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The traditional definition of subject as referring to the 'doer of an action' (or agent), though it is adequate for central or typical cases, will not work for all cases. For example, in passive sentences, such as John was attacked, the subject is John, but John is certainly not the 'doer' of the attacking. Again, not all sentences, even those with transitive verbs, express any action. Examples are This book cost fifty francs and I loathe relativism. But such sentences have always traditionally been held to have subjects (in these cases, this book and I)."
(James R. Hurford, "Grammar: A Student's Guide". 1994)