How did William Rowan Hamilton and Augustus De Morgan shape the modern idea of logical form?

The modern logical notion of "form"—specifically the self-conscious separation of formal from nonformal aspects of sentences and arguments—crystallized in the 19th-century work of the Irish mathematician Sir William Rowan Hamilton and the English logician Augustus De Morgan.12

While earlier logicians had explored patterns of reasoning, Hamilton and De Morgan were instrumental in shifting logic toward the modern emphasis on logical form. De Morgan, in particular, contributed through his seminal work Formal Logic; or, the Calculus of Inference (1847), which helped establish the standard discussions of validity and invalidity that define modern symbolic logic.134

By moving beyond the limitations of traditional Aristotelian syllogistic theory, these thinkers helped foster a new era where logic was increasingly linked with mathematics. Their efforts, alongside those of contemporaries like George Boole, laid the groundwork for the formal representation of statements and inferences, ultimately helping to distinguish between surface grammatical structure and underlying logical structure.514