We extend natural deduction for first-order logic (FOL) by introducing diagrams as components of formal proofs. From the viewpoint of FOL, we regard a diagram as a deductively closed conjunction of certain FOL formulas. On the basis of this observation, we first investigate basic heterogeneous logic (HL) wherein heterogeneous inference rules are defined in the styles of conjunction introduction and elimination rules of FOL. By examining what is a detour in our heterogeneous proofs, we discuss that an elimination-introduction pair of (...) rules constitutes a redex in our HL, which is opposite the usual redex in FOL. In terms of the notion of a redex, we prove the normalization theorem for HL, and we give a characterization of the structure of heterogeneous proofs. Every normal proof in our HL consists of applications of introduction rules followed by applications of elimination rules, which is also opposite the usual form of normal proofs in FOL. Thereafter, we extend the basic HL by extending the heterogeneous rule in the style of general elimination rules to include a wider range of heterogeneous systems. (shrink)

A diagrammatic logical calculus for the syllogistic reasoning is introduced and discussed. We prove that a syllogism is valid if and only if it is provable in the calculus.

The evolution of Euler diagrams is examined from Euler's original system through the modifications made by Venn and Peirce. It is shown that these modifications were motivated by an attempt to increase the expressivity of the diagrams, but that a side effect of these modifications was a loss of the visual clarity of Euler's original system. Euler's original system is reconstructed from a modern, logical point of view. Formal semantics and rules of inference are provided for this reconstruction of Euler's (...) system, and basic logical properties are proved. (shrink)

Que tipo de objetos são os diagramas, os quais podem ser manipulados lógica e matematicamente e também servem para obter conhecimento? Neste artigo, proporei uma maneira de responder a esta pergunta. Sustentarei a hipótese, inspirada por Kant e Peirce, de que os diagramas são uma classe de objetos híbridos com um aspecto intelectual e outro sensível. Os lógicos e matemáticos estão interessados e estudam uma certa estrutura que se exemplifica em um diagrama, embora não de maneira perfeita. Devido a seu (...) caráter sensível e a sua acessibilidade, tais diagramas permitem a manipulação e a experimentação, mas unicamente em virtude do tipo de relações por eles implicadas tornam possíveis juízos absolutamente universais. (shrink)

In Volume 33:1 of the Notre Dame Journal of Formal Logic, a system for diagramming syllogistic inferences using straight line segments is presented by Englebretsen. In light of recent research on the representational power of diagrammatic representation systems by the authors, we point out some problems with the proposal, and indeed, with any proposal for representing logically possible situations diagrammatically. We shall first outline the proposed linear diagrammatic system of Englebretsen, and then show by means of counterexamples that it is (...) inadequate as a representation scheme for general logical inferences (the task for which the system is intended). We also show that modifications to the system fail to remedy the problems. The considerations we present are not limited to the particular proposal of Englebretsen; we thus draw a more general moral about the use of spatial relations in representation systems. (shrink)

ArgumentThe use of diagrams is pervasive in theoretical physics. Together with mathematical formulae and natural language, diagrams play a major role in theoretical modeling. They enrich the expressive power of physicists and help them to explore new theoretical ideas. Diagrams are not only heuristic or pedagogical tools, but they are also tools that enable developing the content of models into novel implications.

We present a reading of the traditional syllogistics in a fragment of the propositional intuitionistic multiplicative linear logic and prove that with respect to a diagrammatic logical calculus that we introduced in a previous paper, a syllogism is provable in such a fragment if and only if it is diagrammatically provable. We extend this result to syllogistics with complemented terms à la De Morgan, with respect to a suitable extension of the diagrammatic reasoning system for the traditional case and a (...) corresponding reading of such De Morgan style syllogistics in the previously referred to fragment of linear logic. (shrink)