States on polyadic MV-algebras (Q965911)

The paper generalises Gaifman's approach to probabilistic models of first-order classical logic to the case of first-order Łukasiewicz infinite-valued logic. The paper is structured as follows: Section 2 recalls fundamental notions of the theory of MV-algebras, which constitutes the algebraic semantics of propositional Łukasiewicz logic, while Section 3 recalls the definition and basic properties of polyadic MV-algebras, which constitute the algebraic semantics of first-order Łukasiewicz logic, in the sense that Lindenbaum algebras of first-order Łukasiewicz logic are polyadic MV-algebras. New results are given in Sections 4 and 5. In particular, in Section 4, the author introduces the notion of polyadic MV-state and proves that for any MV-subalgebra \(D\) of a polyadic MV-algebra \(A\), and for any state \(\mu\) (the MV analog of a probability) there is a polyadic MV-state \(m\) extending \(\mu\) in the sense that the restriction of \(m\) to \(D\) coincides with \(\mu\). In Section 5, this result is carried over from algebra to the first-order Łukasiewicz logic \(\L\forall\) as follows: Given a set of new constant symbols \(U\), an MV-state \(m\) over the language of \(\L\forall\) extended with \(U\) is called a Gaifman state if for any formula \(\varphi(x)\) it holds that \[ m(\exists x\varphi(x))= \sup\{m(\bigvee_{i=1}^n \varphi(c_i)) \colon n \in \omega,\, c_1,\ldots,c_n \in U\}\,. \] Then a Gaifman MV-model of a state \(\mu\) over an MV-algebra \(D\) is a pair \((U,m)\) for \(U\) as before and \(m\) a Gaifman state over \(\L\forall\) extended with \(U\) such that the restriction of \(m\) to \(D\) coincides with \(\mu\). The final theorem then proves that any MV-state \(\mu\) has a Gaifman MV-model.