See also: informal description of first order predicate logic
formal description of predicate logic
and What is Logic?.
This is a semiformal specification of something close to the logic described by Gottlob Frege in [Frege1879].
I call the specification semiformal because I am not supplying a definition of the metalanguage in which it is written.
The concrete syntax of the formal system described by Frege in Begriffsschrift is graphical (for details see: ). I have not attempted to replicate this, but have confined myself to describing an equivalent system in more conventional linear syntax. Though the Begriffsschrift marked the beginning of a new epoch in Logic it falls short of perfection in some aspects of the definition of the formal system, and it has thefore been necessary (for the purpose of presenting a fully defined formal system) to make some definite choices which are not drawn from the text of the Begriffsschrift. 
The main areas in which this has been necessary are:


