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

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].

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: \[ \left ((e\overline{\vee} \overline{f})(f\overline{\vee}\infty) (e \vee \infty) \right ) \overline{\vee} .\]