But, on the other hand, I can remember how to say "Wo yao yi ping Pijiu." Da ping.
The occasion for all of this is that a poor school here in Georgia is making Mandarin mandatory. Why? Because China is offering instructors for about half what they'd have to pay an instructor in any other language: $16,000 a year.
Now, how useful will this be to the students? Well, in theory it could be quite useful: Chinese is one of the most different languages from English, in structure, in terms of being tonal, and in terms of having a character-based writing system. Studying it even a bit will help you see that many things you take for granted about how thoughts should be formed and ordered is not, in fact, logically necessary but a mere consequence of the language in which you learned to think.
That is also true, by the way, of artificial languages. Bertrand Russell and others hoped to eliminate this tendency to confuse logic with grammar in part by instituting formalized ways of writing. The problem turns out to be that you just introduce new errors of grammar, but now believe that you have said something logically necessary because you are writing in "the formal language of logic."
For example, I recently mentioned D. M. Armstrong's What is a Law of Nature? He makes a great deal -- by which I mean that he goes on for many pages -- out of a "paradox" that he believes is a serious problem. It's really just a case of mistaking grammar for logic. The problem arises here:
(∀x)(Fx⊃Gx)
Fx: "x is a raven"
Gx: "x is black"
Now, what that says in plain language is, "All ravens are black." But what it says literally is more like "For every x, if x is a raven then x is black." The material conditional -- "⊃" -- is a logical function. It has a truth table so that you can determine when a given proposition is true.
For the material conditional, which links two terms, the truth table says that it is true any time the antecedent is false ("this is not a raven") or the consequent is true ("it is black"). Thus, if a given raven is black, the statement is true; if we find a white raven it is false. If we find something that isn't a raven, the statement is satisfied because this is only a rule about ravens.
Dr. Armstrong was greatly concerned by the fact that things that are not ravens have to be taken as helping to prove the rule that all ravens are black. (Nor is he the only one to treat this as if it were a serious problem.) He wasn't so concerned about cases of not-black things, because they seem to help reinforce the idea of a link between the categories of "raven" and "black." But what about black things that are not ravens? That seems to trouble him quite a bit.
In fact, though, this is just a convention of language. What we really have here is a rule about ravens: "All ravens are black." It's only the form of the logical language that requires us to express it as a universal truth about all things ("For every x"). We aren't talking about all things. We're talking about ravens.
What the formal language forces us to do is to say something purely formal and empty: "Every not-raven either is or is not black." In any natural language we would omit this formality because it's entirely irrelevant. Those logicians who take this as a serious problem -- something that might, for example, seriously inform our understanding about the laws of nature -- have fooled themselves. They don't realize that they're doing the very thing that they set up this system to avoid doing.
I must not understand that kind of logical notation, which isn't surprising, since I don't know anything about it. But to me, if you say "For all x," you're talking about all things, but you quit that as soon as you add "if x is a raven." Who cares if you were talking about all things before you even got to the end of the sentence?
ReplyDeleteWait, isn't 'for every x' a limiting statement? Unless x = all things, the statement limits the following proposition to instances of x, right? So 'for every x' immediately establishes that, except in one case (x = all things), we are NOT talking about 'all things' but about one specific type of thing.
ReplyDeleteOr am I confused?
Unless x = all things...
ReplyDeleteThat's just what it does do, though. That upside-down A is technically called a 'universal quantifier.' You have to specify which universal set you want, or else you're talking about "every x," that is, every thing of any kind.
You can do this in several ways; the linked Wikipedia article gives 'For all natural numbers n, n times 2 = n plus n>.' But you can also write that without using words (those dastardly holdovers from natural language):
UD: Set of natural numbers
(∀n)(n*2=n+n)
UD stands for "universe of discourse," that is to say in this case, we've specified that we are only talking about natural numbers. So "every n" is really "every natural number," and not things like ravens. You can also do it the way that Dr. Armstrong is doing it, "Fn: n is a natural number." Then your statement would read:
(∀n)(Fn*2=Fn+Fn)
Fn: n is a natural number.
You can understand why a universal quantifier would come up when a logician tries to talk about laws of nature. Most moderns have an intuition that a true law of nature should apply to everything at all times (like the 'law of gravity,' for example -- although, like all so-called laws of nature, it turns out to be a bad example because there are at least some exceptions related to the initial and very early conditions of the universe).
So x is all things until we define it, and by convention we define it after we give the formula.
ReplyDeleteIs that why you are saying this is a grammar issue? To resolve it, all we have to do is define our terms first, before we give the formula.
(Or so it seems to me, though I'm often wrong on these points.)
That's half of why. The other half is that formalizing it in this way obscures the irrelevance of the not-raven claims. Those claims are "true," but in a purely empty way: they're just an assertion of the law of non-contradiction. ('For any quality x, everything either does or does not have that quality.') This is a logical truth, so in that sense the claim is true. But it's not a truth at all relevant to the issue of whether all ravens are, or are not, black.
ReplyDeleteThis point is totally obvious when the claim is phrased in English. It's completely obscured by the grammar of formal logic. Serious philosophers have spilled a ton of ink over it.
I wonder, was any of that ink proposals to change the language of formal logic? Or have people decided any language is just going to have some kind of problem?
ReplyDeleteAlso, I'm curious about something I've been studying recently. For the statement, "For every x," I can use eduction to restate it with something like "Excluding all non-x." Does modern formal logic do that sort of thing? If so, wouldn't that also present a solution to the problem?
Well, you can't exclude 'not-x' from the universal, unqualified 'every x,' because this x just means "any thing at all." So, nothing is left to exclude; there is no not-x.
ReplyDeleteYou can certainly exclude the not-x from a statement about a qualified x. If Fx is 'x is a raven,' then you can say:
(∀x)(Fx & (Fx⊃Gx))
That says something strange, though: it says "everything that exists is a raven, and all ravens are black." This isn't actually true, since you and I are not ravens.
You could use the UD to limit the proposition. This says something much closer to what we want to say.
UD: ravens
(∀x)(Gx)
Gx: "x is black."
This says simply "all ravens are black." We no longer need Fx because we've limited the scope of the universal in the UD line.
Another interesting fact about the language of formal logic is that this proposition does not guarantee that there are any black ravens, because it doesn't guarantee ravens. For example:
UD: Centaurs
(∀x)(Gx)
Gx: 'x is hairy'
This says "all centaurs are hairy." Is that true? Since there are no centaurs, we can't find any evidence.
But! Notice what happens when you do it the original way:
(∀x)(Fx⊃Gx)
Fx: 'x is a centaur'
Gx: 'x is hairy'
Now, since x is never a centaur (since there are none), the proposition will be true every time. Since ~Fx is always the case, in other words, the material conditional gives you a "true" on the truth table for any thing at all you take from the universe to serve as your "x".
Is that a problem? Yes, it is! Because I can give you the exact opposite claim: "All centaurs are not hairy."
((∀x)(Fx⊃~Gx)
And this is true every time too. So, it is a logically necessary truth that centaurs both are and are not hairy -- and now you have a violation of the law of non-contradiction.
For this reason, there's another quantifier called the existential quantifier. It serves to ensure that there is at least one example of the thing in the universe, to prevent this kind of horrifying result.
But, again, we get the result only because we're operating in this formalized grammar: it's clear from the meaning of the word "centaur," in English, that it would be hairy. It's clear from the meaning of the word "unicorn" that it will have exactly one horn. We are able to talk about things that exist only in our imagination without falling into contradiction in natural language, but it's very hard to do it in formal logic.
Sorry, for some reason there's an extra opening paren mark in that last one. You get the idea, though.
ReplyDeleteInteresting.
ReplyDeleteThere are programming languages that allow you to set a variable without assigning it a value, in which case the default value is null. (When you wanted to use it later, you would then assign it a value.) That makes more sense to me than automatically setting it at 'everything.' Is there a reason for it defaulting to everything?
Well, for one thing, classical syllogisms are built around statements about all members of a class. We've been talking about these since Aristotle, so it was natural to start there:
ReplyDelete"All men are mammals,
All mammals have hair,
Therefore, all men have hair."
Likewise:
"All men are mammals,
No mammals are reptiles,
Therefore, no men are reptiles."
These are all easily framed in the language we've been discussing:
(∀x)(Fx⊃Gx) (All men are mammals)
(∀x)(Gx⊃Hx) (All mammals have hair)
Therefore,
(∀x)(Fx⊃Hx) (All men have hair)
Or:
(∀x)(Fx⊃Gx) (All men are mammals)
(∀x)(Gx⊃~Hx) (No mammals are reptiles)
Therefore,
(∀x)(Fx⊃~Hx) (No men are reptiles.)
Now, if you symbolize "therefore" as "∴", the dream of having dispensed with natural language is complete.
ReplyDelete(∀x)(Fx⊃Gx)
(∀x)(Gx⊃Hx)
∴
(∀x)(Fx⊃Hx)
The theory was that this new, rational, mathematical language would eliminate errors arising from the vagaries of natural language.
I was first exposed to that notation in my high school freshman math class, courtesy of the New Math movement some decades ago.
ReplyDeleteWhile many did not like New Math,New Math changed my indifference towards math into a love of math. If you can write proofs when a mere 14 year old, you feel empowered. At least I did.
Symbolic Logic and Set Theory was a required class at my University for all Math and Comp Sci majors. And it was the class set to weed out folks not suited to those majors. There was routinely a first time failure rate of about 30% of the class. What set logic is VERY good at is telling us about numbers and why they are the way they are. What it is very bad at is telling us about the real world. Mostly because the real world doesn't operate on a strictly true/false basis. All men are hairy. Which is likely true... save for someone with ichthyosis or some other horrible disease. Are they not men? Set theory says they cannot be. Else your premise (which holds true for almost all of the adult human population) is flawed.
ReplyDeleteThen, these logicians compound the problem of attempting to apply set theory to the real world by attempting to force English (or any other naturally formed human language) to comply with the world of logic. Square peg in a round hole puts it too mildly. So they're trying to force human language to conform to rules of logic and use that to describe a world which does not conform to set logic. What makes me laugh about this is that I'm sure they don't see why this will never work (since for every X, it will work or it will not... and if it does not work, to them that just means they haven't found the correct formula to describe it yet).
Gringo:
ReplyDeleteIt's fun to play with, I agree. And if it makes you enjoy math -- or just logic -- it's a great tool. You just have to be mindful of its limitations.
Mike:
That's how I learned it, too -- from a weed-out course designed to filter out people who couldn't do it. It sounds like you came to very similar conclusions about its applicability.
Grim: Well, for one thing, classical syllogisms are built around statements about all members of a class.
ReplyDeleteYes, but they don't begin with "everything". The terms come with specific meanings, so we don't face the same problem with the classical syllogism as these formulas do today.
I'm just curious about the benefit of beginning with everything. Or, the reason it's done that way, in any case.
Maybe because math is done that way?
Math was certainly the main model they were thinking of when trying to construct these modes of expression.
ReplyDeleteThere are programming languages that allow you to set a variable without assigning it a value, in which case the default value is null. (When you wanted to use it later, you would then assign it a value.) That makes more sense to me than automatically setting it at 'everything.' Is there a reason for it defaulting to everything?
ReplyDeleteI would guess it has something to do with human nature. People have a very strong desire to find simple rules that make a complex world seem more orderly and predictable than it is.
This is why I have so much trouble with philosophy (or merely overthinking things in general). Why even bother to argue about whether all ravens are black? Is blackness an essential quality of a raven - is it what makes a bird a raven, rather than an owl?
Of course not. And can we imagine albino ravens? Easily. Would they be 'not ravens' simply because their feathers lack pigment? No.
A more cautious framing of the statement that admits uncertainty and variation would eliminate all of this woolgathering: "Most ravens are black", or "The natural coloring of ravens is black (which admits that some medical condition might result in a white raven or a speckled one)."
But we want the absolute, which is artificial, rather than the qualified rule, which reflect reality.
"Most" is not a quantity that set theory is prepared to handle. Something is inside of a set, or it is not. Language is well suited to deal with the real world, because it evolved to handle real world situations (re: "most"). Set theory, as Grim and I have already stated, is poorly designed to handle the realities of the world. I can't speak to philosophy, as I only ever got through the Allegory of the Cave (mostly understood) then turned my attentions elsewhere as it didn't interest me. But set theory is math, or a language to talk about math perhaps. Ravens are not math.
ReplyDeleteYou're exactly correct that ravens MAY not be black, and yet be ravens. The mathematician would need to build an equation that would contain a set of all ravens, a subset for black ravens, and a subset for albino ravens. Thus, the set of ravens would be the union of the sets of black ravens and albino ravens. And if they encountered some other color of raven, they would need to include that subset as well. In the mathematician's mind, if his/her set definition does not account for everything in the world, then it's because his/her equation was incomplete.
The truly foolish thing is, set theory is VERY well designed for math. There are NO exceptions in math; things are true or false, not mostly true or mostly false. ANY even number may be expressed in the form 2k (where k is a whole number), and ANY odd number may be expressed in the form 2k+1. I could go through the logical proof to prove it to you, but the obviousness of the statement will bear out from observation. But precisely because math is a language of absolutes, at one point I was required to do that proof myself (proving for all values of k). Those kinds of proof is what symbolic logic and set theory is all about. And it's very good at it.
The folks who are trying to make language conform to logic are just missing the most basic point. The world is not math. But ironically, in their field, the explanation is just that their proof is incomplete, they just need to refine it further. But that premise is itself flawed. Thus the failure of mathematicians to describe reality.
This is why I have so much trouble with philosophy (or merely overthinking things in general). Why even bother to argue about whether all ravens are black? Is blackness an essential quality of a raven - is it what makes a bird a raven, rather than an owl?
ReplyDeleteI have a similar problem taking contemporary analytic philosophy -- the kind of philosophy that relies on symbolic logic to make its points -- very seriously. However, it's what most American and British philosophers are doing these days, so it's worth making the rather grim effort to suppress the laugh factor.
As for why they would ask if "all ravens are black," what was really at issue in Armstrong's book is the laws of nature. Moderns chiefly want these to apply without exceptions -- gravity, for example, is something we expect to hold in all circumstances. We really do believe (as Mike says we shouldn't) that there is an essentially mathematical relationship between objects with a given mass, that can be described with a sufficiently developed formula.
If it failed to hold, we'd want to know why. What happened? Gravity -- and other "laws" -- are supposed to hold without exception. If it doesn't hold, we'd need to find out why and adjust our understanding of the law accordingly.
Ancient philosophers would have been satisfied with a law that held "for the most part." Aristotle describes the rules governing the world in just that way: in his Physics Book II, he says that we know there is a sort of rule when we observe things that happen "always, or for the most part." That something in matter might make a behavior "usual" rather than "always" was something he accepted from empirical experience.
Medieval philosophers, of course, generally believed in exceptions to the laws of nature -- not just Aristotle's exceptions, but also exceptions resulting from miracles. They believed that the laws could be suspended.
So when we find our contemporary, the analytic philosopher, standing on other ground than his predecessors, it's kind of interesting to see if he can do it. Can we describe the laws of nature in such a way that they truly do not admit exceptions? Many scientists would like to say that we can; and many in the popular culture believe that we can. Yet here we find a basic problem with the logic required to do it.
That's interesting, isn't it?
...when we find our contemporary, the analytic philosopher, standing on other ground than his predecessors, it's kind of interesting to see if he can do it. Can we describe the laws of nature in such a way that they truly do not admit exceptions? Many scientists would like to say that we can; and many in the popular culture believe that we can. Yet here we find a basic problem with the logic required to do it.
ReplyDeleteI think part of the problem is that moderns have elevated logic to the level of faith. But faith in supernatural forces necessarily includes recognition of the limits of human reasoning... and logic.
Imperfect beings, using imperfect tools of their own devising...
What could possibly go wrong? :)
"What could possibly go wrong? :)"
ReplyDeleteUninitialized variables notwithstanding, a Theory of Everything (TOE for short), maybe?
//grabs a fresh beer and lowers the cone of silence such that the hun's carcass is, once again, returned to a hermetically sealed read-only state//
I would guess it has something to do with human nature. People have a very strong desire to find simple rules that make a complex world seem more orderly and predictable than it is.
ReplyDeleteActually, we went a couple thousand years in Western logic before people jumped on the 'start with everything' bandwagon.
Why even bother to argue about whether all ravens are black? Is blackness an essential quality of a raven - is it what makes a bird a raven, rather than an owl?
For what it's worth, medieval logicians would agree with you. Color was not considered a 'necessary predicate,' so, while you could use the proposition 'all ravens are black,' it was understood that being black wasn't necessary to being a raven, and therefore there might be non-black ravens somewhere.
What is strange to me is that something that was basic to the medieval philosopher has become a problem to the contemporary one.
I think part of the problem is that moderns have elevated logic to the level of faith.
ReplyDeleteA certain subset of moderns, but I think Western civilization in general is hostile to logic. We no longer teach it in our schools, most of our academics don't understand it and many denigrate it in favor of political ideology, and our media is saturated with logical fallacies that every child should be able to point out, but no one takes them to task for it.