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Russell defines the famous Barber paradox in the POM: "You can define the barber as: one who shaves all those, and those only, who do not shave themselves. Does the barber shave himself?" Lepetic, Vladimir. Principles of Mathematics: A Primer. Hoboken: John Wiley & Sons, Incorporated, 2016. https://www.youtube.com/watch?v=GpVRePLMLbU
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Russell defines the theory of denoting phrases. What that basically means is that he explains a "complex expression as a subject of a sentence." Russell states: "a phrase may be denoting and yet not denote anything" and is further explained in mathematical applications.
Russell, Bertrand, II.—ON DENOTING, Mind, Volume XIV, Issue 4, 1905, Pages 479–493, https://doi.org/10.1093/mind/XIV.4.479
https://academic.oup.com/mind/article/XIV/4/479/964157
https://www.youtube.com/watch?v=91NuS0VU2-k -
The significance of this may at first seem understated in that it is 1+1=2. "Russell worked on the philosophical, including the no-class theory (in which set or class terms become meaningful only when placed in well-defined contexts)"
Linsky, Bernard and Irvine, Andrew David, "Principia Mathematica", The Stanford Encyclopedia of Philosophy (Fall 2019 Ed), Edward N. Zalta (ed), https://plato.stanford.edu/archives/fall2019/entries/principia-mathematica.
https://www.youtube.com/watch?v=JMUydTJdXu4 -
Russell "introduced logical atomism to describe his philosophy." It is defined as an "endorsement of analysis."
Klement, Kevin, "Russell’s Logical Atomism", The Stanford Encyclopedia of Philosophy (Spring 2020 Ed), Edward N. Zalta (ed), https://plato.stanford.edu/archives/spr2020/entries/logical-atomism.
https://www.youtube.com/watch?v=uz9_u_06QsA
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