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Published online by Cambridge University Press: 22 January 2016
The purpose of this paper is to present a propositional calculus whose decision problem is recursively unsolvable. The paper is based on the following ideas:
(1) Using Löwenheim-Skolem’s Theorem and Surányi’s Reduction Theorem, we will construct an infinitely many-valued propositional calculus corresponding to the first-order predicate calculus.
(2) It is well known that the decision problem of the first-order predicate calculus is recursively unsolvable.
(3) Thus it will be shown that the decision problem of the infinitely many-valued propositional calculus is recursively unsolvable.
page 145 note 2) In the first-order predicate calculus, the semantical decision problem is equivalent to the syntactical one by the completeness theorem.
page 146 note 1)
Using those logical operations, we define 
page 147 note 1) Of course, h(F(y)) → ◊ Y F, h(F(z)) → ◊ZF, h(G(y, z)) → ◊(YG1 Λ ZG2), ….
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