## On second order intuitionistic propositional logic without a universal quantifier.(English)Zbl 1163.03010

IPC$$^2$$ is the intuitionistic second-order propositional calculus. The author proves that sentences (closed formulas) $$\phi$$ without $$\forall$$ constitute a Glivenko class: $$\phi$$ is provable in IPC$$^2$$ iff $$\phi$$ is a classical tautology with $$\exists p\alpha(p)$$ interpreted as $$\alpha(\top)\vee\alpha(\bot)$$. For formulas with free variables, like $$p\vee\neg p$$, this is false. The result is shown by a modification of Kalmar’s completeness proof for classical propositional logic. For every $$\phi(p_1,\dots,p_n)$$ with free variables $$p_1,\dots,p_n$$ without $$\forall$$ induction on $$\phi$$, the author shows that intuitionistically $$p_1^{\varepsilon_1}\&\dots\&p_n^{\varepsilon_n}\to \phi^{\varepsilon}$$, where $$\varepsilon_1,\dots,\varepsilon_n$$ are truth-values of $$p_1,\dots,p_n$$ and $$\varepsilon$$ is the corresponding classical truth-value of $$\phi$$. In the (crucial) $$\exists$$-case, one uses the formula $$\phi(\tau)\to\exists p\phi(p)$$ for $$\varepsilon=1$$ and the formula $$(\forall p((p\vee\neg p)\to\neg\phi(p)))\to \neg\exists p\phi(p)$$ for $$\varepsilon=0$$. As a consequence, $$\forall$$ is not intuitionistically definable in terms of the remaining connectives.

### MSC:

 03B20 Subsystems of classical logic (including intuitionistic logic)
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### References:

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