In Euclid's Elements, the famous five planar geometry axioms are formulated in common language (ancient greek in this case) and use ambiguous terms.

On the other hand, modern theories like ZFC or Peano arithmetic are formulated using symbols (as part of first order logic) and reasoning can be formulated in a mechanical way using syntactic manipulations.

My question is: when and how did people first thought of using a purely symbolic logic?

  • 2
    $\begingroup$ Use of symbols is just a matter of abbreviation, it is conceptually irrelevant. First formal logic was Aristotle's syllogistic, which predated Euclid, soon followed by Stoic version. Symbolic abbreviations for syllogism figures were introduced by medieval scholastics. Attempts to "syllogize" Euclid started in the late Renaissance, but were unsuccessful because syllogistic had limited expressive means. Formalization of mathematics only became possible after introduction of relations and quantifiers into logic in late 19th century. $\endgroup$
    – Conifold
    Commented Apr 13 at 22:29
  • $\begingroup$ After the development of symbolic algebra during the Renaissance. First attempts with Leibniz. $\endgroup$ Commented Apr 14 at 6:50
  • $\begingroup$ Perhaps George Boole, using $+$ for "or" and $\times$ for "and". His book was called The Laws of Thought. His system is now called "Boolean algebra". $\endgroup$ Commented Apr 16 at 1:50
  • $\begingroup$ @Conifold I agree that a word can be used as a symbol. In the case of Euclide, this does not seem to be the case as he used words from the common language, which he did not specify formally (at least, that's not mentioned in the axiomatization presented on Wikipedia). $\endgroup$
    – Weier
    Commented Apr 16 at 12:16


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