Natural language statements have quantifiers such as, “most”, “many”, “few”, and “only”. How could ordinary first-order predicate logic with identity (hereafter, FOPL) treat statements containing these vague quantifiers? It seems that FOPL, with only the existential and universal quantifiers at its disposal, is insufficient. I read somewhere that ‘restricted quantification’ notation can ameliorate such problems. Is this true, or are there difficulties with the restricted quantification treatment of vague quantifiers?
What are some of the inference rules for restricted quantification notation? For example, in FOPL you have the existential instantiation and universal instantiation inference rules. Are there analogue inference rules for the quantifiers, "many", “most” and “few”? Can you recommend any books or articles that outline, critique or defend restricted quantification?
I also read that there are issues with FOPL regarding symbolizing adverbs and events from natural language. Is this true or just a superficial problem?
Another complaint about FOPL, (especially Russell’s treatment of statements in the form of “The so and so...”), is that, often there are no obvious correspondences between the grammatical structure of the natural language and its logical notation counterpart. For example, in the English statement, “All men are mortal” to the logical notation, (x)(Mx->Rx), there seems to be no obvious correspondence to the connective ‘->’ from anything in its natural language grammatical structure. In other words, the logical notation seems too contrived. What is the common response to this complaint if any?
These seem to be grave problems for the applicability and effectiveness of FOPL to natural language arguments. (I am not referring to the “limits” of FOPL where extensions such as modal, tense, or second-order logic might accommodate the richer parts of natural language, but rather to the apparent inability of any logic(s) dealing with these problems.)
Note: Much of these concerns I have come from an article I read by Kent Bach in “A Companion to Philosophical Logic” by Blackwell Publishing.
Thanks Kindly for your reply,