## 1. From Monadic to Polyadic Logic



Peirce and Frege, independently of each other, took us from the
traditional Aristotelian logic to modern logic--a large leap.
Nobody could deny the power of formalization which has led early
twentieth century mathematicians to surprising achievements and
 results.[1]
 What is the essence of the leap made by Peirce and Frege? Is it just
a matter of introducing new formal notations, i.e., quantifiers and
variables, so that we may easily formalize our reasoning? If so,
modern logic would be just dressing up Aristotelian logic with
quantifiers/variables. This would equate one of Peirce's main
contributions in logic to the increase in formal vocabulary.



The enormous impact of the adoption of quantifiers/bound variables on
the world of logic and mathematics cannot be denied. However, that
should not overshadow Peirce's insight behind the new extended
formalism. This section will explore how Peirce's conviction
about the novelty of the logic of relations led him to the
introduction of quantifiers/variables. Hence, quantification theory,
according to Peirce, is not a matter of a linear extension of formal
vocabulary, but an expansion into territory that is qualitatively
different from what Aristotelian logic covers. At the same time, we
should not forget that Peirce extended the territory of logic in the
spirit of Boole's algebra of logic.



In "An Improvement in Boole's Calculus of Logic"
(1867) Peirce hints at the need for an improvement of Boolean logic,
not in the context of predicate logic, but in its inability to express
existential statements in the context of term logic. His 1870 paper
"Description of a Notation for the Logic of Relatives, Resulting
from an Amplification of the Conceptions of Boole's
Calculus" (DNLR) reveals his ambition to marry Boole's
algebraic notation with De Morgan's effort at relational
representation. Many agree that this paper introduces essential
vocabulary of first-order predicate logic for the first time in
history. Subsequently, in "On the Algebra of Logic" (1880)
Peirce investigates two kinds of operations over
relations--relative sum and relative product--and "The
Logic of Relatives" (known as "Note B") published in
his edited book *Studies in Logic by Members of the Johns Hopkins
University* (1883) shows a major progress in quantification,
influenced by the work of O. H. Mitchell (who was his student).
Finally, Peirce's 1885 paper "On the Algebra of Logic: A
Contribution to the Philosophy of Notations" has been considered
to be the place where Peirce fully presented his quantification
theory.



Starting with DNLR, the first subsection examines Pierce's
subsequent steps until he presented the final form of his first-order
logic in his 1885 paper "On the Algebra of Logic: a Contribution
to the Philosophy of Notations". (For a number of manuscripts
written between these two papers, refer to Beatty 1969; Dipert 2004:
297-299; and Merrill 1978.) The second subsection locates
Peirce's first-order predicate logic work in a larger
context.


### 1.1 Relations and quantification formalized



Peirce's quantification theory is presented in a comprehensive
way together with axiomlike "icons" in his 1885 paper
"On the Algebra of Logic: A contribution to the Philosophy of
Notations". Peirce's not-short journey to modern logic
started with his attempt to extend the territory of formalization. In
this, Peirce was inspired by De Morgan's struggle for the
representation of relations, and at the same time Peirce was empowered
with Boolean calculus which formalizes Aristotelian term logic. That
is, Peirce took De Morgan's ambition as a road map for the
direction while being equipped with Boole's method and notation
for getting there. This subsection will follow Peirce's journey
to see how he reached the destination, by checking in at his main
stops.



The title of Peirce's 1870 paper "Description of a
Notation for the Logic of Relatives, Resulting from an Amplification
of the Conceptions of Boole's Calculus" (DNLR) is spelled
out at the beginning of the paper in the following way:



> 
> 
> [I]t is interesting to inquire whether it [Boole's logical
> algebra] cannot be extended over to the whole realm of formal logic,
> instead of being restricted to that simplest and least useful part of
> the subject, the logic of absolute terms,...The object of this
> paper is to show that an affirmative answer can be given to this
> question. (DNLR [CP 3.45])
> 
> 
> 



Boolean logic needs to be "extended" if we want to cover
the entire realm of formal logic, Peirce states. What does Peirce mean
by "the whole realm of formal logic"? Peirce answers:
"Deductive logic can really not be understood without the study
of the logic of relatives" (1911a [CP
 3.641]).[2]



Being encouraged by Boole's algebra of logic, but at the same
time taking his father, Professor Benjamin Peirce's negative
view of logic
 seriously,[3]
 Peirce explored a way to apply Boole's method to a larger
domain of our reasoning so that relations may be formalized.



What are relations and why are they so special? Let's compare
three sentences: "John is an American", "John is
taller than Tom", and "John is between Tom and
Mary". The first sentence has a unary predicate "is an
American", the second sentence a binary predicate "is
taller than", and the third sentence a ternary predicate
"is between...and...". A unary predicate stands
for a property or quality, while a binary or ternary predicate stands
for a relation. If a first-order logical system has only unary
predicates, then we say it is monadic. Otherwise, predicate logic is
assumed to have binary or other higher predicates.



When we move from monadic to polyadic logic, substantial changes take
place. The following three changes are at the top of the list.
*First* of all, a move from property to relation is a territory
expansion. Noting that Aristotelian syllogisms are limited to unary
predicates, one expects polyadic logic to represent more than the
reasoning involved in Aristotelian syllogism, that is, term logic.
*Second*, monadic logic is decidable while polyadic logic is
not decidable, as Church's theorem proved. In some sense, as the
territory is expanded, we are losing its grip. *Third*, a
change in notation is inevitable, which necessitates modern
quantification theory. How are these three important aspects are
handled by Peirce?



The realm of relations was a frontier where De Morgan did much of his
creative and novel work on
 logic.[4]
 However, his inquiry on the topic resides within the patterns of
traditional
 syllogisms.[5]
 Even more importantly, De Morgan did not have enough tools to
formalize this newly extended
 realm.[6]
 Hence, not surprisingly De Morgan's relations are rather
limited to a certain group which fit in syllogistic reasoning. As
Merrill points out, 



> 
> 
> De Morgan develops the general logic of relations only to the point
> where it can be used for his familiar syllogistic purposes. This means
> that he is especially interested in relations which are convertible
> and/or transitive,.... (Merrill 1990: 113) 
> 
> 
> 



It is somewhat unclear and controversial whether Peirce's
interest in the logic of relations started independently of De
Morgan's work on the
 subject.[7]
 Regardless of the origin of Peirce's inquiry into relations,
many have come to agree that it is Peirce (not De Morgan) who
successfully formalized the logic of relations. Merrill, a De Morgan
scholar, puts the matter in the following way:



> 
> 
> The most obvious problem with this view of the proposition [De
> Morgan's way of handling relational arguments] is that it does
> not seem general enough. If we can unite two terms into a proposition
> by relating them, why not three or four or ten terms? De
> Morgan's concern with the relational syllogism seems to have
> precluded this generalization; but there is no reason in principle why
> it could not be made. For this, though, we must wait for Frege and
> Peirce. (Merrill 1990: 110)
> 
> 
> 



Interestingly enough, Peirce's writings before 1870 DNLR show
Peirce also attempted to solve relational arguments by traditional
syllogistic reasoning
 rules,[8]
 but the approach taken in DNLR is totally different--not within
a syllogistic frame but by introducing Boolean algebra notation.
Peirce must have realized the power of generalization that Boolean
notation could provide. Boole's algebra formalized Aristotelian
categorical syllogisms and opened a way for generalization of term
 logic.[9]
 Peirce, who was impressed with Boole's mathematical treatment
of Aristotelian syllogisms, not surprisingly aimed to apply this
method to relations. In that sense, modern predicate logic started in
Peirce's 1870 pioneering work. Hence, the goal of Peirce's
project--that is, to broaden the scope of formalization in
logic--was a main motivation for the introduction of new
vocabulary for quantifiers and bound variables. If so, Peirce's
insight early on as to the importance of reasoning involving relations
is a key element in understanding a difference between Peirce's
and Frege's developments of first-order
 logic.[10]
 Furthermore, the next section will show how Peirce's obsession
with the logic of relations led him to the invention of Existential
Graphs.



The logic of relations formalizes a larger territory than monadic
logic, but there is a price to pay for obtaining the additional
expressive power: While monadic logic is decidable, polyadic logic is
not. Even though we need to wait until Church's theorem to see
the undecidability of first-order predicate logic, Peirce intuited a
fundamental difference between the logic of non-relatives versus the
logic of relations. Here are Peirce's suggestive ideas over the
comparisons between monadic and relational logic: 



> 
> 
> The logic of relatives is highly multiform; it is characterized by
> innumerable immediate inferences, and by various distinct conclusions
> from the same set of premises. (1883a [CP 3.342]) 
> 
> 
> 



And: 



> 
> 
> [T]he old syllogistic inference can be worked by machinery, but
> characteristic relative inferences cannot be performed by any mere
> mechanical rule whatever. (1896: 330) 
> 
> 
> 



As Dipert correctly points out, Peirce's remarks reveal his
"*understanding* of the richness and difficulty which
relations introduce into logic" (Dipert 1984a:
 63).[11]



In order to increase expressive power, Peirce left the traditional
syllogistic pattern and brought in Boolean algebra of logic. The
following comment emphasizes that Peirce's choice of notation
marks a clear departure from De Morgan's pursuit of relational
logic: 



> 
> 
> De Morgan's methodology is governed by the logic of syllogism
> while Peirce's methodology is entirely algebraic. This algebraic
> model taken over from Boole is foreign to De Morgan's methods.
> This difference in methodology reflects a significant difference at
> the level of definition. (Brunning 1991: 36)
> 
> 
> 



And after realizing the complicated nature of the logic of relations,
Peirce explored new notation *beyond* Boolean calculus. That
move is predicted in the following passage: 



> 
> 
> The effect of these peculiarities [the non-mechanic nature of relative
> logic] cannot be subjected to hard and fast rules like those of the
> Boolian calculus. (1883a [CP 3.342]) 
> 
> 
> 



Here is the third aspect of a transition from monadic to polyadic
logic: The complication that relations bring in our reasoning,
obviously, pushed Peirce to develop a new notational system. As the
rest of this subsection shows, the process made over the course of 15
years--from DNLR to "On the algebra of
logic"--is rather complicated. Importantly, Peirce's
introduction of quantifiers and bound variables could be seen as an
inevitable outcome of his ambitious goal to expand the scope of
formalization to cover relations, as Merrill says "The
quantification complexities of many relational statements cried out
for quantifiers" (1997: 158).



The third section of DNLR, as the title "Application of the
Algebraic Signs to Logic" says, is one of the first places where
Boolean algebraic notation and relational logic joined each other. In
the first subsection Peirce makes it clear that the territory he aims
to cover is relational, by including polyadic predicates in the
following way: 



(DNLR [CP 3.63-64]; the entry adopts our modern terminology
instead of Peirce's.)





| Predicates | Peirce's terminology | Letters | Examples |
| --- | --- | --- | --- |
| unary | absolute terms | \(\unary{a}, \unary{b}, \unary{c},\)... (Roman
alphabet) | Frenchman \((\unary{f})\), violinist
\((\unary{u})\),... |
| binary | simple relative terms | \(\binary{a}, \binary{b}, \binary{c}\),... (italics) | wife \((\binary{w})\)), lover \((\binary{l})\), owner \((\binary{o})\),... |
| ternary | conjugative terms | \(\ternary{a}, \ternary{b}, \ternary{c}\)... (Kennerly
[Kennerley]) | giver to -- of -- \((\ternary{g})\) |




For the rest of the third section, four kinds of algebraic signs are
introduced to be applied on these predicate letters: the inclusion
sign (\(\inclusion\)), the addition sign (\(\cunion\)), the
multiplication sign (juxtaposition or ","), and the
involution sign (exponentiation).



First, he combines the sign equality "=" and the sign
\(<\) for "less than" to come up with the sign
"\(\inclusion;\)" to represent inclusion:



> 
> 
> Thus,
> 
>  
> 
> \[\unary{f} \inclusion \unary{m}\]
> 
> 
> means "every Frenchman is a man", without saying whether
> there are any other men or not. So,
> 
>  
> 
> \[\binary{m} \inclusion \binary{l}\]
> 
> 
> will mean that every mother of anything is a lover of the same thing;
> although this interpretation in some degree anticipates a convention
> to be made further on. (DNLR [CP 3.66])
> 
> 
> 



Note that "\(\unary{f} \inclusion \binary{m}\)" (unlike
"\(\unary{f} \inclusion \unary{m}\)") would be
ungrammatical since \(\binary{m}\), being a binary predicate, cannot
have an inclusion relation with a unary predicate \(\unary{f}\).



For the sign of addition, Peirce brings in Boolean sign \(+\), but
with slight variation:



> 
> 
> The sign of addition is taken by Boole, so that
> 
>  
> 
> \[x + y\]
> 
> 
> denotes everything denoted by \(x\), and *besides*, everything
> denoted by \(y\)....But if there is anything which is denoted by
> both the terms of the sum, the latter no longer stands for any logical
> term on account of its implying that the objects denoted by one term
> are to be taken *besides* the objects denoted by the other. For
> example,
> 
>  
> 
> \[\unary{f} + \unary{u}\]
> 
> 
> means all Frenchmen besides all violinists, and, therefore, considered
> as a logical term, implies that all French violinists are *besides
> themselves*. For this reason alone...I preferred to take as
> the regular addition of logic a non-invertible process, such that
> 
> 
> 
> \[\unary{m} \cunion \unary{b}\]
> 
> 
> stands for all men and black things, without implication that the
> black things are to be taken besides the men. (DNLR [CP 3.67])
> 
> 
> 



Hence, Peirce's slightly modified addition sign, \(\cunion\) ,
denotes inclusive disjunction. "\(\unary{f} \cunion
\unary{u}\)" denotes all those who are either a Frenchman or a
violinist. The notation does not imply that no Frenchman is a
violinist or no violinist is a Frenchman. Even though Peirce's
example is limited to unary predicates, we can extend the idea to
binary. Using modern notation, 

\[\binary{l} \cunion \binary{s} = \{\langle x, y\rangle \mid lover(x, y) \lor servant(x,y)\}.\] 

 That is, it corresponds to
a union of relations.



When the multiplication sign enters the picture, the logic of
relations becomes powerful, and here is Peirce's hallmark for
the interpretation of multiplication:



> 
> 
> I shall adopt for the conception of multiplication *the application
> of a relation*, in such a way that, for example,
> \(\binary{l}\unary{w}\) shall denote whatever is a lover of a
> woman....\(\binary{s}(\unary{m} \cunion \unary{w})\) will, then,
> denote whatever is a servant of anything of the class composed of men
> and women taken together. (DNLR [CP 3.68])
> 
> 
> 



When polyadic predicates are in the picture, how to form a new
relation becomes more interesting and complicated. This is why the
multiplication operation of relative product is extremely important
for further work on relational logic. A product between two predicates
is much more interesting than an addition between two, depending on
what kinds of predicates are involved: 


1. The product between two properties is another property, being the
intersection between two properties,
2. the product of a relation and a property is another new property,
and
3. the product between relations produces a new relation.



Let's try to understand Peirce's concept of relative
product in terms of modern terminology:




Let


* "\(\unary{w}\)" be a unary predicate, being a
woman,
* "\(\unary{u}\)" be a unary predicate, being a
violinist,
* "\(\binary{l}\)" a binary predicate, being a lover
of, and
* "\(\binary{s}\)" be a binary one, being a servant of.



Then,


* \(\unary{w}\bcomma\unary{u} = \{x \mid \textit{woman}(x) \land
 \textit{violinist}(x)\}\).[12]
* \(\binary{l}\unary{w} = \{ x \mid \exists y (\textit{lover}(x, y)
\land \textit{woman}(y))\}\).
* \(\binary{l}\binary{s} = \{ \langle x, z\rangle \mid \exists y
(\textit{lover}(x, y) \land \textit{servant}(y, z))\}\).




In this modern translation, the existence of an existential quantifier
is noticeable, even though Peirce himself did not mention it at all in
DNLR.



Hidden quantifier implication becomes even more obvious in the
operation of involution below.



> 
> 
> I shall take involution in such a sense that \(x^y\) will denote
> everything which is an \(x\) for every individual of \(y\). Thus
> \(\binary{l}^{\unary{w}}\) will be a lover of every woman. (DNLR [CP
> 3.77])
> 
> 
> 



That is, \(\binary{l}^{\unary{w}} = \{ x \mid \forall y
(\textit{woman} (y) \rightarrow \textit{lover} (x,y))\}\). Here, a
universal quantifier is present!



Before we go into the details of Peirce's quantifiers,
let's summarize algebraic signs Peirce adopted to handle
polyadic predicates:




| Algebraic

Signs | Meanings/

Operations | Examples |
| --- | --- | --- |
| \(\inclusion\) | inclusion |  \(\unary{w} \inclusion \unary{u}\)

\(\quad\forall x (\textit{woman} (x) \rightarrow
\textit{violinist}(x))\)

\(\binary{l} \inclusion \binary{s}\)

\(\quad\forall x \forall y (\textit{lover}(x, y) \rightarrow
\textit{servant}(x, y))\) |
| \(\cunion\) | union | \(\unary{w} \cunion \binary{u}\)

\(\quad\{ x\mid \textit{woman}(x) \lor \textit{violinist}(x)\}\)

\(\binary{l} \cunion \binary{s}\)

\(\quad\{ \langle x, y \rangle \mid \textit{lover}(x,y) \lor
\textit{servant} (x, y)\}\) |
| \(\bcomma\) | intersection | \(\unary{w}\bcomma \unary{u}\)

\(\quad\{ x\mid \textit{woman}(x) \land \textit{violinist}(x)\}\) |
| (no comma) | relative product | \(\binary{l}\unary{w}\) (a lover of *some* woman)

\(\quad\{ x \mid \exists y (\textit{lover}(x, y) \land
\textit{woman}(y))\}\)

\(\binary{l}\binary{s} = \{ \langle x, z \rangle \mid \exists y
(\textit{lover}(x, y) \land \textit{servant}(y, z)\}\) |
| \(x^y\) | \(x\) of every \(y\) |  \(\binary{l}^{\unary{w}}\) (a lover of *every* woman)

\(\{ x \mid \forall y (\textit{woman} (y) \rightarrow \textit{lover}
(x,y)\}\) |



Let's focus on hidden but assumed presence of quantifiers in the
case of multiplication and exponentiation: \(\binary{l}\unary{w}\) is
interpreted as "a lover of some woman" and
\(l^{\unary{w}}\) as "a lover of every woman".
Interestingly, in the process of introducing polyadic predicates,
Peirce ends up bringing in quantifiers, *some* and
*every*. On the other hand, considering that Aristotle's
syllogisms have two quantifiers, Peirce's algebraic notation for
quantifiers--*some* and *every*--should not
surprise us. However, a crucial aspect of this development is that
Boole's unsatisfactory representation of existential
propositions (as opposed to universal propositions) pushed Peirce and
his student O. H. Mitchell to go beyond Boole's
 logic.[13]
 The way Peirce interprets a relative
product--"\(\binary{l}\unary{w}\)" meaning
"lover of some woman"--allows an existential
quantifier to be expressed implicitly in terms of multiplication.



Let us see several different ways Peirce pursued to represent
existential statements in a more explicit way. In the above for
"\(\binary{l}\unary{w}\)", the existential quantifier is
carried out in the way relation \(\binary{l}\) is applied to a unary
predicate "\(\unary{w}\)", but not explicitly. There are
several different explicit ways Peirce represents existential
statements. One method: borrowing exponentiation (in which a binary
predicate is applied to a unary predicate, e.g.,
\(\binary{l}^{\unary{w}}\)), Peirce expresses an existential statement
as a contradiction of a universal statement:



> 
> 
> Particular [existential] propositions are expressed by the
> consideration that they are contradictory of universal propositions.
> Thus, as \(\unary{h}\bcomma(1-\unary{b})=0\) means every horse is
> black, so \(0^{\unary{h}, (1-\unary{b})} = 0\) means that some horse
> is not black; and as \(\unary{h}\bcomma \unary{b}= 0\) means that no
> horse is black, so \(0^{\unary{h}, \unary{b}} = 0\) means that some
> horse is black. (DNLR [CP 3.141])
> 
> 
> 



The number 1 represents the universe class and 0 the null class.
However, Peirce's notation of exponentiation whose base is 0 has
some slightly different nuance. Let's recall Peirce's
exponentiation operation:

 

\[
\binary{l}^{\unary{w}} = \{ x \mid \forall y (\textit{woman} (y) \rightarrow \textit{lover}(x,y))\}.
\]


Then,

 

\[
0^{0} = \{ x \mid \forall y (\textit{null-class}(y) \rightarrow \textit{null-relation}(x,y))\}.
\]


[Note: The base 0 denotes a relation--the null relation, while
the exponent 0 a class--the null class.]



There is no \(y\) such that \(\textit{null-class}(y)\), since noting
could be in the *null-class*. Hence, vacuously, every object in
the domain gets in \(0^{0}\). That is, the class of things which has
no relation to the null class is the universe class which is
represented by 1. Hence, \(0^{0} = 1\).




Suppose \(\unary{m}\not = 0\).

 

\[
0^{\unary{m}} = \{ x \mid \forall y (\textit{non-null-class}(y) \rightarrow \textit{null-relation}(x,y))\}.
\]



Since nothing can bear the null-relation to every member of any
non-null class, \(0^{\unary{m}} = 0\). Hence, we get the following
exponentiation notation:




\[\tag{\*} 
\begin{align}
0^x = 0 &\quad \textrm{if } x \not =0\\
0^x = 1 &\quad \textrm{if } x =0\\
\end{align}
\]



Using this result, let's unpack Peirce's above
quotation:


1. 1 represents the universe class and 0 the null class.
(Boolean symbols)
2. "\(\unary{h}\bcomma(1-\unary{b})\)" denotes
the class of non-black horses. (Multiplication operation for
intersection)
3. "\(\unary{h}\bcomma(1-\unary{b})=0\)" says
that there is nothing that is non-black horse. I.e., every horse is
black. (by
 1
 and
 2)
4. "\(\unary{h}\bcomma(1-\unary{b}) \not = 0\)"
means it is not the case that every horse is black. I.e., some horse
is not black. (by
 3)
5. Since \(\unary{h}\bcomma (1-\unary{b}) \not = 0,
0^{\unary{h},(1-\unary{b})} = 0\) (by
 (\*)
 above)



Similarly, "\(\unary{h}\bcomma\unary{b}=0\)" means no
horse is black. Hence, "\(\unary{h}\bcomma \unary{b} \not =
0\)" says some horse is black. Therefore,
"\(0^{\unary{h}\bcomma\unary{b}} = 0\)" (being the
exponent being not-zero) means that some horse is black.



The method presented in DNLR [CP 3.141] is interesting for several reasons.
First of all, Peirce maintains Boole's theme that all
propositions are represented as equations. Another is that Peirce
utilizes the contradictory relation between universal and existential
propositions. Even more interestingly and importantly, Peirce brings
in his exponentiation notation, exponentiation between relation to
property (that is, between binary to unary predicates), to express
existential statements.



After the rather complicated presentation of existential statements,
focusing on the exponent part, Peirce suggests another, much simpler,
way to express existential statements by using the inequality
sign:



> 
> 
> Particular [Existential] propositions may also be expressed by means
> of the signs of inequality. Thus, some animals are horses, may be
> written \(\unary{a}\bcomma\unary{h} > 0\). (DNLR [CP 3.143])
> 
> 
> 



Another method Peirce adopts is to utilize the sign \(\inclusion\) for
inclusion and the sign \(\bar{ \ \ }\) for complement. That is,




|  |  |
| --- | --- |
|  All \(a\) is \(b\). |  \(a \inclusion b\) |
|  No \(a\) is \(b\).  |  \(a \inclusion \bar{b}\) |
|  Some \(a\) is \(b\).  |  \(\overline{[a \inclusion \bar{b}]}\) |
|  Some \(a\) is not \(b\).  |  \(\overline{[a \inclusion b]}\) |



However, these explicit ways to handle existential statements are
limited to unary predicates or Aristotelian syllogisms, and cannot go
beyond them. Instead, we would like to examine Peirce's
multiplication and exponential expressions between relations and
properties more carefully. Let's recall \(\binary{l}\unary{w}\)
means "a lover of some woman" while
\(\binary{l}^{\unary{w}}\) "a lover of every woman". Here
Peirce's proposal for individual terms gets in the picture so
that existential and universal quantifiers may become more explicitly
represented. Peirce suggests individuals be denoted by capitals (DNLR
[CP 3.96]). For example, for unary predicate \(\unary{w}\), if
\(\unary{w} > 0\), then \(\unary{w} = \unary{W}' \cunion
\unary{W}'' \cunion \unary{W}''' \cunion \cdots\),
where each of \(\unary{W}'\), \(\unary{W}''\),
\(\unary{W}'''\),...denotes an individual woman.
Hence,

 

\[
\begin{align}
\binary{l}\unary{w} & = \binary{l}(\unary{W}' \cunion \unary{W}'' \cunion \unary{W}''' \cunion \cdots)\\
&= \binary{l}\unary{W}' \cunion \binary{l}\unary{W}'' \cunion \binary{l}\unary{W}''' \cunion \cdots\\
\binary{l}^{\unary{w}} &= \binary{l}^{(\unary{W} ' \cunion \unary{W}'' \cunion \unary{W}''' \cunion \cdots)}\\
& = \binary{l}^{\unary{W}'}, \binary{l}^{\unary{W}''}, \binary{l}^{\unary{W}'''},\ldots\\
& = \binary{l}\unary{W}', \binary{l}\unary{W}'', \binary{l}\unary{W}''',\ldots (\binary{l}^{\unary{W}}=\binary{l}\unary{W}, \unary{W} \textrm{ being an individual term.})
 \end{align}
 \]


At this point, Peirce makes a connection (i) between existential
statements and the sign \(\Sigma\) (as logical addition) and (ii)
between universal statements and \(\Pi\) (as logical multiplication),
and the following passage is a precursor for the notations which
appear in subsequent papers:

 

\[\Pi' \inclusion \Sigma',\]


where \(\Pi'\) and \(\Sigma'\) signify that the addition and
the multiplication with commas are to be used. From this it follows
that

 

\[s^{\unary{w}} \inclusion s\unary{w}. \qquad \textrm{(DNLR [CP 3.97])}\]


We have now come quite close to a modern notation of quantifiers.
Dipert emphasizes the significance of Peirce's notations found
in DNLR:



> 
> 
> C. S. Peirce was the first person in the history of logic to use
> quantifier-like variable binding operators (briefly in 1870, W2, 392f,
> predating Frege's *Begriffsschrift* (1879)). (Dipert
> 2004: 290)
> 
> 
> 



Ten years later, in a more comprehensive and more survey-style paper,
"On the algebra of logic" (1880), we find Peirce's
ideas of DNLR emerge more tightly and more systematically. Below we
will now summarize the developments of quantification presented in
"Brief description of the algebra of relatives" (1882a),
"The logic of relatives" (1883a), and "On the
algebra of logic: A contribution to the philosophy of notations"
(1885a plus 1885b).



First, he modifies the previous idea of representing a property in
terms of individual terms and extends it to relations. In DNLR (1870),
"\(\unary{w}\)", which stands for the property
"being a woman", is expressed as

 

 \[\unary{w} = \unary{W}' + \unary{W}'' + \unary{W}''' + \cdots\] 


(where \(\unary{W}'\),... denotes each individual woman).
However, in 1882a, Peirce expresses a unary predicate using
coefficients: For each unary predicate \(x\) and for each object \(a\)
in the domain, Peirce defines the coefficient \((x)\_a\) in the following way.



Continuing the example for the unary predicate \(\unary{w},\) suppose
the domain has objects \(\unary{A},\) \(\unary{B},\)
 \(\unary{C},\)....[14]
 Some object is a woman and some is not. A coefficient
\((\unary{w})\_a\) is defined as follows:

 

\[
\begin{align}
(\unary{w})\_{a} A & = 1 \textrm{ if } A \textrm{ is woman.}\\
& = 0 \textrm{ if } A \textrm{ is not a woman.}\\
\end{align}
\]


Then,

 

\[
\begin{align}
\unary{w} & = (w)\_{a} A + (w)\_{b} B+ (w)\_{c} C + \cdots\\
& = \Sigma\_{i} (w)\_{i} \binary{I}.\\
\end{align}
\]


A unary predicate is successfully represented as a sum of individuals.
Moving on to a binary predicate, a relation is modeled by a pair of
 objects:[15]



> 
> 
> A dual relative term [binary predicate], such as "lover",
> "benefactor", "servant", is a common name
> signifying a pair of objects. (1883a [CP 3.328])
> 
> 
> 



And he expresses a pair of objects as "\(A:B\)", where
\(A\) and \(B\) are individual objects. Let \(\binary{l}\) stand for
"a lover". Peirce defines a coefficient for each ordered
pair of objects in the following way:

 

\[
\begin{align}
(\binary{l})\_{i, j} (A\_i: A\_j) & = 1 \textrm{ if } A\_i \textrm{ is a lover of } A\_j.\\
& = 0 \textrm{ if } A\_i \textrm{ is not a lover of } A\_j.\\
\end{align}
\]


Then,

 

\[
\begin{align}
\binary{l} & = (l)\_{1,1}(A\_{1}: A\_{1}) + (l)\_{1,2}(A\_{1}: A\_{2}) + (l)\_{2,1}(A\_{2}: A\_{1}) + (l)\_{2,2}(A\_{2}: A\_{2}) + \cdots\\
& = \Sigma\_{i} \Sigma\_{j}(l)\_{i,j}(A\_i: A\_j).\\
\end{align}
\]


Peirce's generalization for a unary predicate \(\unary{x}\) and
a binary predicate \(\binary{l}\) goes like this (1883a [CP
3.329]):

 

\[
\begin{align}
\unary{x} & = \Sigma\_{i} (\unary{x})\_i \binary{I} & \textrm{ (1882a [CP 3.306])}\\
\binary{l} & = \Sigma\_{i} \Sigma\_{j}(\binary{l})\_{i,j}(I: J) &\textrm{ (1883a [CP 3.329])}\\
\end{align}
\]


This is what Peirce had in mind when he wrote "Every term
[predicate] may be conceived as a limitless logical sum of
individuals" (1880 [CP 3.217]). We take only objects whose
coefficients are 1. Suppose \(A\), \(B\), and \(D\) are women.
Applying Peirce's sign "+" as inclusive-or,
\(\unary{w} = \{ A, B, D\}\). Suppose \(A\) is a lover of \(C\), and
\(B\) is a lover of \(D\). Then, \(l = \{( A: C),
(B:
 D)\}\).[16]



The next task is how to utilize this tool to express an existential
proposition. If there is at least one woman, say \(K\), in the domain,
there is at least one coefficient \(\unary{w}\_k\) such that it is 1.
Hence, the sum of coefficients of the individuals in the domain is
greater than 0. That is, \(\Sigma\_{i} w\_i > 0\). If nobody is a
woman, we will get \(\Sigma\_{i} w\_i = 0\). In the case of the
universal proposition that everybody is a woman, the product of
coefficients is 0 as long as one coefficient is 0, that is, there is
one person who is not a woman. That is, \(\Pi\_{i} w\_i = 0\). When
everybody is a woman, \(\Pi \_{i} w\_i = 1\).



While the Boolean approach for quantifiers is limited to term logic,
this way of handling quantifiers is so general that we can apply it
directly to relations as well, as the following passage states:



> 
> 
> Any proposition whatever is equivalent to saying that some complexus
> of aggregates [sums] and products of such numerical coefficients is
> greater than zero. Thus,
> 
>  
> 
> \[\Sigma\_{i}\Sigma\_{j} \binary{l}\_{ij} > 0\]
> 
> 
> means that something is a lover of something; and
> 
>  
> 
> \[\Pi\_{i}\Sigma\_{j} \binary{l}\_{ij} > 0.\]
> 
> 
> means that everything is a lover of something. (1883a [CP 3.351])
> 
> 
> 



And Peirce proposes to drop the \(">0"\) part:



> 
> 
> We shall, however, naturally omit, in writing the inequality, the
> \(">0"\) which terminates them all; and the above two
> propositions will appear as
> 
> 
> 
> 
> \(\Sigma\_{i}\Sigma\_{j} \binary{l}\_{ij}\) and \(\Pi\_{i}\Sigma\_{j}
> \binary{l}\_{ij}.\qquad\) (1883a [CP 3.351])
> 
> 
> 
> 



Getting to Peirce's 1883 paper, we witness two significant steps
taken: One is the use of index-subscripts in a crucial way, and the
other is the alternation of \(\Sigma\) and \(\Pi\). After dropping
"\(>0\)" Peirce draws our attention to the role of
subscripts:



> 
> 
> The following are other examples:
> 
>  
> 
> \[\Pi\_i \Sigma\_j (l)\_{ij} (b)\_{ij}\]
> 
> 
> means that everything is at once a lover and a benefactor of
> something.
> 
>  
> 
> \[\Pi\_i \Sigma\_j (l)\_{ij} (b)\_{ji}\]
> 
> 
> means that everything is a lover of a benefactor of itself. (1883a [CP 3.352])
> 
> 
> 



The order of index-subscripts is crucial to make a distinction between
these two propositions. On the other hand, a difference between
"Everybody loves some woman" and "There is a woman
everybody loves" relies on the order of \(\Pi\) and
\(\Sigma\):

 

\[\Pi\_i \Sigma\_j (\binary{l})\_{ij} ({\unary{w}})\_j \quad \textrm{ vs.}\quad \Sigma\_j \Pi\_i (\binary{l})\_{ij }({\unary{w}})\_j.\]


Finally in "On the Algebra of Logic: A contribution to the
philosophy of notation" (1885a) all of these
developments--(i) \(\Sigma\) (sum) for *some* and \(\Pi\)
(product) for *all*, (ii) utilizing coefficients and their
subscripts to drop denotation of individuals (e.g., from \((l)\_{ij}
(A\_{i}: A\_{j})\) to \((l)\_{ij}\)), (iii) mixing \(\Sigma\) and
\(\Pi\), and (iv) omitting the "\(>0\)" part--have
become official:




In general, according to 1885a [CP 3.393]:


\(\Sigma\_i x\_i\) means that \(x\) is true of some one of the
individuals noted by \(i\) or

 

 \[\Sigma\_{i}x\_i = x\_i + x\_j + x\_k + \textit{etc.}\]


In the same way, \(\Pi\_i x\_i\) means that \(x\) is true of all these
individuals, or

 

 \[\Pi\_{i}x\_i = x\_i x\_j x\_k, \textit{etc.}\]


And if \(x\) is a simple relation [binary predicate],


* \(\Pi\_i\Pi\_j x\_{i,j}\) means that every \(i\) is in this relation
to every \(j\),
* \(\Pi\_j\Sigma\_i x\_{i,j}\) that every \(j\) some \(i\) or other is
in this relation,
* \(\Sigma\_i\Sigma\_j x\_{i,j}\) that some \(i\) is in this relation
to some \(j\).

 
For example, according to 1885a [CP 3.394]:


Let \(l\_{ij}\) mean that \(i\) is a lover of \(j\) and \(b\_{ij}\) that
\(i\) is a benefactor of \(j\).



Let \(g\_i\) mean that \(i\) is a griffin, and \(c\_i\) that \(i\) is a
chimera.



Then \(\Pi\_i\Sigma\_j (l)\_{ij }(b)\_{ij}\) means that everything is at
once a lover and a benefactor of something; and


* [I.e., \(\forall x \exists y [\textit{Lover} (x, y) \land
\textit{Benefactor} (x, y)]\), meaning that everybody is a lover and a
benefactor of someone.]



And \(\Pi\_i\Sigma\_j (l)\_{ij}(b)\_{ji}\) means that everything is a
lover of a benefactor of itself.


* [I.e., \(\forall x \exists y [\textit{Lover} (x, y) \land
\textit{Benefactor} (y, x)]\), meaning that everybody is a lover of a
benefactor of himself/herself.]

 


And we find ourselves arriving at the land of modern logic. At the
same time, we realize that the key concepts and vocabulary of
first-order logic were already formed in his previous work discussed
above. We also note that the goal outlined in the first section of the
1885 paper is more or less the same as the proposal made in his 1870
paper: "The first is the extension of the power of logical
algebra over the whole of its proper realm" (1885a [CP 3.364]).
Also, he almost reiterates the limit of the project which he
"regrets" in 1870: "I shall not be able to perfect
the algebra sufficiently to give facile methods of reaching logical
conclusions" (1885a [CP 3.364]). That is, we should not expect a
full-blown deductive system in this paper, but "I can only give
a method by which any legitimate conclusion may be reached and any
fallacious one avoided" (1885a [CP 3.364]). He carries out his
promise in section 3 of the paper titled as "SS3.
First-intentional logic of
 relatives",[17]
 by suggesting a list of methods of transformation. He did not mean to
claim this list is exhaustive, but "the one which seems to me
the most useful tool on the whole" (1885a [CP 3.396]). The
following are some of the rules involving
 quantifiers:[18]




\[
\begin{align}
\forall x \phi(x) \land \forall y \phi(y) & = \forall x\forall y (\phi(x) \land \phi(y))\\
\exists x \phi(x) \land \forall y \phi(y) & = \exists x\forall y (\phi(x) \land \phi(y))\\
\exists x \phi(x) \land \exists y \phi(y) & = \exists x\exists y (\phi(x) \land \phi(y))\\
\end{align}
\]

 

\[
\begin{align}
\forall x \forall y \chi(x,y) & = \forall y\forall x \chi(x,y)\\
\exists x \exists y \chi(x,y) & = \exists y\exists x \chi(x,y)\\
\forall x \exists y (\phi(x) \land \psi(y)) & = \exists y\forall x (\phi(x) \land \psi(y))\\
\forall x \exists y \chi(x,y) & \not = \exists y\forall x \chi(x,y), \textrm{ but}\\ 
\exists x\forall y \chi(x,y) & \Rightarrow \forall y \exists x \chi(x,y)\\
\end{align}
\]

 

\[\exists x \forall y \chi(x,y) = \exists x \forall y (\chi(x,y)\land \chi(x,x))\]


In spite of the six-year interval between Frege's
*Begriffsschrift* (1879) and Peirce's quantification
theory in 1885, credit has been given to both logicians. We call them
both the founders of modern logic, since Peirce was not aware of
Frege's work on the topic. Also, it should be noted that Frege
presented a logical system equipped with axioms and rules, which was
not pursued in Peirce's work.


### 1.2 Boolean tradition--algebraic and model-theoretic



Boole's aspiration to capture logic in terms of an algebraic
system has inspired many mathematicians and logicians who are
interested in connecting the two disciplines, logic and mathematics.
As seen in the previous section, Peirce is clearly one of them. He
pushed the Boolean idea of an algebraic system further in two ways:
One is to improve Boole's representation of particular
proposition (i.e., "Some *A* is *B*") so that
traditional Aristotelian syllogisms may fit in an algebraic system.
The other is to represent not only qualities but also relations so
that a new algebraic system may reach beyond traditional syllogisms.
In that process, new notations for quantifiers and variables were
invented.



Peirce's two-decade work made a major contribution to the
algebra of logic tradition in two important ways. First,
Peirce's introduction of quantifiers and variables itself is a
significant advance in formal logic, close to the predicate logic we
know. Second, subsequent momentous work on mathematical logic was
built on the new notation and extended logic by Peirce and his
students, O. H. Mitchell and C. Ladd (later Ladd-Franklin). A decade
after Peirce's "On the Algebra of Logic", (1885)
Ernst Schroder published three volumes of mathematical logic
*Vorlesungen uber die Algebra der Logik*
 (1890-1905).[19]
 His work was carried out squarely in the Boolean algebraic tradition,
and two important aspects of the book reflect Peirce's
influence: He adopted Peirce's notation (over Frege's),
and the third volume is devoted to the logic of relations.
Goldfarb's insightful paper expresses these two aspects of
Peirce's influence in the following way:



> 
> 
> Building on earlier work of Peirce, in the third volume of his
> *Lectures on the algebra of logic* [1895] Schroder
> develops the calculus of relatives (that is, relations). Quantifiers
> are defined as certain possibly infinite sums and products, over
> individuals or other relations. (Goldfarb 1979: 354)
> 
> 
> 



As seen in the previous subsection, including relations (not just
qualities) in algebraic expressions and representing the universal
quantifier as products (i.e., \(\Pi\)) and the existential quantifier
as sums (i.e., \(\Sigma\)) were the main output of Peirce's two
decade tireless work. Considering Schroder's book was the
most popular logic textbook for mathematical logic students during
that era, we can easily say Peirce's legacy has lived on.
Peckhaus, working on a delicate relation between Frege's and
Schroder's quantification theory, locates where
Schroder's modern quantification theory originates from:




> 
> 
> This [Schroder's *Vorlesungen uber die Algebra der
> Logik* was the result of his learning modern quantification from
> Frege's *Begriffsschrift*] is a simple and plausible
> answer, but it is false. Schroder never claimed any priority for
> his quantification theory, but he did not take it from Frege.
> Schroder himself gives the credit for his use of \(\Sigma\) and
> \(\Pi\) to Charles S. Peirce and Peirce's student Oscar Howard
> Mitchell (Schroder 1891, 120-121). (Peckhaus 2004: 12) 
> 
> 



Afterwards well-known mathematicians and logicians, Lowenheim,
Skolem and Zermelo, all used Peirce-Schroder notation. Peano was
also very much familiar with Peirce-Schroder algebraic logic.
Putnam includes Whitehead in this tradition as well:



> 
> 
> This [Whitehead's *Universal Algebra*] is a work squarely
> in the tradition to which Boole, Schroder, and Peirce belonged,
> the tradition that treated general algebra and logic as virtually one
> subject. (Putnam 1982: 298)
> 
> 
> 



Interestingly enough, Putnam points out that this portion of
Whitehead's work was prior to his collaboration with Russell,
and during this early period Whitehead's work, especially on
quantifiers, mentions Peirce and his students, but not Frege. Clearly
we needed to wait until Russell drew our attention to Frege, but
"it was Peirce who seems to have been known to the entire world
logical community" (Putnam 1982: 297). Putnam's label of
the Peirce group as "effective" discoverers of the
quantifier and Frege as a discoverer could be a resolution to the
Frege-first versus Peirce-first debate.



While many have focused on the development of quantifiers, it is quite
noteworthy that Tarski drew our attention to the importance of the
algebra of relations in his 1941 paper "On the Calculus of
 Relations".[20]
 Just as Peirce did in his DNLR (1870) paper, Tarski acknowledges De
Morgan's contribution: It is De Morgan who first realized the
necessity of representing relations as well as qualities and struggled
over the limits of traditional logic. And Tarski gives full credit to
Peirce in terms of solid advance over the calculus of relations.



> 
> 
> The title of creator of the theory of relations was reserved for C. S.
> Peirce. In several papers published between 1870 and 1882a, he
> introduced and made precise all the fundamental concepts of the theory
> of relations and formulated and established its fundamental
> laws....In particular, his investigations made it clear that a
> large part of the theory of relations can be presented as a calculus
> which is formally much like the calculus of classes developed by G.
> Boole and W. S. Jevons, but which greatly exceeds it in richness of
> expression and is therefore incomparably more interesting from the
> deductive point of view. (Tarski 1941: 73)
> 
> 
> 



This passage not only situates Peirce's logical achievement in
the context of the algebra of logic tradition but also characterizes
Peirce's work as an extension of Boole's and Jevons'
monadic logic. (For more details about Peirce's position in
Boole's tradition, see the entry
 the algebra of logic tradition.)
 



Some Peirce scholars have also claimed that Peirce's invention
of quantifiers is a product of Peirce's own philosophy of logic,
which is different from Frege's (Brady 1997; Burch 1997; Iliff
1997; Merrill 1997). Hintikka's proposal (1997) to explain the
main difference between Frege's and Peirce's contributions
to modern logic is quite intriguing. Tracing back to Frege's own
distinction between *calculus ratiocinator* versus *a lingua
characterica*, van Heijenoort adds a new dimension to these two
opposing views of logic beyond what Frege alluded to (van Heijenoort
1967: footnote 1, p.
 329).[21]
 While Frege emphasized a difference between propositional and
quantification logic, van Heijenoort located a difference in what is
taken as a totality. Boole's tradition does not make any
ontological commitment about a totality, but it "can be changed
at will" (1967: 325). On the other hand, Frege's language
is about *the* universe. Borrowing van Heijenoort's
distinction between Boole's logic as a calculus and
Frege's universality of logic, Hintikka locates Peirce in
Boole's camp, calling it the model-theoretic tradition. Unlike
Frege's view of the universe, the model-theoretic tradition
allows us to reinterpret a language and thus assign different
universes to quantifiers. According to Hintikka, Peirce's
development of modal logic is a good piece of evidence to show how
fruitful Peirce's way of understanding quantifiers could be
(Hintikka 1997). In the next section where Peirce's graphical
systems are introduced, we will revisit this issue.


## 2. From Symbolic to Iconic Representation



So far, we have argued that Peirce's insight on relations pushed
him to extend the territory of logic *from* monadic,
non-relational, propositional logic *to* polyadic, relational,
quantification logic. This is the beginning of modern logic as we know
it. In this section, taking up a different angle of Peirce's
adventure--to extend forms of representation *from*
symbolic systems *to* diagrammatic systems, we present a story
where his two different kinds of extension--one from
non-relations to relations and the other from symbolic to
diagrammatic--are connected with each other.



Peirce presented propositional logic, quantification logic, and modal
logic in a graphical way, and invented three systems of Existential
Graphs (EG)--Alpha, Beta, and Gamma, respectively. In spite of
Peirce's own evaluation of Existential Graphs as "my chef
d'oeuvre", EG had to wait to be understood for a half
century until two philosophers--Don Roberts and Jay
Zeman--produced their impressive work. In the 1980s, EG was
receiving attention from new disciplines--computer science and
artificial intelligence--thanks to John Sowa's novel
application of EG to knowledge representation in *Conceptual
Structure* (1984). More recently, toward the end of the twentieth
century interdisciplinary research on multi-modal reasoning has drawn
our attention to non-symbolic systems (see, e.g., Barwise &
Allwein [eds] 1996 and Barwise & Etchemendy 1991) and EG, not
surprisingly, occupied the top of their list. In that context, Shin
(2002) focused on differences between symbolic versus diagrammatic
systems and suggested a new way of understanding the EG system, though
this was criticized in Pietarinen 2006.



While Peirce mainly presented linear expressions in his official
writings from 1870 to
 1885,[22]
 the notation adopted in Frege's 1879 *Begriffsschrift*
is more iconic; it is at least not as linear as Peirce's in the
above period. However, it is Peirce, not Frege, who invented a
full-blown non-symbolic system for first-order logic--Existential
Graphs. It is Peirce's EG, not his linear first-order notation,
which is presented as a deductive system with inference rules. As the
EG system has been investigated more rigorously, philosophical
questions involving Peirce's invention of the system have been
raised as well. The discovery of EG's power and novelty has
naturally led us to other parts of Peirce's philosophy. Why and
how did the invention of EG come about? What does EG reveal about
Peirce's view of logic and representation?



Many have pointed to Peirce's theory of signs, which classifies
signs as being of three kinds--symbols, indices, and
icons--as the foremost theoretical background for Peirce's
 EG.[23]
 For example, as will be shown below, ovals and lines, along with
letters, are the basic vocabulary of Peirce's EG. It is natural
to connect Peirce's interest in icons with his invention of
graphical systems, and the connection is real (Shin 2002:
22-35). However, to pinpoint the features of icons and the
iconic nature of Peirce's graphical systems requires much more
work than our intuition provides. Moreover, there is a big gap between
Peirce's discussions of
 icons[24]
 and his invention of full-blown graphical systems; something else has
to be brought into the picture to explain how Peirce got from his
initial ideas about icons all the way to his EG.



In a slightly different and bigger picture, van Heijenoort's
distinction between Boole's *calculus ratiocinator*
versus Frege's *lingua characteristica* could be related
to the topic. Agreeing with both Hintikka's and Goldfarb's
evaluation that Peirce belongs to Boole's tradition, Shin finds
a connection between the model-theoretic view of logic (where Boole
and Peirce are placed) and EG's birth (see Shin 2002:
14-16 and Pietarinen 2006). However, Peirce's awareness of
the re-interpretation of language is necessary, but not sufficient,
for his pursuit of a different form of representation. While the
acknowledgment of the possibility of different models of a given
system was presupposed by Peirce's project for various kinds of
systems, not every Boolean has presented multiple systems. Boole
himself 



> 
> 
> was quite conscious of the idea of *disinterpretaion*, of the
> idea of using a mathematical system as an algorithm, transforming the
> signs purely mechanically without any reliance on meanings. (Putnam
> 1982: 294) 
> 
> 
> 



On the other hand, Burris and Legris's entry shows us how
Boole's algebra of logic tradition has led us to the development
of model theory (see the entry on
 the algebra of logic tradition).


### 2.1 Pragmatic maxim applied to the logic of relations



Without challenging these existing explanations involving
Peirce's EG, in this entry we would like to bring in one
overlooked but crucial aspect of Peirce's journey to EG so that
our story may fill in part of the puzzle of Peirce's overall
philosophy. Peirce's mission for a new logic started with how to
represent relations, which led him to invent quantifiers and bound
variables, as we discussed in the previous section. The same
commitment, that is, to represent relations in a logical system, we
claim, was a main motivation behind Peirce's search for a new
kind of sign systems--iconic representation of relations.
Peirce's work on Euler/Venn diagrams provides us with another
piece of evidence to support our claim that the main motivation behind
EG was to represent relations. While improving Venn systems, Peirce
realizes that the following defect cannot be eliminated:



> 
> 
> [T]he system [Venn's] affords no means of exhibiting reasoning,
> the gist of which is of a relational or abstractional kind. It does
> not extend to the logic of relatives. (Peirce 1911b [CP 4.356])
> 
> 
> 



Again, we do not think this is *the* crucial ingredient for the
creation of EG, but one key element which works nicely together with
his theory of signs and his model-theoretic view of logic.



Peirce's graphical representation first appears in his 1897
paper "The Logic of Relatives". After his own new linear
notation came out in 1885 as seen above, why did Peirce revisit the
logic of relations? The first paragraph of the paper provides a direct
answer:



> 
> 
> I desire to convey some idea of what the new logic is, how two
> "algebras", that is, systems of diagrammatical
> representation by means of letters and other characters, more or less
> analogous to those of the algebra of arithmetic, have been invented
> for the study of the logic of relatives, and.... (1897a [CP
> 3.456])
> 
> 
> 



Two things should be noted. One is that diagrammatic systems are also
called "algebra" by Peirce. That is, according to Peirce,
algebra is not limited to symbolic systems. The other is that Peirce
makes it clear that two different forms of algebra carry out
*the* new logic, not new logic*s*.



In thinking about the scope of the logic of relations, the question
arises: Why did Pierce feel the need for another form of
representation different from the 1885 notation? "I must clearly
show what a relation is" (1897a [CP 3.456]). The clear
understanding of "relations", Peirce believes, is a guide
for his excursion into different forms of logical systems. Here we
would like to draw reader's attention to Peirce's
well-known paper "How To Make Our Ideas Clear " (1878),
where three sections are devoted to the three grades of meaning (see
the entry on
 Peirce's theory of signs).



The first grade of understanding the word "relation" comes
from our ordinary experience, and the second grade is to have a more
abstract and general definition-like understanding. According to
Peirce, that is not enough to achieve a full understanding of the word
"relation". Finally, Peirce's hallmark of the
pragmatic maxim leads us to the third grade of clarity:



> 
> 
> It appears, then, the rule for attaining the third grade of clearness
> of apprehension is as follows: Consider what effects, which might
> conceivably have practical bearings, we conceive the object of our
> conception to have. Then the whole of our conception of those effects
> is the whole of our conception of the object. (1878 [CP 5.402])
> 
> 
> 



In order to understand what a relation is, we need to know what
follows from it. Then, the question is how we know what its
consequences are. Here is one answer given by Peirce in the 1897a
paper, as far as the term "relation" goes:



> 
> 
> The third grade of clearness consists in such a representation of the
> idea that fruitful reasoning can be made to turn upon it, and that it
> can be applied to the resolution of difficult practical problems.
> (1897a [CP 3.457])
> 
> 
> 



Therefore, how a relation is represented is crucial in figuring out
what follows from a relational state of affairs. Better
representations will yield more "fruitful reasoning" and
hence, will be more helpful for solving practical problems. It is
obvious that in the paper Peirce intends to search for more
*desirable* representations. Importantly, in section 4 when the
third grade of clearness of the meaning "relation" is
discussed, diagrammatic representation of relations makes its first
appearance.



Influenced by A. B. Kempe's graphic
 representation,[25]
 Peirce finds an analogy between relations and chemical compounds:



> 
> 
> A chemical atom is quite like a relative in having a definite number
> of loose ends or "unsaturated bonds", corresponding to the
> blanks of the relative. (1897a [CP 3.469])
> 
> 
> 



A chemical molecule consists of chemical atoms, and how atoms are
connected with one another is based on the number of loose ends of
each atom. For example, chemical atom H has one loose end and chemical
atom O has two. So, the following combination is possible, and it is a
representation of the water molecule, H2O:



![The letter O with lines connecting it to two different letter Hs](Atom.svg)


An analogy to the logic of relations runs like this: A sentence
consists of names (proper names or indices) and predicates, and each
predicate has a fixed arity. For example, the predicate
"love" needs two names and "give" three.
Hence, the following diagrammatic representation is grammatical and it
is a representation of the proposition "John loves
Mary".



![The word Love with lines connecting it to the word John and the word Mary](Love.svg)


Peirce created a novel and productive analogy in representation
between chemistry and the logic of relation by adopting the doctrine
of valency as the key element for the analogy, as shown in the above
two diagrams. Believing that this graphic style of representation
would help us conceive the consequences or effects of a given relation
in a more efficient
 way,[26]
 Peirce presents Entitative Graphs, which is a predecessor of
 EG.[27]



EG keeps the representation of a relation developed here, and remains
as Peirce's final and the most cherished notation for the logic
of relations (1903a). EG consists of three parts, Alpha, Beta, and
Gamma, which correspond to propositional, first-order, and modal
logic, respectively. After presenting the Alpha system in a formal
way, we discuss the Beta system of EG focusing on Peirce's novel
ideas in expanding a propositional graphic system to a
quantificational graphic system. For more details, we recommend works
on EG by Roberts, Zeman, Sowa, and Shin.


### 2.2 Alpha system



Peirce's Alpha graphs may be drawn on a blackboard, on a
whiteboard, or on a sheet of paper. The basic unit is a simple
sentence without any sentential connectives, that is, negation,
conjunction, disjunction or conditional, etc. The following is an
example of a basic Alpha graph, asserting that it is sunny.



![the sentence: It is sunny.](Assertion.svg)


When we would like to assert that it is sunny and windy, we juxtapose
two basic Alpha graphs in the following way:



![the sentence 'It is sunny.' next to the sentence 'It is windy.'](Juxtaposition.svg)


In order to make the Alpha Graph Boolean-functionally complete, all we
need is to represent negation. The following Alpha graph says that it
is not the case it is sunny, by enclosing the above graph with a
cut:



![In an oval the sentence 'It is sunny.'](Cut_1.svg)


When we have negation and conjunction, it is important to keep the
order in the right way. "It is not sunny and it is not
windy" is different from "It is not the case that it is
sunny and windy". Hence, the sentence "It is not the case
that it is sunny and it is windy" is ambiguous, depending on the
scope of "it is not the case". In the case of sentential
logic, parentheses get in to prevent this ambiguity: \(\neg (S \land
W)\) versus \(\neg S \land W\). Peirce's warning follows:



> 
> 
> The interpretation of existential graphs is *endoporeutic*,
> that is proceeds inwardly; so that a nest sucks the meaning from
> without inwards unto its centre, as a sponge absorbs water. (Peirce
> 1910a: 18, Ms 650)
> 
> 
> 



Hence, the following Alpha graph should be read *not* as
"\(\neg P \land \neg Q\)", *but* as "\(\neg(P
\land \neg Q)\)":



![The letter Q in an oval with another oval enclosing both the first oval and the letter P.](Endo.svg)


This way of understanding Alpha Graphs is not incorrect, but has given
a wrong impression that the Alpha system is equivalent to a sentential
system with two connective symbols, negation and conjunction. We all
prefer having more connectives than these two, especially when we use
the language. The section explores an alternative reading of Alpha
diagrams, beyond negation and conjunction only, without introducing
any new syntactic device.



Below we introduce Alpha Graphs as a formal system equipped with its
syntax and semantics. These tools not being available to Peirce, the
presentation aims to show Peirce's EG is not intrinsically
different from other formal systems. At the same time, in order to
place Peirce's graphical systems in the traditional
well-developed discourse of logic, there will be an intermediate
stage, that is, to read off Peirce's graphs into symbolic
language. This will make Peirce's graphs more accessible, and at
the same time support our claim that Peirce extended forms of
representations with the same scope of logic as symbolic
representation.


#### Syntax


##### Vocabulary


1. Sentence symbols: \(A\_{1},\) \(A\_{2},\)...
2. Cut



![an oval](Cut_2.svg)


##### Well-formed diagrams


1. An empty space is a well-formed diagram.
2. A sentence symbol is a well-formed diagram.
3. If \(D\) is a well-formed diagram, then so is a single cut of
\(D\) (we write "\([D]\)]").
4. If \(D\_{1}\) and \(D\_{2}\) are well-formed diagrams, then so is
the juxtaposition of \(D\_{1}\) and \(D\_{2}\) (write "\(D\_{1}\
D\_{2}\)").
5. Nothing else is a well-formed diagram.



Here we present two equivalent reading methods for the system. The
Endoporeutic reading algorithm, formalized based on Peirce's own
suggestion (as quoted above), is a traditional way to understand EG.
An alternative reading method, the Multiple reading algorithm, was
more recently presented to approach EG in a more efficient
 way.[28]


#### Endoporeutic Reading Algorithm


1. If \(D\) is an empty space, then it is translated into
\(\top\).
2. If \(D\) is a sentence letter, say \(A\_{i}\), then it is
translated into \(A\_{i}\).
3. Suppose the translation of \(D\) is \(\alpha\). Then, \([D]\) is
translated into \((\neg \alpha)\).
4. Suppose the translation of \(D\_{1}\) is \(\alpha\_{1}\) and the
translation of \(D\_{2}\) is \(\alpha\_{2}\).



Then, the translation of \(D\_{1}\ D\_{2}\) is \((\alpha\_{1} \land
\alpha\_{2})\).


#### Multiple Readings Algorithm


1. If \(D\) is an empty space, then it is translated into
\(\top\).
2. If \(D\) is a sentence letter, say \(A\_{i}\), then it is
translated into \(A\_{i}\).
3. Suppose the translation of \(D\) is \(\alpha\). Then,
\([D]\) is translated into \((\neg \alpha)\).
4. Suppose the translation of \(D\_{1}\) is \(\alpha\_{1}\)
and the translation of \(D\_{2}\) is \(\alpha\_{2}\).
	1. the translation of \(D\_{1} D\_{2}\) is \((\alpha\_{1}
	\land \alpha\_{2})\),
	2. the translation of \([D\_{1} D\_{2}]\) is
	\((\neg\alpha\_{1} \lor \neg \alpha\_{2})\),
	3. the translation of \([D\_{1}\ [D\_{2}]]\) is \((\alpha\_{1}
	\rightarrow \alpha\_{2})\), and
	4. the translation of \([[D\_{1}]\ [D\_{2}]]\) is
	\((\alpha\_{1} \lor \alpha\_{2})\).



Each of these two readings has its own 
 strength.[29]
 The Endopreutic
reading assures us that the Alpha system is truth-functionally
complete, since it has power to express conjunction and negation.
However, this traditional method has been partly responsible for the
following two incorrect judgments about Alpha graphs: 


1. There is not much difference between the Alpha system and a
propositional language with only two connectives, \(\land\) and
\(\neg\), except that Alpha graphs have cuts instead of symbolic
connectives.
2. When it comes down to practical use, just as we do not want to
use only two connectives in a language, we have no reason to adopt the
Alpha system over propositional languages with more connectives.



Challenging these misconceptions, the Multiple readings algorithm
shows that Alpha diagrams do not have to be read off as a sentence
with "\(\land\)" and "\(\neg\)" only, but can
be directly read off in terms of other connectives as well. Two
questions may be raised: 


1. Is there a redundancy in the Multiple readings method? For
example, is
 clause 4(b)
 above dispensable in terms of
 clause 3
 and
 clause 4(a)?
2. Does this new reading show that the Alpha system is just like a
propositional language with various connectives?



Let us answer these questions through the following example.


##### Example



The following graph is translated into the following four
formulas:



![an oval enclosing two other disjoint ovals; one containing the letter R and the other the letter S](AlphaT.svg)


|  |  |  |
| --- | --- | --- |
| 1. | \(\neg(\neg R \land \neg S)\) | Endoporeutic Reading |
| 2. | \(R \lor S\) | 4(d)
 of Multiple Readings |
| 3. | \(\neg R \rightarrow S\) | 3
 and
 4(c)
 of Multiple Readings |
| 4. | \(\neg\neg R\lor \neg\neg S\) | 3
 and
 4(b)
 of Multiple Readings |




The Endoporeutic reading allows us to get the first reading only, but
we may obtain different sentences by the Multiple Readings. Of course,
all of these sentences are logically equivalent. Here is an
interesting point: In the case of symbolic systems, we need to prove
the equivalence among the above sentences by using inference rules.
But, derivation processes are dispensable in the case of the Alpha
system when the Multiple readings are
 adopted.[30]
 Hence, having the
 clause 4(b)
 above in addition to
 clause 3
 and
 clause 4(a)
 is not redundant, but instead highlights a fundamental difference
between the Alpha system and a symbolic language with various
connectives (see Shin 2002: SSSS4.3.2, 4.4.4, and 4.5.3).



Since we have the semantics for propositional logic and our reading
methods translate Alpha diagrams into a propositional language, we can
live without the direct semantics. However, if one insists on the
direct semantics:


#### Semantics



Let \(v\) be a truth function such that it assigns t or f to each
sentence letter and t to an empty space. Now, we extend this function
to \(\overline{v}\) as follows:


1. \(\overline{v}(D) = v(D)\) if \(D\) is a sentence symbol or an empty
space.
2. \(\overline{v} ( {\bf [}D{\bf ]})\) = t iff \(\overline{v}(D)\) =
f.
3. \(\overline{v}(D\_{1} D\_{2})\) = t iff \(\overline{v}(D\_{1}) =\)t and
\(\overline{v}(D\_{2})\) = t.



We also would like to emphasize that this is not the only way to
approach Peirce's EG. For example, some claim that
game-theoretic semantics were foreshadowed by Peirce, and thus argue
for a more dynamic understanding of EG from the game-theoretic point
of view (Burch 1994; Hilpinen 1982; Hintikka 1997; Pietarinen
2006).



Peirce makes it clear his EG is a deductive system equipped with
inference rules:



> 
> 
> *The System of Existential Graphs* is a certain class of
> diagrams upon which it is permitted to operate certain
> transformations. (1903a [CP 4.414])
> 
> 
> 



The inference rules for the Alpha system are presented as follows:
(1903a [CP
 4.415])[31]




**Code of Permissions**


* Permission No. 1. In each special problem such graphs may be
scribed on the sheet of assertion as the conditions of the special
problem may warrant.
* Permission No. 2. Any graph on the sheet of assertion may be
erased, except an enclosure with its area entirely blank.
* Permission No. 3. Whatever graph it is permitted to scribe on the
sheet of assertion, it is permitted to scribe on any unoccupied part
of the sheet of assertion, regardless of what is already on the sheet
of assertion.
* Permission No. 4. Any graph which is scribed on the inner area of
a double cut on the sheet of assertion may be scribed on the sheet of
assertion.
* Permission No. 5. A double cut may be drawn on the sheet of
assertion; and any graph that is scribed on the sheet of assertion may
be scribed on the inner area of any double cut on the sheet of
assertion.
* Permission No. 6. The reverse of any transformation that would be
permissible on the sheet of assertion is permissible on the area of
any cut that is upon the sheet of assertion.
* Permission No. 7. Whenever we are permitted to scribe any graph we
like upon the sheet of assertion, we are authorized to declare that
the conditions of the special problem are absurd.




Emphasizing the symmetry both in erasure versus insertion and in even
versus odd number of cuts, Shin rewrote the rules (Shin 2002:
84-85):


#### Reformulated Transformation Rules


1. RR1: In an
 E-area,[32]
 say, area \(a\),


	1. we may *erase* any graph, and
	2. we may *draw* graph \(X\), if there is a token of \(X\)
	
	
		1. in the same area, i.e., area \(a\), or
		2. in the next-outer area from area \(a\).
2. RR2: In an
 O-area,[33]
 say, area \(a\),


	1. we may *erase* graph \(X\), if there is another token of
	\(X\)
	
	
		1. in the same area, i.e., area \(a\), or
		2. in the next-outer area from area \(a\), and
	2. we may *draw* any graph.
3. RR3: A double cut may be erased or drawn around any part of a
graph.



For examples of deduction sequences, refer to Roberts (1973:
45-46) and Shin (2002: 91).


### 2.3 Beta system



In
 SS1.1,
 we showed that formalizing relations was a key motivation behind
Peirce's new logic--first-order logic. In
 SS2.1,
 we established a connection between Peirce's own pragmatic
maxim and his graphic representation of relations. Peirce did not aim
to present a new logic by inventing a graphic system, but rather to
present another new notation for the logic carried out by quantifiers
and bound variables. He almost took it for granted that a graphic
representation of relations helps us observe their consequences in a
more efficient way. Hence, the Beta system may be considered to be the
final stop of Peirce's long journey to search for better
notation for the logic of relations, which started in 1870 at the
 latest.[34]



We will not go into the formal details of the Beta system in this
entry but will instead refer to Chapter 5 of Shin, where three
slightly different approaches to Beta graphs--Zeman's,
Roberts', and Shin's--are discussed at a full length.
While Zeman's reading is comprehensive and formal,
Roberts' method seems to appeal to a more intuitive
understanding of the system. Taking advantage of the merits of these
two existing works, Shin developed a new reading method of Beta graphs
and reformulated the transformation rules of the
 system.[35]
 Her approach focuses on visual features of Beta graphs and highlights
fundamental differences between symbolic versus diagrammatic systems.
In the remaining part of the entry, we would like to examine how the
essence of the logic of relations is graphically represented in the
Beta system so that the reader may place EG in the larger context of
Peirce's enterprise.



The introduction of quantifiers and bound variables is believed to be
one of the key steps of first-order logic in symbolic systems. This is
why some logicians take Peirce's 1885 paper "On the
Algebra of Logic: A contribution to the Philosophy of Notations"
to be the birthplace of modern logic. If this is the case, then how
does Peirce represent quantifiers and bound variables in Beta
graphs?



Interestingly enough, when Peirce considered a graphic system his
first concern was representation of relations, not representation of
quantifiers. As we said in SS3.1, Peirce presented diagrammatic
representation based on an analogy to chemical molecules for a full
understanding of relations. Hence, the arity of a predicate is
represented by the number of lines radiating from the predicate term.
Next, Peirce extends the use of a line to connect predicates:



> 
> 
> In many reasonings it becomes necessary to write a copulative
> proposition in which two members relate to the same individual so as
> to distinguish these members.... [I]t is necessary that the signs
> of them should be connected in fact. No way of doing this can be more
> perfectly iconic than that exemplified in [the following graph]:
> 
> 
> 
> ![The phrase 'A is greater than' connected with a line to another phrase 'is greater than B'](Line.svg)
> 
> 
> (1903b [CP 4.442])
> 
> 
> 



The line connecting two predicates, representing *one and the
same* object, is called a *line of identity* by Peirce.
That is, the sameness is represented visually in Beta
 diagrams.[36]
 In the case of a symbolic language, we may adopt one and the same
quantified variable-type to represent the identity. For example, the
above diagram says \(\exists x(x <A \ \land \ B < x)\), and
hence, the variable-type \(x\) (roughly) corresponds to the identity
line. However, the same variable-type is not sufficient for expressing
the sameness in other cases, e.g., \(\exists x(x <A \ \land \ B
< x) \rightarrow \exists x (x < C)\).



The way universal and existential statements are represented in the
Beta system highlights a difference between graphic and symbolic
systems. Rather than adopting one more syntactic device for
quantification, Peirce relies on the following visual features:



> 
> 
> [A]ny line of identity whose outermost part is evenly enclosed refers
> to **something**, and any one whose outermost part is
> oddly enclosed refers to **anything** there may be.
> (1903b [CP
>  4.458][37])
> 
> 
> 



Let us borrow the two following graphs from Roberts (1973:
 51):[38]



![The first graph consists of the the phrases 'is good' and 'is ugly' connected by a line. The second graph is the same as the first except the second phrase is surrounded by an oval and the whole is surrounded by another oval.](Quantifier1.svg)


The first graph (where the outermost part of the line is evenly, zero,
enclosed) says that something good is ugly, and the second graph
(where the outermost part is enclosed once) says that everything good
is
 ugly.[39]



How about the scope problem which arises when multiple quantifiers are
used? In the case of a symbolic system, the linear order takes care of
the problem. Peirce's solution for EG is to read off another
kind of visuality: The less enclosed the outermost part of a line is,
the larger the scope that the line gets.



Roberts' following example illustrates the scope matter nicely
(1973: 52):



![first graph has three phrases, 'adores' which is connected by a line to 'is a woman' and both enclosed in an oval and the phrase 'is Catholic' connected by a line to 'adores' with all three phrases enclosed in a second oval. Second graph is the same as the first minus the ovals and with an oval enclosing 'adores' and a second oval enclosing 'is Catholic' and the first oval. The phrase 'is a woman' is outside both ovals.](Quantifier2.svg)


The first graph says 

 

\[\forall x (\textit{Catholic}(x) \rightarrow \exists y [\textit{Adores} (x,y) \land \textit{Woman}(y)])\]


and the second 

 

\[\exists y (\textit{Woman}(y) \land \forall x [\textit{Catholic}(x) \rightarrow \textit{Adores} (x,y)]).\]


In the first graph, the line whose outermost part is oddly enclosed is
less enclosed than the line whose outermost part is evenly enclosed.
Therefore, the universal quantifier has larger scope than the
existential quantifier. In the second graph, it is the other way
around.



Let us summarize three interesting features of the Beta system:


1. Relations are represented graphically, not symbolically, in the Beta
system, in terms of a line. We argued that ultimately Peirce's
pragmatic maxim was behind this alternative way of
representation.
2. A distinction between universal versus existential statements is
represented by the visual fact about whether the outermost part of a
line lies in an area enclosed either by an odd number or by even
number of cuts.
3. The order of quantification is represented by the following visuality:
The less enclosed a line is, the more extensive scope it has.


## 3. From Bivalent to Triadic Logic



Fisch and Turquette (1966) discovered three crucial pages out of
Peirce's *Logic Notebook* (1865-1909, Ms
 339).[40]
 This shows that Peirce's invention of three-valued sentential
logic predates by at least a decade Jan Lukasiewicz's and Emil
Post's achievements on the same topic. The three pages contain
the essential elements of triadic logic and an intriguing passage
about Peirce's motivation behind triadic logic. If we put
Peirce's development of triadic logic in contemporary terms,
Peirce seemed to be branching out to non-standard logic. If so, this
adventure would be qualitatively different from the other two we have
just discussed in the previous sections.



When Peirce developed relational logic, the territory of formalization
was vastly expanded. New vocabulary, hence, new syntactic rules and
semantic rules, were added. Naturally we welcome the territory of
formalization and, hence, theoretical justification is not needed.
From sentential to relational logic--this is an extension in a
literal sense: We do not discard the previous results--hence they
are preserved--but all we do is to expand them.



On the other hand, in the case of extension to non-symbolic languages,
the logic itself stays the same, without addition or subtraction, but
a new form of representation is introduced. That is, what to represent
is not extended, but how to represent is. Some might not see the need
for various forms of representation, and might not be convinced of the
necessity of graphical systems. Nonetheless, at a theoretical level,
Peirce's EG does not demand lengthy theoretical justification.
In some sense, the proof is in the pudding: Can this new graphical
system carry out the same task as existing symbolic systems do? If so,
which system is easier to use? Which system is more efficient? We
might not arrive at a clean consensus, but the discussions are more or
less predictable.



However, when one more semantic value is added to **T** (true) and
**F** (false), logic is not preserved any more. When semantics is
extended or changed, the new logic is neither a monotonic expansion of
the territory of logic nor an alternative syntactic form of
representation to existing symbolic systems. Triadic logic, by
introducing one more semantic value, departs from the standard logic
which is based on bivalence. Here, the status of the principle of
excluded middle ("*Q* or not-*Q*") is shaken. So
is the law of contradiction. It is the burden of any non-standard
logic to justify its being non-standard: Why the third value? What is
the third value? Unknown? If so, is it an epistemological issue?
Indeterminate? If so, does this require a metaphysical
explanation?



The first subsection summarizes Peirce's calculus of triadic
logic and the second briefly discusses Peirce's own motive for
triadic logic.


### 3.1 Truth table of a three-valued system



Three values, **V**, **L**, and **F**, are introduced, where
**V** is true, **L** indeterminate, and **F** false. The
traditional semantic domain for sentential logic, true and false, is
extended to include "indeterminate". Based on this
extended semantic territory, Peirce presents the semantics of several
sentential operators, one unary and the others binary. Modifying
Peirce's presentation slightly to make it more similar to our
conventional truth table style without changing content, we present
the three operators' truth tables.



The semantics of a unary operator, which corresponds to negation:




| \(x\) | \(\bar{x}\) |
| --- | --- |
| **V** | **F** |
| **L** | **L** |
| **F** | **V** |



The semantics for six binary connectives is presented:




| \(x\) | \(y\) | \(\Phi(x, y)\) | \(\Theta(x, y)\) | \(\Psi(x, y)\) | \(Z(x, y)\) | \(\Omega(x, y)\) | \(\Gamma(x, y)\) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| **V** | **V** | **V** | **V** | **V** | **V** | **V** | **V** |
| **L** | **V** | **V** | **V** | **V** | **L** | **L** | **L** |
| **F** | **V** | **V** | **V** | **F** | **F** | **F** | **V** |
| **V** | **L** | **V** | **V** | **V** | **L** | **L** | **L** |
| **L** | **L** | **L** | **L** | **L** | **L** | **L** | **L** |
| **F** | **L** | **F** | **L** | **F** | **F** | **L** | **L** |
| **V** | **F** | **V** | **V** | **F** | **F** | **F** | **V** |
| **L** | **F** | **F** | **L** | **F** | **F** | **L** | **L** |
| **F** | **F** | **F** | **F** | **F** | **F** | **F** | **F** |



Why six? What is the rationale behind the semantics of these
connectives? One way to understand them is to figure out a dominance
hierarchy among the three values.



In the case of \(\Phi\),


1. if at least one is **V**, then \(\Phi(x, y)\) is **V**,
2. else if at least one is **F**, then \(\Phi(x, y)\) is **F**,
and
3. else \(\Phi(x, y)\) is **L**.



That is, **V** is the most dominant, **F** next, and **L** is
the least.



Six patterns of hierarchy emerges:




 
| \(\Phi\) | **V** \(>\) **F** \(>\) **L** |
| \(\Theta\) | **V** \(>\) **L** \(>\) **F** |
| \(\Psi\) | **F** \(>\) **V** \(>\) **L** |
| \(Z\) | **F** \(>\) **L** \(>\) **V** |
| \(\Omega\) | **L** \(>\) **F** \(>\) **V** |
| \(\Gamma\) | **L** \(>\) **V** \(>\) **F** |



Peirce's \(\Theta\) is our familiar disjunction and
Peirce's *Z* conjunction.


### 3.2 Why the third value?



In Peirce's own words:



> 
> 
> Triadic Logic is that logic, which though not rejecting entirely the
> Principle of Excluded Middle, nevertheless recognizes that every
> proposition, *S* is *P*, is either true or false, or else
> has a lower mode of being such that it can neither be determinately
> *P*, nor determinately not-*P*, but is at the limit between
> *P* and not *P*. (Ms 339, copied from Fisch & Turquette
> 1966: 75)
> 
> 
> 



When do we have value **L** (indeterminate) for the proposition
"*S* is *P*"? Sometimes *S*, Peirce says,
has a lower mode of being *P* and is at the limit between
*P* and not *P*. The crux of the matter is how to interpret
the two phrases--"lower mode of being *P*" and
"being at the limit between *P* and not *P*".
Existing literature offers two different explanations--modality
versus continuity.



Fisch and Turquette, upon the discovery of Peirce's notes on
triadic logic, locate the root of indeterminacy in potentiality. That
is, indeterminacy is the semantic value assigned to an unrealized
situation; hence, we can say neither "*S* is
*P*" nor "*S* is not *P*" at this
point. Potentiality, according to this view, cannot be captured by
dyadic logic. If so, Peirce's triadic logic is directly related
to modality talk, which Fisch and Turquette conclude:



> 
> 
> Essentially, Peirce seems to be saying that triadic logic may be
> interpreted as a modal logic which is designed to deal with the
> indeterminacies resulting from that mode of being which Peirce has
> called "Potentiality" and "Real Possibility".
> Under such an interpretation, dyadic logic becomes a limiting case of
> triadic modal logic resulting from removing indeterminacy and being
> determined entirely by "Actuality". (Fisch & Turquette
> 1966: 79)
> 
> 
> 



According to the modality interpretation, Peirce's "lower
mode of being *P*" means *P* not being actual, and
Peirce's third value **L**, being potential, is "at the
limit between *P* (i.e., **T**) and not *P* (i.e.,
**F**)". Later in the paper, suggesting a possible relation
between Peirce's triadic logic and MacColl's implication
(as opposed to material implication), the authors make an interesting
remark: 



> 
> 
> Considering MacColl's rejection of Russell's material
> implication, it is interesting to notice also that MacColl's
> "Def. 13" gives what is now called "C. I.
> Lewis's strict implication". (Fisch & Turquette 1966:
> 83) 
> 
> 
> 



Even though the connection was not pursued further in their paper, one
cannot help realizing that their modality interpretation is boosted by
a relation to MacColl's implication since C. I. Lewis'
strict implication is a beginning of modal logic. However, equating
Peirce's triadic logic with modal logic, the modality view needs
to explain the relation between Peirce's Gamma graph lecture in
1903c (which is about
modality) and the triadic logic notes written in 1909 ([Ms 339] 340v, 341v, 344r). Modal logic explored in Gamma
graphs is the extension of classical logic, which required new
vocabulary, e.g., broken cuts and tincture. Modal logic does not have
to be non-standard. On the other hand, triadic logic does not add any
vocabulary, but brings in different interpretations, and becomes
non-standard logic. On a slightly different note, Fisch and
Turquette's suggests Peirce's tychism (the view that
indeterminacy is part of the reality) as a motivation for
Peirce's invention of triadic logic. If so, Peirce's
triadic logic is a reflection of his own metaphysics.



Challenging the modality view, Robert Lane proposes the continuity
interpretation for Peirce's triadic logic. According to Lane,
Peirce's indeterminate value **L** has nothing to do with
modality, hence, Peirce's development of triadic logic is not
another mechanism for modal logic, but with Peirce's
synechism--the doctrine "that all that exists is
continuous" (c. 1897b [CP 1.172])! How does Peirce's
philosophy of continuity justify the third value?



First, Lane makes a distinction between the principle of excluded
middle (PEM, henceforth) *being false* with regard to a
proposition and PEM *not being applied to* a proposition. If
PEM is true or false, it means the principle is applied to it. And,
Lane claims that PEM is applied to only non-general and non-modal
propositions, citing the following passages from Peirce:



> 
> 
> anything is *general* in so far as the principle of excluded
> middle does not apply to it and is *vague* in so far as the
> principle of contradiction does not apply to it. (1905: 488 [CP
> 5.448])
> 
> 
> 
> an assertion is said to be made in "the mode of necessity"
> if, and only if, the affirmation and the denial that [sic] which is so
> asserted could conceivably be both alike *false*. Thus if a
> person says "It will *certainly* rain tomorrow", it
> may be alike false that it is certain to rain and that it is certain
> not to rain. (1910b: 26-28, Ms 678)
> 
> 
> 



If a proposition is either general or expresses necessity, PEM is not
false, but is not applied. Hence, focusing on individual and non-modal
propositions, Lane draws our attention to a special nature of a
predicate in **L**-propositions. Lane calls the kind of the
property which results in **L**-propositions a
"boundary-property". Here is Peirce's own example of
a boundary-property:



> 
> 
> Thus, a blot is made on the sheet. Then every point of the sheet is
> unblackened or blackened. But there are points on the boundary line,
> and those points are insusceptible of being unblackened or of being
> blackened, since these predicates refer to the area about S and a line
> has no area about any point of it. (Ms 339: 344r, quoted in Lane 1999:
> 294)
> 
> 
> 



Those points on the boundary line are neither black nor non-black.
Consider the propositions "Point *O* is black" and
"Point *O* is not black" (where point *O* is on
the boundary line of a black blot). Neither of them is true, but
false, either. These are prime examples of Peirce's
**L**-proposition. Using Peirce's own truth-tables in the
previous subsection, let's compute the truth value of
"Point *O* is black or point *O* is not
black".



Let \(\alpha\) be the value of "Point *O* is black",
which is **L**.




| \(\alpha\) | \(\bar{\alpha}\) | \(\Theta(\alpha, \bar{\alpha})\) |
| --- | --- | --- |
| **L** | **L** | **L** |



Note that PEM is applied, but not true, period.



Lane's following conclusion would be welcomed by many Peirce
scholars:



> 
> 
> [B]oundary-propositions were important to Peirce because continuity
> was important to him;...*this* [the thought that an actual
> breach of continuity possesses neither of the properties that are the
> boundary properties relative to that breach] led him to think that
> boundary-propositions are neither true nor false. To accommodate such
> propositions, and thus the phenomenon of continuity, within the bounds
> of formal reasoning, was, I contend, the motivation behind
> Peirce's experiments in triadic logic. (Lane 1999: 304)
> 
> 
> 



Regardless of endorsement of the continuity talk, some might not
welcome metaphysics getting into logic. Moreover, if Peirce's
synechism is not embraced, Peirce's triadic logic, which Lane
argues is an attempt to formalize the continuity phenomenon, might
lose its force.



The previous two sections showed that Peirce's relational logic
and graphical systems push us further both in what we do with logic
and how we do logic so that we may formalize more and we may formalize
in more diverse ways. Triadic logic, as explained at the beginning of
this section, is neither just a monotonic extension to get further nor
an alternative to get to the same place. By extending semantic
entities, we have a different logic, for example, PEM being not true.
That is why we call triadic logic a non-standard logic. However,
Peirce's way of introducing the third value gives us some pause.
First of all, unlike with contemporary triadic logic, Peirce does not
discard PEM completely:



> 
> 
> Triadic Logic...not rejecting entirely the Principle of Excluded
> Middle,... (Ms 339: 344r, copied from Fisch & Turquette 1966:
> 75)
> 
> 
> 
> I do not say that the Principle of Excluded Middle is downright
> *false*; (1909: 21-22 [NEM 3/2: 851], quoted in Fisch
> & Turquette 1966: 81)
> 
> 
> 



For certain (not all) properties, because of the way things are, we
find ourselves caught at limits between clearly *P* and clearly
non-*P*. If we want to formalize those cases as well, the third
value, **L**, is needed to express the indeterminacy of boundary
cases. Hence, Peirce himself does not think triadic logic is a new
logic, but an addition or extension of the existing dyadic logic:



> 
> 
> The recognition [that there is an intermediate ground between
> *positive assertion* and *positive negation* which is
> just as Real as they are] does not involve any denial of existing
> logic, but it involves a great addition to it. (1909: 21-22 [NEM
> 3/2: 851], quoted in Fisch & Turquette 1966: 81)
> 
> 
> 



If we accept Peirce's suggestion literally, his triadic logic is
not a typical form of non-standard logic but Peirce's another
way to extend the territory of logic, along with his relational
logic.