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Probabilistic models for intuitionistic predicate logic. (English) Zbl 1250.03013

The language of intuitionistic predicate logic \({\mathbf I}\) (respectively \({\mathbf I}(U)\)) is a first-order language with a non-empty set of constants \(C\) (respectively a set of constants \(U\) including \(C\)) and a non-empty set of predicates; \(E\) (\(E(U))\) denotes the set of sentences of \({\mathbf I}\) (\({\mathbf I}(U)\)). Let \(D\) be a subset of \(E(U)\) such that \(D\) contains the theorems of intuitionistic predicate calculus and is closed under the connectives \(\lor\), \(\land\), \(\to\), and \(\bot\). Intuitionistic probability on \(D\) is a function \(\mu:D\to[0,1]\) such that \(\mu(\phi)=1\) if \(\vdash\phi\); \(\mu(\phi)+\mu(\phi\to\psi)=\mu(\psi)+\mu(\psi\to\phi)\). A pair \((U,m)\), where \(m\) is an intuitionistic probability on \(E(U)\), is called a weak probabilistic structure for intuitionistic predicate logic if \(m(\exists x\,\phi(x))=\sup\,m\left(\bigvee\limits_{i=1}^n\phi(a_i)\right)\) for all \(a_1,\ldots,a_n\in U\). Let \(D\subseteq E\) and \(\mu:D\to[0,1]\) an intuitionistic probability. A weak probabilistic structure \((U,m)\) is called a weak probabilistic model of \(\mu\) if the restriction of \(m\) to \(D\) is equal to \(\mu\). It is proved that any intuitionistic probability \(\mu:D\to[0,1]\) has a weak probabilistic model.

MSC:

03B20 Subsystems of classical logic (including intuitionistic logic)
03F55 Intuitionistic mathematics
06D20 Heyting algebras (lattice-theoretic aspects)
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