Crucial to answering this question is understanding how London presents the dog and the man and how they are compared to each other. The dog is clearly sensitive to nature's signs of danger and follows its instincts. In contrast, the man ignores the warnings, believing that he can master the situation through force of will. London shows in this story that animal instincts are superior to human judgement in such a situation, precisely because animals are part of nature and thus will not think they are better or superior to nature. Note the following description of the dog:
This was a matter of instinct. To permit the ice to remain would mean sore feet. It did not know this. It merely obeyed the mysterious prompting that arose from the deep crypts of its being. But the man knew, having ahiceved a judgment on the subject, and he removed the mitten from his right hand and helped tear out the ice particles.
Consider how the dog is compared to the man. London is here suggesting that the dog's ability to rely on its instincts makes it better suited to survive than the man, who must depend on his limited and partial judgment. Man is presented as being out of sync with nature, having lost the natural connection with nature that once he had. The dog, by constrast, instinctively knows how to survive. Clearly, the chances of survival for the man in this situation are greatly reduced because of his separation from nature.
Therefore I think the dog would try to convince the man to trust more in instincts than in intelligence. Also the dog would try to encourage the man to trust the dog, rather than relying on his own ability to survive.