Contrary to common misconceptions, today's logic is not devoid of existential import: the universalized conditional ∀ x [S→ P] implies its corresponding existentialized conjunction ∃ x [S & P], not in all cases, but in some. We characterize the proexamples by proving the Existential-Import Equivalence: The antecedent S of the universalized conditional alone determines whether the universalized conditional has existential import, i.e. whether it implies its corresponding existentialized conjunction.A predicate is an open formula having only x free. An existential-import predicate (...) Q is one whose existentialization, ∃ x Q, is logically true; otherwise, Q is existential-import-free or simply import-free.How abundant or widespread is existential import? How abundant or widespread are existential-import predicates in themselves or in comparison to import-free predicates? We show that existential-import predicates are quite abundant, and no less so than import-free predicates. Existential.. (shrink)

This entry traces the historical development of the Square of Opposition, a collection of logical relationships traditionally embodied in a square diagram. This body of doctrine provided a foundation for work in logic for over two millenia. For most of this history, logicians assumed that negative particular propositions ("Some S is not P") are vacuously true if their subjects are empty. This validates the logical laws embodied in the diagram, and preserves the doctrine against modern criticisms. Certain additional principles ("contraposition" (...) and "obversion") were sometimes adopted along with the Square, and they genuinely yielded inconsistency. By the nineteenth century an inconsistent set of doctrines was widely adopted. Strawson's 1952 attempt to rehabilitate the Square does not apply to the traditional doctrine; it does salvage the nineteenth century version but at the cost of yielding inferences that lead from truth to falsity when strung together. (shrink)

John Venn and Charles L. Dodgson (Lewis Carroll) created systems of logic diagrams capable of representing classes (sets) and their relations in the form of propositions. Each is a proof method for syllogisms, and Carroll's is a sound and complete system. For a large number of sets, Carroll diagrams are easier to draw because of their self-similarity and algorithmic construction. This regularity makes it easier to locate and thereby to erase cells corresponding with classes destroyed by the premises of an (...) argument, a particularly difficult task in Venn diagrams for more than four sets. Carroll diagrams can represent existential propositions easily, so they are capable of clearly representing more complex problems than Venn's system can. Finally, both Carroll and Venn diagrams are maximal, in the sense that no additional logic information like inclusive disjunctions is able to be represented by them. Carroll's logic diagrams and logic trees constitute his visual logic system. (shrink)

Hypotheticals, conditionals, and their connecting relation, implication, dramatically changed their meanings during the nineteenth and early part of the twentieth century. Modern logicians ordinarily do not distinguish between the terms hypothetical and conditional. Yet in the late nineteenth century their meanings were quite different, their ties to the implication relation either were unclear, or the implication relation was used exclusively as a logical operator. I will trace the development of implication as an inference operator from these earlier notions into the (...) first third of the twentieth century using as the starting point the ideas primarily of R. Whately and W. Hamilton before discussing the work of the transitional logicians, A. De Morgan and G. Boole on these topics. Then we discuss the relevant views of four prominent but relatively unknown nineteenth century British logicians, W.E. Johnson, J.N. Keynes, E.E.C. Jones, and H. MacColl, as well as those of the more influential logicians, W.S. Jevons and J. Venn, closing with a section on ‘implication as inference’ where we explore some key ideas of B. Russell and sketch the work of D. Hilbert, P. Hertz, and G. Gentzen who together are responsible for the development of the modern ideas related to the subjects of this paper. (shrink)

Three distinctly different interpretations of Aristotle’s notion of a sullogismos in Prior Analytics can be traced: (1) a valid or invalid premise-conclusion argument (2) a single, logically true conditional proposition and (3) a cogent argumentation or deduction. Remarkably the three interpretations hold similar notions about the logical relationships among the sullogismoi. This is most apparent in their conflating three processes that Aristotle especially distinguishes: completion (A4-6)reduction(A7) and analysis (A45). Interpretive problems result from not sufficiently recognizing Aristotle’s remarkable degree of metalogical (...) sophistication to distinguish logical syntax from semantics and, thus, also from not grasping him to refine the deduction system of his underlying logic. While it is obvious that Aristotle most often uses ‘sullogimos’ to denote a valid argument of a certain kind, we show that at Prior Analytics A4-6, 7, 45 Aristotle specifically treats a sullogismos as an elemental argument pattern having only valid instances and that such a pattern then serves as a rule of deduction in his syllogistic logic. By extracting Aristotle’s understanding of three proof-theoretic processes, this paper provides new insight into what Aristotle thinks reasoning syllogistically is and, moreover, it resolves three problems in the most recent interpretation that takes a sullogismos to be a deduction. (shrink)

Peter Geach describes the 'doctrine of distribution' as the view that a term is distributed if it refers to everything that it denotes, and undistributed if it refers to only some of the things that it denotes. He argues that the notion, so explained, is incoherent. He claims that the doctrine of distribution originates from a degenerate use of the notion of ?distributive supposition? in medieval supposition theory sometime in the 16th century. This paper proposes instead that the doctrine of (...) distribution occurs at least as early as the 12th century, and that it originates from a study of Aristotle's notion of a term's being ?taken universally?, and not from the much later theory of distributive supposition. A detailed version of the doctrine found in the Port Royal Logic is articulated, and compared with a slightly different modern version. Finally, Geach's arguments for the incoherence of the doctrine are discussed and rejected. (shrink)

It is of common use in modern Venn diagrams to mark a compartment with a cross to express its non-emptiness. Modern scholars seem to derive this convention from Charles S. Peirce, with the assumption that it was unknown to John Venn. This paper demonstrates that Venn actually introduced several methods to represent existentials but felt uneasy with them. The resistance to formalize existentials was not limited to diagrammatic systems, as George Boole and his followers also failed to provide a satisfactory (...) symbolic representation for them. This difficulty points out issues that are inherent to the very nature of existentials. This paper assesses the various methods designed for the representation of existential statements with Venn diagrams. First, Venn’s own attempts are discussed and compared with other solutions proposed by his contemporaries and successors, notably Lewis Carroll and Peirce. Since disjunctives hold an important role in an effective representation of existentials, their representation is also discussed. Finally, recent methods for the diagrammatic representation of existing individuals, rather than mere existence, are surveyed. (shrink)

Contemporary logicians continue to address problems associated with the existential import of categorical propositions. One notable problem concerns invalid instances of subalternation in the case of a universal proposition with an empty subject term. To remedy problems, logicians restrict first-order predicate logics to exclude such terms. Examining the historical origins of contemporary discussions reveals that logicians continue to make various category mistakes. We now believe that no proposition per se has existential import as commonly understood and thus it is unnecessary (...) to restrict first-order predicate logics to non-empty classes. After introducing the problem, we trace some nineteenth century treatments of the issue to locate a source of misconstruing propositional import in misconceptions of ‘implies’ and ‘affirms’ and name the process/product fallacy, along with the translation of categorical sentences using quantifiers and accommodating an empty class. Next we treat some metalogical matters to orient our discussion by which we provide a more precise nomenclature about ‘sentence’ and ‘proposition’ to correct previous misconceptions; here we uncover a common category mistake in respect of a proposition’s efficacy. The semantic distinction between agent and force is helpful in this connection. We conclude by showing that logicians have reinserted existence as a predicate, a position previously excised by Kant, and that the Frege-Russell ambiguity thesis applies only to relationships within a categorical sentence between grammatical predicate and subject. (shrink)