival analogical inferences
and not analogical assertions by themselves.
Our analysis expresses that analogy is an inference such that the degree of belief in the
conclusion is defined coherently with the degree of belief in a background over-hypothesis
supporting it. The value of this kind of reasoning cannot be established on a purely syntactical
basis but is linked to externally defined beliefs, which should be made explicit within analogical
Hence, our analysis reconciles the opposite ideas that analogical reasoning is a useful method,
especially in science, and that it is a fanciful reasoning, especially in current reasoning. It may
be both, in science or in current reasoning, while using always the same inference scheme:
the plausibility of the conclusion depends only of the plausibility of the back-ground over-
hypothesis used in this scheme. This analysis departs from those which state that performing
good or bad analogical inferences depends of the fact to fulfil or not a good reasoning method,
or to rely or not on good analogies.
The background over-hypothesis used in an analogical reasoning is a meta-level hypothesis in
that it concerns upper level belief about properties. Consideration of this upper level avoids
to trivialise the reasoning into a focusing redundant one and explains in a coherent way the
possible – though multiple – origins of those background belief, especially in an inductive way.
Our analysis departs finally from usual analysis which considers analogical reasoning as a
specific reasoning mode different from deduction, induction and abduction. On the contrary,
it relies on the idea that it is a category which includes one case induction, but with two
specific pragmatic contexts, one of them corresponding to the more traditional examples of
one case induction.
Further work can be done in some theoretical directions. First, one could try to build
quantitative types of similarity indices which could be useful for establishing a similarity
assessment. Second, one could analyse more closely the different ways to empirically build
the kind of belief inherent in the over-hypothesis. Third, one could examine how the usual
paradoxes of induction such as Hempel’s and Goodman’s apply to general analogical
Further investigations can also be done in empirical domains. On the one hand, one could
precise on historical examples how analogy is used in the process of science, in combination
with other reasoning modes, especially in the “context of discovery” as opposed to the
“context of proof”. On the other hand, one could examine how scientific or popular analogies
are revised through time, under the pressure of new information and hot debates, and
become widely accepted or definitely discarded.
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