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

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

Identifying "famous firsts" in history is often a messy endeavor, in part requiring that achievements be stated in precise terms. When I came across the case of Christine Ladd-Franklin (1847–1930), the first American woman to write a doctoral dissertation in mathematics at an American university but not the first American woman to be formally awarded the PhD degree for mathematics,1 I could not help being hooked by her life story of creating opportunities and addressing challenges. While other scholars have provided more detailed accounts of Ladd-Franklin's biography [Green and LaDuke 2009; Riddle 2016; Johnson 2008], I used the technique of explication (see Delaware 2019) to analyze the content of her 1883 dissertation, "On the Algebra of Logic." In this contribution to symbolic logic, she created a test for the validity of syllogisms that she later named "antilogism". In the explication, Ladd shows that all valid syllogisms can be reduced to a single form. It had been long believed that this was the case, but mathematicians and logicians conjectured that the perfect form to which all syllogisms could be reduced was one of affirmative, or positive, statements. Ladd showed, however, that the single form is actually formed out of a contradiction, which is why she coined the term “antilogism” to represent the final product of her work.

Opening paragraph of Ladd's dissertation "On the Algebra of Logic"

##### Notes:

[1] Winifred Edgerton Merrill (1862–1951) earned a PhD in mathematics at Columbia University in 1886 [Kelly and Rozner 2012; Riddle 2016], an event that was extraordinary enough in its day to be considered newsworthy by the New York Times. When Clara L. Bacon (1866–1948) became the first woman to earn a PhD in mathematics from Johns Hopkins in 1911, she was one of only a dozen women in the US to hold a doctorate in the field.

# An Explication of the Antilogism in Christine Ladd-Franklin's "Algebra of Logic" – Early Life and Education

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

1885 photograph of Hopkins Hall,
on the original downtown Baltimore campus of The Johns Hopkins University, Public Domain

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

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

Before this article discusses the specific work in Ladd’s dissertation, a brief introduction to the study of syllogisms, an area of interest to mathematicians and scholars from ancient times, is provided. Several terms used when discussing syllogism are defined:

 Statement A sentence, either affirmative or negative, that has a truth value. Example: “All clouds produce rain” is an affirmative statement. Term A portion (subject or predicate) of a statement describing sets or classes or objects or properties. Example: The subject term from above is “clouds” (a class of objects) and the predicate term is “produce rain” (a property that this class of objects is claimed to have). Quantifier An indication of how many objects in the class possess a property. Quantifiers may be universal (“all” or “none”) or particular (“some”). Premise A statement from which another statement is inferred. Conclusion The inference statement made from the premise(s). A syllogism has one conclusion.

One specific form of logical argument is the syllogism. A syllogism is made up of three statements, each of which consists of a quantifier, a subject term, and a predicate term. For each statement, the quantifiers can be universal or particular, and the statement can be affirmative or negative. Because statements can be affirmative and negative, and quantifiers can be universal or particular, the individual statements in a syllogism can take four forms, also referred to as moods. These are shown in the following table using variables a and b to represent the classes described by the following terminology:

 Statement Mood All a is b. universal affirmative No a is b. universal negative Some a is b. particular affirmative Some a is not b. particular negative

The syllogism, then, is a logical argument with the structure of two premises and a conclusion, each of which takes on one of these four moods. While any argument consisting of three such statements may be a syllogism, not all syllogisms are valid. A valid syllogism is an argument form that is truth-preserving, in that the truth of the conclusion follows necessarily from the truth of premises. For instance, a classic example (obviously written in a male-dominant culture) of a valid syllogism is:

All men are mortal.
All Greeks are men.
Therefore, All Greeks are mortal.

The following syllogism has the same form, and is therefore also valid, even though one of its premises and its conclusion are false:

All flowers are blue.
All roses are flowers.
Therefore, All roses are blue.

Syllogisms can also be valid even when the premises are false and the conclusion is true or when the premises and conclusions are all false. The only way a syllogism is invalid is when it follows a general form that allows the premises to be true while the conclusion to be false.1

An important feature of syllogisms is that, though composed of ordinary sentences, they cannot simply be rearranged or reworded. The statements in a syllogism are taken as a whole, with subject and predicate inseparable and non-transposable, meaning the statements are not generally symmetric. As an example, the statement “all men are mortal” has a different meaning when changed to read “all mortals are men.” Additionally, syllogistic logic does not allow for negation of statements in a straightforward way. Again, in the statement “all men are mortal,” we might propose either “not all men are mortal” or “all men are not mortal” as possible negations, but these two statements do not have the same meaning. This shows that the negation or rearrangement of terms is not a simple process and leaves people at risk of considering an incorrectly converted syllogism to be logically sound [Shen 1927, p. 55].

The study of syllogisms began in ancient times, when the ancient Greek philosopher Aristotle (384–322 BCE) began to write on the subject of logic. Aristotle recognized that in addition to each statement having one of the four possible moods, the syllogism as a whole can be constructed in different ways, which he called figures. To give a more precise definition, a syllogism is an argument consisting of three statements concerning three terms:

• P, the major term found in the predicate of the conclusion,
• S, the minor term found in the subject of the conclusion, and
• M, the middle term linking the two

In the example of the classic syllogism above, “mortal” is the major term (P), “Greeks” is the minor term (S), and “men” is the middle term (M). Any single statement within a syllogism contains two of these three terms. Aristotle discovered that there are fourteen valid forms of syllogisms, depending on the order in which the terms are combined to make up the statements of the premises and conclusion, and the mood of these three statements. To show all fourteen is beyond the scope of this paper; however, the most important of these involved what Aristotle called the first figure,2 in which the terms are arranged in the following form:

MP
SM
SP

This figure is read as a series of statements: the first (MP) indicates a relationship between the middle term and the major term, the second (SM) indicates that the minor term also has a relationship to the middle term, then the third (SP) draws a conclusion that the minor term and major term must have a relationship because these have the middle term in common. As an example, we have:

MP: All men (M) are mortal (P).
SM: All Greeks (S) are men (M).
SP: Therefore, all Greeks (S) are mortal (P).

Aristotle believed, but was unable to conclusively show, that all valid syllogisms could be reduced to the first figure, which he considered to be the “perfect” figure, with all universal affirmative statements. He therefore attempted to formulate rules that would allow for the conversion of any (valid) syllogism to the first arrangement in order to demonstrate its validity. However, due to the complexities of rearrangement and negation discussed above, he was unable to provide a complete treatment of syllogistic argument that accomplished his goal [Russinoff 1999, pp. 453–454]. Although the study of syllogism remained a focus of logicians from the time of Aristotle through the middle nineteenth century, no one was able to show how all figures of syllogism could be converted to a single perfect figure [Russinoff 1999, p. 454]. In other words, there was no simple test or rule to apply that would identify a syllogism as either valid or not. It was precisely this question to which Ladd proposed a solution two thousand years after Aristotle, in what should have been her PhD dissertation, “On the Algebra of Logic” [Ladd 1883].

##### Notes:

[1] For instance, the following general form leads to invalid syllogisms, as the reader can check by taking, for instance, P to be “roses,” M to be “flowers” and S to be “marigolds.”

All P are M.
Some M are S.
Therefore, Some P are S.

Other values of M, P and S could result in true statements for the two premises and the conclusion, but that would be purely accidental, rather than necessitated by the syllogistic form itself.

[2] Aristotle’s treatment of syllogistic logic included three distinct figures:

 1st 2nd 3rd MP MP PM SM MS SM SP SP SP

Medieval logicians added a fourth figure to the three recognized by Aristotle:

 4th PM MS SP

# 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.

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

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

Editors' Note: The following 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 this page, all comments in square brackets were added to Ladd's text by the author.

From “On the Algebra of Logic” by Christine Ladd [1883, pp. 37–39]: [Note: Throughout, Ladd used “proposition” to refer to a relationship between any number of terms, what has been defined as a “statement” above. Also, throughout, parentheses have been added for clarity, particularly to make clear groupings on either side of the copula.]

On Elimination. – In (24ʹ)

[derived in the section above and accompanied by an example quoted here without proof. The symbol $$\therefore$$ is read “therefore.”

“… if the premises are both universal,

$$\begin{array}{ll} (24ʹ) \,\,\,\,& a \overline{\vee} b \\ & c \overline{\vee} d \\ & \therefore ac \overline{\vee} (b + d) \end{array}$$

If no bankers [$$a$$] have souls [$$b$$] and no poets [$$c$$] have bodies [$$d$$], then no banker-poets [$$ac$$] have either souls or bodies [$$b + d$$]."]

there is no elimination [all terms in both of the premises, namely $$a$$, $$b$$, $$c$$, $$d$$, appear in the conclusion], and in (24°)

[derived earlier in the thesis and quoted here with example:

“…if the premises are one universal and one particular, $\begin{array}{ll} (24^{\circ}) \,\,\,\,& a \overline{\vee} b \\ & ac \vee (b + d) \\ & \therefore c \vee d \end{array}$ If no Africans [$$a$$] are brave [$$b$$] and some African chiefs [$$ac$$] are either brave or deceitful [$$b + d$$], then some chiefs [$$c$$] are deceitful [$$d$$].”]

there is elimination of the whole of the first premise [no Africans are brave, $$a \overline{\vee} b$$] and part of the second [some African chiefs are brave, $$ac\vee b$$ ]. The most common object [goal] in reasoning is to eliminate a single term at a time—namely, one which occurs in both premises [as seen in the classic "All Greeks are mortal" example of a valid syllogism, with the elimination of the middle term “men”]. Each of these inferences gives rise to a form of argument, as a special case, by which that object [elimination of a single term at a time] is accomplished,—the premises being on the one hand both universal [as in (24ʹ)], and on the other hand one universal and the other particular [as in (24°)]. The inconsistency $$I$$

[derived by Ladd-Franklin earlier in the thesis (p. 34), stated with large parentheses added: $I. \,\,\, (a \overline{\vee} b)(c\overline{\vee}d)\overline{\vee}\left (ac \vee (b+d)\right )$ Meaning, the conjunction of statements “no $$a$$ is $$b$$” and “no $$c$$ is $$d$$” is inconsistent with the conclusion that “some $$ac$$ is $$b$$ or $$d$$.” Or, as Ladd-Franklin explains, “it is not possible that a combination of several qualities should be found in any classes from each of which some one of those qualities is absent. If, for example, culture [$$a$$] is never found in business men [$$b$$] nor respectability [$$c$$] among artists [$$d$$], then it is impossible that cultured respectability [$$ac$$] should be found among either business men or artists [$$b + d$$]” [Ladd 1883, p. 35]]

becomes when $$d$$ is equal to $$\overline{b}$$ [$$d$$ is not-$$b$$] and hence $$b + d$$ [what is either $$b$$ or $$d$$, means, with the substitution, what is $$b$$ or not-$$b$$] equal to [$$b +\overline{b} =$$] $$\infty$$. $\left ( (a \overline{\vee} b)(c\overline{\vee}\overline{b})(ac\vee \infty)\right )\overline{\vee}$

[large parentheses added. This formula indicates that the combination of statements, “$$a$$ is not $$b$$” ($$a \overline{\vee} b)$$ and “$$c$$ is not not-$$b$$” ($$c\overline{\vee}\overline{b})$$ and “the conjunction of $$a$$ and $$c$$ exists” ($$ac \vee\infty$$) is inconsistent ($$\overline{\vee}$$ ), meaning that given the first two statements as premises, the third statement does not follow as a valid conclusion.]

or

$II. \left ( (a \overline{\vee} b)(\overline{b}\overline{\vee}c)(c \vee a )\right ) \overline{\vee}$

[large parentheses added. The argument consisting of the first premise statement "$$a$$ is not $$b$$” and the second premise statement “not-$$b$$ is not $$c$$” ($$\overline{b}\overline{\vee}c$$) equivalent to ($$c\overline{\vee}\overline{b}$$) above, and the conclusion statement “some $$c$$ is $$a$$” ($$c \vee a$$) is inconsistent ($$\overline{\vee}$$) because, since by the first premise no $$a$$ is $$b$$, then not-$$b$$ would include $$a$$, and by the second premise not-$$b$$ is inconsistent with $$c$$, so the conclusion statement ($$c \vee a$$) does not follow ($$\overline{\vee}$$), meaning $$a$$ cannot coexist with $$c$$, previously written ($$ac\overline{\vee}\infty$$).]

Given any two of these propositions, the third proposition, with which it is inconsistent, is free from the term common to the two given propositions; $$a$$, $$b$$, and $$c$$, are, of course, expressions [statements] of any degree of complexity. The propositions $$m a\overline{\vee}(x+y)$$  [what is both $$m$$ and $$a$$ is inconsistent with $$x$$ or $$y$$], $$\overline{x}\,\overline{y}\overline{\vee}(c+n)$$ [what is both not-$$x$$ and not-$$y$$ is inconsistent with $$c$$ or $$n$$], for instance, are inconsistent with $$ma\vee(c+n)$$ [what is both $$m$$ and $$a$$ is consistent with $$c$$ or $$n$$; this cannot be a valid conclusion because $$ma$$ is not consistent with $$x$$ or $$y$$, meaning what is not-$$x$$ and not-$$y$$, including $$ma$$, must be inconsistent with $$c$$ or $$n$$.]; any number of terms may be eliminated at once by combining them in such a way that they shall make up a complete universe [$$\infty$$, the universe of discourse].

When any two of the [three mutually] inconsistent propositions in II. are taken as the premises, the negative of the remaining one is the [valid] conclusion.

There are, therefore, two distinct forms of inference with elimination of a middle term, special cases of (24ʹ) [when the premises are both universal] and (24°) [when one premise is universal and the other particular]. If we write $$x$$ for the middle term, we have

 (25ʹ) $\mbox{[i]}\,\, a\overline{\vee}x$  [$$a$$ is inconsistent with $$x$$] $\mbox{[ii]}\,\, b \overline{\vee} \overline{x}$ [$$b$$ is inconsistent with not-$$x$$, so $$b$$ is consistent with $$x$$] $\mbox{[iii]}\,\, \therefore ab \overline{\vee}$ [therefore, there is no $$a$$ and $$b$$, or $$a$$ is inconsistent with $$b$$] The premises are [can be rewritten as] $\mbox{[i*]}\,\, a(b+\overline{b})x\overline{\vee}$ [$$a=a(b+\overline{b})=a\infty$$] $\mbox{[ii*]}\,\, (a+\overline{a})b\overline{x}\overline{\vee}$  [$$b=(a+\overline{a})b=\infty b$$] and together they affirm that $\mbox{[iv]}\,\, (ab(x+\overline{x})+a\overline{b}x+\overline{a}b\overline{x})\overline{\vee}$ or [equivalently] $\mbox{[iv*]}\,\, (ab+a\overline{b}x+\overline{a}b\overline{x})\overline{\vee}$ [Applying the distributive laws, known to Ladd, to [i*] and [ii*], we obtain [i*] $$a(b+\overline{b})x\overline{\vee}=(abx+a\overline{b})x\overline{\vee} = (abx+a\overline{b}x)\overline{\vee}$$, and [ii*] $$(a+\overline{a})b\overline{x}\overline{\vee}= (ab\overline{x}+\overline{a}b\overline{x})\overline{\vee}$$. Then, adding, we obtain ($$abx + ab\overline{x} +a\overline{b}x+\overline{a}b\overline{x})\overline{\vee}$$, which simplifies to [iv]. Since $$x+\overline{x} = \infty$$ ,  it can be removed, so we finally reach [iv*].] Dropping the information concerning $$x$$ [in [iv*]], there remains $ab\overline{\vee}. \,\,\,\, \mbox{[the conclusion, [iii].}]$ The information given by the conclusion is thus exactly one half of the information given by the premises (Jevons) [Ladd is referencing a condition for elimination given by a developer of an existing algebra of logic, William Stanley Jevons (1835–1882). The condition that has been satisfied is the removal of one of the two terms (one half of the information), namely $$x$$, from each premise, leaving only $$a$$ and $$b$$ in the conclusion]. (25°) $\mbox{[i]}\,\, a\overline{\vee}x$  [$$a$$ is inconsistent with $$x$$] $\mbox{[ii]}\,\, b \vee x$ [$$b$$ is consistent with $$x$$] $\mbox{[iii]}\,\,\therefore b \overline{a}\vee$ [therefore, $$b$$ and not-$$a$$ are consistent] The second premise is [can be rewritten as] $\mbox{[ii*]}\,\, bx(ax+\overline{ax})\vee$ [because $$ax+\overline{ax} =\infty$$] Which becomes, since there is no $$ax$$, [as known from [i]] $\mbox{[ii*]}\,\, bx(ax +\overline{ax})\vee = bx (\overline{ax})\vee = bx(\overline{a}+\overline{x})\vee$ [By De Morgan’s law which states that the negation of ($$a$$ and $$x$$) is (not-$$a$$ or not-$$x$$)] or [equivalently] $bx\overline{a}\vee.$  [In [ii*] $$bx(\overline{a}+\overline{x})\vee = (bx\overline{a}+bx\overline{x})\vee$$. Since $$x\overline{x}=0$$, that term can be removed, leaving only $$bx\overline{a}\vee$$.] Dropping the information concerning $$x$$ there remains $b\overline{a}\vee \,\,\,\, \mbox{[the conclusion, [iii].}]$ This conclusion is equivalent to $b\overline{a}\vee(x+\overline{x})$ [because $$x+\overline{x}=\infty$$] but the [rewritten and simplified] premises permit the conclusion $b\overline{a} \vee x;$ [if $$b\overline{a}$$ is consistent with $$x$$ or not-$$x$$, then $$b\overline{a}$$ is consistent with $$x$$ is a valid conclusion] hence the amount of information retained is exactly one half of the (particular) information given by the premises.

Elimination is therefore merely a particular case of dropping irrelevant information.

# An Explication of the Antilogism in Christine Ladd-Franklin's "Algebra of Logic" – From Elimination to Antilogism

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

While impressive, this treatment of syllogism in Ladd’s algebraic notation in our explication was not the end of, or even the most important part, of Ladd’s work. After some further discussion of how figure and mood affect the structure of her inconsistency, Ladd arrived at the following conclusion, for which she provided no rigorous proofs or examples:

Those syllogisms in which a particular conclusion is drawn from two universal premises [as in (25ʹ)] become illogical [inconsistent] when the universal proposition is taken as not implying the existence of its terms. The argument of inconsistency [defined as II. above], $II. \left ((a\overline{\vee} b)(\overline{b}\overline{\vee}c) (c \vee a) \right ) \overline{\vee}$ is therefore the single form [what Aristotle called “figure”] to which all the ninety-six valid syllogisms [referring to the possible combinations of the figures and moods which give rise to syllogisms that are logically sound. How the number ninety-six was calculated is not described] (both universal and particular) may be reduced. [Ladd 1883, pp. 39-40]

This claim highlights the importance of Ladd’s work. With it, she provided an answer to the question first posed by Aristotle, but in a surprising way. Contrary to Aristotle’s unproven belief that all valid syllogisms could be reduced to the single perfect figure of all universal affirmative statements, Ladd showed that by using her algebra of logic based on the idea of exclusion (or logical inconsistency), all valid syllogisms can be reduced to a single symbolic argument–but not one of all universal affirmative statements. She later coined the term “antilogism” [Ladd-Franklin 1928] for this argument, which has the following symbolic representation: $II. \left ((a\overline{\vee} b)(\overline{b}\overline{\vee}c) (c \vee a) \right )$

In other words, when given a syllogism, if the statements can be rewritten so that the premises and conclusion are in the form of II., the antilogism, then the syllogism is valid. This test was presented by Ladd as a rule in her dissertation:

Rule of Syllogism. – Take the contradictory [negation] of the conclusion, and see that the universal propositions [statements] are expressed [rewritten] with a negative copula [ $$\overline{\vee}$$ ] and particular propositions [statements] with an affirmative copula [$$\vee$$]. If two of the propositions [statements] are universal and the other particular, and if that term only which is common to the two universal propositions [meaning the middle term, or the one being eliminated from both of those universal statements] has unlike signs [the middle term must be $$x$$ in one premise and $$\overline{x}$$ in the other], then, and only then, the syllogism is valid [Ladd-Franklin 1883, p. 41].

 Ladd's first example of the Rule of Syllogism, "On the Algebra of Logic," p. 41.

As explained by Russinoff, by developing a rule or test that could be used to determine the validity of syllogisms, Ladd made a significant contribution to the study of syllogistic logic, but one which was incomplete by today's standards [Russinoff 1999, p. 463]. After providing this rule, Ladd did not give a formal proof of its correctness, though in the remainder of her paper she did provide a number of examples that illustrate its correctness. If Ladd’s rule is taken as a theorem, what she did was, in a sense, to prove only one direction of an “if and only if” statement. As Russinoff writes, “although it is obvious that all triads [arguments made up of three statements] with the form she describes [triads in the form of Ladd’s II.] are inconsistent, it is not at all obvious that every inconsistent triad has that form” [Russinoff 1999, p. 463]. In other words, Ladd did not show that all inconsistent arguments will take the form of the antilogism.

Although it may be that Ladd simply left the proof of the reverse direction out of her dissertation, it is instead more likely that, at the time of her writing, what she showed in her dissertation was considered a sufficiently rigorous proof in symbolic logic. Russinoff [1999, pp. 463-467] has further argued that a complete proof of Ladd’s theorem is not possible without certain results and tools of modern logic that were not part of symbolic logic in the late nineteenth century. For instance, Ladd used the ideas of consistency and inconsistency in her work in the following way: $$a \overline{\vee} b$$ means “$$a$$ is inconsistent with $$b$$” or “if $$a$$ is true, $$b$$ is false and if $$b$$ is true, $$a$$ is false” whereas to say that $$a \vee b$$, or “$$a$$ is consistent with $$b$$” means that the truth of one does not imply the falseness of the other. As Russinoff explains, however, the definitions of inconsistency and consistency have changed to include possible interpretations, so the modern understanding is that a set of statements is said to be inconsistent only when there is no possible interpretation that would allow for all members of the set to be simultaneously true. Applying this current idea of interpretation and using modern notation, Russinoff was able to provide a contemporary proof for the reverse direction of the antilogism rule that Ladd’s insights allowed her to formulate in her 1883 dissertation.

Though Ladd’s work may appear incomplete by today’s standards, this does not diminish the importance of her contribution to the study of logic. Her work showed that every valid syllogism can be reduced to a single argument of the antilogism. Thus, Ladd solved a problem that logicians from the time of Aristotle had failed to answer satisfactorily. Her antilogism, then, offered a very powerful tool to logicians, allowing the study of syllogisms to be greatly simplified. As she stated in a later paper, another benefit of using antilogism instead of syllogism is that it is more natural than formal syllogistic arguments, being a form of reasoning commonly used in rebuttal or discussion when speaking [Ladd-Franklin 1928, p. 532]. Ladd supported her claim that antilogism is a more natural form of reasoning by providing an example that she claimed was a real occurrence:

A little girl of four years of age was making, at her dinner, the interesting experiment of eating her soup with a fork. Her nurse said to her, “Nobody eats soup with a fork, Emily,” and Emily immediately replied, “But I do, and I am somebody” [Ladd-Franklin 1928, p. 532].

This can be seen to be an antilogism by letting $$f$$ represent the class of people who eat soup with a fork, and $$e$$ represent the class of people like Emily. Then we have that Emily, as a member of $$e$$, is not consistent with the idea that people do not eat soup with a fork ($$e \overline{\vee}\overline{f}$$ ), and also that Emily (or people like her) do actually exist ($$e \vee\infty$$), which is inconsistent with the conclusion that no people who eat soup with a fork exist ($$f \overline{\vee} \infty$$). Meaning, when we combine these three statements, we arrive at an antilogism: $\left ((e\overline{\vee} \overline{f})(f\overline{\vee}\infty) (e \vee \infty) \right ) \overline{\vee} .$

# An Explication of the Antilogism in Christine Ladd-Franklin's "Algebra of Logic" – Later Career and Historical Significance

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

Despite not having received the degree for her work in 1882, Ladd-Franklin’s accomplishments did not go unrecognized. Her work in optics was so well-regarded that in 1887 she was awarded an honorary doctorate by Vassar, the only person to ever have been given this honor [Green and LaDuke 2016, p. 341]. In 1926, during celebrations of the institution's 50th anniversary, Johns Hopkins offered to bestow upon her an honorary doctorate in a special ceremony. Her response, as described by historian Judy Green, was to remind those at Johns Hopkins that she had already been awarded an honorary doctorate by Vassar. She went on to explain that, instead of another honorary degree, she believed she should be given the PhD she earned for the work done there [Green 1987, p. 124]. Thus it was in 1926, forty-four years after her completion of the degree requirements, Christine Ladd-Franklin was finally awarded her PhD by Johns Hopkins University for her work in symbolic logic [Green 1987, p. 124]. As the first female to earn, but not be awarded, a PhD in mathematics in the United States, Christine Ladd-Franklin deserves to be recognized for what she was: a pioneer in mathematics, physiology, and women’s rights, contributing significant and impressive work across all of these areas.

Christine Ladd’s contributions to the fields of mathematics and science did not end with the completion of her doctoral-level studies at Johns Hopkins. Shortly after completing her studies in 1882, she married Fabian Franklin (1853–1939), a fellow student at Johns Hopkins who received his PhD in  mathematics in1880. She continued her work in symbolic logic and began to work in the field of physiological optics, where she contributed to a theory of color vision, eventually publishing a book of her work in optics [Ladd-Franklin 1929]. She also became an advocate for women’s education, spending both her time and money to help women obtain graduate education [Green 1987, pp. 122–124].

Acc. 90-105 - Science Service, Records, 1920s–1970s, Smithsonian Institution Archives

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

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

Delaware, R. (2019). More than just a grade: The HOM SIGMAA student contest fosters writing excellence at UMKC. MAA Convergence 16.

Ending Life at College; Commencement Exercises at Columbia College. Awarding of Honors at the Academy of Music—Miss Winifred Edgerton Receives a Degree. (1886, June 10). New York Times.

Goldin, C. (1980). The work and wages of single women, 1870–1920. Journal of Economic History 40(1): 81–88.

Green, J. (1987). Christine Ladd-Franklin (1847-1930). In Women of Mathematics, edited by L. S. Grinstein and P. J. Campbell, 121–128. New York: Greenwood Press.

Green, J., LaDuke, J. (2009). Pioneering Women in American Mathematics: The Pre-1940 Ph.D.’s. Providence, RI: American Mathematical Society.

Green, J., LaDuke, J. (last revised in 2016). Supplementary Material for Pioneering Women in American Mathematics: The Pre-1940 PhD’s. American Mathematical Society.

Johnson, C. (2008). Christine Ladd-Franklin. Vassar Encyclopedia.

Kelly, S. E., Rozner, S. A. (2012). Winifred Edgerton Merrill: "She Opened the Door". Notices of the American Mathematical Society 59(4): 504–512.

Ladd, C. (1883). On the algebra of logic. In Studies in Logic by Members of the Johns Hopkins University, edited by Eschbach, A., Fisch, M. H., Pierce, C. S., 17–71. Baltimore, MD: Johns Hopkins University.

Ladd-Franklin, C. (1928). The antilogism. Mind 37(148): 532–534.

Ladd-Franklin, C. (1929). Colour and Colour Theories. New York, NY: Harcourt, Brace, and Co.

Rossiter, M. W. (1980). “Women’s Work” in science, 1880–1910. Isis 71(3): 381-398.

Russinoff, I. S. (1999). The syllogism’s final solution. The Bulletin of Symbolic Logic 5(4): 451–469.

Shen, E. (1927). The Ladd-Franklin formula in logic: The antilogism. Mind 36(141): 54–60.

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

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

Julia M. Parker initially wrote this paper as a part of a History of Mathematics class at the University of Missouri – Kansas City, where she was pursuing teacher certification in the field of mathematics. (Her BA degree is in Biology from William Jewell College.) By the time the paper appeared in Convergence, she was enrolled at Avila University as a Graduate Student in Education, completing the requirements for both a middle school math teaching certification and a Master’s in Education.She works at Longview Community College as a Learning Specialist, working with tutors and Supplemental Instruction (SI) programs to provide in-classroom support for students in typically difficult college courses. With her husband of almost 19 years, she has five wonderful children, four boys ages 16, 14, 8, and 8, and a 12-year-old daughter. After supporting her husband through his BS, MDiv and PhD degrees, it was Julia's turn to go back to school. She is also busy with the kids’ band and school activities and helping to care for the family's various pets: a dog, a guinea pig, a rabbit, a bearded dragon, a Russian tortoise, and an assortment of fresh- and salt-water fish in several aquariums. She is a voracious reader who also crochets and cross-stitches when life gives her anything resembling free time.