If 'If' Weren't So Iffy

[Laure]

Thom, that's a lot of words, there. (So here are a lot of words back at ya!) I still don't see your point with the M statements. If I paraphrase them to be more exact, they all say "if H was true, I just added (or will add) a dollar". That statement is always true if we accept that the person will do what he says. So, the 2nd line in the truth table, "H was true but I didn't add a dollar", does not occur. So, I don't see a problem. Unless I'm still missing your point.

I re-read what you wrote about the "paradox of entailment". I don't think that modern logicians would conclude "the U.S. economy negotiates one trillion decisions per microsecond" from a false premise (P & not P). Modern logicians would say that "(P & not P) --> Q" is always true, because the antecedent is always false, but that doesn't tell you whether Q is true or false. It could be either. When we say that the argument is "valid," all we are saying is that Q is derivABLE from "P and not P." Yeah, sure, in a world where P and not P are both true, it would follow that Q was true. Now I dare you to prove to me that "P and not P" (i.e. Check Your Premises!), and if you can do it, I'll grant you that Q is true! I think it just means that "if contradictions exist, then everything's true!"

It has been a while since I read The Analytic Synthetic Dichotomy, and I guess I have a little trouble with understanding the word "dichotomy." So I went and looked it up again. :-) "A division into two especially mutually exclusive or contradictory groups or entities." They give the example of "theory and practice" (which of course I don't think of as a dichotomy). I've also heard of the "mind-body dichotomy", I guess meaning that the mind can survive independent of the body (which of course I also reject).

When Peikoff discusses the Analytic-Synthetic Dichotomy, if I recall, his complaint was that philosophers who buy into this dichotomy believe that if something is provable, it has nothing to do with reality, and if it has to do with reality, that is, with something we actually see with our own eyes, that it can't be proven. I reject that, too.

However, with symbolic logic (as with all of mathematics), what we are doing is abstracting away the particulars, ignoring the semantics, and dealing with pure symbol manipulation or syntax. I don't think there is anything wrong with doing that. If we adopt the language from that Wikipedia article, we can determine if an argument is "valid" just by looking at the syntax, the symbol manipulation, without worrying about the semantics. We also have to look at the semantics to determine if an argument is "sound." But, I think it makes the job easier if we can ignore semantics in the checking of "validity," and I don't see anything wrong with ignoring semantics at that stage.

Just like with an algebra problem -- once I encode it in a formula, I don't have to think about the meaning until I get my final answer and then check it for reasonableness. Say I'm trying to calculate the average speed of a car, and I get 1000 mph -- then I need to go back and check my work! But while doing the calculation, I don't need to keep anything in mind about the meaning of the problem I'm solving. I start out with my "story problem," encode it into a string of symbols, pop down into the realm of pure syntax, manipulate the symbols, then pop back up to the realm of semantics to check the result, keeping in mind the meaning of what I was trying to calculate. So, while it's not a dichotomy, I think it is very useful to be able to examine the analytic or syntactic aspect of a problem, independent of the meaning.

[thomtg]

Thom, that's a lot of words, there. (So here are a lot of words back at ya!) I still don't see your point with the M statements. If I paraphrase them to be more exact, they all say "if H was true, I just added (or will add) a dollar". That statement is always true if we accept that the person will do what he says. So, the 2nd line in the truth table, "H was true but I didn't add a dollar", does not occur. So, I don't see a problem. Unless I'm still missing your point.

[...]

Laure, my point with the M statements is that, the person reading them always judges and acts on them as hypothetical statements regardless of how he judges and acts on the H statements in the opposite role.

Notice from the evidence of your coded program that you processed the H hypothetical statements by means of translating them into material implicated statements. You converted each H statement into an either-or+(not) function, and you judged its truth piecemeal, and then you acted by calling it either "true" or not. But when you switched role to evaluate the M statements, you initially did not believe the M statements to be hypothetical statements, because you could not seem to evaluate them truth functionally. And yet, the dollars kept being added on the table. So, implicitly you were not processing these statements as material implicated statements but as fully hypothetical statements.

Thus what I would like to know is, what if you bring back that same process of judging and acting on hypothetical statements in the manner of processing the M statements to the original task of judging and acting of the H statements? Then given what I said in Post #25 about the truth and falsity of hypothetical statements, which statements among the H statements would you call out "true"?

[...]

I re-read what you wrote about the "paradox of entailment". I don't think that modern logicians would conclude "the U.S. economy negotiates one trillion decisions per microsecond" from a false premise (P & not P). Modern logicians would say that "(P & not P) --> Q" is always true, because the antecedent is always false, but that doesn't tell you whether Q is true or false. It could be either. When we say that the argument is "valid," all we are saying is that Q is derivABLE from "P and not P." Yeah, sure, in a world where P and not P are both true, it would follow that Q was true. Now I dare you to prove to me that "P and not P" (i.e. Check Your Premises!), and if you can do it, I'll grant you that Q is true! I think it just means that "if contradictions exist, then everything's true!"

[...]

I am not claiming anything more than the simple fact that modern logicians take the conclusion from any contradiction to be "valid." My point is about the prescription that follows from the paradox of entailment. For you see, as logicians understands it, since man's deductive method is entailment, and because they take entailment to be truth functional, which turns out to be paradoxical, their prescription becomes: don't trust reason, don't trust knowledge.

[...]

Just like with an algebra problem -- once I encode it in a formula, I don't have to think about the meaning until I get my final answer and then check it for reasonableness. Say I'm trying to calculate the average speed of a car, and I get 1000 mph -- then I need to go back and check my work! But while doing the calculation, I don't need to keep anything in mind about the meaning of the problem I'm solving. I start out with my "story problem," encode it into a string of symbols, pop down into the realm of pure syntax, manipulate the symbols, then pop back up to the realm of semantics to check the result, keeping in mind the meaning of what I was trying to calculate. So, while it's not a dichotomy, I think it is very useful to be able to examine the analytic or syntactic aspect of a problem, independent of the meaning.

It is very interesting to me that you take the view that symbolization is a shortcut method to deduction because it allows you to blank out meaning in the intervening process. I can see it being useful in some contexts, but I cannot take it as a general principle. For even in arithmetics and algebra we normally make sure we don't divide by a variable that may turn out to be zero. This act, to me, is a counterexample to your general principle. It shows that no matter how much complex symbolization we build our math or logical statements, we should always take cognizant of the material meanings of the symbols. If we don't, though we may make all the deductions and simplications and applying all the rules of equivalences, we may come out with errors. And I believe this is exactly what happened with the logician's evaluations of the H statements.

Edited June 1, 2009 by Thom T G

[thomtg]

Rising star James Taranto at The Wall Street Journal has a problem with certain "if" statements. He doesn't understand them. See the Friday segment entitled "Dept. of Big Ifs" in his daily "Best of the Web Today" column. Even when he issues the "Homer Nods" correction in today's column, he takes the interpretation from modern formal logic. It goes to show that an error needs to be caught early at the beginning of the transmission belt; otherwise, it will be repeated down the line in new and varied ways.