Abduction and Conjecturing in Mathematics

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In: Magnani, L., Nersessian N.J. & Thagard, P. (eds.),
Abduction and Scientific Discovery, Special Issue of Philosophica (in print)
Abduction and conjecturing in mathematics
Ferdinando Arzarello, Valeria Andriano,
Federica Olivero, Ornella Robutti
Dipartimento di Matematica, Universita’ di Torino,
via Carlo Alberto, 10 I-10123 ITALIA
e-mail: arzarello@dm.unito.it, andriano@polito.it,
olivero@dm.unito.it, robutti@dm.unito.it
Summary. The logic of discovering and that of justifying have been a permanent source
of debate in mathematics, because of their different and apparently contradictory features
within the processes of production of mathematical statements. In fact, a fundamental
unity appears as soon as one deeply investigates in concrete examples the
phenomenology of conjecturing and proving. In this paper it is shown that abduction, in
the sense of Peirce, is an essential unifying activity, which rules such phenomena.
Abduction should be the major ingredient in a theoretical model that can describe the
transition from the conjecturing phase to the proving one. In the paper such a model is
introduced and worked out to test Lakatos' machinery of proofs and refutations from a
new point of view. Abduction and its categorical counterpart, adjunction, allow to
explain most of the phenomenology of conjectures and proofs, encompassing also the
method of Greek analysis-synthesis, within a unifying framework.
Introduction. The contrast between the logic of discovering and that of justifying has
been a permanent source of debate in the whole history of mathematics; in fact, the
literature on the subject is very wide: it includes contributions of different people
(mathematicians, philosophers, psychologists, didacticians, etc.) and embraces more than
two thousand years papers, from the pre-Euclidean mathematics up to nowadays. For
surveys from different points of view, see: Cellucci (1998), Hanna (1989) and (1996),
Lolli (1996), Rav (1999), Thurston (1994), Tymoczko (1985).
In the literature, there is a continuum of positions from those who have
underlined the central role of formal proof in mathematics, insofar it means something
positively different from the heuristic used in the process of research, to those who have
stressed the latter more, denying any value to the former in some cases. Moreover, many
papers, written by people of different cultural areas in different times (from Pappus or
Plato to Poincaré, Polya or Lakatos, through Descartes or Arnaud and Nicole), underline
that in solving a problem, in conjecturing an hypothesis and in proving (or disproving) a

result, a crucial point consists in the dialectic between an explorative, groping phase and
an organising strategy which converges towards some piece of validated knowledge.
In this paper we show that abduction, in the sense described in Peirce's Logic
(Peirce, 1960, Vol.II, Book III, Chap.5, pp.372-388) plays an essential role in this
dialectic: abduction reveals as an essential resolutive move, after which the conjectures
are formulated and it allows the transition to the proving modality, which remains in any
case deeply intertwined with it. This is partially in conformity with Peirce's claim that of
the three logic operations, namely deduction, induction, abduction (or hypothesis), the
last is the only one which introduces new ideas (Peirce, 1960, 5.171).
Our thesis is based both on a theoretical analysis and on empirical observations of
subjects who solve problems aimed at conjecturing and proving theorems. Our data
concern: (i) about 60 (higher school and college) students, who have been involved in a
teaching experiment about elementary geometry for three years (working in different
environments: paper and pencil, computers); (ii) the performances of experts
(mathematics teachers in higher schools and at University) dealing with elementary but
non trivial problems, who accepted to speak aloud, while solving the problems; (iii) the
(rare) papers of professional mathematicians who have written about their processes of
thought, while discovering new results.
The paper is divided into three chapters. In the first, some major features of the
epistemological debate concerning the dialectic discovering Vs/ justifying are
summarised; in particular as regards the topic analysis Vs/ synthesis. The aim of this part
is to put forward the framework within which our researches are embedded; no
originality is in it. In the second chapter, we sketch out our research on the role of
abduction in mathematics; namely, we show that abduction is a resolutive move in the
dialectic conjecturing/proving. In fact, it is crucial in producing conjectures. We will
describe a theoretical model that is suitable for describing the transition from the
conjecturing to the proving phase. This model, which fits with the empirical data of our
protocols, will also be used to present a fresh analysis of the well known example by
Lakatos (1976), concerning Euler’s conjecture about edges, faces and vertices of a
polyhedron. In the third chapter, the theoretical model is attacked from the point of view
of categorical logic. In fact, the way abduction is used in our model hides an interesting
phenomenon, namely the so called adjunction, which can be revealed using such logical
tools. Abduction and adjunction are deeply connected; hence it seems to us that the
above dichotomies (logic of discovery Vs/ the logic of justifying, conjecturing Vs/
proving etc.) are like the two sides of the same coin and that underpinning their contra
position may be an ill way of posing problems. We find a common root, which has both

In: Magnani, L., Nersessian N.J. & Thagard, P. (eds.),
Abduction and Scientific Discovery, Special Issue of Philosophica (in print)
cognitive and logical features, namely abduction and adjunction; the above contra
position appears only if one remains at the surface of phenomena. Our result is still
stronger, since analogous conclusions can be found also investigating the Greek analysis
and synthesis with logical tools, as will be sketched at the end of the paper.
1. The historical & epistemological framework of the dialectic conjecturing-
proving. The creative and intuitive aspects of mathematicians' activity vs. the most
rigorous ones have been scrutinised with different tools and ideas in the course of
centuries, particularly concentrating on the relationships between logic(s) and
mathematics, conceived both as a product and as a process. In this order of ideas,
roughly speaking, we can see different streams of thought (see Feferman, 1978, and
Cellucci, 1998).
a. Many scholars have distinguished between a scientific logic and a natural
logic: Descartes, Frege, Peirce, Dedekind.
b. Some have argued in favour of a scientific and formal logic which does
capture the essence of mathematics, namely its justificative aspects as well as its creative
features: Aristotle (partially), Leibniz, Hilbert, Gentzen, Hintikka;
c. Some have seen the formal logic only as a justificative tool, claiming that
intuitive and creative aspects of mathematicians' work elude a logical scaffolding and
generally they leave them to psychology: Frege, Feferman;
d. Some have tried to investigate the natural logic, possibly as a distinct
'discipline' from formal logic; they have investigated both the origin of mathematical
ideas (Dedekind) as well as the features of mathematical discovering: Plato, Descartes,
Peirce, Polya, Lakatos, Hintikka and Remes.
We sketch here only some example to give the flavour of what is meant. For
Frege, the formal logic is his Begriffschrift, that is the science which studies the laws of
correct inference, whilst the natural logic concerns the ways after which inference is
concretely performed. As such it pertains more to psychology than to logic and is based
upon empirical principles and not upon necessary and universal rules (Frege, 1969,
Logik, p.4 and Grundgesetze der Arithmetik, vol.1, p. XIV). The relationship between
the acknowledgement of the truth, which is a thought, therefore not purely formal, and
the developing of the proving process, is very complex. Frege, in a letter to Hilbert in
1895 (Frege, 1976, p.58) uses the metaphor of lignification: a proof to develop must be
built upon the truth acknowledgement, in the same way as a tree to develop must be
soft and juicy in those points in which it lives and grows up; but inference must become

something mechanical, that needs to be strongly developed, as a tree must lignify its
juices in order to grow up. In this sense Frege develops Leibniz ideas about a
characteristica universalis and a calculus ratiocinator (as he explicitly says in the
Introduction to his Begriffschrift). The difference between Frege and Leibniz consists
mainly of the fact that Leibniz believes that the formal logic concerns also the
discovery of new results, while Frege leaves this part to psychology: for Leibniz logic
is useful "not only for judging what is proposed but also for discovering what is
hidden" (Leibniz, 1965, VII, p.523, letter to G.Wagner, 1696); the formal axiomatic
method is a mechanical substitute of thought, insofar it "discharges imagination". In
this sense Leibniz is similar to Hilbert, according to whom the problem of discovering
concerns logic and not psychology (Hilbert, 1926, p.170).
Many scholars point out that Descartes illustrates the needs of a new logic of
discovery, which can no longer be embodied in the formal logic of Scholastics: the
Aristotelian logicians "cannot skilfully form a syllogism, which entails the truth, if they
have not previously had its matter, that is, if they have not already known in advance the
very truth which is deduced in it" (Descartes, 1998, p.47; Regulae ad directionem
ingenii, Regula X). The Aristotelian logic is "useless for investigating the truth of things,
but it can only be useful for exposing to the others the reasons which are already known,
hence it must be shifted from philosophy to rhetoric" (Ibid.). To ascend to the top of
human knowledge people need a new logic (Regula II). The new logic has different roots
from the one of Aristotle and Euclid; in fact it goes back to Pappus and Diophantus
(Regula IV), namely to the so called analytic method.
The same root, namely the analytic method of Pappus, is invoked by many people
who belong to our category d. For ex., the last chapter of Lakatos’ Dissertation at
Cambridge (1956-1961) is devoted to the method of analysis-synthesis, as well as a
presentation at a Conference in Finland (1973), in reply to a paper by Hintikka on this
subject (all together, they constitute the chapter 5 of Lakatos, 1978). Lakatos uses
Pappus' and Proclos' definition of analysis to describe the process of discovering in
mathematics, in particular that of criticising proofs and improving conjectures (Lakatos,
1976, p. 9 and 75). The method, that Lakatos calls of thought-experiment or quasi-
experiment along with Szabó (1958), consists of decomposing "the original conjecture
into sub conjectures or lemmas, thus embedding it in a possible quite distant body of
knowledge" (ibid.). Polya in Polya (1990) and (1954) rephrases Pappus' method (he was
called a ‘second’ Pappus by Hintikka and Remes). According to Polya, analysis is not a
method upon which to build up criticism on the way mathematical truths are presented in
books after Euclid, namely with "finality-certainty requirements (which) survive in

In: Magnani, L., Nersessian N.J. & Thagard, P. (eds.),
Abduction and Scientific Discovery, Special Issue of Philosophica (in print)
mathematics until today as the requirement of necessary and sufficient conditions"
(Lakatos, 1978, p. 75). On the contrary analysis is an auxiliary method, which helps in
building up the rigorous proof: it helps in generating "a better understanding of the
mental operations which typically are useful to solve problems" (ibid.), and as such it is
studied as a useful pedagogical tool.
The most widely known formulation of the so-called method of analysis-
synthesis, at least in mathematics, is the introduction to the Book 7 of the Collection by
Pappus of Alexandria (Pappus, 1986). According to Pappus, "analysis is the path which
starts from what one is seeking, considered as if it was given for granted; one draws
consequences (akolouqon) from it, and consequences from consequences, till one
arrives to something that is established by the synthesis. That is to say, in analysis we
assume what is sought as already found; we inquire from what antecedent the desired
result could be derived; then we inquire again what could be the antecedent of that
antecedent, and so on, until we eventually come upon something already known or
admittedly true. (...) In synthesis, reversing the process, we start from the point which we
reached last of all in the analysis, from the thing already known or admittedly true. We
derive from it what preceded in the analysis, and go on making derivations until,
retracing our steps, we finally succeed in arriving at what is required" (ibid.) This text
poses several problems of interpretation, concerning the direction of analysis-synthesis,
as it is well known (for a wider discussion see Pappus, 1986). The second part of the
passage seems to exclude that analysis is a downstream process (drawing logical
conclusion from the desired theorem). On the other hand, if analysis is an upstream
process (looking for the premises from which the conclusions can be drawn), then the
synthesis is not the reverse of analysis, and it is useless. The problem is how can these
two different concepts coexist in Pappus. The current interpretation is that they can
coexist if we assume that Pappus thought of all steps of deduction, as being convertible,
that is the normal situation in geometry. By means of analysis we come to something
already known. Then what is sought will be known if, by means of the synthesis, we test
the reversibility of each step.
Hintikka and Remes propose another interpretation of the text, based on a
different translation of the term akolouqon. They suggest that "to akolouqon... does
not mean a logical consequence, but is a much more vague term for whatever
'corresponds to', or better, 'goes together with'" (Hintikka & Remes, 1974). Hence they
translate 'concomitant' instead of 'consequence'. Outlining the central role of auxiliary
constructions in Greek geometry, they say that "the very purpose of analysis is to find
the desired construction which is executed in the synthesis (...) If analysis is a series of

steps which start from those parts of the figure which illustrate the desired theorem, and
which establish connections between these and certain pre-existing entities, we of course
do not obtain a synthesis in the sense of construction by simply reversing the order of
these steps"(ibid.). The distinction between analysis and synthesis is then no longer a
difference in direction. Also Descartes' methodological description of his algebraic
method of analysis seems to agree with this interpretation. Instead of seeking a
deductive connection between what is done and what is sought, he suggests to look for
the dependencies between known and unknown quantities.
In this interpretation, called configurational in contrast with the directional one,
the two ways, analysis and synthesis, could be really different and would represent two
dual approaches to mathematical truths, the one complementary to the other. It is
interesting to observe that these two ways have been interpreted by Scandinavian
scholars (Hintikka, Remes, Mäenpää) within the machinery of Natural deduction (as
formulated by Genzen, Prawitz, Martin-Löf) in a very suggestive way, which will result
one of the major ingredients in our investigation.
2. Abduction. We shall show that abduction is a resolutive move in the dialectic
conjecturing-proving. First, we will discuss how it fits into the theoretical framework of
chapter 1; then we will give a model (based upon our empirical observations) which
describes the way conjectures are produced by experts and how they manage the
transition from the conjecturing to the proving phase. Afterwards we will use it to give a
fresh and detailed analysis of Lakatos model of proofs and refutations. The main points
which emerge from the discussion in chapter 1 can be summarised as follows:
1. There are two complementary ways of producing mathematical statements and
theorems, e.g. the logic of discovering or that of justifying, analysis or synthesis.
2. The two aspects seem to be always present in the activity of the working
mathematicians and can be distinguished only for the benefit of a theoretical analysis;
hence mathematics as a human product must be explained not only taking into account
both of them but a model is needed where the two are integrated within a framework of
continuity and complementarity, both from an epistemological and from a cognitive
point of view.
3. Instead of a directional analysis of the process of research, it is more suitable to
look at processes of conjecturing-proving within a configurational framework.
4. As a consequence, the analysis of the interactions between the two opposite
streams mentioned above becomes crucial. It is necessary to give a theoretical reason

In: Magnani, L., Nersessian N.J. & Thagard, P. (eds.),
Abduction and Scientific Discovery, Special Issue of Philosophica (in print)
why in the real life of mathematicians such apparently contradictory approaches can live
together. A fresh analysis is needed, insofar a lot of existing research seems to underline
more the contradictions and the disconnection between the two: for some different
examples see Tymoczko (1985), Davis & Hersh (1981) or Cellucci (1998).
We shall elaborate point 4 above, by describing a model which exhibits a fine dynamic
structure of the two streams within a framework, which stresses more the continuity and
the configurational aspects. This model has an experimental basis, but it has been
elaborated also through a theoretical analysis of problems under items 1, 2, 3 above.
Interested readers can find more details in Arzarello et al. (1998a) and (1998b).
We shall expose the major features, showed by experts while solving elementary
(but not routine) geometrical problems. Typically, a geometrical situation is given to be
explored, together with some very open questions to be answered (e.g.: under which
hypothesis, does such and such a property concerning the drawn figure hold? make
conjectures and prove them).
The subjects show successively two main modalities of acting, namely:
exploring/selecting a conjecture and concatenating sentences logically. In fact, any
process of exploring-conjecturing-proving is featured by a complex switching from the
one modality to the other and back, which requires a high flexibility in tuning oneself to
the right one.
Our aim is to analyse carefully how the transition from the one modality to the
other does happen. What a typical (clever) solver does, can be sketched as follows.
PHASE 1: The process of solution begins with an exploring modality, which
includes for example the use of some heuristic to guess what happens working on some
particular examples; hence the subject selects a conjecture. The conjecture in reality is a
working hypothesis to be checked: generally its form is far from a conditional statement;
to confirm it new explorations are made by using some heuristic. Slowly, the solver
detaches himself from the exploration process; generally, the situation is described by the
subject in a language which has a logic flavour, but perhaps it is not phrased in a
conditional form (if...then) nor it is crystallised in a logical form; on the contrary, the
subject expresses his hypothesis not yet as a deductive sentence, but as an abduction,
namely a sort of reverse deduction. In fact, generally the subject sees what rule it is the
case of, to use Peirce language. Namely, she/he selects the piece of her/his knowledge
she/he believes to be right. The conditional form is virtually present: its ingredients are
all alive, but their relationships are still reversed, with respect to the conditional form;
the direction after which the subject sees things is still in the stream of exploration: the

control of the meaning is ascending (we use this term as in Saada-Robert, 1989, and
Gallo, 1994).
PHASE 2. A switching to the deductive form takes place, thanks to the
abduction. Now the control is descending and there is a new exploration of the situation,
in which things are looked at in the opposite way: not in order to have hints for
producing a conjecture, but in order to see why the regularity discovered through
abduction does work. The reversed way of looking at figures leads the subject to
formulate the conjecture in the conditional form. Now the modality is typically that of a
logical concatenation.
PHASE 3. In this phase, suitable, possibly fresh, heuristic is used, in order to
prove the conjecture. Here the descending control is crucial: it allows the detached
subject to interpret in the 'right' way what is happening, namely to produce logical
concatenations. At first they are local, then they are organised in a more global and
articulated way. Here the detachment has increased: the subject has become a true
rational agent (see Balacheff, 1982), who controls the products of the whole exploring
and conjecturing process from a higher level, selects from this point of view those
statements which are meaningful for the very process of proving and rules possible new
explorations. In this last phase, conjectures are possibly reformulated in order to combine
better logical concatenations and new explorations are possibly made to test them.
We observe that the exploration/selection modality is a constant in the whole
conjecturing and proving process; what changes is the different attitude of the subject
towards her/his explorations and the consequent type of control with respect to what is
happening in the given setting. It is the different type of control that changes the
relationships among the geometrical objects, both in the way they are 'drawn' and in the
way they are 'seen'. This is essential for producing meaningful arguments and proofs.
Also the detachment of the subject changes with respect to control: there are two types of
detachment. The first one is very local and marks the switching from ascending to
descending control through the production of conjectures formulated as conditional
statements (that is, local logical concatenations) because of some abduction. The second
one is more global and we used the metaphor of the rational agent to describe it: in fact it
is embedded in a fully descending control, produces new (local) explorations and
possibly proofs (that is, global logical combinations). The transition from the ascending
to the descending control is promoted by abduction, which puts on the table all the
ingredients of the conditional statements: it is the detachment of the subject to reverse the
stream of thought from the abductive to the deductive (i.e. conditional) form, but this can
happen because an abduction has been produced. The consequences of this transition are

In: Magnani, L., Nersessian N.J. & Thagard, P. (eds.),
Abduction and Scientific Discovery, Special Issue of Philosophica (in print)
a deductive modality and new relationships among the geometrical objects of the figures.
The inverse transition from descending to ascending control is more 'natural': in fact as
soon as a new exploration starts again, control may change and become again ascending,
even if at a more local level (with the rational agent who still control the global situation
in a descending way).
In short, the model points out an essential continuity of thought which rules the
successful transition from the conjecturing phase to the proving one, through exploration
and suitable heuristic, ruled by the ascending/descending control stream. The most
delicate cognitive point is the process of abduction, that is crucial for switching the
modality of control. One further relevant aspect is the change in the mutual relationships
among geometrical objects, which is the essential product of such a switching. A further
observation: as far as analysis-synthesis methods are concerned, our model is more in
accordance with the configurational than with the directional approach; in fact abduction
is active insofar there is no pre-determined direction, the useful known and unknown
quantities and their mutual dependencies are not completely fixed in advance (to
rephrase Descartes), but must be looked for through an open exploration.
Let us now apply our model to Lakatos method of proofs and refutations (see
Lakatos, 1976) concerning the Euler formula V-E+F=2 on the number of vertices, edges
and faces of a polyhedron.
Lakatos does not analyse the conjecturing phase: his work starts immediately
after that a first conjecture has been produced. As he says: "The phase of conjecturing
and testing in the case V-E+F=2 is discussed in Polya (1954). Polya stopped here, and
does not deal with the phase of proving....Our discussion starts where Polya stops."
(Lakatos, 1976, p.7). It is a pity that Lakatos, as well as Polya, discusses only half of the
story; in fact, the analysis of the two sides reveals strong elements of continuity; in the
game of counter-examples of Lakatos, there are lots of abductions, which mark for
example the switching from a global counter-example to new definitions (a global
counter example produces a criticism of the conjecture, not of the proof, according to
Lakatos). In fact such abductions allow to bound and refine better the domain of validity
of a proposition: it is the case when the counter-example shows that the proposition is "in
principle true, but admits exceptions in certain cases"(ibid., p.24).
But if abduction is a main ingredient also in Lakatos logic of discovery, still there
is a big difference with the example of the conjecturing phase, which illustrates the
different modalities of the refinement of conjectures discussed in Lakatos. The main
point is the general control of the rational agent with respect to the situation: whilst in
our model there is a positive logic (namely, something is first conjectured, then proved),

in Lakatos, so to say, there is a logic of not: abduction does happen within this new
frame, dual in a certain sense to the one seen before: we shall discuss this duality very
precisely in chapter 3.
Let us explain the differences in a more explicit way.
In our model (producing a conjecture and proving it) we can distinguish:
(i) a context, more precisely a fragment of a theory of reference, let us say P,
within which explorations and conjectures are drawn (e.g., some pieces of elementary
geometry);
(ii) A surprising or interesting situation, let us say E, worthwhile to be explained
by a conjecture, namely by saying the/a reason why E holds within P (for example, think
about the surprising situation of the square built on the diagonal of another square, the
first one having twice the area of the first one, described in Plato's Meno). We can
represent the resulting problematic situation in the following diagram (it does not
necessarily mean that one is operating within a formal system, but only that one is
looking for a reasonable hypothesis for E within a certain mathematical domain of
discourse P, see Rav (1999, p.11):
P |— ( ? ) --> E ;
(iii) Dynamic explorations (like in Meno), with ascending control, allow the
subject to find such an hypothesis P', as a 'possible cause' of E within that context;
namely P' is produced with an abduction. Then the descending control starts, possibly
producing the final proof in the end, within a logic of discovering/proving, which result
so deeply intertwined:
P |— P' --> E .
Lakatos’ example and consequently his theory of proofs and refutations, which
consists in refining an existing conjecture and proving it, can be pictured in the following
way:
(i) At the beginning, there is a conjecture, namely some sentence E which
possibly holds within a context P:
P |— E ;

In: Magnani, L., Nersessian N.J. & Thagard, P. (eds.),
Abduction and Scientific Discovery, Special Issue of Philosophica (in print)
(ii) Because of some thought experiment, a surprising or interesting situation
appears, namely a counterexample c, which has a global character (see above):
P |— ¬E(c) ;
Hence the problematic situation now consists of explaining the reason why the
counterexample holds, that is which rule this counterexample is the case of, within the
given context. Hence by a new abduction , we get such a reason, let us say P-:
P |— P-(c) ;
(iii) Now, a new resolutive move starts; it is typical of what we have called the
logic of not. The new move consists of investigating the connections between the cause
of the counterexample, namely P-, and the conjecture P |— E. In fact, P- is possibly a
reason why the conjecture does not hold; hence it is reasonable to look within the context
P for some new hypothesis, let us say P+, which eliminates P- and consequently, the
counterexample. Lakatos' paper describes how the subject can find such a P+, within the
context P, through a dynamic exploration: namely how P+ is produced with a new
abduction, of the type described in the preceding example:
P |— P+ --> ¬P- .
Generally, with the new hypothesis P+, one has all the ingredients for producing
a proof of the conjecture E, within the enriched theory P& P+:
P & P+ |— E .
The process of proving consists of combining the pieces of information and local
logical connections produced in the previous phases logically; in this last phase the
control typically switches from an ascending to a descending modality.
It is also interesting to observe that such a dynamic is sketched by Lakatos
himself in Lakatos, 1978, p. 93 and followings, where he describes the machinery of
proofs and refutations within the framework of Pappusian analysis-synthesis: "The
analysis provides the hidden assumptions needed for the synthesis. The analysis contains
the creative innovation, the synthesis is a routine task for a schoolboy.....However, the
hidden lemmas are false. ....But nevertheless we can extricate from the analysis (or from

the synthesis) a 'proof-generated theorem' by incorporating the conditions articulated in
the lemmas." (ibid., p.95).
A last remark: the two types of searching hypotheses that we have illustrated, can
be analysed and evaluated also from the point of view of economy, as Peirce does in is
discussion of abduction.
3. The theoretical model from the point of view of Natural Deduction. The
discussion in chapter 2 shows essentially two ways of attacking problems:
P |— ( ? ) --> E [1]
P & ( ? ) |— E [2]
In both cases abduction plays an essential role in reversing the course of thought
(from ascending to descending control); in the second case the use of counterexamples
seems to be at the origin of a more involved course of thought, which we have called the
logic of not.
In this chapter, we will show that the two modalities are dual each other, in a
precise technical sense. First, let us approach the question in an intuitive way; later on,
we shall give a few technicalities. The two problems [1] and [2] have analogies and
differences: namely, in both cases one is looking for some missing (or hidden)
hypothesis; but in the first case, the context is fixed and the hypothesis is looked for
within that context, whilst in the second one, it is the context, that is the (implicit or
explicit) domain of discourse, to be challenged because of the counterexample. In fact,
the concrete examples we have observed in our experiments and the example of Lakatos
corroborate this observation. In the former, which are non-routine problems, no
challenge is made to the (implicit) theory and the new hypothesis can be got within the
theory (a typical trivial example could be: "under the hypothesis that the two sums of the
opposite sides of a quadrilateral are equal, a circle can be inscribed into the
quadrilateral"); in the latter, it is the implicit theory to be challenged and a new
hypothesis within it must be formulated (a typical example being theorems concerning
trapezia, where the hypothesis of the convexity of figures must be made explicit; a
similar case is precisely the example discussed by Lakatos).
Now Natural Deduction (see Prawitz, 1971) illustrates in a precise and deep way
this phenomenon, provided one takes into account an approach to logic which shows also
the dynamic aspects of thought, like the categorical approach to Natural Deduction (see

In: Magnani, L., Nersessian N.J. & Thagard, P. (eds.),
Abduction and Scientific Discovery, Special Issue of Philosophica (in print)
Martin- Löf, 1984). In such an approach the above relationship between conjunction and
implication (now we are working within a formal system, for example in the way
described in Prawitz, 1971):
P & Q |— E P |— Q --> E
is described in terms of a fundamental concept in category theory, namely the so called
adjunction (see Crole, 1993). We can say that the implication is the right adjoint of
conjunction.
Hence, adjunction illustrates from the logic point of view (provided you approach
logic according to category theory) a phenomenon which has a cognitive counterpart in
abduction.
To be more precise, solving such problems like
P & __ |— E P |— __ --> E [3]
requires to find two dual relationships j(_) and y(_) (two adjoined functors in the
language of categories); to do that, one must go backwards, from the conjecture E towards
the unknown hypothesis _ , with a thinking style which is 'opposite' to the deductive one
and that has more a configurational than a directional pattern. The backwards direction,
which is mirrored at a cognitive level by what we have called ascending control, has been
emphasised also by Lakatos: "Whether a deductive system is Euclidean or quasi-empirical
is decided by the pattern of truth value flow in the system. The system is Euclidean if the
characteristic flow is the transmission of truth from the set of axioms 'downwards' to the
rest of the system -logic here is an organ of proof; it is quasi-empirical if the characteristic
flow is retransmission of falsity from the false basic statements 'upwards' towards the
'hypothesis' -logic here is an organ of criticism." (Lakatos, 1978, p.29).
Backwards direction and the logic of not, typical of finding solutions to the j(_)
in [3], correspond to what Lakatos calls retransmission of falsity; it is however
questionable that their nature implies quasi-empiricism in mathematics.
On the contrary, the relationships between abduction and adjunction which live
behind the solution processes to equations [3] are similar to the relationships between
informal and formal proofs. The distinction has been made by many scholars (see
Kreisel, 1970, pp. 445-467): the former are those "of customary mathematical discourse,
having an irreducible semantic content" (Rav, 1999, p.11) and the latter are syntactic
objects of a formal system. In fact, abduction and backwards strategies seem to exhibit

essentially two typologies (corresponding to [1] and [2], respectively), whose
psychological and epistemological features, essentially embodied in abduction, have
been described in chapter 2. The formalisation within the systems of Natural Deduction
and the categorical interpretation given in chapter 3, essentially based upon adjunction,
seem to represent the formal counterpart of such processes. Of course it is questionable
that adjunction incorporates everything, as well as it is perhaps debatable that every
informal proof has a formal counterpart (as the so called Hilbert's Thesis says, see Rav,
1999, p.11); something similar happens when one compares the intuitive notion of
effectively computable functions with the technical definition of partial recursive
functions (Church's Thesis says that the technical definition capture all of the intuitive
concept). We do not suggest here to invent a new Thesis, to support the adequacy
between the two notions. What we underline is that, like in the previous two examples
(computable functions, proofs) the existence of an intuitive notion has brought to deepen
its analysis through mathematical investigations without claiming a new epistemology, in
the same way a lot of things done by people who make conjectures and proofs can be
explained using suitable mathematical (and traditional) tools of analysis.
We wish to conclude this paper with two more observations.
First, the works of Hintikka and Remes (1974) and Mäenpää (1997) have used
deductive logic to interpret the configurational aspects of analysis; they used essentially
the systems of Natural Deduction with (in the last papers) the machinery of types of
Martin- Löf, see Mäenpää (1997), that is in the version which shows more connections
with computer science. They showed that "the Greek method of geometrical analysis can
be generalised into a method of solving all kinds of mathematical problems in type
theory by taking into account inductively defined problems, which are characteristic of
programming. The method known as top-down programming turns out to be a special
case of analysis." (Mäenpää, 1997, p.226). It is intriguing (and corroborating the ideas
above) that we find also from another approach a reduction to the same formal
machinery!
Secondly -as a further corroboration- the same strategy (of looking for suitable
economic hypothesis in a backward approach) that we have analysed above is at the base
of many programs for automatic theorem proving in elementary geometry, e.g., those
which use the Grîbner-Buchberger algorithm, see Chou (1988): the algebraic varieties,
which are the geometric counterpart of the algebraic symbolism incorporated into the
software, are the natural model of a dynamic logic, where the method of resolution
means to look for a new economic hypothesis, which allows to restrict the domain of

In: Magnani, L., Nersessian N.J. & Thagard, P. (eds.),
Abduction and Scientific Discovery, Special Issue of Philosophica (in print)
validity of a supposed conjecture, in order that this really becomes true, see Cox et al.
(1992), pp.280-296. The strategy is very similar to that illustrated in problem [2],
modulo the algebraic translation.
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