Abstract of a Term Paper on Birnbaum's Proof

I'm writing a term paper that I hope will serve as a preliminary step toward my philosophy comp.  Here's the abstract.

Frequentist methods of statistical inference violate the likelihood principle (L). However,  Birnbaum [4] proved that (L) follows from specific versions (S) and (C ) of two principles—the sufficiency principle and the conditionality principle, respectively—to which frequentists appear to be committed.   In a recent publication [15],  Mayo notes that Birnbaum’s proof  “has  generally  been  accepted  by  frequentists,  likelihoodists,  and Bayesians alike” (p. 307). Nevertheless, she argues that the proof is fallacious (chapter 7(III)). Mayo’s critique involves replacing Birnbaum’s (S) and (C ) with different formulations of the principles of sufficiency and conditionality (S’) and (C’). Mayo shows that (S’) and (C’) do not entail
(L) but gives no reason to doubt Birnbaum’s theorem that (S) and (C) entail (L). While Mayo thus fails to show that Birnbaum’s proof is fallacious, her critique does raise the important question whether (S) and (C) or (S’) and (C’) are better formulations of the principles of sufficiency and conditionality.  I canvas a few arguments that have been offered on either side of this issue.   On balance,  these arguments appear to favor Birnbaum’s position.   However,  they are not sufficiently compelling to declare the issue resolved.

I think it's a good start.  For the comp, I would like to address other responses to Birnbaum's argument in addition to Mayo's and to have something more definite to say about how we should interpret the sufficiency and conditionality principles.