Axiomatizing first-order consequences in independence logic

Abstract

Independence logic, introduced in [8], cannot be effectively axiomatized. However, first-order consequences of independence logic sentences can be axiomatized. In this article we give an explicit axiomatization and prove that it is complete in this sense. The proof is a generalization of the similar result for dependence logic introduced in [15].

Dependence and Independence in Social Choice: Arrow’s Theorem

“…However, the first-order consequences of IndS are axiomatizable. A complete natural deduction system for the first-order consequence relation over sentences of IndS was given in [16]. More recently, Kontinen [23] generalized this result to open formulas by adding an extra predicate symbol to the signature.…”

“…That is, Â D is derivable in independence logic using the assumptions in Arrow . Due to its strong expressive power, the full independence logic is not axiomatizable (see [16] and also [25]). However, the first-order consequences of IndS are axiomatizable.…”

“…More recently, Kontinen [23] generalized this result to open formulas by adding an extra predicate symbol to the signature. Our main goal in this section is to demonstrate that Arrow's Theorem not only can be formalized in IndS, but also can be derived syntactically using the system of [16] and [23].…”

“…If 'ˆ and ˆ', then we say that ' and are logically equivalent, in symbols ' Á . We refer the reader to [11,16,24,25,48] for other properties of the logic. In our formalization of Arrow's Theorem, most formulas will have free variables only from the set V C D fx 1 ; : : : ; x n ; yg of the distinguished variables that we fixed.…”

“…In Section 5, we present a version of independence logic that we use to formalize Arrow's Theorem. We show in Section 5.3 that Arrow's Theorem can be derived in a natural deduction system for this logic [16,25]. Section 6 contains some concluding remarks.…”