Logicians have often suggested that the use of Euler-type diagrams has influenced the idea of the quantification of the predicate. This is mainly due to the fact that Euler-type diagrams display more information than is required in traditional syllogistics. The paper supports this argument and extends it by a further step: Euler-type diagrams not only illustrate the quantification of the predicate, but also solve problems of traditional proof theory, which prevented an overall quantification of the predicate. Thus, Euler-type diagrams can (...) be called the natural basis of syllogistic reasoning and can even go beyond. In the paper, these arguments are presented in connection with the book Nucleus Logicae Weisaniae by Johann Christian Lange from 1712. (shrink)

This paper undertakes a re-examination of Sir William Hamilton’s doctrine of the quantification of the predicate . Hamilton’s doctrine comprises two theses. First, the predicates of traditional syllogistic sentence-forms contain implicit existential quantifiers, so that, for example, All p is q is to be understood as All p is some q . Second, these implicit quantifiers can be meaningfully dualized to yield novel sentence-forms, such as, for example, All p is all q . Hamilton attempted to provide a deductive system (...) for his language, along the lines of the classical syllogisms. We show, using techniques unavailable to Hamilton, that such a system does exist, though with qualifications that distinguish it from its classical counterpart. (shrink)

The Aristotelian syllogistic cannot account for the validity of certain inferences involving relational facts. In this paper, we investigate the prospects for providing a relational syllogistic. We identify several fragments based on (a) whether negation is permitted on all nouns, including those in the subject of a sentence; and (b) whether the subject noun phrase may contain a relative clause. The logics we present are extensions of the classical syllogistic, and we pay special attention to the question of whether reductio (...) ad absurdum is needed. Thus our main goal is to derive results on the existence (or nonexistence) of syllogistic proof systems for relational fragments. We also determine the computational complexity of all our fragments. (shrink)

The numerically definite syllogistic is the fragment of English obtained by extending the language of the classical syllogism with numerical quantifiers. The numerically definite relational syllogistic is the fragment of English obtained by extending the numerically definite syllogistic with predicates involving transitive verbs. This paper investigates the computational complexity of the satisfiability problem for these fragments. We show that the satisfiability problem (= finite satisfiability problem) for the numerically definite syllogistic is strongly NP-complete, and that the satisfiability problem (= finite (...) satisfiability problem) for the numerically definite relational syllogistic is NEXPTIME-complete, but perhaps not strongly so. We discuss the related problem of probabilistic (propositional) satisfiability, and thereby demonstrate the incompleteness of some proof-systems that have been proposed for the numerically definite syllogistic. (shrink)

. In logic, diagrams have been used for a very long time. Nevertheless philosophers and logicians are not quite clear about the logical status of diagrammatical representations. Fact is that there is a close relationship between particular visual (resp. graphical) properties of diagrams and logical properties. This is why the representation of the four categorical propositions by different diagram systems allows a deeper insight into the relations of the logical square. In this paper I want to give some examples.

It has long been a standard practice for the natural sciences to classify things. Thus, it is no wonder that, for two and a half millennia, philosophers have been reflecting on classifications, from Plato and Aristotle to contemporary philosophy of science. Some of the results of these reflections will be presented in this chapter. I will start by discussing a parody of a classification, namely: the purportedly ancient Chinese classification of animals described by Jorge Luis Borges. I will show that (...) many of the mistakes that account for the comic features of this parody appear in real-life scientific databases as well. As examples of the latter, I will discuss the terminology database of the National Cancer Institute (NCI) of the United States, the NCI Thesaurus. (shrink)

The paper outlines the advantages and limits of the so-called ‘Calculus CL’ in the field of ontology engineering and automated theorem proving. CL is a diagram type that combines features of tree, Euler-type, Venn-type diagrams and squares of opposition. Due to the simple taxonomical structures and intuitive rules of CL, it is easy to edit ontologies and to prove inferences.

This paper presents the first use of a computational model of natural logic—a system of logical inference which operates over natural language—for textual inference. Most current approaches to the PAS- CAL RTE textual inference task achieve robustness by sacrificing semantic precision; while broadly effective, they are easily confounded by ubiquitous inferences involving monotonicity. At the other extreme, systems which rely on first-order logic and theorem proving are precise, but excessively brittle. This work aims at a middle way. Our system finds (...) a low-cost edit sequence which transforms the premise into the hypothesis; learns to classify entailment relations across atomic edits; and composes atomic entailments into a top-level entailment judgment. We provide the first reported results for any system on the FraCaS test suite. We also evaluate on RTE3 data, and show that hybridizing an existing RTE system with our natural logic system yields significant performance gains. (shrink)

This paper is a brief history of natural logic at the interface of logic, linguistics, and nowadays also other disciplines. It merely summarizes some facts that deserve to be common knowledge.

The paper presents the diagrammatic calculus CL, which combines features of tree, Euler-type, Venn-type diagrams and squares of opposition. In its basic form, `CL' (= Cubus Logicus) organizes terms in the form of a square or cube. By applying the arrows of the square of opposition to CL, judgments and inferences can be displayed. Thus CL offers on the one hand an intuitive method to display ontologies and on the other hand a diagrammatic tool to check inferences. The paper focuses (...) mainly on the adaptation of the square of opposition in CL and offers an algebraic notation, which corresponds to the diagrammatic representation. (shrink)