Axiomatizing first order consequences in inclusion logic

Abstract

Inclusion logic is a variant of dependence logic that was shown to have the same expressive power as positive greatest fixed‐point logic. Inclusion logic is not axiomatisable in full, but its first order consequences can be axiomatized. In this paper, we provide such an explicit partial axiomatization by introducing a system of natural deduction for inclusion logic that is sound and complete for first order consequences in inclusion logic.

Complete Logics for Elementary Team Properties

“…Let us end this section with a remark that the rule with respect to (the usual) negation of first-order formulas is not in general sound in independence logic (which is extended with independence atoms) or in inclusion logic (which is extended with inclusion atoms). In the systems of these two logics (introduced in [23, 42] the corresponding rule has an extra side condition that all formulas in the context set have to be first-order. For this reason, the same argument as in Corollary 5.13 for dependence logic does not go through for the other two logics.…”

“…For this reason, the same argument as in Corollary 5.13 for dependence logic does not go through for the other two logics. Nevertheless, the system of independence logic in [23] together with the rule with respect to was proved in [41] to be complete (over formulas) for first-order consequences, and the system of inclusion logic in [42] (in which the rule with respect to and classical formula is derivable) is indeed also complete (over formulas) for first-order consequences.…”

“…More precisely, a system of natural deduction for dependence logic was introduced in [34] for which the completeness theorem holds whenever is a set of D -sentences (with no free variables) and is an -sentence (with no free variables). This result has been, subsequently, generalized to, e.g., allow also open formulas [31], independence logic [23] and inclusion logic [42], and extensions of dependence logic by generalized quantifiers [12]. A recent new generalization given in [41] extends the known systems for D and Ind to cover the case when in (1) is not necessarily an -formula but merely a formula defining a first-order team property.…”