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Reflexive intermediate propositional logics. (English) Zbl 1105.03025
The author modifies the formulation and proof of a theorem due to D. C. McCarty [Notre Dame J. Formal Logic 43, 243–248 (2002; Zbl 1050.03041)] to get the following statement: Let $$T$$ be a second-order theory based on a superintuitionistic logic. If $$T$$ proves the completeness theorem (consistency implies existence of a model) for some superintuitionistic propositional logic, then $$T$$ proves every instance of $$\neg\varphi\vee\neg\neg\varphi$$. The proof uses very few properties of $$T$$ except comprehension. Defining $$A= \{p\mid\neg\varphi\}\cup \{\neg p\mid\neg\neg\varphi\}$$ and $$B= A\cup\{p\vee\neg p\}$$, $$B$$ is consistent since both $$\{p\}$$ and $$\{\neg p\}$$ are. By completeness, there is a model $$M\models B$$, hence $$M\models p\vee\neg p$$, so $$\neg\varphi\vee\neg\neg\varphi$$.

##### MSC:
 03B55 Intermediate logics
##### Keywords:
intermediate logic; weak excluded midlle
Full Text:
##### References:
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