Showing papers by "Neil Tennant published in 2022"

The Logic of Number

Abstract: This book defines and develops the program of Natural Logicism for the natural, rational, and real numbers. The central method is to formulate rules of natural deduction governing variable-binding number-abstraction operators and other logico-mathematical expressions such as zero and successor. The introduction and elimination rules for a number-abstraction operator @ allow one to infer to, and away from, identity statements in the canonical form ‘t=@xΦ‎(x)’. These enable ‘single-barreled’ abstraction, in contrast with the ‘double-barreled’ abstraction effected by principles such as Frege's Basic Law V, or Hume's Principle. The logical system used for the foundational reasoning is free Core Logic. It handles non-denoting singular terms and allows only constructive and relevant reasoning. Natural Logicism imposes upon its account of the numbers four conditions of adequacy. First, one must show how it is that the various kinds of number are applicable in our wider thought and talk about the world. One does this by deriving all instances of three respective schemas: Schema N for the naturals, Schema Q for the rationals, and Schema R for the reals. These provide truth-conditions for statements deploying terms referring to numbers of the kind in question. Second, one must show how it is that the naturals sit among the rationals as themselves again, and the rationals likewise among the reals. Third, one should reveal enough of the metaphysical nature of the numbers to be able to derive the mathematician's basic laws governing them. Fourth, one should be able to demonstrate that there are uncountably many reals.