Fragments of first order logic, I: universal Horn logic

Abstract

Let L be any finitary language. By restricting our attention to the universal Horn sentences of L and appealing to a semantical notion of logical consequence, we can formulate the universal Horn logic of L. The present paper provides some theorems about universal Horn logic that serve to distinguish it from the full first order predicate logic. Universal Horn equivalence between structures is characterized in two ways, one resembling Kochen's ultralimit theorem. A sharp version of Beth's definability theorem is established for universal Horn logic by means of a reduced product construction. The notion of a consistency property is relativized to universal Horn logic and the corresponding model existence theorem is proven. Using the model existence theorem another proof of the definability result is presented. The relativized consistency properties also suggest a syntactical notion of proof that lies entirely within the universal Horn logic. Finally, a decision problem in universal Horn logic is discussed. It is shown that the set of universal Horn sentences preserved under the formation of homomorphic images (or direct factors) is not recursive, provided the language has at least two unary function symbols or at least one function symbol of rank more than one.This paper begins with a discussion of how algebraic relations between structures can be used to obtain fragments of a given logic. Only two such fragments seem to be under current investigation: equational logic and universal Horn logic. Other fragments which seem interesting are pointed out.

Dynamic Epistemic Logics

“…A sentence of first order logic is preserved under restriction and product if and only if the sentence is universal Horn, where a universal Horn sentence of FOL is the universal closure of a disjunction with at most one atom disjunct, and with the remaining disjuncts negations of atoms [52]. Reflexivity, transitivity and euclidity are universal Horn, but seriality is not.…”

“…It is folklore from model theory that a sentence of first order logic is perserved under restriction and product iff the sentence is universal Horn. A universal Horn sentence of FOL is the universal closure of a disjunction with at most one atom disjunct, while the remaining disjuncts are negations of atoms (see, e.g., [52]). The classes of S5 models or S5n models (multi-modal logics where all modalities are S5) are univeral Horn: the formulas for reflexivity, symmetry and transitivity can be written as Horn formulas.…”

Some characterization theorems for infinitary universal Horn logic without equality

“…The universal Horn fragment of first-order logic (with equality) has been extensively studied; for references see [10], [7] and [8]. But the universal Horn fragment without equality, used frequently in logic programming, has received much less attention from the model theoretic point of view.…”

Axiomatisability and hardness for universal Horn classes of hypergraphs

“…

We characterise finite axiomatisability and intractability of deciding membership for universal Horn classes generated by finite loop-free hypergraphs.A universal Horn class is a class of model-theoretic structures of the same signature, closed under taking ultraproducts (P u ), direct products over nonempty families (P) and isomorphic copies of substructures (S); see [9,17,30,32] for example. Equivalently they are classes axiomatisable by way of universal Horn sentences: universally quantified disjunctions α 1 ∨ · · · ∨ α k , where each α i is either an atomic formula of the language, or a negated atomic formula, and all but at most one of the α i are negated.

…”