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My prior is as follows: it is very possible to go from mathematics or computer science to specific philosophy PhD programs, including (but not limited to) CMU, UC Irvine LPS, Pitt HPS, Berkeley’s logic group, and (maybe) Notre Dame’s logic track. But, I would guess that most successful students have taken a number of relevant philosophy classes. For such programs it is probably less important to have taken, say, an ancient philosophy course, whereas philosophy of science courses might be more helpful.

Of additional note are specific master’s degrees that offer a bridge: LMU’s MCMP, LSE’s master’s degrees, Bristol’s master’s degrees, Amsterdam’s master’s degree in logic, and CMU’s master’s degree. All of these cost money, with the exception of LMU.

From computer science specifically, it would probably behoove the applicant to note how her CS background is applicable to what philosophers at the relevant school are working on. For example, CMU has a number of philosophers that work with computer scientists on issues in causality, Bayesian decision theory, and computability theory. 
 

In summary I would say that yes, it is possible, provided the applicant has some background in formal philosophy coursework. A well-done writing sample should also go a long way (but, CMU’s admissions page does state that some students have instead submitted a specific proof or piece of code, if I remember correctly). These programs are all quite competitive, and often niche in the kinds of philosophy that they do. 

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