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REASONING BY ANALOGY

Bernard Walliser1

Denis Zwirn

Hervé Zwirn2

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

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Introduction

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,

France)

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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-

hypothesis (§3).

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).

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2.

Analogical statement

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

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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

stimulus).

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

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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

identity.

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

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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

target.

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

economic influence.

-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

Freud himself.

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:

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(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) &

R(C,D)

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.

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For instance, Paul is to Ana what Bob is to Julia, with regard to family relationship, he is her

son.

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

instance:

- 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.

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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,

Then:

“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

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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)

or

(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

developed into:

(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.

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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

candidate.

(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

explanation":

(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

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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,

(P(X) →-Q(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

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“to be of color k”. The over-hypothesis is the fact that each bag is associated with only one

color.

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

already defined:

- 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

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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

antecedents.

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

analysis.

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

HE4.

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.

Situation 1:

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

conclusion.

5 See Walliser et al. (2005) for a formalisation of the abductive reasoning.

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Situation 2:

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.

Situation 3:

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.

Situation 4:

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

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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.

4.

Comparison with other reasoning modes

4.1. Induction

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

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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).

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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

be applied.

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.

5.

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

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- 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

question.

- 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

scientific examples.

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”.

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Bartha’s analysis could in fact be simplified and casted in our own analysis in the following

way:

- 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

Redundancy.

- 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

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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

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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

propositional framework.

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

this one:

23

(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

(2011), is:

(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

implausible worlds

- 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.

6.

Conclusion

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

reasoning disputes.

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

24

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

reasoning.

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.

References

Alchourron, C.E.., Gärdenfors, P., Makinson, D. (1985): On the logic of theory change: partial meet

contraction and revision functions, Journal of Symbolic Logic, 50, 510-530.

Amgoud, L., Ouannani, Y., Prade, H. (2014): Arguing by analogy-Towards a formal view, in ECAI

12.

Aragones, E., Gilboa,I., Postelwaite, A., Schmeidler, D. (2014): Rhetoric and analogies,

Research in Economics, 68, 1 :10.

Bartha, P. (2010): By Parallel Reasoning, the construction and evaluation of analogical

arguments, Oxford University Press.

Bartha, P. (2013): Analogy and Analogical Reasoning, in Edward N. Zalta (ed.) The Stanford

Encyclopedia of Philosophy.

Bicchieri, C. (1988): Should a scientist abstain from metaphors, in Klamer,M., Mc Closkey, D.,

Solow, M.: The consequences of economic rhetoric, Cambridge University Press.

25

Bird, A., Tobin, E. (2015): Natural Kinds, in Edward N. Zalta (ed.) The Stanford Encyclopedia of

Philosophy.

Bouveresse, J. (1999): Prodiges et vertiges de l’analogie, Raisons d’agir.

Boyer, A. (1995): Cela va sans le dire, éloge de l’enthymème, Hermès, La Revue, 15.

Davies, T.R., Russel, S.J. (1987): A logical approach to reasoning by analogy, in IJCAI-87, 364-

270, Morgan Kaufman.

Dupin, J.J., Joshua, S. (1994): Analogies et enseignement des sciences: une analogie thermique

pour l’électricité, INRP.

Dupré, J. (1993): The Disorder of Things: Metaphysical Foundations of the Disunity of Science,

Harvard University Press.

Gentner, D. (1983): Structure mapping, a theoretical framework for analogy, Cognitive Science.

Gentner, D., Holyoak, K.J., Kokinov, B.N. eds (2001): The analogical mind, MIT Press.

Goodman, N., (1947): Fact, Fiction, Forecast, (Fourth Edition). Harvard University Press, 1983,

First published in The Journal of Philosophy, 44, 113-28.

Guarini et alii (2009): Resources for research on analogy, Informal Logic, 29(2), 84-

197.Hempel, C.G. (1965a): Aspects of scientific explanation, in Aspects of Scientific Explanation

and Other Essays in the Philosophy of Science, Free Press, 331-496.

Hempel, C.G. (1965b): Inductive-Statistical explanation, in Aspects of Scientific Explanation

and Other Essays in the Philosophy of Science, Free Press, 331-496.

Hesse, M. (1966): Models and analogies in science, University of Notre-Dame Press.

Hofstader, D., Sander, E. (2013): Surfaces and Essences: analogy is the fuel and fire of thinking,

trad. franc. L’analogie, cœur de la pensée, Odile Jacob.

Jackson, F. (1991): Conditionals, Oxford University Press.

Jarvis Thomson, J. (1971): “A defense of abortion”, Philosophy and Public Affairs, 1, 47-66.

Juthe, A. (2005): Argument by Analogy, Argumentation, 19, 1-27.

Keynes, J.M. (1921): A treatise on probability, Macmillan.

Kraus S., Lehmann D., Magidor, M. (1990): Non monotonic reasoning, preferential models and

cumulative logics, Artificial Intelligence, 44,167-208, 1990.

Lichnerowicz, A., Perroux, F., Gadoffre, G. (1980-81) : Analogie et connaissance, Maloine.

Musgrave, A. (1989) : Deductivism versus Psychologism, Perspectives on Psychologism,

Notturnoed.

Miller, D. (1995): How Little Uniformity Needs an Inductive Inference Presuppose?, in

Festchrifft, in honour of J. Agassi. In Jarvie and Laor (eds.), Boston Studies for the Philosophy

of Science, Kluwer.

Norton, J.D. (2005): A little survey on induction, in P. Achinstein ed. Scientific Evidence, John

Hopkins University Press, 9-34.

Norton, J.D. (2014): A material dissolution of the problem of induction, Synthèse, 191, 671-90.

Norton, J.D. (2018): The Material Theory of Induction, in preparation.

26

Pearl, J. (1988): Probabilistic Reasoning in Intelligent Systems, Networks of Plausible Inference,

Morgan Kaufman Publishers Inc.

Polya, G.. (1945): How to solve it. Princeton University Press.

Prade, H., Richard, G. (2011): Cataloguing / analogizing: a non-monotonic view, International

Jounal of Intelligent Systems, 26, 117-1195.

Prade, H., Richard, G. (2012a): Homogeneous logical proportions: Their uniqueness and their

role in similarity-based prediction.13th International Conference on the Principles of

Knowledge Representation and Reasoning, KR 2012, 2012, 402 – 412.

Prade, H., Richard, G. (2012b): Logical Proportions – Further Investigations. International

Conference on Information Processing and Management of Uncertainty in Knowledge-Based

Systems, IPMU 2012: Advances on Computational Intelligence , 208-218.

Prade, H., Richard, G. (2014): From analogical proportions to logical proportions, Logica

Universalis, 7, 441-505.

Quine, W.V. (1969): Ontological relativity and other essays, Columbia University Press.

Shaw, W.H., Ashley, L.R. (1983): Analogy and inference, Dialogue, Canadian Journal of

Philosophy, 22, 415-32.

Voskoglou, M.Gr., Salem, A.M. (2014): Analogy-Based and Case-Based Reasoning : Two sides

of the same coin. International Journal of Applications of Fuzzy Sets and Artificial Intelligence,

Vol.4, 5-51.

Walliser, B. (1994): Three generalization processes for economic models, in B. Hamminga, N.

de Marchi eds.:Idealization VI, Idealization in Economics, Poznan Studies in the Philosophy of

Science and Humanities, Rodopi.

Walliser, B., Zwirn, D., Zwirn, H. (2005): Abductive Logic in a Belief Revision Framework;

Journal of Logic, language and Information, Vol 14, n°1.

Wolton, D. (2010): Similarity, precedent and argument from analogy, Artificial Intelligence and

Law, 18(3), 217-46.

Zwirn, D., Zwirn, H. (1996): Metaconfirmation, Theory and Decision, Nov. 96, Vol.41, 195-228.