Are there first order theories where every sentence or its negation is a theorem of the theory? I know there are many examples of theories without this property, such as fields and statements such as (4+1)=0. In the rational numbers this is false while in a 5 element finite field it is true. I am curious as to whether there exist any other theories where all logical sentences or their negations are theorems.
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3Many, among them the theory of algebraically closed fields of any given characteristic, and the theory of real-closed fields. – André Nicolas Jun 14 '15 at 07:24
