The Logic of 'Almost All'
Adams (Ernest)
Source: Journal of Philosophical Logic, Vol. 3, No. 1/2 (Mar., 1974), pp. 3-17
Paper - Abstract

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Author’s Introduction

  1. We wish to study deductive relations among propositions of the form 'Almost all A's are B's', which assert that almost all members of a class of instances have a given property. Let us call these statements imperfect generalizations, in contrast to perfect generalizations of the form 'All A's are B's' to which formal logic is properly applicable. The motivation for studying imperfect generalizations arises from the fact that many generalizations of empirical science actually admit of exceptions, although they are symbolized as though they were perfect generalizations for the purpose of analyzing their deductive interconnections. It will prove that in fact it is not always safe to ignore the existence of exceptions or the qualification 'almost' when ascertaining logical relations among generalizations.
  2. In what follows we will take explicit account of the vagueness of the quantifier 'almost all' by introducing a continuous-valued measure of truth for statements involving it, rather than ascribing dichotomous truth-values to them (this is analogous to proposals of Zadeh [6] and Goguen [3] in their work on fuzzy concepts). Our measure of the degree of truth of 'Almost all A's are B's' will be the proportion of A's which are B's (suitably relativised to a model, and to a continuous probability measure over the domain of the model). What we shall be interested in is the relation of measure-entailment holding between sets of imperfect generalizations and another imperfect generalization, which holds if it is possible to assure an arbitrarily high degree of truth (short of 'perfect') for the conclusion by requiring all of the premises to be 'sufficiently true' (short of being perfectly true). The basic definitions and formalism are given in Section 2, and detailed results concerning the measure-entailment relation are presented in Sections 3 and 4.

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