  To explain these puzzles, Frege suggested (1892a) that in addition to having a denotation, names and descriptions also express a  sense.[5]   The sense of an expression accounts for its cognitive significance—it is the way by which one conceives of the denotation of the term. The expressions ‘\(4\)’ and ‘\(8/2\)’ have the same denotation but express different senses, different ways of conceiving the same number. The descriptions ‘the morning star’ and ‘the evening star’ denote the same planet, namely Venus, but express different ways of conceiving of Venus and so have different senses. The name ‘Pegasus’ and the description ‘the most powerful Greek god’ both have a sense (and their senses are distinct), but neither has a denotation. However, even though the names ‘Mark Twain’ and ‘Samuel Clemens’ denote the same individual, they express different senses. (See May 2006b for a nice discussion of the question of whether Frege believed that the sense of a name varies from person to person.) Using the distinction between sense and denotation, Frege can account for the difference in cognitive significance between identity statements of the form ‘\(a=a\)’ and those of the form ‘\(a=b\)’. Since the sense of ‘\(a\)’ differs from the sense of ‘\(b\)’, the components of the sense of ‘\(a=a\)’ and the sense of ‘\(a=b\)’ are different.  Frege can claim that the sense of the whole expression is different in the two cases. Since the sense of an expression accounts for its cognitive significance, Frege has an explanation of the difference in cognitive significance between ‘\(a=a\)’ and ‘\(a=b\)’, and thus a solution to the first puzzle.