A study of truth predicates in matrix semantics
| Autor: | Tommaso Moraschini |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Logic
Computer science Semantics (computer science) 010102 general mathematics Mathematics - Logic 02 engineering and technology Propositional calculus 01 natural sciences Algebra Philosophy Matrix (mathematics) Mathematics (miscellaneous) Algebraic semantics Leibniz operator Computer Science::Logic in Computer Science ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION 0202 electrical engineering electronic engineering information engineering FOS: Mathematics Truth predicate 020201 artificial intelligence & image processing Abstract algebraic logic 0101 mathematics Algebraic number Logic (math.LO) |
| ISSN: | 1755-0203 |
| DOI: | 10.48550/arxiv.1908.01661 |
| Popis: | Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic ${\cal L}$ is associated with a matrix semantics $Mo{d^{\rm{*}}}{\cal L}$. This article is a contribution to the systematic study of the so-called truth sets of the matrices in $Mo{d^{\rm{*}}}{\cal L}$. In particular, we show that the fact that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ can be defined by means of equations with universally quantified parameters is captured by an order-theoretic property of the Leibniz operator restricted to deductive filters of ${\cal L}$. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ are implicitly definable if and only if the Leibniz operator is injective on deductive filters of ${\cal L}$ over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of ${\cal L}$ to its deductive filters over arbitrary algebras. We show that this is the case for logics expressed in a countable language, and that it need not be true in general. Finally we consider an intermediate condition on the truth sets in $Mo{d^{\rm{*}}}{\cal L}$ that corresponds to the order-reflection of the Leibniz operator. |
| Databáze: | OpenAIRE |
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