Grice's Proportion: and:or::every:some

Relations to logical conjunction and disjunction For a finite domain of discourse D = {a1,...an} the universal quantifier is equivalent to a logical conjunction of propositions with singular terms ai having the form Pai for monadic predicates.  The existential quantifier is equivalent to a logical disjunction of propositions having the same structure as before. For infinite domains of discourse the equivalences are similar.  Infinite domain of discourse Consider the following statement:  1 · 2 = 1 + 1, and 2 · 2 = 2 + 2, and 3 · 2 = 3 + 3, ..., and 100 · 2 = 100 + 100, and ..., etc. This has the appearance of an infinite conjunction of propositions. From the point of view of formal languages this is immediately a problem, since syntax rules are expected to generate finite words.  The example above is fortunate in that there is a procedure to generate all the conjuncts. However, if an assertion were to be made about every irrational number, there would be no way to enumerate all the conjuncts, since irrationals cannot be enumerated. A succinct equivalent formulation which avoids these problems uses universal quantification:  For each natural number n, n · 2 = n + n. A similar analysis applies to the disjunction,  1 is equal to 5 + 5, or 2 is equal to 5 + 5, or 3 is equal to 5 + 5, ... , or 100 is equal to 5 + 5, or ..., etc. which can be rephrased using existential quantification:  For some natural number n, n is equal to 5+5.