There is no evidence that Shakespeare was illiterate. But Cutler is also misguided in calling him a “brilliant scholar.” His contemporaries certainly didn’t think of him that way. Ben Jonson felt he “wanted art,” i.e. lacked erudition; Francis Beaumont considered his best lines “clear” of “all learning.” And that view prevailed: later, John Milton praised Shakespeare’s “native wood-notes wild” – the opposite of scholarship.
The notion that Shakespeare was extraordinarily erudite is a 20th-century fiction, an effect of historical distance. Even now, though, it is easy to identify a truly learned writer: just read Jonson. His Sejanus bursts with classical footnotes; the Venice of his Volpone, unlike Shakespeare’s, is pieced together from a meticulous study of authoritative sources; his scenes debating literary theory are incomprehensible to modern readers. (Jonson was a bricklayer’s son who hadn’t gone to university.)
Jonson’s learnedness, however, makes his works a hard sell nowadays. Unlike Cutler, I would identify Shakespeare’s very lack of erudition, his limitations, as the qualities that make his works enduringly powerful; his thoughts, and especially their expression, can be startlingly simple. Shakespeare’s language shows more familiarity with rural England than with any field of learning, although he clearly could reference the worlds of law, of alchemy, or of sports like hawking and tennis – he lived in London for most of his life, after all, cheek by jowl with courtiers, and performed for aristocratic audiences every year.