By pure calculus of names we mean a quantifier-free theory, based on the classical propositional calculus, which defines predicates known from Aristotle’s syllogistic and Leśniewski’s Ontology. For a large fragment of the theory decision procedures, defined by a combination of simple syntactic operations and models in two-membered domains, can be used. We compare the system which employs `ε’ as the only specific term with the system enriched with functors of Syllogistic. In the former, we do not need an empty name (...) in the model, so we are able to construct a 3-valued matrix, while for the latter, for which an empty name is necessary, the respective matrices are 4-valued. (shrink)

Aristotle's syllogistic is extended to include denumerably many quantifiers such as 'more than 2/3' and 'exactly 2/3.' Syntactic and semantic decision procedures determine the validity, or invalidity, of syllogisms with any finite number of premises. One of the syntactic procedures uses a natural deduction account of deducibility, which is sound and complete. The semantics for the system is non-classical since sentences may be assigned a value other than true or false. Results about symmetric systems are given. And reasons are given (...) for claiming that syllogistic validity is relevant validity. (shrink)

Fred's semantics for McCall's syntactic presentation of Aristotle's assertoric and apodeictic syllogistic is altered to free it from Thom's objections that it is unAristotelian. The altered semantics rejects Baroco-XLL and Bocardo-LXL, which Thom says Aristotle should have accepted. Aristotle's proofs that use ecthesis are formalized by using singular sentences. With one exception the (acceptance) axioms for McCall's system L-X-M are derivable. Formal proofs are shown to be sound.

Syllogisms with or without negative terms are studied by using Gergonne's ideas. Soundness, completeness, and decidability results are given.

Parry discusses an extension of Aristotle's syllogistic that uses four nontraditional quantifiers. We show that his conjectured decision procedure for validity for the extended syllogistic is correct even if syllogisms have more than two premises. And we axiomatize this extension of the syllogistic.

One semantic and two syntactic decision procedures are given for determining the validity of Aristotelian assertoric and apodeictic syllogisms. Results are obtained by using the Aristotelian deductions that necessarily have an even number of premises.

"However that may be, Aristotelian syllogistic concerned itself exclusively with monadic predicates. Hence it could not begin to investigate multiple quantification. And that is why it never got very far. None the less, the underlying grammar of Aristotle's logic did not in itself..