 # An Explication of the Antilogism in Christine Ladd-Franklin's "Algebra of Logic" – Symbolic Notation in "Algebra of Logic"

Author(s):
Julia M. Parker (University of Missouri – Kansas City)

While the syllogism had been studied by philosophers for centuries, George Boole (1815–1864) and others combined the disciplines of logic and mathematics in the latter half of the nineteenth century to undertake a renewed examination of logical deduction. Ladd’s dissertation advisor C. S. Peirce was especially important in the development of Boole’s notion of an “algebra of logic,” a system in which the statements of a logical argument are reduced to mathematical notation such as variables and symbols. In her dissertation, Ladd developed her own algebra of logic and described how it differed from those already in existence at the time of her writing. Before doing so, she first provided definitions of the symbols employed in her algebra of logic and other useful preliminaries, which will be presented here in brief. As is the case in other works on symbolic logic, the terms of statements are represented by variables throughout Ladd’s work. Further, in much of her work, an algebraic argument was paired with examples in which the terms of the symbolic statements were replaced by an object, a set of objects, a quality, or a set of qualities. Additionally, Ladd often used the term “proposition” to mean “statement”. The following table lays out Ladd’s notation, with a third column added to provide examples for clarity.

 Notation Meaning Examples $a = b$ $a$ and $b$ are equivalent “there is no $a$ which is not $b$ and no $b$ which is not $a$” Define $a$ to be the names of months that end in “r.” Define $b$ to be September, October, November, December. $a$ and $b$ are equivalent classes. $\overline{a}$ The negation of a proposition [statement] or term “what is not $a$” Using the same $a$ as above, $\overline{a}$ consists of the names of all the months that do not end in “r.” Alternately, for an example of a term as a quality, let $a$ represent the color blue (say paired with any set of colored objects, $b$). Then $\overline{a}$ would refer to any of those objects in question that are not blue. $a \times b$ or $ab$ What is common to the classes  $a$ and $b$ “what is both $a$ and $b$” Now let $a$ represent the names of all the months and $b$ be months that end with “r,” then $ab$ is the list: September, October, November, December. Or if $a$ is the set of qualities blue or yellow and $b$ is flowers, then $ab$ is the set of all objects that are flowers and are either blue or yellow. $a+b$ The whole of $a$ together with the whole of $b$ “what is either $a$ or $b$” Again $a$ represents months and $b$ represents months ending with “r,” then $a + b$ is the set of all the names of the months. $\infty$ The universe of discourse, or what is logically possible. If the universe of discourse is the names of months, then months that end with the letters: y, h, l, e, t, and r make up all of the possibilities, or $\infty$. $0$ The negation of $\infty$, or what is impossible or nonexistent. There are no months that end with the letter “p,” so this class would equal $0$.

A foundational element in Ladd’s notation is a specific symbol used in her algebra of logic that is not found in those algebras that were developed before hers. Ladd called the symbol the “wedge” or “sign of exclusion,” and denoted it as $\vee$, to indicate an affirmative statement, and $\overline{\vee}$ to indicate a negative statement. The symbols $\vee$ and $\overline{\vee}$ were placed as connectors, or copulas,  between two variables to create a statement. The various specific usages of these copulas are given in the following table (adapted from page 26 of Ladd’s dissertation). In the dissertation, Ladd used capital letters, $A$ and $B$, for the variables in this table without explanation; for continuity with the remainder of her work, variables representing terms of statements have been changed to $a$ and $b$. Also, notes in the third column have been added for clarity.

The new copula $\vee$ and its negation were a distinguishing element of Ladd’s algebra of logic. One advantage of this notation, as described by Ladd, is that $\vee$ and $\overline{\vee}$  are symmetrical, so that statements may be read either forward or backward without a change in meaning. The argument $a \overline{\vee} b$, then, can be considered an inconsistency, stating that the two classes $a$ and $b$ cannot coexist, or that $a$ and $b$ have no elements in common. Hence, the statement  $a \overline{\vee} \infty$ indicates that $a$ cannot, under any circumstances, exist. Ladd then introduced one more convention involving her new copula: when indicating a relationship between a class and $\infty$, the $\infty$ may be left off, leaving the copula as the end of the statement. This gave rise to the notation $a \overline{\vee}$ meaning “there is no $a$” [Ladd 1883, p. 29]; similarly, $a \vee \infty$ was denoted simply as $a \vee$, meaning “$a$ exists.”

 (1)     $a \overline{\vee} b$ $a$ is not $b$. No $a$ is $b$. $\forall t \in a$ we have $t\not\in b$ $\Leftrightarrow$ $\forall t \in b$ we have $t \not\in a$ Self-symmetric in $a$ and $b$ Negation of (2) (2)     $a \vee b$ $a$ is in part $b$ Some $a$ is $b$. $\exists \in a$ such that $t\in b$ $\Leftrightarrow$ $\exists \in b$ such that $t \in a$ Self-symmetric in $a$ and $b$ Negation of (1) (3)     $a \overline{\vee} \overline{b}$ $a$ is not not-$b$. All $a$ is $b$. $\forall t \in a$ we have $t\not\in b$ Symmetric in $a$ and $b$ with (5) Negation of (4) (4)     $a \vee \overline{b}$ $a$ is partly not-$b$. Some $a$ is $b$. $\exists t \in a$ such that $t\not\in b$ Symmetric in $a$ and $b$ with (6) Negation of (3) (5)     $\overline{a} \overline{\vee} b$ What is not $a$ is not $b$. $a$ includes all $b$. $\forall t \in b$ we have $t\not\in a$ Symmetric in $a$ and $b$ with (3) Negation of (6) (6)     $\overline{a} \vee b$ What is not $a$ is part $b$. $a$ does not include all $b$. $\exists t \in b$ such that $t\not\in a$ Symmetric in $a$ and $b$ with (3) Negation of (5) (7)     $\overline{a} \overline{\vee} \overline{b}$ What is not $a$ is not not-$b$. There is nothing besides $a$ and $b$. $\forall t \in \infty$ we have $t \in a$ or $t \in b$ Self-symmetric in $a$ and $b$ Negation of (8) (8)      $\overline{a} \vee\overline{b}$ What is not $a$ is in part not-$b$. There is something besides $a$ and $b$. $\exists t \in \infty$ such that $t \not \in a$ or $t \not \in b$ Self-symmetric in $a$ and $b$ Negation of (7)

After the introduction of her notation, Ladd went on to develop an important connection between the study of symbolic logic and syllogistic argument. She first observed that the important subjects in a symbolic logic are uniting and separating propositions; inserting or omitting terms; and eliminating the least possible amount of content. According to Ladd, this third subject, the elimination of content without loss of content, was most closely related to the study of syllogisms since “the essential character of the syllogism is that it effects the elimination of the middle term” [Ladd 1883, p. 35]. She then began a thorough examination of how her algebra of logic could be applied to all three subjects.

As “elimination” is the subject most closely related to syllogism, this is the section chosen for explication here. This explication was done according to the guidelines provided by [Delaware 2019]. In brief, notes, remarks, and explanations are added to an excerpt from a primary source, such as "Algebra of Logic," to foster readers' understanding of the technical content and intellectual context embedded in a writer's text by answering readers' questions before they have thought to ask them. Thus, on the next page, all comments in square brackets were added to Ladd's text by the author.

Julia M. Parker (University of Missouri – Kansas City), "An Explication of the Antilogism in Christine Ladd-Franklin's "Algebra of Logic" – Symbolic Notation in "Algebra of Logic"," Convergence (December 2019)

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