Ramble, Part 59: Predicate Games
Sep. 20th, 2008 07:29 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowWe'll start examining 19th-century developments in analysis shortly, but I want to make a few observations concerning the importance of quantifiers before doing so.
There's a sharp distinction between terms - which, as I've said, must have a single definite reference - and quantified phrases. This is not merely the invention of logicians; ordinary language recognizes it.
Consider negation. In English, to negate a simple statement with no quantifiers is easy: if the verb is "to be" or one of the modals, we just insert "not" after the verb.
The point here is that, from a logical and linguistic standpoint, quantified statements are qualitatively different from statements without quantifiers. (That's not exactly news; the medieval syllogists distinguished both universal and particular statements from "individual" - i.e., unquantified - statements.) But there's more.
Abraham Lincoln is famously said to have said that "you can fool some of the people all of the time". (The evidence that he actually said this is, shall we say, slender.) It's actually not entirely clear what this claim means. Is it
This sort of ambiguity exists, at least potentially, every time a statement contains two quantifiers of different types. In fairness, I have to point out that - at least in English - it is normally dealt with by word order: "There are always people you can fool" vs. "There are people you can always fool", for example. The Lincoln quote is slightly aberrant in this respect, as it gives no clear indication of the resolution of the ambiguity.
Nonetheless, this points to a significant fact. Statements involving two quantifiers can be difficult to interpret; statements with more than that are even harder, unless careful attention is paid - e.g., by the explicit use of the quantifier symbolism devised by Pierce, or something like it. Modern mathematics leans heavily on multiple-quantifier statements; for example, the definition of a statement like "the function f(x) is continuous" involves four quantifiers, three of them universal and one existential. (The related but stronger statement "the function f(x) is uniformly continuous" turns out to differ only in the order of the quantifiers.) Mathematicians began working with these concepts well before Pierce's time, but it seems unlikely that the progress that was made in the twentieth century would have been possible without his work.
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There's a sharp distinction between terms - which, as I've said, must have a single definite reference - and quantified phrases. This is not merely the invention of logicians; ordinary language recognizes it.
Consider negation. In English, to negate a simple statement with no quantifiers is easy: if the verb is "to be" or one of the modals, we just insert "not" after the verb.
NOT(John is here) = John isn't here.Otherwise, we usually insert some form of "do".
NOT(I can do this) = I can't do this.
NOT(I saw him) = I didn't see him.But if quantifiers are involved, more complex manipulations are required.
NOT(I understand this) = I don't understand this.
NOT(Someone is here) = No one is here.And those are the easy ones; quantified statements involving "two", "a few", "exactly one", "most" and the like are notoriously hard to negate.
NOT(Everyone can do this) = Not everyone can do this = Some people can't do this.
The point here is that, from a logical and linguistic standpoint, quantified statements are qualitatively different from statements without quantifiers. (That's not exactly news; the medieval syllogists distinguished both universal and particular statements from "individual" - i.e., unquantified - statements.) But there's more.
Abraham Lincoln is famously said to have said that "you can fool some of the people all of the time". (The evidence that he actually said this is, shall we say, slender.) It's actually not entirely clear what this claim means. Is it
(for some people x) (for all times t) you can fool x at t,i.e., there are people you can always fool, or is it
(for all times t) (for some people x) you can fool x at t,i.e., at any given time there are people you can fool?
This sort of ambiguity exists, at least potentially, every time a statement contains two quantifiers of different types. In fairness, I have to point out that - at least in English - it is normally dealt with by word order: "There are always people you can fool" vs. "There are people you can always fool", for example. The Lincoln quote is slightly aberrant in this respect, as it gives no clear indication of the resolution of the ambiguity.
Nonetheless, this points to a significant fact. Statements involving two quantifiers can be difficult to interpret; statements with more than that are even harder, unless careful attention is paid - e.g., by the explicit use of the quantifier symbolism devised by Pierce, or something like it. Modern mathematics leans heavily on multiple-quantifier statements; for example, the definition of a statement like "the function f(x) is continuous" involves four quantifiers, three of them universal and one existential. (The related but stronger statement "the function f(x) is uniformly continuous" turns out to differ only in the order of the quantifiers.) Mathematicians began working with these concepts well before Pierce's time, but it seems unlikely that the progress that was made in the twentieth century would have been possible without his work.
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no subject
Date: 2008-09-21 12:50 am (UTC)In computer programming (and, IIRC, in mathematics), precedence is defined. The statement x=4*3+7; isn't ambiguous because multiplication takes precedence over addition (in most languages).
Too, you can always use parentheses to override "natural" precedence. In the statement x=4*(3+7);, x takes the value 40, not 19, because sub-expressions in the innermost parentheses (only one set in this example) are interpreted (or compiled :) ) first. Liberal use of parentheses in programming to eliminate ambiguity is considered virtuous, as not everyone remembers the precedence of operators in a language, and some may switch between several languages with different rules.
The statement Q1Q2S would always be interpreted unambiguously and in the same way by a compiler (granted, different compilers might interpret it differently, but any given compiler would always handle the same way).
no subject
Date: 2008-09-21 08:05 pm (UTC)To say that the sentence is ambiguous is precisely to say that it's not clear what the underlying statement is. The problem is, indeed, at the syntax/semantics interface; English syntax has some tools for specifying the order of quantifiers, but the given sentence doesn't use them.
Any language or sublanguage that has to use quantifiers extensively is going to have syntactic devices - precedence, grouping symbols, etc. - to deal with the problem; the jargons of mathematics and computer science do, in particular. But in everyday language, multiple quantifiers come up rarely enough that the tools just aren't there.