[grosz]

I am currently reading Brand Blanshard's "Reason & Analysis." This book is routinely touted by one Scott Ryan (a rationalist Ayn Rand critic who constantly "bumps up" his scathing reviews of Rand's books on Amazon.com, so that they appear at the top of each "review" section).

I've only managed to read the first 4 chapters, which provided a history of philopsophy and rationalism, an analysis of the rise of Logical Positivism, and most importantly - the beginning of a series of impressive (albeit Rationalistic) refutations of Logical Positivism and of its exponents, primarily Russell and Wittgenstein.

Blanshard attacks the rules of inference described in Russell's Principia Mathematica, specifically that of implication. For any two statements p and q, Russell asserts, p implies q according to the following "truth table":
p q (p implies q: p==>q)
------------------------
F F T
F T T
T F F
T T T

He attacks this rule of implication on the grounds that it is impossible to assert implication by merely looking at the "truth-value" of the statements, i.e. by a total disregard to the contents of the statements. This is obvious, because according to the above truth-table, any false statement implies anything whatsoever. For example, if I were to say "I am naked at the moment," which is false, this would imply that I have green hair, or any other looney ideas I might come up with.
Reading the section that dealt with this implication problem, I was transported a year back in time, to my "Introuction to Logic and Set Theory" class, where the professor proved that the empty set is a subset of any set, by using the above rule of implication: x is an element of the empty set ==> x is an element of arbitrary set S. Since x is clearly NOT a member of the empty set (by virtue of it being empty), the above implication is TRUE. It took me ages to come to terms with this outrageuos absurdity, and now Blanshard has re-opened a can of worms.

Obviously, this logic does not apply to reality. One cannot arbitrarily state that something implies something else without knowing what those "somethings" actually are.

So, finally, my questions:
1. Principia Mathematica was published in ~1910, but it must be that this "mathematical logic" is much older than Russell. Who originated it?

2. Since mathematics purports to give correct numerical descriptions of various facts of reality, how is it possible that the basic means of mathematical proof - logic - is so far removed from reality?

3. Obviously there are many applications of mathematics, for instance matrix theory, and these do in fact work. BUT WHY? I mean, mathematicians prove theorems, and they do so by means of mathematical logic, whose rules of implication are nothing but arbitrary, as I've shown above. So how can anything "mathematical" be true IN REALITY, when its truth was proven by illogical means?

4. Can someone recommend good books that cover these issues, specifically the relation of mathematics and logic to the facts of reality, and which are clearly written? Are there any Objectivist philosophers who deal with these issues and can give a clear explanation to a novice?

Thank you
-- Ori

This post has been edited by grosz: 23 March 2005 - 12:35 PM

[Hal]

Logical implication is fine when used in the correct context, the problems only arise when people start thinking it has anything in common with the 'if' statement in natural language (ie when the concept "goes on holiday", to borrow an expression from Wittgenstein). Frege originally used it in a purely mathematical context and here it is fine - the material conditional is strong enough to capture the essence of (formal) logical reasoning. However it is completely unacceptable as a representation of 'normal' implication since it ignores casuality and the like as you point out.

As to your questions:

1) The foundations were laid by Boole and De Morgan, but it was Frege who invented the predicate calculus and gave it the form it has today.

2) Mathematical truth can be described using a small subset of the methods we use in everyday reasoning - for instance we do not need casuality or induction to describe mathematical truths, although obviously they will be used to actually discover them. Mathematical logic aims to be strong enough to capture this subset of our reasoning, not our reasoning as a whole.

3) This is the fundamental question of mathematical philosophy. I do not claim to have an answer

This post has been edited by Hal: 23 March 2005 - 12:45 PM

[Hal]

With regard to 'false statements implying anything', its important to realise why this is the case and what precisely is being claimed.

The conditional X => Y is a truth function of 2 variables (X and Y), both of which can be either true or false. In classic logic, a truth function MUST be either true or false for any combination of input values - ie every row in the truth table must be filled in.

Now, when X is true and Y is true, it makes sense to say that "X=>Y" is true. And when X is true and Y is false, it makes sense to say "X=>Y" is false. Things become less clear when X is false - there doesnt seem to be any 'obvious' truth value for X=>Y when X is false. But as I said, in classic logic a truth function must have a value for every combination of inputs, so it must have a value when X is false and Y is true. Hence, it was conventionally decided to say it is true. There are no grand claims going on here - we could just have easily said it was false. It isnt important what the value is, as long as it has a value.

It IS nonsense to say that "there are unicorns on mars" implies "grass is blue". But it isnt nonsense to say that "there are unicorns on mars => grass is blue". You just need to keep in mind that the symbol => does NOT mean the same as 'if' or 'implies' in English.

This post has been edited by Hal: 23 March 2005 - 12:56 PM

[DavidOdden]

One of the downsides of that approach to logic is that seems to make it impossible to maintain a strong conclusion about reasoning, that no valid inference can be based on a false premise. Now take a true statement "all men are mammals", which would be formalized as

Ax(Man(x)=>Mammal(x))

The problem is that my goldfish is an x but it's neither man not mammal. But this is an unnecessary use of implication, and is better dealt with by somewhat fancier logic where predicates establish a set-theoretic context binding variables. The other intuition behind "=>" is the concept "cause". Nonsense constructions like "There are unicorns on mars => grass is blue" would not express a valid causal relation. The connective "=>" is superfluous, and misleading given how it is conventionally pronounced in English.

[punk]

First off you are misunderstanding the use of implication in formal logic.

Let us take:

A = the sky is green
B = Dick Cheney is president

So the following statements are true:

-A (read as "not A")
-B
A -> B
A -> -B

Your concern is that you can deduce B (a false statement from this). You cannot. Knowing that A is false you can only deduce

A -> B

which is to say:

If I know that the sky is not green

then I can deduce:

If the sky is green then Dick Cheney is president

But none of this allows me to say:

Dick Cheney is president.

The only way I can deduce Dick Cheney is president is if I assert that the sky is green. The sentence "If the sky is green then Dick Cheney is president" isn't obviously false.

People however do see a problem with the fact that the color of the sky and Dick Cheney being president have nothing at all to do with each other, so we shouldn't be able to assert an if-then statement about them. People have constructed various modal logics in an attempt to deal with this.

What formal logic is primarily concerned with is *consistency*. It makes sure that all of your statements are consistent with each other, that is to say that you haven't set things up in a way that allows you to construct a contradiction. This is the point of its application to mathematics. It keeps mathematical statements consistent with each other.

The issue is the application of formal logic to areas beyond mathematics. Logical Postivism wants to say that we can formulate empirical statements about the world in a manner amenable to the application of formal logic. This would really only be a way of saying that all of our empirical statements (when properly formulated) are consistent with each other. Or that we cannot make contradictory empirical statements.

Inference in formal logic procedes along lines of *consistency* and not on lines usually associated with inference in common parlance.

So if I say a statement Y follows from a set of statements {X}

{X} -> Y

What I am really saying is that -Y is inconsistent with the statements {X}, that is that {X} together with -Y allows one to derive a contradiction. So Y is only "deduced" in the sense that it is the statement that wont produce a contradiction. If one cannot deduce Y or -Y from {X}, this is the same as saying that both Y and -Y are consistent with {X} and adding neither to {X} would produce a contradiction.

So the view of logical positivism is that all philosophy will consist of is empirical claims and a logical system establishing consistency among them, but weak enough to not a priori rule out any empirical claims. Only observation could rule out some empirical claims.

This rules out any claims that are not empirical.

The rationalist wants to cay that there are nonempirical statements that are true of the universe, and so logical positivism is ruling out perfectly good statements about the universe.

This post has been edited by punk: 23 March 2005 - 09:02 PM

[punk]

Frege is the guy usually credited with originating formal logic in his "Begriffschrift". But his tabular notation was horridly onwieldly, and no one uses it.

Russell in his "Principia Mathematica", introduced a more amenable (but now not used at all notation), and aimed in there to prove the logicist thesis that arithmetic could be deduced entirely from logical principles. He was unable to do this without introducing an axiom that doesn't have a terribly logical character, so the logicist thesis is considered to fail.

The contemporary notation we use is primarily derived from Hilbert. I forget the book's name.

[nimble]

I don't think you have to worry. I haven't seen or heard of a logical-positivist in about 50 years. It is a self defeating theory. They claim that the only statements that have meaning are statements that are analytical truths or things that are empirically testible. However their that statement is neither an analytical truth nor an empirically testible statement.

[punk]

nimble, on Mar 23 2005, 10:35 PM, said:

I don't think you have to worry. I haven't seen or heard of a logical-positivist in about 50 years. It is a self defeating theory. They claim that the only statements that have meaning are statements that are analytical truths or things that are empirically testible. However their that statement is neither an analytical truth nor an empirically testible statement.
<{POST_SNAPBACK}>

Actually they'd say it is an analytical statement.

That is a proper analysis of language would show that propositions that are not empirical can only be analytic in nature. That is that propositions that are not empirically verifiable can only serve to define the meaning of words. But if you are only defining the words used, you aren't really making claims that go beyond the structure of the language you are using. In fact the claim only expresses the structure of the language itself.

That's all fine.

The problem is with the empirical claims. Logical positivism states that the meaning of an empirical proposition is the means of its verification.

The result of this is that Logical Postivism has to give a verificationist account of science. That is that if we imagine we could articulate all the empirical propositions in the universe it is simply a matter of testing them systematically to arrive at a proper view of the universe.

Unfortunately the scientific method doesn't work this way. Karl Popper gave a better account in his work. That is that scientists propose a theory and then seek to disprove it. Logical Postivism has trouble giving an account of how a theory is to be constructed, as the construction of theories would seem to be a rather rationalist enterprise.

You know, though, constructing a straw man version of a philosophy and then declaring internally contradictory isn't really going to get you anywhere.

[nimble]

Quote

Actually they'd say it is an analytical statement.

They would be wrong. I do not see how the Verification Theory Method is a statement of analytic truth. But I am not going to argue with you over this, because I really don't care about logical positivism enough to waste my time.