  With this description of language, Frege can give a general account of the difference in the cognitive significance between identity statements of the form ‘\(a=a\)’ and ‘\(a=b\)’. The cognitive significance is not accounted for at the level of denotation. On Frege’s view, the sentences ‘\(4=8/2\)’ and ‘\(4=4\)’ both denote the same truth value. The function \((\:)=(\:)\) maps \(4\) and \(8/2\) to The True, i.e., maps \(4\) and \(4\) to The True. So \(d[4=8/2]\) is identical to \(d[4=4]\); they are both The True. However, the two sentences in question express different thoughts. That is because \(s[4]\) is different from \(s[8/2]\). So the thought \(s[4=8/2]\) is distinct from the thought \(s[4=4]\). Similarly, ‘Mark Twain = Mark Twain’ and ‘Mark Twain = Samuel Clemens’ denote the same truth value. However, given that \(s\)[Mark Twain] is distinct from \(s\)[Samuel Clemens], Frege would claim that the thought \(s\)[Mark Twain = Mark Twain] is distinct from the thought \(s\)[Mark Twain = Samuel Clemens].