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

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.