natural density

The draft paper is still under review. There might be something to correct.

All comments are welcome.

Abstract:
This paper considers the quantified simple sentences by \textit{Most}, sometimes referred to as proportional, sometimes the majority. The sentence form: \textit{Most A are B} where \textit{A} and \textit{B} are plural nouns. A and B range over elements of P(N). Moreover, A and B may appear complemented (i.e., as Non−A and Non−B). Two different but equivalent semantics are for \textit{Most A are B} as (i) C(A∩B)>C(A∖B) and (ii) C(A∩B)>C(A)2 where C(X) is the cardinality of the set X. Both semantics work well on finite sets but exhibit problematic behaviors on infinite sets since division is undefined on cardinal arithmetic. Although semantics (i) is more descriptive than semantics (ii), it also produces insensitivity for certain sets. \textquotedblleft Most" has a solid cardinal structure under the interpretation of the majority, and has the more statistical structure with proportional interpretation, and this statistical interpretation provides more flexible range of motion. For all these reasons, we introduce a new semantics with natural density for the sentences ranging over N. We also give an axiomatization of this logic.

This paper proposes a formalization of the class of sentences quantified by most, which is also interpreted as proportion of or majority of depending on the domain of discourse. We consider sentences of the form "Most A are B ", where A and B are plural nouns and the interpretations of A and B are infinite subsets of N. There are two widely used semantics for Most A are B : (i) C(A ∩ B) > C(A \ B) and (ii) C(A ∩ B) > C(A) 2 , where C(X) denotes the cardinality of a given finite set X. Although (i) is more descriptive than (ii), it also produces a considerable amount of insensitivity for certain sets. Since the quantifier most has a solid cardinal behaviour under the interpretation majority and has a slightly more statistical behaviour under the interpretation proportional of, we consider an alternative approach in deciding quantity-related statements regarding infinite sets. For this we introduce a new semantics using natural density for sentences in which interpretations of their nouns are infinite subsets of N, along with a list of the axiomatization of the concept of natural density. In other words, we take the standard definition of the semantics of most but define it as applying to finite approximations of infinite sets computed to the limit.