Philip Jourdain suffered from severe disabilities all his life. He was already crippled when he went up to Cambridge in 1898. His undergraduate years were very difficult and he had great successes and great disappointments. He did very poorly in his degree and had to settle for only a pass degree. However he was awarded the Allen studentship for research in 1904.
Jourdain worked in mathematical logic. He wrote a number of articles explaining and evaluating Cantor's set theory between 1906 and 1913 under the title Development of the Theory of Transfinite Numbers. In his work he clarified a remark by Russell and formulated precisely the paradox of the largest ordinal. In 1913 Jourdain proposed the card paradox. This was a card on one side of which was printed:
The sentence on the other side of this card is TRUE.
On the other side of the card the sentence read:
The sentence on the other side of this card is FALSE.
Jourdain also applied logic to physics in papers such as On
some Points in the Foundation of Mathematical Physics (1908).