Diagrams and
Diagrammatical Reasoning
– the Peircean Approach
Visual Learning Budapest
December 2012
Diagrams in Cognitive
Semiotics
• Charles Peirce (1839-1914), the
father of pragmatism and semiotics
• His mature philosophy around 1900
contains a doctrine of diagrams and
diagrammatical reasoning which
attracts increasing interest
• FS: Diagrammatology. An
Investigation on the Borderlines of
Phenomenology, Ontology, and
Semiotics, Dordrecht: Springer 2007
Diagrams in Peirce’s
taxonomy of signs
Icon
Index
Image Diagram Metaphor
Symbol
Icon subtypes defined by sign
use
Image
Diagram Metaphor
So, the same sign vehicle may in
some cases be used as different
subtypes
• Crescent shape as image of moon
• Crescent shape as diagram for
calculating ration between light and dark
parts
• Crescent shape as metaphor referring
to moon, referring to the bread type of
”croissant” - or crescent shape referring
to moon referring to islam
Non-trivial icon definition
• Icons generally defined by similarity
between sign and object
• ”For a great distinguishing property of the
icon is that by the direct observation of it
other truths concerning its object can be
discovered than those which suffice to
determine its construction” (Peirce 1895 CP 2.279)
• This icon property is especially developed
in diagrams
Algebra of Logic
• ” ... all deductive reason, even simple syllogism,
involves an element of observation: namely,
deduction consists in constructing an icon or
diagram the relations of whose parts shall
present a complete analogy with those of the
parts of the object of reasoning, of
experimenting upon this image in the
imagination, and of observing the result so as to
discover unnoticed and hidden relations among
the parts.” (Peirce 1885, W5 164; CP 3.363
• Developed from his work on the first linear
”formal” representation of modern logic ...
Typicality of diagrams
• Diagram token (on the page, screen,
imagination, etc.)
VERSUS
• Diagram type accessed through the token
• “One contemplates the Diagram, and one at
once prescinds from the accidental
characters that have no significance.” (PAP,
1906, NEM IV, 317)
• So, Diagrams offer the possibility of
observing general relations -> Peirce’s
philosophy of mathematics
Diagrams: skelettal icons
governed by symbols
• Diagrams are types: the generality of diagram sign
maps onto the generality of the object, indicated
by accompanying symbols
• Diagrams provide possibility for deductive thought
experiments:
”A diagram is an icon or schematic image embodying the
meaning of a general predicate: and from the
observation of this icon we are supposed to construct a
new general predicate.” (”New Elements”, NEM IV, 238)
So, deduction is defined by the manipulation of ideal
diagram models
• Here, diagrams form a special case of a
more general tendency, often
overlooked:
• Text and pictures only rarely appear
isolated – in most cases they appear
together, referring to the same object
and supporting the same inference
process regarding that object
• Text and pictures may be isolated, of
course, for special purposes
Fur
Diagram of the diagram
reasoning process
Cross-links in Peirce
• Continuity (of diagram sheet)
• Realism of possibilities (in diagram
manipulation as in reality)
• Diagrams in Abduction-DeductionInduction cycle
• Pragmatic maxim (thought as
experiment)
• ”Symbols grow” ... Peirce as
Enlightenment thinker
Diagrams in reasoning:
Abduction - deduction - induction
• 1. Abduction
Propose an explanatory diagram on the base of
certain facts which are true if the diagram is true
• 2. Deduction
Experiment with the diagram charting some
further necessary implications
• 3. Induction
Test the experiment results against facts support for 1. or revision of 1. -> Recursion
Husserl parallels
• Central notions addressing access and
construction of theoretical objects are
cognate to those around diagrams in
Peirce:
• Categorical intuition (Kategoriale
Anschauung - neutral as to perception/
imagination) - LU, 1900-01
• Eidetic variation - Wesenserschauung,
Ideenschau - E&U, 1938
• Problem: inaccessability of ”inexact
essences”
Venn diagrams
”Existential Graphs” - Peirce
Diagrams as tools for
Gedankenexperimente
• Alpha Graph proof of Modus Ponens
1) Deiteration 2) Removal of ”ring”
The diagram shows the first so-called Gamma Glimpse recorded by NASA’s SWIFTprogram Dec.17th 2004 - the largest energy bursts in the universe next to Big Bang.
The Island of Robinson Crusoe (Daniel Defoe 1719)
London Underground Map
y = x2
• Different
diagrams
referring to
same
object
Extension of diagram category
•
•
•
•
Diagrams proper
Logic
Mathematics
Grammar ...
• Diagrams are all formalisms which allow
for deduction
• So, Diagrams are not confined to visual,
graphical representations
Synthetize facts - idealize
Diagrams in police investigation: Jørn “Gamle” Holm
Diagram reasoning in comics
• The
representation
of events in
Carl Barks:
select
catastrophic
points princ. –
Østergaard &
Stjernfelt 2012
Varieties of diagrammatical
reasoning
• Corollarial vs. Theorematic Reasoning
(terms stemming from Euclid)
• Peirce’s ”first real discovery” (Carnegie
application 1902)
• Corollarial Reasoning fleshes out what
already lay in the premisses (conforming
to Kant’s idea that logic is tautological)
• Theorematical Reasoning requires
something further – a creative step
requiring further description
Corollarial vs. Theorematical
ctd.
Perimeter of square
with side s
Angle sum of
triangle
- requires the addition of auxiliary
lines
Theorematic reasoning
• Requires ”something more”
• The addition of particular auxiliary entities
to the diagram – the selection of which
entities?
• A specific experiment with the diagram
• Seeing the problem from a specific new
perspective ...
Generality of
Corollarial vs. Theorematical
“Peirce’s brilliant insight is that this geometrical
distinction can be generalized to all deductive
reasoning.” (Hintikka 1983), thereby throwing
light upon the problem of logical incontinence or
omniscience. The amount of auxiliary variables to
be added in a problem is a “rough measure of its
intriviality” which explains a large amount of
cases where agents fail to realize the necessary
implications of their assumptions.
Hintikka “C.S. Peirce’s “First Real Discovery” and Its Contemporary
Relevance”, in E. Freeman (ed.) The Relevance of Charles Peirce,
1983
Hintikka on two approaches to
logic and language
Language as Lingua Universalis
(Frege, Russell, Wittgenstein, Quine Heidegger, structuralism, Derrida ...)
Language as Calculus Ratiocinator
(Boole, Peirce, Schröder, Hilbert, Gödel Husserl, Hintikka)
The former leads to linguistic imperialism and
relativism - the latter leads to semiotic
pluralism and realism ...
Cognitive Semiotics in Aarhus
• 2-years International Elite MA Program
• All courses in English
• http://www.hum.au.dk/semiotics/
Experiment with diagram
Eksperiment with diagram II
Experiment with diagram III
Experiment with diagram IV
Dale Myers http://www.jfkfiles.com/jfk/html/
Experiment with diagram V