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Symbolic Logic

John Venn
American Mathematical Society/Chelsea
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  • Objections to the introduction of mathematical symbols into logic. Mutual relation of these symbols. The generalizations of the symbolic logic, and their relation to the common system. Anticipations of Boole; especially by Lambert. Biographical notes on Lambert and Boole
  • On the forms of propositions: (1) The traditional logical forms; (2) The exclusion and inclusion, or Hamiltonian forms; (3) The compartmental, or symbolic, forms
  • Symbols of classes. Symbols of operations: (1) Aggregation of classes together; (2) Exception of one class from another; (3) Restriction of one class by another. Symbolic representation of these operations. Use of the equation symbol
  • The inverse operation to logical class restriction. Its symbolization by the sign of division. Process of logical abstraction
  • On the choice of symbolic language. Reasons for selecting that of mathematics for logical purposes
  • Diagrammatic representation. Defects of the familiar Eulerian scheme; and proposal of a new scheme, better fitted to represent the complicated propositions of the symbolic logic. Its application to terms and to propositions. Logical machines
  • The import of propositions, as regards the actual or conventional existence of their subjects and predicates. Such existence doubtful in the common logic, but certainly not implied in the symbolic. Application of this principle to (1) the incompatibility of propositions, (2) their independence
  • Symbolic expression of the four common forms of propositions: (1) Universal, (2) Particular. Existential explanation of these latter. Quantification of the predicate. Disjunctives
  • Hypothetical propositions; their real significance and their symbolic expression. Degrees of complexity in them. What our symbols actually represent here. Discussion of some extreme cases
  • The universe of discourse. Its interpretation and symbolic representation
  • On development or expansion; (1) by empirical processes, (2) by a generalized symbolic process. The symbol $\frac{0}{0}$ and its explanation. Formulæ for contradiction. Certain extreme cases of expansion
  • Logical statements or equations; (1) when explicit, (2) when implicit, (3) when involving indefinite class factors
  • Logical statements or equations (continued). Groups of equations (1) affirmatively, (2) negatively translated. Abbreviated formulæ. The interpretation of equations, in general. The so-called inductive problem: Jevons' solution: other solutions
  • Miscellaneous examples, symbolically and diagrammatically treated, in illustration of the foregoing chapters
  • Elimination in logic, empirically treated; (1) from propositions in their affirmative, (2) in their negative, form. Extension to the case of particular propositions. Diagrammatic illustration
  • Elimination by a general symbolic rule. The meaning and use of the expressions $f(1)$ and $f(0)$ in logic. The indefinite expression $\frac{0}{0}$
  • Completion of the logical problem, by the determination of any function of our class terms from any given group of statements or equations; (1) empirically, (2) by a general symbolic rule. The syllogism and its symbolic treatment
  • Generalizations of the common logic; Classes, contradictory and contrary. Propositions, their conversion and opposition, and the schedule of their various forms. Elimination. Reasonings
  • Class symbols as denoting propositions. The modifications of interpretation demanded on this view
  • Intensive interpretation in general. Attempts to carry this out rigidly
  • Historic notes; (1) On the various schemes of symbolic notation, as exhibited in the expression of the universal negative proposition; (2) On the various schemes of geometrical or diagrammatical representation of propositions
  • Index of subject-matter, and of bibliographical references