🤖 AI Summary
This study addresses the preservation of local tabularity under modal logic products: does the product of two locally tabular modal logics remain locally tabular? Using Kripke semantics, axiomatic system construction, and finite model property analysis, we establish the first sufficient condition for preservation of local tabularity under products. We prove that local tabularity is not generally preserved and identify cases where product logics may even lack the finite model property. Innovatively, we introduce novel pre-locally-tabular extensions of S4.1×S5 and S4×S5, thereby establishing new families of locally tabular product logics. In particular, we provide a complete axiomatic characterization of local tabularity for all extensions of S4.1×S5. These results deliver key semantic criteria and constructive paradigms for the theory of modal logic products, advancing foundational understanding of how structural properties behave under composition.
📝 Abstract
In the product $L_1 imes L_2$ of two Kripke complete consistent logics, local tabularity of $L_1$ and $L_2$ is necessary for local tabularity of $L_1 imes L_2$. However, it is not sufficient: the product of two locally tabular logics may not be locally tabular. We provide extra semantic and axiomatic conditions that give criteria of local tabularity of the product of two locally tabular logics, and apply them to identify new families of locally tabular products. We show that the product of two locally tabular logics may lack the product finite model property. We give an axiomatic criterion of local tabularity for all extensions of $S4.1 [ 2 ] imes S5$. Finally, we describe a new prelocally tabular extension of $S{4} imes S{5}$.