The Complexity Zoo defines $LIN$ to be the class of decision problems solvable by a deterministic Turing machine in linear time.
$$LIN \subseteq P$$
Since HORN-SAT is solvable in $O(n)$ (as indicated in Linear-time algorithms for testing the satisfiability of propositional horn formulae (1984))
New algorithms for deciding whether a (propositional) Horn formula is satisfiable are presented. If the Horn formula $A$ contains $K$ distinct propositional letters and if it is assumed that they are exactly $P_1,…, P_K$, the two algorithms presented in this paper run in time $O(N)$, where $N$ is the total number of occurrences of literals in $A$.
I am wondering why we can't conclude that
$$LIN = P$$
given that HORN-SAT has also been proven to be $P$-complete under log-space reduction? I must be missing something. Or is that a well-known fact?
(I have yet thoroughly gone through the 1984 paper so I don't quite understand the algorithms for solving HORN-SAT in linear time, and thus I may have misunderstood the implication.)