No. The documents show that he could write: six of his signatures survive, on four legal documents. They appear on a court deposition of 1612, on the purchase and the mortgage of a house in the Blackfriars district of London in March 1613, and in the three signatures on his will, of March 25, 1616, one on each page. The spelling varies (Willm Shakp, William Shakspere, William Shakspeare, among other forms), and all are in the English secretary hand he would have learned at school.
The doubt arises from gaps. No enrollment record of the King's New School in Stratford, the free school that most biographers believe he attended, has survived. His parents signed with a mark, and so did his daughter Judith. No letters or literary manuscripts of his survive, and his will does not mention books. Supporters of other authors cite the irregular appearance of the signatures as evidence of illiteracy.
Supporters of traditional authorship reply that the loss of papers was common at the time and that the absence of proof is not proof of absence. They also remind us that the plays reflect the curriculum of grammar schools, with references to William Lily's Latin grammar, and that Ben Jonson and Francis Beaumont commented on Shakespeare's lack of a classical education, which is not the same as illiteracy.
From this came the so-called authorship question, the thesis that someone else wrote the plays, with candidates such as Francis Bacon, Edward de Vere (Earl of Oxford), and Christopher Marlowe; more than 80 names have been proposed. Nearly all scholars treat it as a fringe theory. For Shakespeare of Stratford speak, among others, Francis Meres's record of 1598, which cites twelve of his plays; the records of 1600 and 1607 that attribute plays to 'mr Shakespere' and 'Mr William Shakespeare'; and the First Folio, organized by company colleagues who knew him. It is a disputed subject, and it is worth knowing both sides.