# Compactness of first-order fuzzy logics

## TL;DR

This paper investigates the compactness property in first-order many-valued logics, especially focusing on $K$-satisfiability, by introducing topological approaches to extend existing results in continuous t-norm based logic.

## Contribution

It extends the understanding of compactness in many-valued logics by applying topological methods and reverse semantics to continuous t-norm based logic.

## Key findings

- Extended compactness results for $K$-satisfiability in basic logic.
- Introduced a topology on [0,1] and [0,1]^2] for logical connectives.
- Connected topological continuity with logical satisfiability properties.

## Abstract

One of the nice properties of the first-order logic is the compactness of satisfiability. It state that a finitely satisfiable theory is satisfiable. However, different degrees of satisfiability in many-valued logics, poses various kind of the compactness in these logics. One of this issues is the compactness of $K$-satisfiability. Here, after an overview on the results around the compactness of satisfiability and compactness of $K$-satisfiability in many-valued logic based on continuous t-norms (basic logic), we extend the results around this topic. To this end, we consider a reverse semantical meaning for basic logic. Then we introduce a topology on $[0,1]$ and $[0,1]^2$ that the interpretation of all logical connectives are continuous with respect to these topologies. Finally using this fact we extend the results around the compactness of satisfiability in basic ogic.