I saw this video: Tautologies and Contradictions; however, the example he claimed to be a tautology, "That dog is a mammal", is actually NOT a tautology, if I refer to the textbook, A Tour Through Mathematical Logic by Wolf:
Definition. A statement P is called a tautology or law of propositional logic if there is a set of substatements of P such that:
(a) P is a propositional combination of those substatements, and
(b) P is true for every combination of truth values that is assigned to these substatements.
Example 3. The equation $2 + 2 = 4$ is not a tautology. Its only substatement is itself, [...] it's not a tautology because its form is simply "P", with no shorter substatements
Similarly, "For every number x, x = x" is not a tautology. Simply put, it cannot be a tautology because it includes no connectives.
I am genuinely confused. I am sure this guy in the video is well-educated in math. (He got his math PhD from University of Toronto). So can "tautology" be defined differently depending on the "version" of mathematical logic theories? Or is the guy just misleading people (which I would be very surprised about, given his credentials)?