**The dialectic plan: an alternative to
the paradigm**

p.franceschi@univ-corse.fr

http://www.univ-corse.fr/~franceschi

English translation of a paper appeared in French

in *Semiotica*
(2003, vol. 146 (1-4), pp. 353-367)

under the title "Le plan dialectique: pour une alternative au paradigme"

In Franceschi (2002), I exposed a theory which aims
to constitute an alternative to the classification proposed by
Greimas in the field of paradigmatic analysis. In the present
article, I proceed to draw the consequences of this latter theory by
applying it to the technique of conception of a plan. Regarding the
dialectic plan, the current paradigm is in effect a plan of the type
*thesis-antithesis-synthesis*. This form of plan is very
widespread and its use proves to be consensual. In what follows, I
shall propose a novel type of dialectic plan as an alternative to the
classical one. It consists of a type of plan which can be qualified
as matrix-based, and which presents several advantages with regard to
the classical dialectic plan.

**The classical dialectic plan**

The current paradigm regarding the dialectic plan is
a plan of the thesis-antithesis-synthesis type. This plan finds its
origin in the *dialectical* approach developed by Hegel. The
association of the three concepts thesis-antithesis-synthesis, which
is now associated with the dialectical line of reasoning, was
elaborated by Hegel and Marx. The dialectical approach constitutes
thus a process of reasoning that proceeds by the statement of two
contradictory theses and by their reconciliation at the stage of the
synthesis. According to Hegel, every thesis presents then an
inherently incomplete and partial nature, which gives then birth to
its contrary, the antithesis. From Hegel's standpoint, the contraries
present, beyond the contradiction underlying them, an indissociable
nature. This last property allows thus to make their final union, at
a thought level which places itself beyond the one where the
contradiction manifests itself. The contraries present thus by
essence a genuine unity, from which it is worth grasping the fecund
principle, allowing thus to reach, at a higher level, a genuine
knowledge. This latter phase constitutes the synthesis, which can
thus be considered as the step of reasoning which reconciliates
veritably, at a greater level, the contradiction observed between the
thesis and the antithesis. The synthesis allows thus to go beyond the
conflict raised between the thesis and the antithesis, by further
unifying the part of truth simultaneously contained in both of them.
However, the process is not limited to that. For the synthesis thus
obtained constitutes in turn a novel thesis, which itself yields a
novel antithesis and then a novel synthesis, and so on... Within the
current language, the dialectical approach designates now the general
methodology which allows to go beyond and to solve the
contradictions. It is in this dialectical approach that the classical
plan of the thesis-antithesis-synthesis type finds its origin.

At this step, it is worth considering in turn each
component of the thesis-antithesis-synthesis plan. Consider, to begin
with, the *thesis*. This latter constitutes a standpoint
expressed by a given author. It consists of the viewpoint on which
the discussion is based, and toward which the structure of the plan
is oriented. For simplicity, let us assimilate here the thesis to a
given proposition. On the other hand, the *antithesis* is a
standpoint which proves to be contrary to that of the thesis. Like
the thesis, it is useful to reduce the antithesis, for the sake of
simplicity, to a given proposition. At this step, the viewpoints
expressed by the thesis and the antithesis are of an antinomical
nature. Lastly, the *synthesis* constitutes the part of the
discourse where the antagonist viewpoints developed in the thesis and
the antithesis are overcome. The synthesis aims thus classically to
go beyond the antinomy existing between the thesis and the antithesis
and to encompass it.

In a general way, the advantage of the dialectic plan of the type thesis-antithesis-synthesis is to allow to apprehend the double aspect of a given problem or reality. By placing oneself alternatively from one side and from the other, by considering successively the thesis and then the antithesis, this type of plan allows to avoid a partial or truncated vision of the particular problem raised by the thesis. The aim of the classical dialectic plan is thus to apprehend the two-faceted nature of a given reality and to go beyond the contradiction which results from a preliminary study.

**Matrices of concepts**

In Franceschi (2002), I described the structure of a
*matrix *of concepts, the scope of which extends to many
concepts. For the sake of the present discussion, it is not necessary
to describe in a detailed way the structure of concepts put forth in
this article. Nevertheless, the type of dialectic plan which will be
proposed later derives directly from the notion of a matrix of
concepts. It proves then necessary to present the main lines of the
basic structure of a matrix of concepts.

Consider first a given *duality*.
Let us denote it by A/Ā. At this step, A and Ā constitute
*dual* concepts. One can
then consider that A and Ā are concepts which characterize
themselves by a *contrary* *component c* Î
{-1, 1} at the level of a given duality
A/Ā, such that *c*[A] = -1 and *c*[Ā]
= 1. One can also consider that A and Ā are neutral concepts
which can thus be denoted by A^{0} and
Ā^{0}.

At this step, we are in a position to define the
class of the *canonical poles*. It suffices to consider an
extension of the preceding class {A^{0},
Ā^{0}}, such that A^{0} and
Ā^{0} respectively admit of both a positive and a
negative concept which are correlative. Such concepts possess a
certain intuitive support. Let us denote them respectively by {A^{+},
A^{-}} and {Ā^{+},
Ā^{-}}. At this
step, for a given duality A/Ā, we get the following concepts:
{A^{+}, A^{0}, A^{-},
Ā^{+}, Ā^{0},
Ā^{-}}, which constitute the *canonical poles*.
It is worth mentioning here that the notation α(A/Ā,
*c*, *p*) could be used alternatively, for a given
*canonical pole*. In all cases, the components of a canonical
pole are: a *duality* A/Ā, a
*contrary component c* Î
{-1, 1} and a *canonical* *polarity p* Î
{-1, 0, 1}. This definition of the canonical poles leads to
distinguish between the *positive *(A^{+}, Ā^{+}),
*neutral *(A^{0}, Ā^{0})
and *negative *(A^{-}, Ā^{-})
canonical poles. Lastly, the class made up of the six canonical poles
of a same matrix can be dubbed the *canonical matrix*: {A^{+},
A^{0}, A^{-}, Ā^{+},
Ā^{0}, Ā^{-}}.

Let us focus now on the nature of the relationships
existing between the canonical poles of a given matrix. Among the
combinations of relationships existing between the six canonical
poles (A^{+}, A^{0}, A^{-},
Ā^{+}, Ā^{0},
Ā^{-}) of a same
duality A/Ā, one will retain the following relations: *duality,
antinomy*, *complementarity*, *corollarity*, *connexity*,
*anti-connexity*. Thus, two
canonical poles α_{1}(A/Ā,
*c*_{1}, *p*_{1})
and α_{2}(A/Ā,
*c*_{2}, *p*_{2}) of a same matrix
are:

(a) *dual* if their contrary components are
opposite and their polarities are neutral

(b) *contrary* (or *antinomical*) if
their components are opposite and their polarities are non-neutral
and opposite

(c) *complementary* if their contrary
components are opposite and their polarities are non-neutral and
equal

(d) *corollary* if their contrary components
are equal and their polarities are non-neutral and opposite

(e) *connex* if their contrary components are
equal and the absolute value of the difference of their polarities
equals 1

(f) *anti-connex* if their contrary
components are opposite and the absolute value of the difference of
their polarities equals 1

To sum up: {A^{0},
Ā^{0}} are *dual*; {A^{+},
Ā^{-}} and {A^{-},
Ā^{+}} are *contraries*; {A^{+},
Ā^{+}} and {A^{-},
Ā^{-}} are *complementary*; {A^{+},
A^{-}} and {Ā^{+},
Ā^{-}} are *corollary*; {A^{0}, A^{+}},
{A^{0}, A^{-}}, {Ā^{0},
Ā^{+}} and {Ā^{0},
Ā^{-}} are *connex*; {A^{0},
Ā^{+}}, {A^{0},
Ā^{-}}, {Ā^{0},
A^{+}} and {Ā^{0},
A^{-}} are *anti-connex*.

To fix ideas, let us take the example of the matrix
{*eclecticism*^{+}, *multi-disciplinarity*^{0},
*dispersion*^{-}, *expertise*^{+},
*monodisciplinarity*^{0}, *compartmentalization*^{-}}.
One has then the following relationships:

(a') {*multi-disciplinarity*^{0},
*monodisciplinarity*^{0}} are dual

(b') {*eclecticism*^{+},
*compartmentalization*^{-}}, {*dispersion*^{-},
*expertise*^{+}} are antinomical

(c') {*eclecticism*^{+}, *expertise*^{+}},
{*dispersion*^{-}, *compartmentalization*^{-}}
are complementary

(d') {*eclecticism*^{+},
*dispersion*^{-}}, {*expertise*^{+},
*compartmentalization*^{-}} are corollary

(e') {*multi-disciplinarity*^{0},
*eclecticism*^{+}}, {*multi-disciplinarity*^{0},
*dispersion*^{-}}, {*monodisciplinarity*^{0},
*expertise*^{+}}, {*monodisciplinarity*^{0},
*compartmentalization*^{-}} are connex

(f') {*multi-disciplinarity*^{0},
*expertise*^{+}}, {*multi-disciplinarity*^{0},
*compartmentalization*^{-}}, {*monodisciplinarity*^{0},
*eclecticism*^{+}}, {*monodisciplinarity*^{0},
*dispersion*^{-}} are anti-connex

**Structure of a thesis**

At this step, it is worth delving more deeply into
the internal structure of the *thesis* to which the plan
dialectical applies. I shall draw a distinction here between *simple*
and *complex* theses.

*Simples theses *

In general, a simple
thesis presents a structure which is that of an appreciation -
negative, neutral or positive - relative to a given concept. Let α
be such a concept; one denotes then by z
^{p}(α) such structure of
*thesis*, where *p* denotes a negative polarity,
neutral or positive such that respectively *p* Î
{-1, 0, 1}. The negative appreciation can be assimilated to a *blame*
and the positive appreciation to a praise. The *blame* of
a given concept α is thus denoted by z
^{-}(α), the
*neutral* *appreciation* by z
^{0}(α) and the *praise*
by z ^{+}(α).
In a general way, the propositions corresponding to the simple theses
present the following structure: z ^{p}(α),
with *p* Î {-1, 0, 1}
and α Î
{A^{+}, A^{0}, A^{-},
Ā^{+}, Ā^{0},
Ā^{-}}. By referring to the *matrix* notion,
one notes that the different theoretical cases are the following,
with regard to the six *concepts* of a given *matrix*: {z
^{-}(A^{+}), z ^{-}(A^{0}),
z ^{-}(A^{-}), z
^{-}(Ā^{+}),
z ^{-}(Ā^{0}),
z ^{-}(Ā^{-}),
z ^{0}(A^{+}), z
^{0}(A^{0}), z ^{0}(A^{-}),
z ^{0}(Ā^{+}),
z ^{0}(Ā^{0}),
z ^{0}(Ā^{-}),
z ^{+}(A^{+}), z
^{+}(A^{0}), z ^{+}(A^{-}),
z ^{+}(Ā^{+}),
z ^{+}(Ā^{0}),
z ^{+}(Ā^{-})}.
At this step, it appears that the *neutral* *appreciation*
is somewhat rarely found. Thus, for the sake of simplicity, we shall
be mainly concerned here with describing more accurately the theses
which present the structure of a *blame* or of a *praise*.

Let us begin with the *blame*. A number of
theses are thus composed of a depreciative appreciation, related to a
behavior, a way of doing or apprehending things, a given situation.
Such statements correspond to propositions that present the structure
of a *blame*. Such propositions can be denoted by z
^{-}(*s*) where *s* designates a way of
apprehending or of doing things.

Let us take, to fix ideas, a few examples. Consider the following thesis:

(1) In the contempt of ambition is to be found one
of the essential principles of happiness on earth. (Edgar Poe, *The
Domain of Arnheim*)

The author considers here the "contempt of
ambition" as a fundamental principle allowing to reach
happiness. Such a viewpoint can be analyzed as a negative,
depreciative judgment toward *ambition*. This latter concept can
be considered as a neutral notion. Hence, such a simple thesis
presents the structure which is that of the *blame* of *ambition*^{0}
and can be thus denoted by z ^{-}(*ambition*^{0}).

Consider also this other thesis:

(2) Love, the scourge of the world, atrocious
folly. (Alfred of Musset, *Premières poésies*)

The content of this latter thesis can be analyzed as
a very pejorative appreciation formulated with regard to *love*^{+}.
Here also, such thesis presents a structure that can be analyzed as a
*blame* of *love*^{+}, that one can thus denote by
z ^{-}(*love*^{+}).

Conversely, one also frequently encounters some
theses which are composed of a flattering appreciation with regard to
a given behavior, a propensity to act, a situation or a way of
apprehending things. The structure of the corresponding proposition
is then that of a *praise*. One denotes such propositions by z
^{+}(*s*) where *s* designates a way of considering
things or a given behavior.

Consider then a few examples. To begin with, the following viewpoint illustrates this type of structure:

The author formulates here a praise related to the
*passion*, considering thus that "nothing of great
importance" ever came true without this latter. One can consider
here the *passion* as a neutral notion. Such a viewpoint
presents thus the structure of a *praise *of *passion*^{0},
i.e. formally z ^{+}(*passion*^{0}).

One also encounters an identical type of structure, regarding the following affirmation:

which can be analyzed as a *blame* of *passion*^{0},
i.e. formally z ^{-}( *passion*^{0}).

Lastly, the following simple thesis:

(5) The worst vice of the fanatic is his sincerity. (Oscar Wilde)

constitutes an example of *praise* of the
*negative* concept of fanaticism, i.e. formally z
^{+}(*fanaticism*^{-}).

At this step, we are in a position to determine the
*truth value* of the simple theses. The truth value of each type
of *praise*, of *neutral* *appreciation *or of *blame*
indicates if the considered affirmation is plausible and coherent or
not, given that the *praise* of a *positive* *concept *is
*true*, in the same way as the *neutral* *appreciation *of
a neutral concept and the *blame* of a *negative concept*.
Conversely, the *praise* of a *non-positive concept, *the
*neutral* *appreciation *of a non-neutral concept or well
the *blame* of a *non-negative concept* are *false*.
Formally, the truth value [*v*] of propositions of the type P =
z ^{p}(α^{q}),
with *p*, *q* Î {-1,
0, 1} and α Î {A^{+},
A^{0}, A^{-}, Ā^{+},
Ā^{0}, Ā^{-}}
can be calculated as follows: [*v*] = 1 (*true*) if *p*
= *q* and [*v*] = -1 (*false*) if *p* ¹
*q*. Hence, among the different cases which have just been
enumerated, those whose truth value is *true* are: {z
^{-}(A^{-}), z ^{-}(Ā^{-}),
z ^{0}(A^{0}), z
^{0}(Ā^{0}),
z ^{+}(A^{+}), z
^{+}(Ā^{+})}.
And those whose truth value is *false* are: {z
^{-}(A^{+}), z ^{-}(A^{0}),
z ^{-}(Ā^{+}),
z ^{-}(Ā^{0}),
z ^{0}(A^{+}), z
^{0}(A^{-}), z ^{0}(Ā^{+}),
z ^{0}(Ā^{-}),
z ^{+}(A^{0}), z
^{+}(A^{-}), z ^{+}(Ā^{0}),
z ^{+}(Ā^{-})}.

*Complex theses*

Whereas simple theses contain a judgment related to
one single concept belonging to a given matrix, *complex* *theses
*are composed of appreciations relative to *several *concepts
of a same *matrix*. A *complex* thesis can thus be defined
in a general way as the *conjunction* of several simple theses.
A *complex* thesis can thus be composed of appreciations
relative to two, three, ..., *n* different concepts. One will
use accordingly the term of *n-complex thesis*. Under these
circumstances, the combinations prove to be numerous, without it
being nevertheless necessary to enumerate them exhaustively. A given
proposition P constituting a complex thesis presents thus the
following structure: P = Q_{1} Ù
Q_{2} Ù ... Ù
Q_{n}, for *n* > 1, and Q_{i} = z
^{pi}(α^{qi}),
with *p*_{i}, *q*_{i} Î
{-1, 0, 1} and α Î
{A^{+}, A^{0}, A^{-},
Ā^{+}, Ā^{0},
Ā^{-}}. We have then the 2-complex, 3-complex, …,
*n*-complex theses.

At this step, it appears necessary to consider first
the 2-complex theses, which constitute, among the complex theses, the
most common case. The 2-complex theses are composed of some
appreciations relative to *two* concepts of a same matrix. They
present the structure: z ^{p}(α_{1}(A/Ā,
*c*_{1}, *q*)) Ù
z ^{r}(α_{2}(A/Ā,
*c*_{2}, *s*)). The following appreciation
constitutes thus an example of 2-*complex* thesis:

This 2-*complex* thesis is in effect composed
of both the *blame* of *theory* ("all theory is gray")
and the *praise* of *pragmatism* ("the golden tree of
life is green"). It proves here that the concepts of *interest
for theory* and of *pragmatism* belong to the following
matrix: {*capacity of abstraction*^{+}, *interest for
theory*^{0}, *dogmatism*^{-}, *pragmatism*^{+},
*interest for practise*^{0}, *prosaicness*^{-}}.
The structure of the thesis is thus z
^{-}(*interest for theory*^{0}) Ù
z ^{+}(*pragmatism*^{+})
i.e. z ^{-}(A^{0}) Ù
z ^{+}(Ā^{+}).

In the same way, the following appreciation constitutes a case of 2-complex thesis:

(7) The art of being sometimes very audacious, sometimes very cautious is the art of success. (Napoleon Bonaparte)

This 2-complex thesis is composed of both the *praise*
of *boldness* ("The art of being (…) very audacious
(…) is the art of success") and the *praise* of the
*cautiousness* ("the art of being (…) very cautious
is the art of success"). It appears that these latter concepts
belong to the following matrix: {*boldness*^{+},
*propensity to take risk*^{0}, *temerity*^{-},
*cautiousness*^{+}, *propensity to avoid the risk*^{0},
*cowardice*^{-}}. The thesis is thus composed here of
the praise of two complementary positive concepts of a same matrix.
The particular structure of this type of complex thesis is thus
composed of the *praise* of A^{+} and the *praise*
of Ā^{+}, i.e.
formally z ^{+}(*boldness*^{+})
Ù z
^{+}(*cautiousness*^{+}).

Consider lastly the following thesis, which also constitutes a case of 2-complex thesis:

This last thesis is in effect composed of both the
blame of *irrationality* ("exclude the reason") and
the blame of *hyper-rationalism* ("to admit nothing else
than reason"). The corresponding reconstituted matrix is the
following: {*imagination*^{+}, *inspiration*^{0},
*irrationality*^{-}, *rationality*^{+},
*reason*^{0}, *hyper-rationalism*^{-}}. As
we see it, we face here a 2-complex thesis whose structure is z
^{-}(*irrationality*^{-}) Ù
z ^{-}(*hyper-rationalism*^{-})
i.e. z ^{-}(A^{-}) Ù
z ^{-}(Ā^{-}).

Lastly, the following 2-complex thesis:

(9) How can we tolerate that passion be placed on
the same level than reason? (Sénèque, *De Ira*)

can be analyzed as a *blame* of *passion*^{0}
and a *praise* of *reason*^{0}, i.e. formally z
^{-}(*passion*^{0}) Ù
z ^{+}(*reason*^{0}),
i.e. z ^{-}(A^{0}) Ù
z ^{+}(Ā^{0})
at the level of the matrix {*motivation*^{+}, *passion*^{0},
*fanaticism*^{-}, *level-headedness*^{+},
*reason*^{0}, *lukewarmness*^{-}}.

It is worth noting here that this last type of
2-complex thesis corresponds to a common case, for motives of
internal coherence. It is in effect logical when one criticizes or
depreciates such or such value or concept, of flattering its
contrary. To blame such or such thing amounts naturally to praising
its opposite, and conversely. For that reason, the 2-complex theses
whose particular structure is z ^{-}(A^{-})
Ù z
^{+}(Ā^{+})
or well z ^{+}(A^{+}) Ù
z ^{-}(Ā^{-})
also constitute, among all possible combinations of 2-*complex*
theses, a common case.

For what concerns the truth value of the 2-complex
theses, it can be determined in the same way as for the simple
theses. Let thus P Ù Q be a
2-complex thesis, such that P = z ^{p}(α^{q})
and Q = z ^{r}(β^{s}),
with *p*, *q*, *r*, *s* Î
{-1, 0, 1} and α, β Î
{A^{+}, A^{0}, A^{-},
Ā^{+}, Ā^{0},
Ā^{-}}. Formally, the truth value [*v*] of a
2-complex thesis P Ù Q is *true*
if *v*[P] = *v*[Q] = true, and *false* in other cases.
It is worth noting that the most common types of 2-complex theses are
those whose truth value are *true*. Such is the case when the
truth-value of each of the two propositions included within the
complex thesis is *true*. Under this hypothesis, the two
propositions reinforce themselves. It consists thus of the cases
corresponding to: {z ^{+}(A^{+})
Ù z
^{-}(A^{-}), z ^{+}(A^{+})
Ù z
^{+}(Ā^{+}),
z ^{+}(A^{+}) Ù
z ^{-}(Ā^{-}),
z ^{-}(A^{-}) Ù
z ^{+}(Ā^{+}),
z ^{-}(A^{-}) Ù
z ^{-}(Ā^{-}),
z ^{+}(Ā^{+})
Ù z
^{-}(Ā^{-})}.

**Dual theses**

At this step, it is worth focusing on the notion of a dual thesis of a given thesis. This last notion applies both to the simple theses and to the complex ones. The dual thesis constitutes here an element of the dialectical discussion, which proves to be important since it is the basis of the discussion related to the thesis under consideration.

Let us focus, to begin with, on dual theses of
simple theses. Let us begin by giving a general definition. Formally,
a simple thesis z ^{p}(α_{1}(A/Ā,
*c*, *q*)) admits of a dual thesis that corresponds
to the following definition: z ^{p}(α_{2}(A/Ā,
-*c*, *q*)). Thus, a dual thesis of a simple thesis
presents the following characteristics: (i) the polarities of the
appreciation of the dual thesis and of the simple thesis are
identical; (ii) the contrary components of the concepts on which bear
the appreciations of the dual thesis and of the simple thesis are
opposite; (iii) the polarities of the concepts on which bear the
appreciations of the dual thesis and of the simple thesis are
identical.

Let us consider first the dual theses of the *true*
simple theses. The types of true simple theses can be thus enumerated
as follows: {z ^{+}(A^{+}),
z ^{0}(A^{0}), z
^{-}(A^{-}), z ^{+}(Ā^{+}),
z ^{0}(Ā^{0}),
z ^{-}(Ā^{-})}.
Formally, a true simple thesis z ^{p}(α_{1}(A/Ā,
*c*, *p*)) presents a dual thesis which responds to
the following definition: z ^{p}(α_{2}(A/Ā,
-*c*, *p*)). Thus, the dual theses of the true
simple theses are respectively: {z ^{+}(Ā^{+}),
z ^{0}(Ā^{0}),
z ^{-}(Ā^{-}),
z ^{+}(A^{+}), z
^{0}(A^{0}), z ^{-}(A^{-})}.

To take an example, consider the following true simple thesis:

(10) What you can do, or dream you can do, begin it. Boldness has genius, power and magic in it. (Goethe)

which presents the structure z
^{+}(*boldness*^{+}) i.e. z
^{+}(A^{+}) at the level of the matrix {*boldness*^{+},
*propensity to take risk*^{0}, *temerity*^{-},
*cautiousness*^{+}, *propensity to avoid risk*^{0},
*cowardice*^{-}}. The thesis below whose structure is z
^{+}(*cautiousness*^{+}) i.e. z
^{+}(Ā^{+})
constitutes thus its dual thesis:

(11) Cautiousness is as much superior to the other virtues as sight is to the other senses. (Bion of Phlossa)

Consider also the dual theses of the *false*
simple theses. The types of false simple theses are: {z
^{-}(A^{+}), z ^{-}(A^{0}),
z ^{-}(Ā^{+}),
z ^{-}(Ā^{0}),
z ^{0}(A^{+}), z
^{0}(A^{-}), z ^{0}(Ā^{+}),
z ^{0}(Ā^{-}),
z ^{+}(A^{0}), z
^{+}(A^{-}), z ^{+}(Ā^{0}),
z ^{+}(Ā^{-})}.
And the dual theses of the false simple theses are respectively: {z
^{-}(Ā^{+}),
z ^{-}(Ā^{0}),
z ^{-}(A^{+}), z
^{-}(A^{0}), z ^{0}(Ā^{+}),
z ^{0}(Ā^{-}),
z ^{0}(A^{+}), z
^{0}(A^{-}), z ^{+}(Ā^{0}),
z ^{+}(Ā^{-}),
z ^{+}(A^{0}), z
^{+}(A^{-})}.

To take an example, the following false simple thesis:

(4) Passion is an illness that abhors all medication. (Kant)

presents the structure z
^{-}(*passion*^{0}) i.e. z
^{-}(A^{0}) at the level of the matrix {*motivation*^{+},
*passion*^{0}, *fanaticism*^{-},
*level-headedness*^{+}, *reason*^{0},
*lukewarmness*^{-}}. The following thesis whose
structure is z ^{-}(*reason*^{0})
i.e. z ^{-}(Ā^{0})
constitutes thus its dual thesis:

It is worth considering now, on the other hand, the
dual theses of the complex theses. These latter are such that the
contrary components of the concepts on which bear the appreciations
of the two simple theses, which are part of the dual thesis and of
the considered thesis, are opposite. Consider then the true 2-complex
theses. Thus, the dual thesis of z ^{+}(A^{+})
Ù z
^{-}(Ā^{-})
is z ^{+}(Ā^{+})
Ù z
^{-}(A^{-}). And also, the dual thesis of z
^{0}(A^{0}) Ù z
^{+}(A^{+}) is z ^{0}(Ā^{0})
Ù z
^{+}(Ā^{+}).
It is worth noting here in particular that the dual thesis of z
^{0}(A^{0}) Ù z
^{0}(Ā^{0})
is z ^{0}(Ā^{0})
Ù z
^{0}(A^{0}), that the dual thesis of z
^{+}(A^{+}) Ù z
^{+}(Ā^{+})
is z ^{+}(Ā^{
+}) Ù z
^{+}(A^{+}) and that the dual thesis z
^{-}(A^{-}) Ù z
^{-}(Ā^{-})
is z ^{-}(A^{-}) Ù
z ^{-}(Ā^{-}).

Let us also give a few examples. Thus, the true 2-complex thesis corresponding to the following proposition:

(6) All theory is gray, but the golden tree of life is green. (Goethe)

presents the structure z
^{-}(A^{0}) Ù z
^{+}(Ā^{+})
i.e. z ^{-}(*interest for
theory*^{0}) Ù z
^{+}(*pragmatism*^{+}) at the level of the
matrix {*capacity of abstraction*^{+}, *interest for
theory*^{0}, *dogmatism*^{-}, *pragmatism*^{+},
*interest for practice*^{0}, *prosaicness*^{-}}.
The following thesis whose structure is z
^{-}(Ā^{0})
Ù z
^{+}(A^{+}) i.e. z
^{-}(*interest for practice*^{0}) Ù
z ^{+}(*capacity of
abstraction*^{+}) constitutes thus its dual thesis:

(13) All practice is vile, but fecund and elevated is the quest of the genuine abstraction.

Similarly, the following proposition:

(8) Two excesses: to exclude reason, and to admit
nothing else than reason. (Pascal, *Thoughts*)

constitutes a true 2-complex thesis whose structure
is z ^{-}(*irrationality*^{-})
Ù z
^{-}(*hyper-rationalism*^{-}) i.e. z
^{-}(A^{-}) Ù z
^{-}(Ā^{-})
at the level of the matrix: {*imagination*^{+},
*inspiration*^{0}, *irrationality*^{-},
*rationality*^{+}, *reason*^{0},
*hyper-rationalism*^{-}}. The thesis below whose
structure is z ^{+}(*imagination*^{+})
Ù z
^{+}(*rationality*^{+}) i.e. z
^{+}(A^{+}) Ù z
^{+}(Ā^{+})
constitutes thus its dual thesis:

(14) The art of being sometimes very imaginative, sometimes very rational is the art of success.

Lastly, it is worth noting that one has also
analogous definitions for 3-complex, 4-complex, etc. theses. To take
then an example, the dual thesis of the 3-complex thesis z
^{+}(A^{+}) Ù z
^{0}(A^{0}) Ù z
^{0}(Ā^{0})
is z ^{+}(Ā^{+})
Ù z
^{0}(Ā^{0})
Ù z
^{0}(A^{0}). In the same way, the dual thesis of the
3-complex thesis z ^{+}(A^{+})
Ù z
^{0}(A^{0}) Ù z
^{-}(A^{-}) is z ^{+}(Ā^{+})
Ù z
^{0}(Ā^{0})
Ù z
^{-}(Ā^{-}).

**The matrix-based dialectic plan **

The preceding developments allow now to describe the
steps of the dialectical reasoning applicable to the analysis of a
given particular thesis, from the above-mentioned principles. The
first step consists thus in the accurate determination of the
structure of the thesis under consideration. The second step, which
results directly from it, is the attribution of a truth-value to this
latter thesis. The following step consists then in the reconstitution
of the whole matrix applicable to the concept(s) which are the object
of the thesis. One is then in a position to determine the dual thesis
of the considered thesis in the same way as the true simple theses
other than the considered thesis and its dual thesis. Lastly, the
final step is the synthesis which consists in the conjunction of the
true simple theses relative to each of the 6 concepts of the
considered matrix: z ^{+}(A^{+})
Ù z
^{0}(A^{0}) Ù z
^{-}(A^{-}) Ù z
^{+}(Ā^{+})
Ù z
^{0}(Ā^{0})
Ù z
^{-}(Ā^{-}).
Such a synthesis allows to encompass a threefold antinomy: the one
existing between A^{+} and Ā^{-},
A^{0} and Ā^{0},
and A^{-} and Ā^{+}.
It should be observed here that one can eventually retain from the
synthesis but a simplified form consisting of the conjunction of the
true simple theses constituting a praise or a blame: z
^{+}(A^{+}) Ù z
^{-}(A^{-}) Ù z
^{+}(Ā^{+})
Ù z
^{-}(Ā^{-}).
In the same way, one may sometimes limit oneself to a truncated form
of synthesis consisting in z ^{+}(A^{+})
Ù z
^{+}(Ā^{+}),
which emphasizes the complementarity between A^{+} and
Ā^{+}.

At this step, we are in a position to present the
*matrix-based dialectic plan*. Such a plan results directly from
the structure of matrix of concepts which has been just described.
The corresponding matrix-based dialectic plan presents thus the
following structure:

(15) 1. From the viewpoint of A^{0}

1.1 Praise of A^{+}

1.2 Blame of A^{-}

2. From the viewpoint of Ā^{0}

2.1 Praise of Ā^{+}

2.2 Blame of Ā^{-}

3. Complementarity between A^{+} and
Ā^{+}

Consider then, to take an example the following true simple thesis:

(16) Success was always a child of audacity.
(Prosper Crebillon, *Catilina*)

whose structure is z
^{+}(*boldness*^{+}) i.e. z
^{+}(A^{+}) at the level of the matrix {*boldness*^{+},
*propensity to take risk*^{0}, *temerity*^{-},
*cautiousness*^{+}, *propensity to avoid risk*^{0},
*cowardice*^{-}}. It results then the following
matrix-based plan:

(17) 1. From the viewpoint of *risk taking*^{0}

1.1 The necessity of *boldness*^{+}

1.2 The dangers of *temerity*^{-}

2. From the viewpoint of *risk avoidance*^{0}

2.1 The advantages of the *cautiousness*^{+}

2.2 The risk of *cowardice*^{-}

3. The necessary complementarity between *boldness*^{+}
and *cautiousness*^{+}

Consider also the following false simple thesis:

(12) If reason dominated on the earth, nothing would happen there. (Bernard Fontenelle)

whose structure is z
^{-}(*reason*^{0}). The corresponding matrix is:
{*level-headedness*^{+}, *reason*^{0},
*lukewarmness*^{-}, *motivation*^{+},
*passion*^{0}, *fanaticism*^{-}}. And the
following matrix-based plan then ensues:

(18) Introduction: (i) structure of the thesis; (ii) truth value; (iii) matrix

1. From the viewpoint of *reason*^{0}

1.1 The pitfall of *lukewarmness*^{-}

1.2 The necessity of *level-headedness*^{+}

2. From the viewpoint of *passion*^{0}

2.1 The dangers of *fanaticism*^{-}

2.2 The necessity of *motivation*^{+}

3. The necessary complementarity between *level-headedness*^{+}
and *motivation*^{+}

Lastly, such a type of plan also proves to be adapted to a true 2-complex thesis such as the following:

(19) In the first place comes your profession, because doing just one thing well will procure a higher development for you than doing one hundred by halves. (Goethe)

This latter thesis can be analyzed as a 2-complex
thesis whose structure is z ^{+}(*expertise*^{+})
Ù z
^{-}(*superficiality*^{-}) i.e. z
^{+}(A^{+}) Ù z
^{-}(Ā^{-})
at the level of the matrix: {*expertise*^{+},
*monodisciplinarity*^{0}, *compartmentalization*^{-},
*eclecticism*^{+}, *multi-disciplinarity*^{0},
*superficiality*^{-}}. And the following matrix-based
plan then ensues:

(20) 1. From the viewpoint of *monodisciplinarity*^{0}

1.1 The advantages of *expertise*^{+}

1.2 The risk of *compartmentalization*^{-}

2. From the viewpoint of *multi-disciplinarity*^{0}

2.1 The necessity of *eclecticism*^{+}

2.2 The dangers of *superficiality*^{-}

3. The necessary complementarity between *expertise*^{+}
and *eclecticism*^{+}

**Conclusion**

From the above developments, it should be noted that the matrix-based dialectic plan presents a number of advantages with regard to the classical dialectic plan. First, the dialectical approach which has just been described performs first an analysis of the structure of the thesis under consideration, which leads then to assign a truth value to it, on objective grounds.

Second, it appears that the matrix-based dialectic plan replaces the thesis or the main proposition in a context that comprises a greater number of concepts than the classical dialectic plan. In effect, the classical dialectic plan usually places the thesis in an environment comprising in general two, or even three concepts. By contrast, the matrix-based dialectic plan replaces the thesis in a context comprising six concepts which are related to this latter.

Third, one of the advantages of the matrix-based dialectic plan is that it also allows to take into account some concepts which are not lexicalized. In effect, a matrix of concepts describes six canonical concepts. But it is rare that the totality of these latter concepts are lexicalized. In effect, the most common situation is that only some concepts - in general two or three - among the six described by the corresponding matrix, are lexicalized. Here also, the advantage of the matrix-based dialectic plan is to allow to take into account exhaustively the six concepts of a same matrix and to incorporate them in the corresponding discussion.

It should also be noted that the step of the
antithesis at the level of the classical dialectic plan is replaced
here by the determination of the dual thesis, which presents an
identical structure to that of the initial thesis. The dual thesis,
which serves here as a basis for dialectical reasoning, presents by
its simple or well *n*-complex structure a more elaborated
nature than the traditional antithesis.

Lastly, it proves that the classical dialectic plan
allows to overcome an antinomy existing between two concepts, which
serve respectively as a support to the thesis and to the antithesis.
It consists most often of A^{+} and
Ā^{-}, of A^{0} and
Ā^{0}, or well of A^{-} and
Ā^{+}. Most of the time, it consists of a dual or
antinomical pair of concepts which present the property of being
lexicalized. Conversely, the matrix-based dialectic plan constitutes
the expression of a dialectical move of the thought which allows to
go beyond a threefold antinomy: the one existing at the same time
between A^{+} and Ā^{-},
A^{0} and Ā^{0},
and finally A^{-} and Ā^{+},
whether these concepts are lexicalized or not.

**References**

Franceschi,
Paul (2002). *Une classe de concepts*. Semiotica 139 (1-4),
211-226.

Hegel, Georg Wilhelm Friedrich (1812-1816). Wissenschaft der Logik. Science de la logique, trad. Bourgeois, Paris, Aubier Montaigne, 1972.

------------- (1817). Die Encyclopädie der philosophischen Wissenschaften im Grundrisse. Précis de l'encyclopédie des sciences philosophiques, trad. J. Gibelin. Vrin, Paris, 1978