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

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:

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:

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:

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