Everyone has heard the sawpit explanation for "underdog." Two sawyers cut a log, one standing on top, one in a pit below in the sawdust. The top man is the top dog, the bottom man the underdog. The trouble is that nobody has found it in the record.
Phrase Finder, which examined the claim, says no one has found evidence for it. The saw-pit descriptions it cites, from the 1600s and 1800s, never use "top dog" or "under dog." Shakespeare's Merry Wives of Windsor mentions a sawpit without them. World Wide Words reached the same place. Its author says he is "extremely suspicious" of the story, and that every early example he traced refers to actual dog fights, not sawing. He does not call it false, only unproven.
What does the record show? Etymonline defines underdog as "the beaten dog in a fight" and dates it to 1887. Phrase Finder cites an earlier verse called The Under Dog In The Fight that ran in several US newspapers in 1859. Both point to a literal animal on the losing end of a fight.
The figurative sense presumably grew from that picture: a person or team expected to lose is like the dog on the ground. Etymonline gives a matching word, overdog, from 1908, and the poet Robert Frost used underdog in his poem "Canis Major" in 1928.
"Top dog" followed a separate path. Phrase Finder links it to the 1840 phrase "tip-top dog," used of hunting dogs in The Sporting Review. It finds the first figurative use in 1900, when The Speaker wrote of Britain becoming "top dog" in South Africa. "Bottom dog" is recorded by 1884 in the Oxford English Dictionary.
The sawpit story is plausible, which is why it spreads. But plausible is not the same as attested, and the written evidence points to the dog ring.