Barthes' Structuralist Method Applied to "The Doctor and the Doctor's Wife"
A sign functions in the system as an index, icon, or symbol, and the iconic or symbolic functions of a sign give a narrative mythic value.
The most obvious mythic signs in the story are the logs (lost from "the steamer Magic"!), the shotgun (potent but destructive masculinity), the book (contemplative spirit), and the black squirrels (active mystery in the forest from which the logs came).
As an index, the logs are wood, the shotgun is a weapon, the book is a document, and the squirrels are animals. Each of the indices might be more precise if we had numbers associated with them, which the first paragraph begins to supply for the logs. Apparently logs are in such profuse supply that they can be wasted this way (logs as index of hardwood over-production). The number of the squirrels also would be an icon of last year's acorn crop (more food = more baby squirrels [squee!]). The squirrels' presence and quantity might be a further index of the relative lack of use of Doc's shotgun.
As icons, the logs represent the property of the Whyte and McNally company (via their symbol stamped into the wood), as well as Doc's property if treated as flotsam, and even as an index of Dick Bolton's strength as he can manipulate them with a cant hook. The shotgun is an icon of Doc as a hunter, perhaps also as a killer in general, and to an NRA member, as a sign of the Second Amendment. The squirrels are an icon of ecodiversity in a forest in which brown squirrels might be the norm.
As symbols, the logs seem to represent control of the natural world by two competing systems of value. Dick mocks "Doc" for pretending he can assume ownership of a natural piece of wood on a beach, pointing out the logging company's competing ownership mark as evidence that Doc has no power to claim ownership. Which system rules the beach, the Native American ecological co-existence model, or the European capitalist ownership/possession model? The result of the threatened violence suggests the former is more powerful.
As a symbol, the shotgun seems to represent the doctor's attempt to reassert his European, capitalist, violence-enforced control over the narrative. The Doctor's Wife challenges the shotgun's mythic power by chanting the biblical passage from Proverbs 16:32 to challenge the shotgun that "taketh a city" with a superior rule of the spirit. This gives "Doc" the power to escape being the diminutive "dear" or "Henry" and to become "the doctor" and "Nick's father" once more.
As a symbol, the black squirrels seem to represent mysterious natural event whose shared witnessing will restore Nick and Nick's father to a harmonious relationship that was threatened by the failed quest for control of the logs, and by the Doctor's Wife's dangerous imposition of her "rule" over the doctor's "spirit." Seeking and finding the squirrels might even be treated as an initiation ritual in which (role reversal!) Nick enables his father to assume patriarchal authority once more.
So what? Hemingway's story compresses a man's crisis of identity into an encounter with three mythic phenomena. They transform him from father to scorned man to female dominated man and back to father again. Male identity is shown to be constructed by a system of challenging encounters with potent objects whose control defines the self.