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| FOL union | |
|---|---|
| Name | FOL union |
| Field | Mathematical logic |
| Related | First-order logic, Model theory, Set theory |
FOL union
FOL union is a construction in formal logic that combines two first-order theories or formulae into a single theory or formula by taking the disjunction of their axioms or by forming the union of their axiom sets. It appears in work on Alfred Tarski, Kurt Gödel, Alonzo Church, Emil Post, Bertrand Russell, David Hilbert, Per Martin-Löf, Henri Poincaré, Leopold Löwenheim, Thoralf Skolem, Santiago Ramón y Cajal (contextual references), and is employed in interactions between Model theory, Proof theory, Computability theory, Category theory, Lambda calculus, Automata theory, Gödel–Dummet logic studies, and analyses related to Zermelo–Fraenkel set theory and Peano arithmetic. The construction is used in investigations of consistency, satisfiability, conservativity, and interpolation in the settings of Ehrenfeucht–Fraïssé games, Löwenheim–Skolem theorems, Compactness theorem, Completeness theorem, and Craig interpolation theorem.
Given two first-order languages L1 and L2 and corresponding theories T1 and T2 (sets of sentences) the FOL union, denoted here by the union symbol applied at the theory level, is the theory T = T1 ∪ T2 with language L = L1 ∪ L2. When T1 and T2 share predicates, functions, or constants, the union merges signatures; when disjoint, the union is the disjoint-sum theory. For formula-level combinations one also considers the disjunction φ ∨ ψ of formulae φ ∈ Sent(L1) and ψ ∈ Sent(L2), or formation of Boolean combinations in the universal algebra of sentences. Technical notation follows conventions used in Alfred Tarski's semantic approach, Leon Henkin-style expansions, and treatments in textbooks by Wilfrid Hodges, Elliott Mendelson, Judith J. Jarvis (pedagogical literature), and monographs referencing Enderton and Chang and Keisler.
The union preserves syntactic properties like being recursively enumerable when T1 and T2 are r.e., and preserves semantic properties such as satisfiability under certain conditions invoked by the Compactness theorem and Löwenheim–Skolem theorems. If T1 and T2 are consistent and their signatures are disjoint, T1 ∪ T2 is consistent by a simple model-theoretic product construction or by application of the Compactness theorem, a phenomenon discussed in texts by Svenonius and R. M. Smullyan. However, if T1 and T2 share nontrivial constraints, the union can be inconsistent even when each component is consistent, a situation illustrated in work related to Hilbert's program and the incompleteness phenomena found in Kurt Gödel's theorems. Conservativity, i.e., whether T1 ∪ T2 proves new sentences in the language of T1, interacts with model amalgamation and interpolation properties like those studied by William Craig, Michael Rabin, and Dana Scott.
Classic examples include the union of arithmetic fragments such as Robinson arithmetic Q with extensions like Peano arithmetic PA, unions of theories axiomatizing algebraic structures like the theory of fields with the theory of ordered sets yielding theories akin to the theory of ordered fields studied in the context of Alfred Tarski's decision procedures for real closed fields and the work of A. J. Wilkie. Disjoint unions occur in combining the theory of groups with the theory of rings yielding a two-sorted theory; amalgamated unions appear in combining theories that share a subtheory like combining extensions of Zermelo–Fraenkel set theory with choice axioms such as Axiom of Choice. Another special case is the union of complete theories: the union of two complete, consistent, and incompatible complete theories is inconsistent; the union of complete theories over disjoint signatures produces a complete theory of the combined signature as in product constructions used by Franzén and in discussions by Poizat.
FOL union relates to syntactic operations (conjunction, disjunction, negation) and to semantic constructions (product models, amalgamation, reduct, expansion). It corresponds to the set-theoretic union of axiom sets, interacts with theory intersection (T1 ∩ T2) and theory-generated closures like Deductive closure Cn(T1 ∪ T2). It also connects to Boolean combinations of theories, to conservative extensions, and to operations in Category theory such as pushouts of presentations and colimits of theories in institutions studied by Goguen and Meseguer. In model-theoretic algebra it parallels free-product constructions for Group theory and coproducts in varieties discussed by Birkhoff.
Decidability of T1 ∪ T2 depends on the components and signature interactions. If T1 and T2 are decidable and their languages are disjoint, the union is decidable by interleaving decision procedures; classical results on combination of decision procedures by Nelson and Oppen examine cases with shared symbols and produce combination algorithms and counterexamples. When one component is undecidable, the union is generally undecidable; conversely, combinations can raise complexity classes, and conservativity or interpretability reductions connect to Post completeness and Turing degrees studied in Recursion theory by Post, Turing, and Emil Post's successors. Satisfiability of finite subsets of T1 ∪ T2 is governed by Compactness and by complexity results like those in Satisfiability Modulo Theories frameworks used in SAT solvers and SMT research by Clark Barrett and Bruno Dutertre.
Use cases include modular specification in formal methods combining theories for arrays, integers, and bitvectors in SMT solvers; ontology integration in Semantic Web technologies combining OWL fragments; modular algebraic specification in Universal algebra and Term rewriting; and constructing models in Model theory for algebraic geometry and real algebra via combinations of field axioms and order axioms as in work by Tarski and Seidenberg. FOL union underpins theory combination in verification tools developed by teams at Stanford University, SRI International, Microsoft Research, IBM Research, and academic groups working on automated reasoning, proof assistants like Isabelle, Coq, and Lean, and on logical frameworks in projects at Carnegie Mellon University and MIT.