Bulletin of the Section of Logic Volume 39:1/2 (2010), pp. 53–55 Piotr Kulicki HALLDEN INCOMPLETE CALCULUS OF NAMES Hallden completeness is a weaker version of "disjunction property" for logical systems, defined as follows: if α ∨ β ∈ L and var(α) ∩ var(β) = ∅, then α ∈ L or β ∈ L; where L is a system, α and β are formulae and for any formula γ, var(γ) denotes a set of free variables contained in γ. The notion is usually used in the context of intermediate or modal logics. In the present paper it is applied to the systems of the calculus of names, which are axiomatisations of syllogistic. A Hallden incomplete system Sl?, which is placed between two known systems classical axiomatisation of J. Lukasewicz (Luk) [1] and its minimal subsystem containing all Aristotelean laws of J. S lupecki (Sl) ([3], also in [2] as system 10.3 on page 310), is considered. Let S, M and P be individual variables and a and i denote predicates forming respectively universal and particular affirmative sentences of syllogistic. (Thus the atomic formula SaP can be read as every S is P and SiP some S are P.) Let further ¬, ∧, ∨ and → denote classical propositional functors of respectively negation, conjunction, alternative and implication. The systems Luk, Sl and Sl? are all based on classical propositional calculus (CPC). Formally they are defined by rules modus ponens and substitution for individual variables (point substitution) and axioms, including substitutions of all theses of CPC into the language of the systems and the following specific axioms for particular systems. 54 Piotr Kulicki Luk: SaM ∧MaP → SaP, (1) MiS ∧MaP → SiP, (2) SaS, (3) SiS. (4) Sl: (1), (2) and SaP → SiP, (5) PiS → SiP. (6) Sl?: (1), (2), (5), (6) and PiP → SiS. (7) It is easy to check that Sl ⊂ Sl? ⊂ Luk. Theorem. System Sl? is Hallden incomplete. Proof. Axiom (7) is equivalent in CPC to the formula ¬PiP ∨ SiS. Obviously var(¬PiP ) ∩ var(SiS) = ∅. Thus it is enough to show that ¬PiP 6∈ Sl? and SiS 6∈ Sl?. Since all of the Sl? axioms are of the form of implication with a conjunction of atomic formulae in the predecessor and an atomic formula in the consequent, they are all true in the model in which all atomic formulae are true and also in the model in which all atomic formulae are false. In the first model the formula ¬PiP is false. In the second one the formula SiS is false. Thus both formulae are not elements of Sl?.  References [1] J. Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Oxford, 1957. [2] A. N. Prior, Formal Logic, Clarendon Press, Oxford 1962. Hallden Incomplete Calculus of Names 55 [3] J. S lupecki, Uwagi o sylogistyce Arystotelesa (Remarks on aristotle's syllogistic), Annales UMCS I (1946), pp. 187–191. Department of Philosophy John Paul II Catholic University of Lublin e-mail: kulicki@kul.pl