=Paper=
{{Paper
|id=Vol-1132/paper4
|storemode=property
|title=Diagrammatic Autarchy: Linear diagrams in the 17th and 18th centuries
|pdfUrl=https://ceur-ws.org/Vol-1132/paper4.pdf
|volume=Vol-1132
}}
==Diagrammatic Autarchy: Linear diagrams in the 17th and 18th centuries==
https://ceur-ws.org/Vol-1132/paper4.pdf
Diagrammatic Autarchy.
Linear diagrams in the 17th and 18th centuries.
Francesco Bellucci (Author) Ahti-Veikko Pietarinen (Author)
Amirouche Moktefi (Author) Department of Philosophy, History, Culture and Art Studies
University of Helsinki &
Ragnar Nurkse School of Innovation and Governance
Ragnar Nurkse School of Innovation and Governance
Tallinn University of Technology
Tallinn University of Technology
Tallinn, Estonia
Tallinn, Estonia & Helsinki, Finland
bellucci.francesco@gmail.com
ahti-veikko.pietarinen@helsinki.fi
amirouche.moktefi@ttu.ee
Abstract—This paper explores the notion of autarchy of respect to its being more or less autarchic. To be precise,
diagrammatic notations for logic debated in the German- therefore, our reconstruction contributes not so much to the
speaking world of the 18th-century, especially as applied to linear history of logic diagrams, but to the history of the ideas about
diagrams invented by G. W. Leibniz and J. H. Lambert. logic diagrams, or to the history of the philosophy of
Keywords— linear diagrams; autarchy; Leibniz; Lambert;
diagrams.
Ploucquet
I. LEIBNIZ
INTRODUCTION
In some of his writings, Leibniz (1646-1716) claimed that
the aim of the characteristic is to find (adhibire) characters
In this paper we explore the notion of autarchy of such that all the consequences can be derived from them. Such
diagrammatic representation that was debated in the German- characters are “autarchic” (αυτάρκεις). Paraphrasing Heinrich
speaking world in the 18th century. What is diagrammatic Hertz’s famous maxim, Leibniz’s ideal of diagrammatic
autarchy? In one of his writings, Leibniz claimed that one of autarchy amounts to this, that the necessary logical
the aims of the characteristica universalis (his big project of a consequences of the diagram are always the diagram of the
general formal and deductive method for science) is to find natural necessary consequences of imagined object [5, p. 75].
“autarchic” (αυτάρκεις) characters: “One must know that In his mathematical and logical works, Leibniz worked out
characters are more perfect the more they are autarchic, in different examples of “autarchic” systems of symbols. For
such a way that all the consequences can be derived from example, the binary notation is said to be more autarchic than
them” [1, pp. 800-801]. We will use this Leibnizian term to the decimal in that in the binary “all that can be affirmed about
indicate an important property of some diagrammatic numbers can be demonstrated from their characters” [1, p.
representations, and we will try to show that much of the 800], which is not true for the decimal. Further, Leibniz
debate about diagrams and iconic representations in the 18th considered algebra as an imperfect instrument for treating
century, largely reported in [2], may be considered as a debate geometry; algebra is only the characteristic of indeterminate
about the notion of autarchy. Of course, there is much more in numbers or magnitudes (grandeurs), but does not express
that debate than the discussion of diagrams for syllogistic [3; places, angles and motion. A more perfect system of
4, ix]. However, we believe that the notion of autarchy is able geometrical notation (characteristica geometrica) is therefore
to capture an important aspect of that debate. We will focus on imaginable in which the simple enunciation of the problem is
the linear diagrams invented by Leibniz and Lambert and already its solution, or one in which the enunciation, the
discussed in the German logical panorama of that time. construction and the demonstration are one and the same thing
We do not attempt to answer the question whether or not [1, p. 910; 6, II, pp. 20-21, 228-229; 6, V, pp. 141ff).
the notion of autarchy might be re-phrased or explained in the The notion of autarchy also applies to logical notations. In
terms of some contemporary theory of diagrammatic his 1686 “Generales Inquisitiones de Analysi Notionum et
reasoning or read through a more sophisticated logical- Veritatum” [7, pp. 356-399] and in other writings of roughly
philosophical conception, although we will mention a couple the same period [7, pp. 206-210, 247-249, 292-321], Leibniz
of interesting parallels in the last section. Our principal aim proposes a system for representing propositions and
here is to understand what these thinkers thought about syllogisms by means of linear diagrams. Such diagrams, as
diagrammatic representations, and especially what their one of these writings says, are expressly intended as a
criteria were to believe that one system of diagrammatic “demonstration of the logical form” (de forma logicae
representation is better than, or preferable to, another with comprobatione per linearum ductus [7, p. 292].
23
In Leibniz’s linear system (Figures 1-4) the extension of Leibniz also proposes a version of these diagrams in which
concepts is represented by parallel straight lines, while the the part of the line which is relevant for the affirmation or
dotted vertical lines indicate the relation of inclusion or negation is doubled [7, pp. 311-312] (see Figure 5). This
exclusion among concepts: when the vertical lines cut off real method – that is, to double the part of the line which is
segments on each parallel, the proposition is affirmative, when affirmed or denied of the other – is important because it
they pass entirely outside of one or both the parallels the represents visually what Leibniz calls the distribution or non-
proposition is negative (cf. [8; 4, viii]). Leibniz claims that this distribution of the terms, that is their quantity. A term is
system is capable of showing which of the four propositional universal if its line is completely doubled; it is particular if its
forms are convertible and which are not. The diagrams of the line is only partially doubled. In the universal affirmative, for
universal negative and of the particular affirmative (Figures 2, example, the line of the subject is completely doubled, and so
3) are symmetrical, and therefore these propositions are the subject is universal, while in the particular the line of the
convertible (conversio simplex: “No B is C” is convertible into subject is only partly doubled, and so the subject is particular.
“No C is B”; the same applies to the particular: “Some B is C” In order to construct the diagram of the syllogism, Leibniz
is convertible into “Some C is B”); The diagrams of the draws the major premise and then, using the line of the middle
universal affirmative and of the particular negative (Figures 1, term already drawn, adds the minor premise. To obtain the
4) are not symmetrical, and therefore these propositions are conclusion, he draws two continuous vertical lines starting
not convertible. Of course the universal affirmative is from the double part of minor term towards the major term. If
convertible into a particular (per accidens: “All A are B”, then these continuous verticals cut off a real segment of the other
“Some B is A”). extremes, then the conclusion is affirmative. If they fall
It is important to note that, besides these linear diagrams, outside it, the conclusion is negative. For example in Barbara
Leibniz draws the correspondent circular diagrams in the way (Figure 6), the two continuous vertical lines from D fall
Euler would do later. To differentiate the circular diagram of entirely on B, and so the conclusion is affirmative. Further, all
the particular affirmative from that of the particular negative, D is taken into consideration – its line is completely doubled -
he uses letters (Figures 3, 4). In the circular diagrams the and so the conclusion is universal: “All D are B”. In
letters are placed in such a way as to indicate the nature of the Camestres (Figure 7) the two continuous lines are again drawn
proposition, whether affirmative or negative. In the linear from D to B, but they fall outside B, and therefore the
diagrams this expedient is not necessary, for the figure shows conclusion is negative. Further, all D is again taken into
by itself whether the particular proposition is affirmative or consideration, so the conclusion is universal: “No D is B”.
negative. Therefore, Leibniz believes, the linear are more
autarchic than the circular diagrams, for in the latter the figure
is not self-sufficient in determining whether the proposition is
affirmative or not: we must use a conventional or symbolical
devise in order to differentiate the two forms. Fig. 5. [7], p. 311-12
Fig. 1. [7], p. 292
Fig. 6. [7], p. 294
Fig. 2. [7], p. 293
Fig. 3. [7], p. 293 Fig. 7. [7], p. 295
Fig. 4. [7], p. 293
24
II. LAMBERT
Johann H. Lambert (1728-1777) calls “scientific” those Lambert claims that his Zeichnungsart not only shows
signs that are so constructed as to serve as perfect substitutes what relations obtain among concepts, but also shows what
for their objects. The more a system of signs can be made other relations may be deduced therefrom by the mere
object of reasoning according to simple rules, the more observation of the figures [8, I, §§ 191, 194]. Like Leibniz,
scientific it will be: “The signs of concepts and things are Lambert claims that his diagrams are capable of distinguishing
scientific in the stricter sense if they not only represent in the different propositional forms one from another; further,
general those concepts and things, but also indicate each of the four propositional forms has its own diagram,
relationships such that the theory of the object and the theory which is different from the others, so that there is no risk that
of its signs can be interchanged” [9, III, § 23]. different propositions might be represented by the same
Lambert’s Zeichnungsart, his system of linear diagrams, diagram, or that different diagrams represent the same
is quite similar to Leibniz’s. It is not clear whether Lambert proposition.
knew Leibniz’s diagrams, as most of the relevant texts have But further, Lambert claims that these diagrams always
been published later. In his Neues Organon [9, I, §§ 173-194], and necessarily indicate what parts of a concept are
Lambert represents concepts by means of lines, propositions undetermined, that is, they express our imperfect knowledge
as relations between two lines, and syllogisms as relations about a concept’s extension, therefore showing whether or not
between three lines. Lines may be either closed or open a proposition is convertible. For example, take the universal
(having dotted extremities), depending on the certainty or affirmative (Figure 8); we may convert it per accidens into the
uncertainty of the distribution of the terms represented by particular affirmative “Some B are A” simply by reading the
them (i.e. depending on the quantity of these terms). The four diagram top-down instead of bottom-up. The dotted part of the
traditional propositional forms are represented as in Figures 8- line indicates that, when converted, the corresponding term is
11. The use of uppercase and lowercase letters at the to be taken particularly (Some B). Likewise, the universal
extremities of the continuous segments is of no use at all, and negative (see Figure 9) may be simply converted (conversio
may be easily ignored. simplex) by reading it from the right to the left (“No B is A”).
This means that the same diagram can express different
propositions depending on the way we read it.
It is not clear how things stand with the particular
affirmative (Figure 10). Reading it top-down, as we do for the
universal affirmative, would not give us the converted
Fig 8. [9], I, § 181 proposition. We would like to read it top-down as “All B are
some A”, which introduces the quantification of the predicate,
but Lambert would not have been happy with that (he
famously opposed the quantification of the predicate
maintained by G. Ploucquet).
Fig 9. [9], I, § 183 If we compare Lambert’s diagrams to Leibniz’s, we see
that while Leibniz’s diagram for the particular affirmative is
symmetrical, thus suggesting simple conversion (Figure 3)
Lambert’s diagram, on the contrary, is not symmetrical, and
does not show whether and how the proposition can be
converted (Figure 10).
It has further to be noted that Lambert proposes different
Fig 10. [9], I, § 184 ways to draw these linear diagrams. Figure 12 represents an
alternative way of diagramming the particular negative Some
M are not C. Lambert marks by an asterisk the limit of the
extension, that is, the point beyond which the extension of a
term cannot go without invalidating the proposition. For
example, if we allow the dotted line of C to surpass the
asterisk, the line C would extend to cover completely the line
Fig. 11. [9], I, § 184 M, and the proposition “Some M are not C” would be false [2,
p. 218]
Lambert however insisted on a point that was of crucial
importance for him. The idea is that those premises from
which something follows should be capable of being
diagrammed, while those from which nothing follows should
not: “I begin by drawing the middle term, and then I draw
Fig. 12. [2], p. 218 either of the other two terms. If the third is capable of being
25
drawn, then the representation gives me anything that follows
immediately from the premises. If the third term cannot be
drawn, then nothing follows therefrom” [2, p. 152].
Let us take the two negative premises “No M is P” and
“No S is M”. I begin by drawing the middle term M (Figure
Fig. 13. Middle term “M”
13). Then I draw the major term P (Figure 14) so as to place it
completely outside M (for no M is P). Now I should represent
that “No S is M”. So I have to represent the third term, the
minor term S, so as to exclude it from M, too. There are at
least two geometrical possibilities here, for I can draw S either
below P or not (see Figure 15). Since I am not entitled to
choose between these possibilities, no conclusion follows
from these two premises.
If, on the other hand, one of the premises were either a
universal affirmative or a particular, things would be different.
For example, if the second premise were “All S are M”, it
could well be represented, for there is just one possible place Fig. 14. “No M is P”
to draw the line of S (see Figure 16). This is the valid
syllogistic form of first figure Celarent.
Figures 17 and 18 represent the diagrams of the first and
second figure according to Lambert [9, I, § 219]. In his
Zeichnungsart, Lambert argues, everything that is relevant for
the syllogistic calculation is represented; once a couple of
propositions is diagrammed, one immediately sees whether
something follows from it or not, and this is all that is required
to have a scientific or autarchic system of notation.
While Leibniz’s linear diagrams were not known in his
times, Lambert’s method was much debated in the scientific Fig. 15. Two geometrical possibilities for “No S is M”
community of 18th century German-speaking world. Georg
Jonathan Holland (1742–1784), in the Anhang to his
Abhandlung über die Mathematik [2, pp. 95-108], compared
Lambert’s logical calculus to that of his Tübingen professor
Gottfried Ploucquet. Holland claims that Lambert’s system of
linear diagrams is not a real characteristic, as it is possible in it
to represent premises from which false conclusions follow.
Let us take the premises: “All P is O”, and “No A is P”.
If we represent them as Holland does in the Anhang (see
Figure 19), then the conclusion seems to follow that “No O is
A”, which is a false conclusion. Lambert’s method of
diagrams seems therefore imperfect, for in it it is possible to
infer a false conclusion. But Lambert replies that Holland’s
diagram for this syllogism is wrong: “The extension of the line Fig. 16. One geometrical possibility for “All S are M”
O is greater than P, but indeterminately greater. And therefore
it must in this case be dotted” [2, p. 151]. When the
proposition “All P is O” is represented as in Figure 20, we see
that it is not the entire line O which is excluded from the line
A, but only the continuous part of it that coincides with P. So
we must conclude not that “No O is A”, but only that “Some
O are not A”, which is the right conclusion and which gives us
the valid syllogistic form Fesapo of the fourth figure. This
indicates why Lambert attaches so much importance to the
expression of the quantification of concepts by means of
dotted lines. Without this graphic device, the system may yield
false conclusions.
26
III. PLOUCQUET
Mention has to be made in this context of Gottfried
Ploucquet (1716-1790), professor of philosophy at Tübingen
and famous for having introduced in logic the quantification of
the predicate. Although he was somehow skeptical about the
idea of a universal characteristic both in the sense of a
universal calculus and in the sense of a universal language, he
nonetheless invented different systems of logical
representation, including graphical and algebraical. His
diagrams for syllogism are quite similar to Euler’s circles [2,
pp. 6-8, 157-158] (Figures 21, 22).
However, Ploucquet’s main interests lie in symbolic
notations. His fundamental idea is that every affirmative
Fig. 17. [9], I, § 219
proposition states an identity between subject and predicate:
“The judgment is not the cognition of two things, but of just
one; and the affirmative proposition reflects this by expressing
one thing by different signs” [2, p. 52]. The theory of the
identity of subject and predicate in an affirmative proposition
is the ground of Ploucquet’s much discussed “quantification of
the predicate”: not only the subject but also the predicate of a
categorical form is qualified by means of a quantifier
expression ‘omne’ (all) or ‘quoddam’ (some). If I affirm, “All
men are animal”, animal is here taken particularly, that is, as
“some animal”, so that the proposition actually affirms that
“all men are some animal”. As a consequence, Ploucquet
claims that each categorical form can be converted: since
conversion consists in nothing else but exchanging subject and
predicate, each categorical form is convertible, provided that
the quantity of the predicate is made explicit by adding the
“quantifiers”. In his symbolic notation, he uses uppercase
letters for universally quantified terms, lowercase letters for
Fig. 18. [9], I, § 219
particularly quantified terms, the symbol > for negation, and
juxtaposition for affirmation (see Figure 23).
In the debate with Lambert, Ploucquet moves several
objections to Lambert’s system of diagrams. First, he claims
that Lambert’s system has no specific sign to show whether a
term is universal or particular (as he does in his own symbolic
notation) [2, pp. 166-167]. Secondly, the diagram in Figure 24
can be read either as “All A are B” or as “Some B are A”,
Fig. 19. [2] p. 104
which latter is the former proposition converted per accidens.
Since Ploucquet does not accept the traditional version of the
doctrine of conversion, these are two different propositions for
him, and each has to have its own diagram. This can be done,
he claims, if we mark graphically whether a term is universal
or particular.
Thirdly, Ploucquet observes that the representation of our
imperfect knowledge about a concept’s extension by means of
Fig. 20. [2] p. 106
dots is of no use at all [2, p. 170]. Again, in the diagram in
Figure 24, the dotted part represents that we do not know
whether there are B that are not A, and that the only relevant
part of the assertion is that all A are B. Since Ploucquet
believes that in this proposition subject and predicate should
be identical, he needs not employ the dots to represent our
imperfect knowledge about B. For him, there is no such a
thing as imperfect knowledge about a concept’s extension.
27
Lambert’s reply is that that which Ploucquet considers as a
fault of the linear system - representing undetermined
concepts by means of open or dotted lines – is on the contrary
a virtue of it. For if we agree that “All A are B” may cover
both the case in which B is greater than A (B>A) and the case
in which B is identical with A (B = A), then the use of the
dotted lines is of the utmost importance: we are obliged to
represent both the determined and the undetermined part of a
concept’s extension. By the device of the dotted lines, this
indetermination is appropriately “made intuitive” (diese
Unbestimmtheiten recht augenscheinlich zu machen) [2, p.
215].
Ploucquet in its turn proposes an amendment of Lambert’s
linear diagrams (Figures 25-27). In these diagrams any Fig. 23. The 4 standard propositional forms according to Ploucquet
concept is expressed by a straight line as in Lambert’s system,
but the quantity of the terms is not expressed by continuous or
dotted lines, but by uppercase letters for universal concepts
and lowercase letters for particular concepts [2, pp.179-181].
As one can easily perceive, Ploucquet’s system is a in fact a
sort of mixture of algebraical notation (the representation of
universal/particular terms with uppercase and lowercase
letters) and geometrical notation (the lines one above the other Fig. 24. [9], I, § 181
to indicate affirmation of identity, and one external and
separated from another to indicate negation). In other words,
this system is neither completely diagrammatic, nor
completely symbolic, but uses both algebraical and
geometrical structures in order to express propositions and
syllogisms.
Fig. 25. [2] p. 179
Fig. 21. [2] p. 6.
Fig. 26. [2] p. 179
Fig. 22. [2] p. 258 Fig. 27. [2] p. 180
28
IV. CONCLUSION rules is needed in order to draw the conclusion desired. All
that which is necessary to reasoning must be expressed
In his correspondence with Holland, Lambert states that in diagrammatically in such a way as to enable the diagram of
Ploucquet’s symbolism it is on the basis of “external” the premises to be, at once, also the diagram of the conclusion.
information (i.e., syllogistic rules) that it is found e.g. that In Lambert’s terms, a corollarial reasoning is one in which
from a given formula nothing follows. It would be better, either the following or the not-following of a conclusion is
according to Lambert, if this “not following” could be shown shown by the diagram itself.
by the diagram itself [10, pp. 192-193]. Lambert believes he The second point worth mentioning concerns current
has provided a rule to detect invalid syllogistic forms simply diagram research. What we call “autarchy of diagrammatic
by the rules of construction of their diagram. As he declares: representations” seems to correspond to the notion of “free
“Ploucquet calculates, while I construct or draw” [2, p. 151]. ride”, or information which arises in a diagram as a by-product
A couple of points may here be mentioned which indicate of its syntax. Already Jon Barwise and John Etchemendy
directions for further research, both historical and theoretical. observed that “Diagrams are physical situations. They must
One century after the debate, the ideal of an autarchic system be, since we can see them. As such they obey their own set of
of signs is still at work in the philosophy of notation of constraints. [...] By choosing a representational scheme
Gottlob Frege (1848-1925). The aim of the Begriffsschrift appropriately, so that the constraints on the diagrams have a
(1879) is expressly that of preventing anything intuitive or good match with the constraints on the described situation, the
extra-logical from penetrating unnoticed in the chain of diagram can generate a lot of information that the user never
reasoning. Accordingly, all that is necessary to deduction has need infer. Rather, the user can simply read off facts from the
to be appropriately represented, so that the inferential chain is diagram as needed” [14, p. 23]. As explained by Atsushi
kept free of gaps, and at the same time anything without Shimojima, in any system of diagrams whatsoever there exists
significance for the inferential sequence has to be omitted a set of operational constraints which may or may not
[11]. intervene in the process of encoding and extracting
However, we believe that the closest explanation available information [15, p. 28]. Under certain conditions, some
of the notion of autarchy is the conception of corollarial operational constraints will give rise to a free ride: “a free ride
reasoning, which is due to Charles S. Peirce (1839-1914). is where a reasoner attains a semantically significant fact in a
Peirce distinguishes between two kinds of deductive diagram site, while the instructions of operations that the
reasoning, which he calls theorematic and corollarial: “every reasoner has followed do not entail the realization of it. Thus,
Deduction involves the observation of a Diagram (whether we can view the process as one in which the reasoner has
Optical, Tactical, or Acoustic) and having drawn the diagram attained the fact without taking any step specifically designed
(for I myself always work with Optical Diagrams) one finds for it” [15, p. 32]. Under different conditions, the operational
the conclusion to be represented by it. [...] My two genera of constraints will produce “overdetermined alternatives” [15, p.
Deductions are 1st those in which any Diagram of a state of 33], that is, will produce pieces of information which do not
things in which the premisses are true represents the follow from the diagram of the premises.
conclusion to be true and such reasoning I call Corollarial In contemporary terms, then, the debate on logic diagrams
because all the corollaries that different editors have added to pictured above may be taken as a debate on operational
Euclid’s elements are of this nature. 2nd Kind. To the Diagram constraints. When Leibniz claimed that the most perfect
of the truth of the Premisses something else has to be added, systems of representations are those that are autarchic he was
which is usually a mere May-be and then the conclusion maintaining that those systems of logical or mathematical
appears. I call this Theorematic because all the most important notation must be preferred in which the operational constraints
theorems are of this nature” [12, pp. 869-870]. In corollarial always give rise to free rides. In his system of linear diagrams,
reasoning, the diagram of the premises already represents the the drawing of the conclusion from the premises is always a
conclusion; in theorematic reasoning, by contrast, the diagram free ride because the conclusion is obtained directly from the
of the premises must be transformed and experimented upon diagram of the premises, without being necessary that any
– in geometry, for example, subsidiary lines or figures are specific step designed for it be taken.
drawn - in order for it to represent the conclusion [13, 2.267]. Likewise, Lambert’s idea that in an adequate system of
Against Kant, Peirce maintains that all deductive reasoning, representation those premises from which nothing follows
not just mathematics, is diagrammatic (constructive in Kant’s should not be capable of being diagrammed is captured by the
sense). We have to distinguish not between constructive and notion of overdetermined alternative. A system which, given
non-constructive forms of reasoning, but among different certain operational constraints, may produce overdetermined
forms of constructive thinking according to the complexity of alternatives is one in which, in Lambert’s terms, the following
the construction (i.e., diagrammatization) involved therein [13, or not-following of a proposition upon another is not a
3.560]. consequence of those constraints, but is the effect of
In Peirce’s terms, an autarchic system of diagrams is one “external” (non-diagrammatical) logical rules. In other words,
in which any reasoning that can be performed is of the if a system of diagrammatic representation is capable of
corollarial kind. In corollarial reasoning neither auxiliary producing overdetermined alternatives (as in the case of
constructions nor the appeal to “extra-diagrammatical” logical Bocardo in Figure 13), then that system is not autarchic in the
29
Leibnzian and Lambertian sense. On the contrary, if the [4] R. Blanché, La Logique et son histoire d’Aristote à Russell, Paris: Colin,
1970.
system is capable of producing all the consequences as free
rides, then that system is autarchic. An autarchic system of [5] E. Cassirer, Philosophie der symbolischen Formen. I. Die Sprache,
diagrammatic representation is therefore one in which a Berlin: B. Cassirer, 1923.
certain set of operational constraints always gives rise to free
[6] G. W. Leibniz, Leibnizens matematische Schriften, C. I. Gerhardt, Ed.,
rides (corollarial reasoning) and never to overdetermined Berlin-Halle,1849-1863; rist. Hildescheim-New York: Olms, 1962.
alternatives.
The picture is no doubt more complicated than that, and [7] G. W. Leibniz, Opuscules et Fragments Inédits de Leibniz, L. Couturat,
new problems may arise which might contribute drawing Ed., Paris: Alcan, 1903.
parallels between old and new problems in logic, and building
[8] M. Baron, A Note on the historical development of logic diagrams:
new bridges between the history of logic diagrams and current
Leibniz, Euler and Venn. The Mathematical Gazette, vol. 53, pp. 113-
trends in diagrams research in computing and cognitive 125, 1969.
sciences.
[9] J. H. Lambert. Neues Organon, Leipzig: Wendler, 1764.
ACKNOWLEDGMENT
[10] J. H. Lambert, Deutsche Gelehrte Briefswechsel, J. Bernoulli, Ed.,
Research supported by Estonian Research Council Project Dessau 1782.
PUT267, “Diagrammatic Mind: Logical and Communicative
[11] G. Frege, Begriffsschrift, eine der arithmetischen nachgebildete
Aspects of Iconicity,” Principal Investigator Prof. Ahti-Veikko Formelsprache des reinen Denkens, Halle: Nebert, 1879.
Pietarinen.
[12] C. S. Peirce, The New Elements of Mathematics by Charles S. Peirce, C.
Eisele, Ed., The Hague: Mouton, 1976.
REFERENCES
[13] C. S. Peirce, Collected Papers of Charles Sanders Peirce, voll. 1–8 C.
Hartshorns, P. Weiss and A.W. Burks, Cambridge (Mass): Belknap
[1] G. W. Leibniz, Sämtliche Schriſten und Briefe. Re. VI, Vol. 4. Berlin: Press, 1931-1958.
Akademie Verlag, 1923 –
[14] J. Barwise and J. Etchmendy, Visual information and valid reasoning, in
[2] G. Ploucquet, Sammlung der Schriften welche den logischen Calcul G. Allwein and J. Barwise, Eds., New York-Oxford: Oxford University
Herrn Prof. Ploucquets betreffen, mit neuen Zusätzen, A. F. Bök, Press, 1996, pp. 3-26.
Tübingen: Cotta, 1766.
[15] A. Shimojima, Operational constraints in diagrammatic reasoning, in
[3] M. Capozzi and G. Roncaglia, “Logic and Philosophy of Logic from Logical Reasoning with Diagrams, G. Allwein and J. Barwise, Eds. ,
Humanism to Kant,” in The Development of Modern Logic, L. New York-Oxford: Oxford University Press, 1996, pp. 27-4
Haaparanta, Ed. Oxford: Oxford University Press, 2009, pp. 78-158
30