One way of illustrating the issue is to use hypothetical examples. Let's suppose that you have a not too remote ancestor called John Doe. You know this ancestor exists because you exist. In other words, you had a mother and father and so forth back to your Great-grandfather Doe. At this point you do not know diddle about Doe. In fact, the name John Doe is made up, because you don't even have a name. His son's name is Ebenezer Doe. So, can you assume his father's name was also Doe? How many pedigrees have you seen with end-of-line designations as Mr. [fill in the blank]? In our case we call this end-of-line ancestor "John Doe."
Assuming that his name is Doe, is an example of using the secondary evidence of the son's surname to "guess" the name of the father. What might have happened in reality?
- The son, Ebenezer, may have changed his name from Kiawakowski or whatever to Doe.
- The son, Ebenezer, may have taken his mother's maiden name.
- The son, Ebenezer, may have never known his father's name and simply took the name of a foster family.
- The son, Ebenezer, may have been adopted and never knew his birth father's name.
- And so forth.
So you decided to do some searching to see if the remote ancestor's name was really Doe. Guess what? It wasn't. The great-grandfather came over from the "old country" and his name was really Dorobrovitch, which he shortened to Doe. But he never legally changed his name. How did I find his name? I found his Naturalization Record. Was the Naturalization Record a primary or secondary source? It is really a mixture of both, but who entered the name into the Naturalization Record? Good question. Could Johann Dorobrovitch write? Did he fill out the application or was it done for him? See, the issue of whether or not a particular document is primary or secondary is a little more problematic than simply looking to the commonly accepted definition of primary -- that is a document created by a person present at the time of the event and created at or near the time of the event recorded.
The point raised in the comment to my recent post, was how do you know if the secondary sources are probative and not simply serial copies of each other? The answer: you don't unless you do the research to see if any of the secondary accounts have some concrete basis in verifiable, reproducible, source documentation. My adage: doubt everything in genealogy and accept no conclusion as absolutely final. But as a practical solution to the quandary of when to stop looking and draw a conclusion, I suggest the three point rule: any location in space can be determined by three points (excluding extra dimensions and all that). So my rule of thumb is that when I have three unrelated documents that concur on a specific fact, I conditionally accept the fact as established, i.e. the law of witnesses.
So, the answer to the question in the title to this post is yes. You can prove relationships with secondary evidence, but the exception is that the proof is never conclusive and is always tentative awaiting additional evidence.