Sign Relations • Examples

Posted on December 18, 2025 by Jon Awbrey

Soon after I made my third foray into grad school, this time in Systems Engineering, I was trying to explain sign relations to my advisor and he, being the very model of a modern systems engineer, asked me to give a concrete example of a sign relation, as simple as possible without being trivial.  After much cudgeling of the grey matter I came up with a pair of examples which had the added benefit of bearing instructive relationships to each other.  Despite their simplicity, the examples to follow have subtleties of their own and their careful treatment serves to illustrate important issues in the general theory of signs.

Imagine a discussion between two people, Ann and Bob, and attend only to the aspects of their interpretive practice involving the use of the following nouns and pronouns.

\{ ``\text{Ann}", ``\text{Bob}", ``\text{I}", ``\text{you}" \}

  • The object domain of their discussion is the set of two people \{ \text{Ann}, \text{Bob} \}.
  • The sign domain of their discussion is the set of four signs \{ ``\text{Ann}", ``\text{Bob}", ``\text{I}", ``\text{you}" \}.

Ann and Bob are not only the passive objects of linguistic references but also the active interpreters of the language they use.  The system of interpretation associated with each language user can be represented in the form of an individual three‑place relation known as the sign relation of that interpreter.

In terms of its set‑theoretic extension, a sign relation L is a subset of a cartesian product O \times S \times I.  The three sets O, S, I are known as the object domain, the sign domain, and the interpretant domain, respectively, of the sign relation L \subseteq O \times S \times I.

Broadly speaking, the three domains of a sign relation may be any sets at all but the types of sign relations contemplated in formal settings are usually constrained to having I \subseteq S.  In those cases it becomes convenient to lump signs and interpretants together in a single class called a sign system or syntactic domain.  In the forthcoming examples S and I are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question.

When it becomes necessary to refer to the whole set of objects and signs in the union of the domains O, S, I for a given sign relation L, we will call this set the World of L and write W = W_L = O \cup S \cup I.

To facilitate an interest in the formal structures of sign relations and to keep notations as simple as possible as the examples become more complicated, it serves to introduce the following general notations.

Display 1

Introducing a few abbreviations for use in the Example, we have the following data.

Display 2

In the present example, S = I = \text{Syntactic Domain}.

Tables 1a and 1b show the sign relations associated with the interpreters \mathrm{A} and \mathrm{B}, respectively.  In this arrangement the rows of each Table list the ordered triples of the form (o, s, i) belonging to the corresponding sign relations, L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.

Sign Relation Twin Tables LA & LB

The Tables codify a rudimentary level of interpretive practice for the agents \mathrm{A} and \mathrm{B} and provide a basis for formalizing the initial semantics appropriate to their common syntactic domain.  Each row of a Table lists an object and two co‑referent signs, together forming an ordered triple (o, s, i) called an elementary sign relation, in other words, one element of the relation’s set‑theoretic extension.

Already in this elementary context, there are several meanings which might attach to the project of a formal semiotics, or a formal theory of meaning for signs.  In the process of discussing the alternatives, it is useful to introduce a few terms occasionally used in the philosophy of language to point out the needed distinctions.  That is the task we’ll turn to next.

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Sign Relations • Signs and Inquiry

Posted on December 16, 2025 by Jon Awbrey

There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many congruences and parallels it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, “thinking” in its best sense, “is a term denoting the various ways in which things acquire significance” (Dewey, 38).

Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between the converging modes of investigation.  Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject the theory of signs is specialized to treat from comparative and structural points of view.

References

  • Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.
  • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  Archive.  Journal.  Online (doc) (pdf).

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Sign Relations • Definition

Posted on December 14, 2025 by Jon Awbrey

One of Peirce’s clearest and most complete definitions of a sign is one he gives in the context of providing a definition for logic, and so it is informative to view it in that setting.

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.

It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non‑psychological conception of logic has virtually been quite generally held, though not generally recognized.

— C.S. Peirce, New Elements of Mathematics, vol. 4, 20–21

In the general discussion of diverse theories of signs, the question arises whether signhood is an absolute, essential, indelible, or ontological property of a thing, or whether it is a relational, interpretive, and mutable role a thing may be said to have only within a particular context of relationships.

Peirce’s definition of a sign defines it in relation to its objects and its interpretant signs, and thus defines signhood in relative terms, by means of a predicate with three places.  In that definition, signhood is a role in a triadic relation, a role a thing bears or plays in a determinate context of relationships — it is not an absolute or non‑relative property of a thing‑in‑itself, one it possesses independently of all relationships to other things.

Some of the terms Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader.

  • Correspondence.  From the way Peirce uses the term throughout his work, it is clear he means what he elsewhere calls a “triple correspondence”, and thus this is just another way of referring to the whole triadic sign relation itself.  In particular, his use of the term should not be taken to imply a dyadic correspondence, like the kinds of “mirror image” correspondence between realities and representations bandied about in contemporary controversies about “correspondence theories of truth”.
  • Determination.  Peirce’s concept of determination is broader in several directions than the sense of the word referring to strictly deterministic causal‑temporal processes.  First, and especially in this context, he is invoking a more general concept of determination, what is called a formal or informational determination, as in saying “two points determine a line”, rather than the more special cases of causal and temporal determinisms.  Second, he characteristically allows for what is called determination in measure, that is, an order of determinism admitting a full spectrum of more and less determined relationships.
  • Non‑psychological.  Peirce’s “non‑psychological conception of logic” must be distinguished from any variety of anti‑psychologism.  He was quite interested in matters of psychology and had much of import to say about them.  But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a normative science where psychology is a descriptive science, and so they have very different aims, methods, and rationales.

Reference

  • Peirce, C.S. (1902), “Parts of Carnegie Application” (L 75), in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.

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Sign Relations • Anthesis

Posted on December 12, 2025 by Jon Awbrey

Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun.

— C.S. Peirce, Collected Papers, CP 2.274

In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or semiosis, Peirce uses the technical term representamen for his concept of a sign, but the shorter word is precise enough, so long as one recognizes its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign.

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Cactus Language • Semantics 8

Posted on November 1, 2025 by Jon Awbrey

The 16 boolean functions on two variables F^{(2)} : \mathbb{B}^2 \to \mathbb{B} are shown in the following Table.

\text{Boolean Functions on Two Variables}
Boolean Functions on Two Variables

As before, all boolean functions on proper subsets of the current variables are subsumed in the Table at hand.  In particular, we have the following inclusions.

  • The constant function 0 ~:~ \mathbb{B}^2 \to \mathbb{B} appears under the name F_{0}^{(2)}.
  • The constant function 1 ~:~ \mathbb{B}^2 \to \mathbb{B} appears under the name F_{15}^{(2)}.
  • The function expressing the assertion of the first variable is F_{12}^{(2)}.
  • The function expressing the negation of the first variable is F_{3}^{(2)}.
  • The function expressing the assertion of the second variable is F_{10}^{(2)}.
  • The function expressing the negation of the second variable is F_{5}^{(2)}.

Next come the functions on two variables whose output values change depending on changes in both input variables.  Notable among them are the following examples.

  • The logical conjunction is given by the function F_{8}^{(2)} (x, y) ~=~ x \cdot y.
  • The logical disjunction is given by the function F_{14}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{)(} ~y~ \texttt{))}.

Functions expressing the conditionals, implications, or if‑then statements appear as follows.

  • [x \Rightarrow y] ~=~ F_{11}^{(2)} (x, y) ~=~ \texttt{(} ~x~ \texttt{(} ~y~ \texttt{))} ~=~ [\mathrm{not}~ x ~\mathrm{without}~ y].
  • [x \Leftarrow y] ~=~ F_{13}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{)} ~y~ \texttt{)} ~=~ [\mathrm{not}~ y ~\mathrm{without}~ x].

The function expressing the biconditional, equivalence, or if‑and‑only‑if statement appears in the following form.

  • [x \Leftrightarrow y] ~=~ [x = y] ~=~ F_{9}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{,} ~y~ \texttt{))}.

Finally, the boolean function expressing the exclusive disjunction, inequivalence, or not equals statement, algebraically associated with the binary sum operation, and geometrically associated with the symmetric difference of sets, appears as follows.

  • [x \neq y] ~=~ [x + y] ~=~ F_{6}^{(2)} (x, y) ~=~ \texttt{(} ~x~ \texttt{,} ~y~ \texttt{)}.

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Cactus Language • Semantics 7

Posted on October 30, 2025 by Jon Awbrey

A good way to illustrate the action of the conjunction and surjunction operators is to show how they can be used to construct the boolean functions on any finite number of variables.  Though it’s not much to look at let’s start with the case of zero variables, boolean constants by any other word, partly for completeness and partly to supply an anchor for the cases in its train.

A boolean function F^{(0)} on zero variables is just an element of the boolean domain \mathbb{B} = \{ 0, 1 \}.  The following Table shows several ways of referring to those elements, for the sake of consistency using the same format we’ll use in subsequent Tables, however degenerate it appears in this case.

\text{Boolean Functions on Zero Variables}
Boolean Functions on Zero Variables

  • Column 1 lists each boolean element or boolean function under its ordinary constant name or under a succinct nickname, respectively.
  • Column 2 lists each boolean function by means of a function name F_j^{(k)} of the following form.  The superscript (k) gives the dimension of the functional domain, in effect, the number of variables, and the subscript j is a binary string formed from the functional values, using the obvious coding of boolean values into binary values.
  • Column 3 lists the values each function takes for each combination of its domain values.
  • Column 4 lists the ordinary cactus expressions for each boolean function.  Here, as usual, the expression ``\texttt{(( ))}" renders the blank expression for logical truth more visible in context.

The next Table shows the four boolean functions on one variable, F^{(1)} : \mathbb{B} \to \mathbb{B}.

\text{Boolean Functions on One Variable}
Boolean Functions on One Variable

  • Column 1 lists the contents of Column 2 in a more concise form, converting the lists of boolean values in the subscript strings to their decimal equivalents.  Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable.  The constant functions are thus expressible in the following equivalent ways.

\begin{matrix} F_0^{(1)} & = & F_{00}^{(1)} & = & 0 ~:~ \mathbb{B} \to \mathbb{B}. \\[4pt] F_3^{(1)} & = & F_{11}^{(1)} & = & 1 ~:~ \mathbb{B} \to \mathbb{B}. \end{matrix}

  • The other two functions in the Table are easily recognized as the one‑place logical connectives or the monadic operators on \mathbb{B}.  Thus the function F_1^{(1)} = F_{01}^{(1)} is recognizable as the negation operation and the function F_2^{(1)} = F_{10}^{(1)} is obviously the identity operation.

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Cactus Language • Semantics 6

Posted on October 27, 2025 by Jon Awbrey

If one takes the view that PARCEs and PARCs amount to a pair of intertranslatable representations for the same domain of objects then denotation brackets of the form \downharpoonleft \ldots \downharpoonright can be used to indicate the logical denotation \downharpoonleft s_j \downharpoonright of a sentence s_j or the logical denotation \downharpoonleft C_j \downharpoonright of a cactus C_j.

The relations connecting sentences, graphs, and propositions are shown in the next two Tables.

\text{Semantic Translation} \stackrel{_\bullet}{} \text{Functional Form}
Semantic Translation : Functional Form

\text{Semantic Translation} \stackrel{_\bullet}{} \text{Equational Form}
Semantic Translation : Equational Form

Between the sentences and the propositions run the graph‑theoretic data structures which arise in the process of parsing sentences and catalyze their potential for expressing logical propositions or indicator functions.  The graph‑theoretic medium supplies an intermediate form of representation between the linguistic sentences and the indicator functions, not only rendering the possibilities of connection between them more readily conceivable in fact but facilitating the necessary translations on a practical basis.

In each Table the passage from the first to the middle column articulates the mechanics of parsing cactus language sentences into graph‑theoretic data structures while the passage from the middle to the last column articulates the semantics of interpreting cactus graphs as logical propositions or indicator functions.

Aside from their common topic, the two Tables present slightly different ways of drawing the maps which go to make up the full semantic transformation.

Semantic Translation • Functional Form
The first Table shows the functional associations connecting each domain with the next, taking the triple of a sentence s_j, a cactus C_j, and a proposition q_j as basic data, and fixing the rest by recursion on those ingredients.
Semantic Translation • Equational Form
The second Table records the transitions in the form of equations, treating sentences and graphs as alternative types of signs and generalizing the denotation bracket to indicate the proposition denoted by either type.

It should be clear at this point that either scheme of translation puts the triples of sentences, graphs, and propositions roughly in the roles of signs, interpretants, and objects, respectively, of a triadic sign relation.  Indeed, the roughly can be rendered exactly as soon as the domains of a suitable sign relation are specified precisely.

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Cactus Language • Semantics 5

Posted on October 14, 2025 by Jon Awbrey

Last time we reached the threshold of a potential codomain or target space for the kind of semantic function we need at this point, one able to supply logical meanings for the syntactic strings and graphs of a given cactus language.  In that pursuit we came to contemplate the following definitions.

Logical Conjunction
The conjunction \mathrm{Conj}_j^J q_j of a set of propositions \{ q_j : j \in J \} is a proposition which is true if and only if every one of the q_j is true.

\mathrm{Conj}_j^J q_j is true  \Leftrightarrow  q_j is true for every j \in J.

Logical Surjunction
The surjunction \mathrm{Surj}_j^J q_j of a set of propositions \{ q_j : j \in J \} is a proposition which is true if and only if exactly one of the q_j is untrue.

\mathrm{Surj}_j^J q_j is true  \Leftrightarrow  q_j is untrue for unique j \in J.

If the set of propositions \{ q_j : j \in J \} is finite then the logical conjunction and logical surjunction can be represented by means of sentential connectives, incorporating the sentences which represent the propositions into finite strings of symbols.

If J is finite, for instance, if J consists of the integers in the interval j = 1 ~\text{to}~ k, and if each proposition q_j is represented by a sentence s_j, then the following forms of expression are possible.

Logical Conjunction
The conjunction \mathrm{Conj}_j^J q_j can be represented by a sentence which is constructed by concatenating the s_j in the following fashion.

\mathrm{Conj}_j^J q_j ~\leftrightsquigarrow~ s_1 s_2 \ldots s_k.

Logical Surjunction
The surjunction \mathrm{Surj}_j^J q_j can be represented by a sentence which is constructed by surcatenating the s_j in the following fashion.

\mathrm{Surj}_j^J q_j ~\leftrightsquigarrow~ \texttt{(} s_1 \texttt{,} s_2 \texttt{,} \ldots \texttt{,} s_k \texttt{)}.

If one opts for a mode of interpretation which moves more directly from the parse graph of a sentence to the potential logical meaning of both the PARC and the PARCE then the following specifications are in order.

A cactus graph rooted at a particular node is taken to represent what that node represents, namely, its logical denotation.

Denotation of a Node
The logical denotation of a node is the logical conjunction of that node’s arguments, which are defined as the logical denotations of that node’s attachments.
The logical denotation of either a blank symbol or empty node is the boolean value \underline{1} = \mathrm{true}.
The logical denotation of the paint \mathfrak{p}_j is the proposition p_j, a proposition regarded as primitive, at least, with respect to the level of analysis represented in the current instance of \mathfrak{C} (\mathfrak{P}).
Denotation of a Lobe
The logical denotation of a lobe is the logical surjunction of that lobe’s arguments, which are defined as the logical denotations of that lobe’s appendants.
As a corollary, the logical denotation of the parse graph of \texttt{()}, also known as a needle, is the boolean value \underline{0} = \mathrm{false}.

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Cactus Language • Semantics 4

Posted on October 12, 2025 by Jon Awbrey

Words spoken are symbols or signs (symbola) of affections or impressions (pathemata) of the soul (psyche);  written words are the signs of words spoken.  As writing, so also is speech not the same for all races of men.  But the mental affections themselves, of which these words are primarily signs (semeia), are the same for the whole of mankind, as are also the objects (pragmata) of which those affections are representations or likenesses, images, copies (homoiomata).  (Aristotle, De Interp. i. 16a4).

At this point we have two distinct dialects, scripts, or modes of presentation for the typical cactus language \mathfrak{C} (\mathfrak{P}), each of which needs to be interpreted, that is to say, equipped with a semantic function defined on its domain.

\textsc{parce} (\mathfrak{P})
There is the language of strings in \textsc{parce} (\mathfrak{P}), the painted and rooted cactus expressions collectively forming the language \mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*.
\textsc{parc} (\mathfrak{P})
There is the language of graphs in \textsc{parc} (\mathfrak{P}), the painted and rooted cacti themselves, a family of graphs or species of data structures formed by parsing the language of strings.

Those two modalities of formal language, like written and spoken natural languages, are meant to have compatible interpretations, which means it is generally sufficient to give the meanings of just one or the other.

All that remains is to provide a codomain or target space for the intended semantic function, that is, to supply a suitable range of logical meanings for the memberships of those languages to map into.  One way to do that proceeds by making the following definitions.

Logical Conjunction
The conjunction \mathrm{Conj}_j^J q_j of a set of propositions \{ q_j : j \in J \} is a proposition which is true if and only if every one of the q_j is true.

\mathrm{Conj}_j^J q_j is true  \Leftrightarrow  q_j is true for every j \in J.

Logical Surjunction
The surjunction \mathrm{Surj}_j^J q_j of a set of propositions \{ q_j : j \in J \} is a proposition which is true if and only if exactly one of the q_j is untrue.

\mathrm{Surj}_j^J q_j is true  \Leftrightarrow  q_j is untrue for unique j \in J.

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Cactus Language • Semantics 3

Posted on October 10, 2025 by Jon Awbrey

The task before us is to specify a semantic function for the cactus language \mathfrak{L} = \mathfrak{C}(\mathfrak{P}), in other words, to define a mapping from the space of syntactic expressions to a space of logical statements which “interprets” each expression of \mathfrak{C}(\mathfrak{P}) as an expression which says something, an expression which bears a meaning, in short, an expression which denotes a proposition, and is in the end a sign of an indicator function.

When the syntactic expressions of a formal language are given a referent significance in logical terms, for example, as denoting propositions or indicator functions, then each form of syntactic combination takes on a corresponding form of logical significance.

A handy way of providing a logical interpretation for the expressions of any given cactus language is to introduce a family of operators on indicator functions called propositional connectives, to be distinguished from the associated family of syntactic combinations called sentential connectives, where the relationship between the two realms of connection is exactly that between objects on the one hand and their signs on the other.

A propositional connective, as an entity of a well‑defined functional and operational type, can be treated in every way as a logical or mathematical object and thus as the type of object which can be denoted by the corresponding form of syntactic entity, namely, the sentential connective appropriate to the case at hand.

There are two basic types of connectives, called the blank connectives and the bound connectives, respectively, with one connective of each type for each natural number k = 0, 1, 2, 3, \ldots.

Blank Connective
The blank connective of k places is signified by the concatenation of the k sentences filling those places.

For the initial case k = 0, the blank connective is an empty string or a blank symbol, both of which have the same denotation among propositions.

For the generic case k > 0, the blank connective takes the form s_1 \cdot \ldots \cdot s_k.  In the type of data called a text, the use of the center dot “⋅” is generally supplanted by whatever number of spaces and line breaks serve to improve the readability of the resulting text.

Bound Connective
The bound connective of k places is signified by the surcatenation of the k sentences filling those places.

For the initial case k = 0, the bound connective is an empty closure, an expression taking one of the forms \texttt{()}, \texttt{(~)}, \texttt{(~~)}, \ldots with any number of spaces between the parentheses, all of which have the same denotation among propositions.

For the generic case k > 0, the bound connective takes the form \texttt{(} s_1 \texttt{,} \ldots \texttt{,} s_k \texttt{)}.

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