Graphs for Concepts
Lutz Geldsetzer © 2013
1. Explanations of the graphs: 1. Letters represent the intensions of regular concepts. Circles represent the
concept-position in a pyramidal hierarchy of genus, species and individuals. This 3-level hierarchy may be
extended for subspecies. The pyramid visualizes the extensional domaines of every concept.
2. The pyramidal position expresses the definition of every concept: the left-side letter is the “generic” intension
of the (highest) genius or category (aristotelian “genus proximum”) in every included lower concept. The rightside letter is the aristotelian “differentia specifica”. Nota bene: axiomatic categories become definable by the
induced common intensions of their immediate subspecies. Undefined categories can’t be genuine concepts!
2. Two types of conceptual pyramids
Figures 1 and 2 Extensional-dihaeretic and extensional-multiple conceptual pyramid with
inscribed intensions
A
AB
ABD
A
AC
ABE
ACF
AB
ACG
ABF
AC
ABG
AD
ADH
AE
ADI
ADJ
3. Different ways for the induction (aristotelian eisagoge) or abstraction of higher level concepts from the same
individual instances
The various modes of induction explain the occurence of concurrencing theories about the same instances.
Figure 3
Four possibilities of complete induction from one individual instance
1.
2.
AB
4.
A
C
B
A
ABC
3.
BA
AC
CB
CBA
BAC
ACB
4. Construction of contradictory concepts by fusing dihairetic species into one concept which contain the specific
differences of both of the dihairetically distinguished species. Contradictory and contrary concepts are represented in
quadrangels
Figure 4
Establishing contradictory concepts by fusion of dihaeretic species
A
ABC
AB
Example:
AC
A = organism; AB = living organism; AC = dead organism; ABC = mortal organism
5. Construction of “dispositional” concepts as contrary concepts in a multiple-species pyramid
Figure 6
Establishing contrary or dispositional concepts from multiple species
Example:
A = State of Aggregation
AB = solid
AC = liquid
AD = gaseous
ABC = meltable
A
ABC
AB
AC
AD
6. Applications of conceptual pyramids as “hard-cores” of theories in the deductive mode
Figure 5
Deduction of possible worlds
AB
A
B
ABX
Example: A = Being;
possible world Z
ABY
B = Non-Being;
ABZ
AB = Possibility; ABX = possible world X; ABY = possible world Y; ABZ =
7. Pyramid of the proposition building logical connectors convaying thruth values
Reading of the lines in the pyramid in the indicated direction produces true assertions. Other readings produce
false assertions. Negation is described by the empty spaces between concepts in horizontal relations.
Figure 8:
Pyramid of the proposition forming connectors
General
implication ↕↔
If...then
Gen. Aristotelian
attribution
↕
Correlation
Belongs to; is
↔
Copula
Material Implic.↑
Arist. belonging
Formal implic. ↓
is, if...then
Belongs to
to,includ
If...then
Negation
not
Existence
−
there is
→←
8. Deduction of the expression building logical connectors which have no truth values
Figure 9:
Pyramid of both the proposition forming and the expression forming connectors
General
implication ↕↔
if...then
Gen. Aristotelian
attribution ↕
General disjunction
and/or
↕↔
Correlation
Adjunction
and
↕↔
Quantifier
one, some, all
none
↕
↔
if...then
belongs to; is
Alternative
Equivalence
either...or
equal with
↕↔
Copula
Material implic.↑
Arist. belonging
Formal implic. ↓
is, if...then
belongs, inclus.
═
Negation
not
Existence
−
there is
10. The three basic types of (aristotelian) syllogistic relations between concepts in parts of the pyramid
Figure 10:
The three schemata of Aristotelian syllogisms
1. ladder
2. split
3. summit
Example: A
A
A
AB
AC
ABD
ABD
= living being
AB
AB = animal
AC = plant
ABD = dog
AB
AC
→←
11. Deduction of the concepts of numbers from logical quantification-categories
Figure 11:
Deduktion of the concept of numbers by fusion of the general and the individual quantor
Logical
quantifier
Concept of
number
Natural number
Logical one
Element
Logical all
Totality
Letter-number
Variable
Commensurate
number, i. e.
Rational number
Incommensurate
number, i. e.
Complex number
Zero
Positive number
Integer and rational fractals
Non standard
number
Negative
Neg. number
number
Integer
and rational
Integer
and
fractals
Imaginary number
Radices from negative numbers
Irrational number
Periodical fractals
and irrat. radices
ra-
tional fractals
12. Representation of Hegels “Phaenomenology of Mind” in the pyramid showing Hegel’s “dialectical” concepts
in the quadrangles and regular concepts in the circles. Readings following the connection rules delivers the true
und the contradictory assertions in Hegel’s theory
A
Erinnerung
absoluter Geist
Vern. Wissen
nmber
AB
Selbstbewußt
sein,Person
ABD
unglückl. Bewußtsein
ABDF
Totes Äußeres, Sinne
ABDFHI
Wahrnehmung
ABDFH
Täuschung
Allgemeines
Dieses sagen
ABDFHJ
Ding, Objekt
ABDFHJK
Erfahrung
ABDFG
Begreifen
Begriff
ABDFI
Einzelnes
Meinen
ABDE
Handeln
Begierde
ABDG
Inneres Leben
Verstand
ABDGL
Kraft
ABC
Schaffen
Taten
nmber
AC
Institutionen
Normen
ABE
freies HerrenBewußtsein
ACP
Sitte
ABEN
"stoisches"
Bewußtsein
ABDGLM
Erklärung
ACQ
Recht
ABEO
"sekptisches"
Bewußtsein
ABDDGM
Gesetz
ABDFHK
Ich, Subjekt
Hegel begins with the induction of “Erfahrung” (experience) by fusing the dihäretic individual instances of object and
subject. Experience is a contradictory concept containing the specific differences of both of the instances. Only after this
he induces the common concept of “Allgemeines” (the general) as the regular next higher concept. Contraposing (by
negation) the concept of “Einzelnes” (the particular) he continues in the same way to construct all higher concepts of
praxis as contradictory fusions. Under way he deduces the “dialectical” concept of “Erklärung” (explanation) as
contradictory fusion of “Kraft” (force) and “Gesetz” (law). At last he induces the “principle” of “Er-Innerung” (reminding) as his highest metaphysical principle expressing the action of the mind.