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What are logicians?

What are logicians? Here are some definitions.

Noun
  1. plural of logician
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You do not want linguistic theory to analyze all coordinate structures the way logicians do, in terms of binary coordinators.
Venn extended Boole's mathematical logic and is best known to mathematicians and logicians for his diagrammatic way of representing sets, and their unions and intersections.
Less is known about the Megarian logicians, but they seem to have been particularly interested in conditionals, and also in logical paradoxes.
How, exactly, does the terminology usually regarded by logicians as logical work in making it the case that one sentence follows from others?
There are certainly mathematical logicians who are formalists, even in the light of the incompleteness theorems.
Any one of the ordinary, everyday proofs mathematicians deal with in their work can be re-written in a formal way that fits into this framework the logicians have axiomatized.

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