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I have just started learning logic, and was wondering is there any difference between symbolic and formal logic, or are they the same thing? And I would also like to know what the relationship of mathematical logic and these two logics are

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  • $\begingroup$ They are basically he same discipline. When Aristotle "discovered" formal logic, he did not used mathematical symbols to study it. Math logic born in 19th Century, with Boole, through the "application" of math symbols (mainly: algebraic ones) to formal logic. $\endgroup$ – Mauro ALLEGRANZA Jun 4 '16 at 19:05
  • $\begingroup$ See math.stackexchange.com/a/1666842/21820. $\endgroup$ – user21820 Jun 5 '16 at 2:15
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"Symbolic logic" means writing things using symbols rather than prose. Most mathematics more than 400 years old was done using prose. Many logic arguments are still done using prose, rather than the more common grammar of something like first order logic. There is a balance between the readability of prose and the precision of symbols that has to be balanced for humans to be able to easily read a mathematical argument.

"Formal logic" means defining things to the point where a computer can verify them. Every single step of a proof is so well defined that it requires absolutely no human intuition at all to verify. Most formal logic proofs are not easily human readable, although each individual step may be.

Formal logic is always symbolic since natural language isn't precise enough to be formalized. However, symbolic logic is not always formal. It is common to leave mundane details out of mathematical proofs, leaving behind a proof that is possibly symbolic but not formal.

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