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To appear in Semiotica
PEIRCE AND THE LOGIC OF IMAGE
Ahti-Veikko Pietarinen
8 December 2006
Abstract
Peirce divided hypoicons into images, diagrams and metaphors. For diagrams, he developed a logical theory of graphs:
many-dimensional linguistic expressions analysing meaning by virtue of iconicity of logical form. He neglected 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
interpretation 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 favoured 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.
Key words: Peirce, image, logic, diagrams, existential graphs, picture theory, language.
Short Bio: Dr. Ahti-Veikko Pietarinen (b.1971), MPhil in Computer Science from the University of
Turku, Finland, in 1997, DPhil in Theoretical Philosophy from the University of Helsinki, Finland
in 2002. Currently Professor of Semiotics at the University of Helsinki, Department of Philosophy,
History, Culture and Art Studies, Pietarinen‟s research includes logic, semiotics, game theory,
Peirce and Wittgenstein. He has published on logic, semiotics and philosophy in scientific journals
and collections. His recent monograph is Signs of Logic: Peircean Themes on the Philosophy of
Language, Games, and Communication (Synthese Library, Springer, 2006).
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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 reasoning. He felt a weighty need for diagrammatising 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 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 underpinnings 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 first is images
and the third is metaphors. I will consider here hypoicons, which Peirce took to be icons that are
pure representamens. They are signs capable of representation independently, without the
involvement of other classes of signs such as indices or symbols. Hypoicons are independent of
demonstrative or conventional representation. He notes that the various “modes of representation”
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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 classifications of signs, the relationship here is that of an inclusion:
diagrams have images and metaphors diagrams as their constituents.
Concerning diagrams, and to be able to construct a logical theory of diagrammatic icons, Peirce
coined the trichotomy between graphs, logical graphs and existential graphs. A graph “is a
superficial diagram” (CP 4.419, c.1903, On Existential 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 bounded regions of the sheet qualitatively distinct from other regions, (iii) lines of
connection (lines of identity) drawing links between spots, and (iv) enclosures or cuts, which is an
operation that removes the content of an area from the sheet. A logical graph “is a graph
representing logical relations iconically”, which for 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 recognized universe, real or fictive”, and graphs represent “some fact existing in
that universe” (CP 4.421). We can note here a smooth passage from syntax to semantics, “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 first-order
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predicate logic, as well as with a number of systems of modal, quantificational modal, and higherorder logics (Pietarinen 2006b).
Might anything comparable be attempted for the other two classes of hypoicons, 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 modalities. 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 directly 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 symbolic mode of expression; Peirce sometimes terms them as
words or rhemas. They are the firstness 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 important. Peirce took it that to
interpret images, the use of symbols is indeed necessary (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, indeed happens. According to Peirce, one of the defining characters of symbols is that
they grow and evolve: “every symbol is a living thing” (CP 2.222, 1903, The Ethics of
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Terminology). But they also possess a certain original, „core‟ or „stable‟ meaning which Peirce took
to be iconic in its essential form. He writes that “every symbol is, in its origin, either an image of
the idea signified, 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 trichotomy of
hypoicons noticeably similar to that of how he elsewhere (and also in 1903) had characterised 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 discourse of some
fact, event, object, collection or logical possibility (CP 1.214, c.1902, A Detailed Classification 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 occurrences are diagrams. What Peirce effectively 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 effectuate 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 to articulate his insight a little further by stating that “whatever
account there is to be given of it is the psychologist‟s affair”. 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
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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 psychological, not logical, laws. Unlike the
intellectual signs in his general theory of meaning, or pragmaticism, symbol-image interpretations
are not effectuated using self-controlled habits of acting in a certain way whenever a certain kind of
situation is confronted. Hence such interpretations cannot be general and universal in the same way
as interpretations of logical and diagrammatic signs, 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.
However, just as in the interpretation of symbols, the symbol-image processes do rely on some
essential linguistic features. Images serve similar purposes as symbols do even though their outward
appearance is quite different from natural languages. Moreover, despite being uncontrollable, such
interpretations have a significant role to play in our logical theories. For symbolic interpretations 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 justification. Let us recall the key elements 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 which “iconizes the Universe
of Discourse” (CP 4.553n2, 1906, The Bed-Rock beneath Pragmaticism). The sheet is the pure
representamen of what is in the “field of attention” of the discourse participants, the Utterer and the
Interpreter, who undertake to draw and interpret the graphs (Hilpinen 1982, Pietarinen 2006a). In
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other words, “in representing the field 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 interpreted 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 precisely
how something becomes a „system‟ with explanatory value serving some specific 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 fixed and the conditions in which propositions hold in such fixed and isolated
models are studied. Existential graphs are perhaps the first instance in logic where the possibilities
of what much later was baptised model theory were realised in depth (Pietarinen 2006a).
Second, Peirce notes that the phemic sheet “is an image of the universal field of interconnected
Thought” (CP 4.553n2, 1906, The Bed-Rock beneath Pragmaticism). 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 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 qualities distinguishing it from the replica of any other spot” (CP
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4.416, 1903, Existential 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 emphasises 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 colours refers among other things to different modalities in Peirce‟s “tinctured” gamma
graphs.
Moreover, spots, together with the topological operations of juxtaposition (conjunction) and the cut
(negation), define the language of the alpha part of existential graphs. That language is an iconic
counterpart to the symbolic language of propositional logic. We need not delve into the details of
how these graphs are constructed or how some more expressive graphs might be constructed 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 identity can be attached” (CP 4.416, 1903, Pure
Mathematical Definition 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 diagrammatic
counterpart to fragments of first-order predicate logic with identity.
A couple of philosophically and logically fundamental repercussions concerning the ways in which
Peirce sets up his system of logical diagrams are imminent. He intended the phemic sheet to be the
iconic counterpart to what logicians 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
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constituents of such interpreted languages, complex assertions subject to semantic interpretation are
then constructed by applying different logical constants 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 significant for several reasons. First, what my
interpretation of Peirce‟s remarks implies is that how images 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
singular, 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 appearance. 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
phemic sheet partake of what he termed the “universal field 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
behaviour constitutive of the meaning of intellectual signs. And that general mode of constitution is
what his philosophy of pragmaticism is all about (Pietarinen & Snellman 2006). Thus, he was left
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 occurrence” (CP 2.222), in
other words, of interpreted diagrams on the phemic sheet that are composites of images and other
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iconic signs. Such processes are logical and semantic, and are constituted by the activities of the
Utterer and the Interpreter, who act according to the given spatial and inductive structure of the
diagram. (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 for choosing right subgraphs to proceed and for finding 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 profiles 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 unconscious elements of thought in that study.
Fourth, images are isolated spots that are not connected to occurrences before 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, without anything that occupies those hooks, without anything
could make them interpretable. The symbolic counterpart of such an unoccupied spot is a predicate
term or an unsaturated rhema that has some arity but no variables in argument 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 justification 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 justification is that it subsequently turns out to be
useful. (CP 1.538, 1903, The Firstness of Firstness, Secondness, and Thirdness)
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One can make a predication only after attaching dots or lines to the hooks of spots that hit upon
some object in the universe of discourse. That operation is secondary 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 neurodevelopmental disorder in
which patients tend to think entirely in terms of firstnesses of hypoicons. Patients suffering from
autism have their minds devoid of the secondness 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.
Fifth, the previous passage also contains a key to what Peirce is after in stating 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 images 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 fifth 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 represents at any particular location of the space singled out to compose the spot in
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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” 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 theories of language, or what more accurately should
be called Wittgenstein‟s picture theory of elementary propositions. Two main clauses from
Wittgenstein‟s Tractatus 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 final 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 elementary 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 topological 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 proposition. This
goes some way towards the Peircean idea of a cut or an incision around those regions of the phemic
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sheet that need to be negated. But in Peirce‟s theory, all propositions, including any truth-functional
composites of propositions, are at root iconic and thus pictorial.
To remark on some of the most notable differences between the thinking of these two logicians, the
atomicity of elementary propositions is an example of an assumption we do not find 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 finding 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 nonsymbolic language just as diagrams and metaphors are, (ii) symbolic interpretations of images
instantiate non-habitual „physiognomic‟ processes, and (iii) the elementary constituents of logical
diagrams are images (spots) subject to such physiognomic interpretations closely connected with
interpretations of non-logical vocabularies of logical languages.
Moreover, these points entail a picture theory of language of a broadly Wittgensteinian 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 character, precisely how to
consider „images as a language of some sort‟ is bound to be largely arbitrary. Any symbolic
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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.
Acknowledgments
Presented at the meeting “Peirce and Image” held during the Semiotics Summer School in Urbino,
Italy, July 2006. My thanks to the organisers 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.
References
Grandin, Temple (1995). Thinking in Pictures: My Life with Autism. New York: Vintage Books.
Hilpinen, Risto (1982). On C. S. Peirce‟s theory of the proposition: Peirce as a precursor of gametheoretical semantics. The Monist 65, 182–188.
Peirce, Charles S. (1931–1958). Collected Papers of Charles Sanders Peirce. 8 volumes, vols. 1–6
edited by Charles Hartshorne and Paul Weiss, vols. 7–8, edited by Arthur W. Burks.
Cambridge, Mass.: Harvard University Press.
Peirce, Charles S. (1967). Manuscripts in the Houghton Library of Harvard University, as identified
by Richard Robin, Annotated Catalogue of the Papers of Charles S. Peirce (Amherst:
University of Massachusettes Press, 1967), and in The Peirce Papers: A supplementary
catalogue, Transactions of the C. S. Peirce Society 7 (1971), 37–57.
Peirce, Charles S. (1998). The Essential Peirce. Selected Philosophical Writings. Vol. 2 (1893–
1913). Edited by the Peirce Edition Project. Bloomington and Indianapolis: Indiana University
Press.
14
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Pietarinen, Ahti-Veikko (2006a). Signs of Logic: Peircean Themes on the Philosophy of Language,
Games, and Communication. Synthese Library 329, Dordrecht: Springer.
Pietarinen, Ahti-Veikko (2006b). Peirce‟s contributions to possible-worlds semantics, Studia
Logica 82, 345-369.
Pietarinen, Ahti-Veikko (2008). Iconic logic for metaphors, Journal of Cognitive Science.
Pietarinen, Ahti-Veikko and Lauri Snellman (2006). On Peirce‟s late proof of pragmaticism, in
Tuomo Aho and Ahti-Veikko Pietarinen (eds), Truth and Games. Helsinki: Acta Philosophica
Fennica 78, 275-288.
Wittgenstein, Ludwig (1922). Tractatus Logico-Philosophicus. London: Kegan Paul.
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