Peirce and the logic of image

Abstract

Peirce divided hypoicons into images, diagrams, and metaphors. For diagrams, he developed a logical theory of graphs: many-dimensional linguistic expressions analyzing meaning by virtue of iconicity of logical form. He neglect-ed the logic of images as well as metaphors, however. Metaphors relate to non-standard meanings that combine complex diagrammatic representations. Images are elementary constituents of qualitative space. I will argue that the interpreta-tion of images corresponds to the interpretation of non-logical vocabularies. This raises the question of whether images are also linguistic, in other words whether the simple qualities they partake of are the simple qualities of some propositional content. I will argue that Peirce favored a picture theory of language that takes images to interpret elementary characters of objects that constitute propositions. He did not ascribe images with properties of propositions, as that would have rendered them non-hypoiconic signs. Peirce was a visual interpreter of language. This led him to a lifelong search for methods and systems that would assist him in doing " logical analysis " (CP 3.443, 1896, " The Regenerated Logic "). For him, logical analysis was a methodology that analyses meaning and rigorously captures formal structures of thought and rea-soning. He felt a weighty need for diagrammatizing and animating the inferential content of thought, and frequently complained of having a singular incapacity to think within the confines of the verbal or written, linear structure of language. " I do not think I ever reflect in words, " he writes in a 1909 manuscript. " I employ visual diagrams, firstly, because this way of thinking is my natural language of

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DOI ./sem--Semiotica ; : –261
Ahti-Veikko Pietarinen
Peirce and the logic of image*
Abstract: Peirce divided hypoicons into images, diagrams, and metaphors. For
diagrams, he developed a logical theory of graphs: many-dimensional linguistic
expressions analyzing meaning by virtue of iconicity of logical form. He neglect-
ed the logic of images as well as metaphors, however. Metaphors relate to non-
standard meanings that combine complex diagrammatic representations. Images
are elementary constituents of qualitative space. I will argue that the interpreta-
tion of images corresponds to the interpretation of non-logical vocabularies. This
raises the question of whether images are also linguistic, in other words whether
the simple qualities they partake of are the simple qualities of some propositional
content. I will argue that Peirce favored a picture theory of language that takes
images to interpret elementary characters of objects that constitute propositions.
He did not ascribe images with properties of propositions, as that would have
rendered them non-hypoiconic signs.
Keywords: Peirce; image; logic; diagrams; existential graphs; picture theory;
language.
Ahti-Veikko Pietarinen: University of Helsinki. E-mail: ahti-veikko.pietarinen@helsinki.
Peirce was a visual interpreter of language. This led him to a lifelong search for
methods and systems that would assist him in doing “logical analysis” (CP 3.443,
1896, “The Regenerated Logic”). For him, logical analysis was a methodology that
analyses meaning and rigorously captures formal structures of thought and rea-
soning. He felt a weighty need for diagrammatizing and animating the inferential
content of thought, and frequently complained of having a singular incapacity to
think within the connes of the verbal or written, linear structure of language. “I
do not think I ever reect in words,” he writes in a 1909 manuscript. “I employ
visual diagrams, rstly, because this way of thinking is my natural language of
* Presented at the meeting “Peirce and Image” held during the Semiotics Summer School in
Urbino, Italy, July 2006. My thanks to the organizers and commentators. Supported by the
University of Helsinki “Excellence in Research” Grant No. 2023031, “Peirce’s Pragmatistic
Philosophy and Its Applications,” 2006–2008, Principal Investigator A.-V. Pietarinen.
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252Ahti-Veikko Pietarinen
self-communion, and secondly, because I am convinced that it is the best system
for the purpose” (MS 619: 8, 1909, “Studies in Meaning. The Import of Thought:
An Essay in Two Chapters”).
The struggles Peirce experienced in attempting to unearth the logical under-
pinnings of natural language led him to develop quite unprecedented systems
of diagrammatic logics. They turned out to be of paramount importance not
onlyfor his own logical investigations but also for later generations of logicians
and cognitive and computer scientists. Especially worth closer study are Peirce’s
existential graphs. In regard to them, he stated that the visual representation
ofassertions by means of such graphs puts before us “a moving picture of the
action of the mind in thought” (MS 298: 1, 1905, “Phaneroscopy”; Pietarinen
2006a).
Diagrammatic signs constitute the second class of the trichotomy of iconic
signs. The rst is images and the third is metaphors. I will consider hypoicons
here, which Peirce took to be icons that are pure representamens. They are signs
capable of representation independently, without the involvement of other class-
es of signs such as indices or symbols. Hypoicons are independent of demonstra-
tive or conventional representation. He notes that the various “modes of repre-
sentation” in icons may well involve conventional considerations, but that “in
itself” icons are to be called hypoicons (CP 2.276, 1903, “A Syllabus of Certain
Topics of Logic”; EP 2:273).
Hypoicons can, according to Peirce, “be roughly divided according to the
mode of Firstness of which they partake.” There are three such modes. First, says
Peirce, “those which partake of simple qualities, or First Firstnesses, are images.”
Second, “those which represent the relations, mainly dyadic, or so regarded, of
the parts of one thing by analogous relations in their own parts, are diagrams.”
Third, “those which represent the representative character of a representamen by
representing a parallelism in something else, are metaphors” (CP 2.277; EP 2:273).
Just as with the other well-known trichotomic classications of signs, the rela-
tionship here is that of an inclusion: diagrams have images and metaphors dia-
grams as their constituents.
Concerning diagrams, and to be able to construct a logical theory of diagram-
matic icons, Peirce coined the trichotomy between graphs, logical graphs, and
existential graphs. A graph “is a supercial diagram” (CP 4.419, c.1903, “On Exis-
tential Graphs, Euler’s Diagrams, and Logical Algebra”) composed of four signs:
(i) the sheet of assertion upon which a graph is drawn, (ii) spots, which are bound-
ed regions of the sheet qualitatively distinct from other regions, (iii) lines of con-
nection (lines of identity) drawing links between spots, and (iv) enclosures or
cuts, which are operations that remove the content of an area from the sheet. A
logical graph “is a graph representing logical relations iconically,” which for
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Peirce and the logic of image253
Peirce was “to be an aid to logical analysis” (CP 4.420). Third, an existential graph
“is a logical graph governed by a system of representation” concerning “one rec-
ognized universe, real or ctive,” and graphs represent “some fact existing in that
universe” (CP 4.421). We can note here a smooth passage from syntax to seman-
tics, “a system of representation” that relies on his universes of discourse idea
from the antedating algebraic investigations of logic. He worked out the theory of
existential graphs in extraordinary proportions, coming up, among other things,
with sound and complete systems of propositional and rst-order predicate logic,
as well as with a number of systems of modal, quanticational modal, and
higher-order logics (Pietarinen 2006b).
Might anything comparable be attempted for the other two classes of hypoi-
cons, namely, images and metaphors? I will discard the question of metaphors,
which I have discussed in Pietarinen (2008) and which concerns non-standard
meanings of language arising from non-standard use of diagrams with modali-
ties. I will instead focus on the role of images in Peirce’s logical theory of graphs.
What is there to be found in Peirce’s conception of images from a logical point of
view? The following couple of remarks are meant to clarify the philosophical
grounds for the future study of the logic of images.
According to Peirce, as noted, images are “First Firstnesses,” hypoicons
“which partake of simple qualities.” Simple qualities are described regardless of
anything else, independently of other signs, and so have a certain immediacy
that objects might lack, such as sense and feeling. They are comprehended di-
rectly or immediately, without mediation. They may be “tones of consciousness,”
as Peirce once put it (CP 7.530, “Consciousness,” undated). For those who fancy
Peirce’s semeiotic lingo, they are evoked by “iconic sumisigns” (CP 2.317, 1903,
“Syllabus”). I take iconic sumisigns to correspond to predicate terms in the sym-
bolic mode of expression; Peirce sometimes terms them words or rhemas. They
are the rstness of symbols. The other two classes of symbols are propositions or
sentences and arguments or text (CP 2.369, 1901, “Propositions”).
The interplay between symbols and hypoicons is thus particularly impor-
tant.Peirce took it that to interpret images, the use of symbols is indeed neces-
sary(CP 4.479, c.1903, “On Existential Graphs”). But that runs the risk of images
losing their presumed character of being hypoicons. And that, I shall argue, in-
deed happens. According to Peirce, one of the dening characters of symbols
is that they grow and evolve: “every symbol is a living thing” (CP 2.222, 1903,
“The Ethics of Terminology”). But they also possess a certain original, “core”
or“stable” meaning that Peirce took to be iconic in its essential form. He writes
that “every symbol is, in its origin, either an image of the idea signied, or a
reminiscence of some individual occurrence, person or thing, connected with
its meaning, or is a metaphor” (CP 2.222). What Peirce is asserting here is a
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254Ahti-Veikko Pietarinen
trichotomy of hypoicons noticeably similar to that of how he elsewhere (and
alsoin 1903) had characterized them but without using anything like the term
“hyposymbol.”
A comment is needed regarding the second class, namely, a symbol being
a reminiscence of “some individual occurrence [that is] connected with its
meaning.” I suggest that there is a logical correlate to this in what existential
graphs are taken to assert, namely, the existence of some thing or entity, since
existence is for Peirce nothing more than an occurrence in the universe of dis-
course of some fact, event, object, collection or logical possibility (CP 1.214,
c.1902, “A Detailed Classication of the Sciences”; CP 1.358, c.1890, “A Guess at a
Riddle”). The existence of something of which a meaningful assertion can be
made is what logical diagrams are intended to capture. Hence, individual occur-
rences are diagrams. What Peirce eectively claims is that the “core meanings” or
“the origins of symbols” are actually constituted by hypoicons of these three
kinds.
But the central task at hand here is not to convince ourselves once again of
the well-argued point that symbols involve iconicity but to address the question:
How do symbols interpret images? What are the processes, mechanisms, and
practices that eectuate such interpretations? Here is the crucial passage from
“On Existential Graphs.” According to Peirce, the “step of thought, which consists
in interpreting an image by a symbol, is one of which logic neither need nor can
give any account, since it is subconscious, uncontrollable, and not subject to
criticism.” He does not greatly elaborate on this point, though he goes on to
articulate his insight a little further by stating that “whatever account there is
tobe given of it is the psychologist’s aair.” He does admit that “it is evident that
the image must be connected in some way with a symbol if any proposition is to
be true of it” (CP 4.479), but this does not change the fact that the study of these
connections is not at bottom a logician’s concern.
Any conventional interpretation of images is for Peirce uncontrollable and
subconscious, in a word, a singular, process. Such a process is subject to psycho-
logical, not logical, laws. Unlike the intellectual signs in his general theory of
meaning, or pragmaticism, symbol-image interpretations are not eectuated
byself-controlled habits of acting in a certain way whenever a certain kind of
situation is confronted. Hence such interpretations cannot be general and univer-
sal in the same way as interpretations of logical and diagrammatic signs are, not
even if logical signs were symbolic or heterogeneous, namely, combinations of
symbols and diagrams. In this fundamental sense, then, how symbol-image links
are established is unrelated to Peirce’s anti-psychologistic conception of the
meaning and interpretation of intellectual signs, such as thoughts, purports, and
generalities.
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Peirce and the logic of image255
However, just as in the interpretation of symbols, the symbol-image pro-
cesses do rely on some essential linguistic features. Images serve similar pur-
poses as symbols do even though their outward appearance is quite dierent
from natural languages. Moreover, despite being uncontrollable, such interpreta-
tions have a signicant role to play in our logical theories. For symbolic interpre-
tations of images are, I contend, closely connected with various ways in which we
go about interpreting and assigning meanings to non-logical vocabularies of our
logical languages.
That last sentence needs a detailed justication. Let us recall the key ele-
ments of Peirce’s diagrammatic logics. First, the sheet of assertion is the surface
onto which diagrammatic assertions are drawn. Peirce coined the “Phemic Sheet”
to be that spatial entity that “iconizes the Universe of Discourse” (CP 4.553n2,
1906, “The Bed-Rock beneath Pragmaticism”). The sheet is the pure representa-
men of what is in the “eld of attention” of the discourse participants, the Utterer
and the Interpreter, who undertake to draw and interpret the graphs (Hilpinen
1982, Pietarinen 2006a). In other words, “in representing the eld of attention,”
the phemic sheet “represents the general object of that attention, the Universe of
Discourse” (CP 4.561n1, 1906).
At the outset, the sheet in question is not empty. For example, if nothing is
scribed on the phemic sheet, a tautology is asserted by the empty graph, which is
the sheet itself. If I have not asserted anything, I have asserted all truths and any
truth. The sheet articulates both the boundary conditions and a readily interpret-
ed background theory relative to which the discourse in question is understood to
run. It expresses the “natural history of logic.” Peirce’s idea is that this is pre-
cisely how something becomes a “system” with explanatory value serving some
specic purpose: through depicting some isolated and regimented parts of nature
and then “considering how the diagram is to be connected with nature” (CP 3.423,
1892, “The Critic of Arguments”).
This idea is very much in line with what Alfred Tarski, Rudolf Carnap, and
others came to think of as the preferred way that logical languages ought to be
setup. Later, model theory was born in which models are xed and the conditions
in which propositions hold in such xed and isolated models are studied. Exis-
tential graphs are perhaps the rst instance in logic where the possibilities of
what much later was baptized model theory were realized in depth (Pietarinen
2006a).
Second, Peirce notes that the phemic sheet “is an image of the universal eld
of interconnected Thought” (CP 4.553n2, 1906, “The Bed-Rock beneath Pragmati-
cism”). If the phemic sheet is an image of “interconnected thought,” any part of it
is an image of some thought. What are these images in the realm of the phemic
sheet? The answer is found in the second key component of Peirce’s theory of
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256Ahti-Veikko Pietarinen
graphs. It is the bounded, non-overlapping regions of a surface of the sheet or a
space qualitatively distinct from other bounded regions of the surface. Peirce
calls these regions spots. According to him, spots are graphs, “any replica of
which occupies a simple bounded portion of a surface, which portion has quali-
ties distinguishing it from the replica of any other spot” (CP 4.416, 1903, “Existen-
tial Graphs”). Thus they have the character of possessing those “simple qualities”
which Peirce attributed to images.
The bounded regions of spots should not be confounded with closed areas
marked by cuts denoting negations of a graph. Spots are the iconic counterparts
of what in symbolic languages are expressed by predicate terms. Peirce empha-
sizes that “spots (or their equivalents)” have “various visible qualities (as colors,
etc.)” (MS 491: 4, c.1903, “Logical Tracts”). The use of certain visible qualities
such as colors refers among other things to dierent modalities in Peirce’s “tinc-
tured” gamma graphs.
Moreover, spots, together with the topological operations of juxtaposition
(conjunction) and the cut (negation), dene the language of the alpha part of
existential graphs. That language is an iconic counterpart to the symbolic lan-
guage of propositional logic. We need not delve into the details of how these
graphs are constructed or how some more expressive graphs might be construct-
ed by extensions of these ideas; what is essential is that, in Peirce’s terms, “upon
the boundary of the surface occupied by the spot are certain points, called the
hooks of the spot, to each of which, if permitted, one extremity of one line of iden-
tity can be attached” (CP 4.416, 1903, “Pure Mathematical Denition of Existential
Graphs, Regardless of Their Interpretation”). He also writes that, “when all the
hooks have received such attachments, the spot with these attachments becomes
a graph signifying a proposition” (MS 491). Such additions give rise to the dia-
grammatic counterpart to fragments of rst-order predicate logic with identity.
A couple of philosophically and logically fundamental repercussions con-
cerning the ways in which Peirce sets up his system of logical diagrams are im-
minent. He intended the phemic sheet to be the iconic counterpart to what logi-
cians call an interpreted language. The phemic sheet consists of a potentially
uncountable number of images, the spots, which are isolated regions of space.
Out of the constituents of such interpreted languages, complex assertions subject
to semantic interpretation are then constructed by applying dierent logical con-
stants according to the given permissions and conventions. In diagrams, such
conventions are topological and they follow from the properties of the manifolds
upon which graphs are scribed. In this manner conventions, too, can preserve the
iconicity of the logical form as far as possible.
The point about non-logical vocabularies is signicant for several rea-
sons.First, what my interpretation of Peirce’s remarks implies is that how im-
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Peirce and the logic of image257
agesare conceived through symbols is closely correlated with the processes of
how non-logical vocabularies are interpreted in logic. But unlike interpretations
of symbols and intellectual signs, these processes are uncontrollable and singu-
lar, and do not go by way of self-controlled habits. Instead, they go by way of what
might be called physiognomic processes, such as judging character by appear-
ance. An example from Peirce himself is “that a large and prominent nose is
associated with push and energy” (CP 7.256, 1901, “Notes on Science”). Such
processes determine what “simple qualities” the images contained in the phe-
mic sheet partake of what he termed the “universal eld of interconnected
Thought.”
Second, Peirce rejected such physiognomic processes (which may include
conceptions, fears, hopes, desires, expectations, and so on) as having any role to
play in general modes of action and behavior constitutive of the meaning of intel-
lectual signs. And that general mode of constitution is what his philosophy of
pragmaticism is all about (Pietarinen and Snellman 2006). Thus, he was le with
non-physiognomic, general processes, which from the point of view of logical
theories are the processes connected with constructing symbolic interpretations
of complex diagrams and complex logical graphs.
Third, a symbolic interpretation is a reminiscence of “some individual occur-
rence” (CP 2.222), in other words, of interpreted diagrams on the phemic sheet
that are composites of images and other iconic signs. Such processes are logical
and semantic, and are constituted by the activities of the Utterer and the Inter-
preter, who act according to the given spatial and inductive structure of the dia-
gram. (This is the so-called “endoporeutic” interpretation involving numerous
pragmatic factors, see Pietarinen 2006a: ch. 6.) The activities are general and
guided by stable, self-controlled tendencies in choosing right subgraphs to pro-
ceed and in nding right objects from the universe of discourse to be the values
for the occurrences and identities. Essentially, these tendencies are functions
from possible situations to actions. In the contemporary parlance of game theory,
they are the strategy proles of the players playing the game of interpretation.
Peirce’s pragmaticism is indeed closely connected with the contemporary theory
of games. It provides both the philosophical basis as well as the logic for the study
of the meaning of intellectual signs. There is no room for psychology or uncon-
scious elements of thought in that study.
Fourth, images are isolated spots that are not connected to occurrences be-
fore the appearance of identity lines that could be attached to the hooks of the
spots. The diagrammatic counterpart of images is spots with empty hooks, with-
out anything that occupies those hooks, without anything could make them inter-
pretable. The symbolic counterpart of such an unoccupied spot is a predicate
term or an unsaturated rhema that has some arity but no variables in its argument
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258Ahti-Veikko Pietarinen
places. However, symbols do not capture the essence of what it is to “see images”
that are devoid of propositional content. Witness the following passage:
For example, you look at something and say, “It is red.” Well, I ask you what justication
you have for such a judgment. You reply, “I saw it was red.” Not at all. You saw nothing in
the least like that. You saw an image. There was no subject or predicate in it. It was just one
unseparated image, not resembling a proposition in the smallest particular. It instigated
you to your judgment, owing to a possibility of thought; but it never told you so. Now in all
imagination and perception there is such an operation by which thought springs up; and its
only justication is that it subsequently turns out to be useful. (CP 1.538, 1903, “The First-
ness of Firstness, Secondness, and Thirdness”)
One can make a predication only aer attaching dots or lines to the hooks of spots
that hit upon some object in the universe of discourse. That operation is second-
ary to the one of seeing an image of a quality. Should the attachments be absent,
the assertion will be incomplete and lack a subject and predicate altogether.
Let me speculate that something like this happens in autism, a neurodevel-
opmental disorder in which patients tend to think entirely in terms of rstnesses
of hypoicons. Patients suering from autism have their minds devoid of the sec-
ondness of the relationships such as indexical signs that could cater icons with
concrete information to hook their inferential thoughts with reality. They see and
think in terms of iconic “pictures” (Grandin 1995), images in their phaneron that
nevertheless lack the semantic component of the universes of discourses that
could exhibit the representational patterns of those vital relationships.
Fih, the previous passage also contains a key to what Peirce is aer in stat-
ing that “any image is a ‘composite photograph’ of innumerable particulars”
(CP2.441, c.1893, “The Grammatical Theory of Judgment and Inference”). Peirce
frequently alluded to the notion of a percept as a point of comparison with im-
agesthat is ominously psychologistic (“res percepta,” CP 7.619, 1903, “Telepathy
and Perception”): A percept “is an image or moving picture or other exhibition”
and has an “uncontrollable” operation of “judging what it is that the person
perceives” following its formation (CP 5.115, 1903, “The Reality of Thirdness”).
Like percepts, images are not representations: they do not stand for or intend
anything.
We can contrast this h point with the idea of a spot in logical diagrams.
Spots are isolated regions of the phemic sheet that are supposed to have certain
distinguishing qualities. A property is a topological entity in the manifold of a
phemic sheet. But at the same time, spots are collections of all that the sheet rep-
resents at any particular location of the space singled out to compose the spot in
question. Like percepts, spots themselves do not represent or stand for anything.
It is the sheet upon which spots are drawn that represents those “innumerable”
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Peirce and the logic of image259
particular entities that there may be in the areas of the universe of discourse
marked out by spots.
I hope that the role and nature of images in Peirce’s diagrammatic logic has
thus been tolerably explained. Finally, I would like to note a couple of similarities
as well as dissimilarities of the iconic character of Peirce’s logic with picture theo-
ries of language, or what more accurately could be called Wittgenstein’s picture
theory of elementary propositions. Two main clauses from Wittgenstein’s Tracta-
tus are of particular interest here, namely, that “A logical picture of facts is a
thought” (Proposition 3) and that “A proposition is a truth-function of elementary
propositions” (Proposition 5). From the perspective of Peirce’s iconic logic, what
a picture of facts is, namely, what interpretation a given assertion yields as its
nal cause, is a “picture of the action of a mind in thought” (MS 298: 1). In actual
fact, a picture of a fact is a dynamic, moving picture of such mental action and
not a static snapshot or immutable image. Those pictures are mediated by the
phemic sheet, representing the evolution of thought via its simple constituents,
that is, via images. And images themselves are the constituents that correspond
to Tractarian elementary propositions. Thus the pictures of Wittgenstein’s ele-
mentary propositions are deeply connected with Peirce’s notion of spots in the
phemic sheet.
That propositions are truth functions of elementary propositions preserves
even greater pictorial character in Peirce’s theory than in Wittgenstein’s, since the
truth-functionally complete operations themselves are pictorial, that is, are topo-
logical and iconic. For instance, negation-as-an-incision and conjunction-as-
juxtaposition is precisely what Wittgenstein was lacking in his picture theory. He
considered negation, for instance, as a process of switching the polarity of a prop-
osition. This goes some way towards the Peircean idea of a cut or an incision
around those regions of the phemic sheet that need to be negated. But in Peirce’s
theory, all propositions, including any truth-functional composites of proposi-
tions, are at root iconic and thus pictorial.
To remark on some of the most notable dierences between the thinking of
these two logicians, the atomicity of elementary propositions is an example of an
assumption we do not nd in Peirce’s theory. The continuity of the phemic sheet
and the continuous connectivity between spot-images make any hard-and-fast
mutual independence of atomic assertions impossible. Recall also that Peirce
was motivated in his diagram logic by nding an iconic basis for all reasoning,
especially necessary (deductive) reasoning, much more than Wittgenstein was.
I have argued for the following points: (i) Images are components of a wide
conception of a non-symbolic language just as diagrams and metaphors are, (ii)
symbolic interpretations of images instantiate non-habitual “physiognomic” pro-
cesses, and (iii) the elementary constituents of logical diagrams are images
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260Ahti-Veikko Pietarinen
(spots) subject to such physiognomic interpretations closely connected with in-
terpretations of non-logical vocabularies of logical languages.
Moreover, these points entail a picture theory of language of a broadly Witt-
gensteinian stripe. But it is not a picture theory in Wittgenstein’s limited sense of
elementary pictures. Given the essential role of icons in Peirce’s mature logic, his
theory enjoys greater pictoriality for both elementary and complex propositions
(graphical assertions), while Wittgenstein’s propositions are expressed in the
idiom of symbolic logic.
Some outstanding matters inevitably remain. Due to its physiognomic char-
acter, precisely how to consider “images as a language of some sort” is bound to
be largely arbitrary. Any symbolic interpretation produces an uncountable list of
sentences of the form “An image a shows that p,” “An image a shows that q,” “An
image a shows that r,” and so on. What is more, images that necessitate captions
of this sort cease to be hypoicons. It is thus evident that we are still a far cry from
a theory of the logic of image quâ image, namely, a theory of what images say
themselves as images, what they sagen sich selbst.
References
Grandin, Temple. 1995. Thinking in pictures: My life with autism. New York: Vintage.
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Peirce and the logic of image261
Bionote
Ahti-Veikko Pietarinen (b. 1971) is full professor at the University of Helsinki
〈ahti-veikko.pietarinen@helsinki.〉. His research interests include logic, philos-
ophy, pragmatics, and semiotics. His publications include Game theory and lin-
guistic meaning (ed., 2007).
Authenticated | ahti-veikko.pietarinen@helsinki.fi author's copy
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