I regard “laws of thought” like the Law of the Excluded Middle the same way as I do the Law of Non-contradiction (and the “Action Axiom” for that matter) - as redundant and unhelpful responses to failed communication attempts.
I know of no formal logic that validates “both A and not A” (contradiction), but quite a few that do not validate “either A or not A” (excluded middle). There are different motivations for these logics (quantum mechanics, computability, or philosophical constructivism), but it seems that these two “laws” are of quite different nature.
They are of a different nature of you define truth differently, sure, which as I said is a pretty contentious issue. I don’t find words like true or false useful at a deep level of analysis, nor terms like objective reality. They have an everyday usefulness because language is inefficient, but they become meaningless when you are trying to be extremely precise. All there is at bottom is utility. Beliefs are either reliable guides for my actions, or they aren’t.
Well, it seemed like you were using Schrödinger’s cat as evidence that the law of the excluded middle is invalid. I wanted to refute that argument. What I wrote doesn’t refute the statement that some people nevertheless reject the law of the excluded middle.
Of course. I’ll look into that, thanks.
Not at all. But I’m wondering if you’re implying that I shouldn’t ask so many questions myself.
Thought precedes communication. I would say that the laws of thought are at least guidelines for clear thinking (though I think they’re stronger than that).
No, the Law of the Excluded Middle comes from the Law of Non-Contradiction. If something cannot be both A and not-A, then it must be either A or not-A. This is equivalent to De Morgan’s Laws.
It’s not that Godel sentences are unprovable, as such: they can be proven in a metalanguage. People are kind of doing it right here. We use English as a metalangauge to prove that a Godel sentence is true. The “problem” is that each metalanguage capable of proving a Godel sentence generates its own Godel sentences which can only be proven in a meta-metalangauge, and so on, ad infinitum. Some true sentences always remain unproveable.
I use scare quotes around “problem” above, because whether it is a problem just depends on your goals. Personally, I don’t think being unable to prove true statements is a big deal. Most of the things that I believe are true I cannot prove in any strict mathematical sense. Some I might be able to with study, but others, such as scientific hypotheses, are just not possible to prove, in principle. It’s not really a big deal. Doubt is not to be feared and defeated. What is important is getting the facts right, not our degree of conviction that we’re right.
Since people have constructed formal languages where the law of noncontradiction and excluded middle do not hold (so-called paraconsistent and intuitionist logics, respectively), they cannot be mere “outworkings” of the law of identity. That said, any formal language where the law of identity, noncontradiction, and excluded middle do not hold shouldn’t really be called a “logic,” because they do not concern truth in the conventional sense. The use of truth values seems like an artifact. They are formal languages that are similar to logic, but they are not contenders. They can only serve alternate purposes. For example, in classical logic all sets of contradictory premises have the same logical content, i.e. there is only one contradictory hypothesis. In paraconsistent logic this is not true: one can distuinguish different sets of contradictory premises by thier logical content. I don’t really know why this would be very interesting, but each to his own, I suppose.
Just an addendum… the second theorem basically states that a given formal system is consistent if and only if a statement of the form “this formal system is consistent” cannot be proven within the given system. Without a proof for something, you are only left with empirical study, that is, wait-and-see. So, this is actually bad news from the point of view of mathematicians and other formal system-builders who want to know that they chose the “right” axioms, that is, axioms that will never lead to contradiction. Godel’s second theorem says “If I can do arithmetic in your formal system, then your formal system is either consistent or it can prove its consistency but not both” which means that the only way to know if your axioms are consistent is to wait and see, that is, empirical search for contradictions. Appealing to meta-systems is no help because they are regressively trapped by the same problem ad infinitum.
Indeed. This is mostly a philosophical issue. Does one need to prove that a system is consistent to know that it is consistent? If I do not know that a system if consistent, do I have any epistemic right to believe it? How can I be certain of my conclusions if I cannot prove that my reasoning is consistent? Can I believe anything that I am uncertain of? etc.
For me, these are not big problems. I don’t think we need “epistemic rights” to believe stuff, and I am quite comfortable being uncertain of my conclusions. If knowing requires proving in the strongest sense, then knowing is impossible, untinteresting, and irrelevant. Being right matters, not knowing that your right. Billions of years of evolution attest to the problem solving power of not knowing but being right anyway.
Yep. You and I are on the same page. And it’s also important to look at the history behind Godel’s theorem, it didn’t just spring out of a vacuum, it came out of Hilbert’s program and the “Entscheidungsproblem” or “decision-problem”. This, in turn, was partly motivated by the failure of Cantorian set-theory to construct consistent sets (Russell’s paradox) … how can a mathematical theory so beautiful be so flawed in such a basic way?
“Being right matters, not knowing that your right.” - Great quote.
The “problem” is that each metalanguage capable of proving a Godel sentence generates its own Godel sentences which can only be proven in a meta-metalangauge
It is only a problem in a fixed, “frozen” formal system. Living natural languages evolve to include more and more metalevels as these become useful for solving problems.
This. The only thing I’d add is that there can be no such thing as “ultimate truth” in any system that allows reflection or self-reference. When it comes to logic, though, empirical truth (i.e. facts) is out of the picture - all logic cares about is consistency. That’s why I define “logical truth” to be the same as “logical consistency”.
On another note, Goedel’s Incompleteness Theorems have led me to conclude that there’s no way to know whether what we call “reality” is the “ultimate reality”. In order to know that, we’d have to somehow look outside of reality. But doing that would imply that there’s some other reality outside what we call “reality”. How, then, do we know whether that reality is the “ultimate reality”? Etc., etc.
I think there’s a more succinct way to put this: it’s turtles all the way down.
That may work for aggregates resulting from markets or evolution, but not for individuals making informed choices. An individual entrepreneur would prefer knowing which method is “right” and bet on it, instead of comforting himself that the user of the right method would survive and proliferate even before knowing that he is using the right method.
Otherwise, I appreciate your constructive (and intuitive, no pun intended) post.
“Truth”* is an attribute of an individual proposition… every proposition is either true or not true. Consistency is an attribute of a group of propositions.
Clayton -
*Truth is in quotes here because we could use any word here … we are not talking about truth as in the “do you swear to tell the truth, the whole truth and nothing but the truth” kind of truth
The fact that you struck through that question implies that one or more of the questions above are trick questions. I have to wonder why you’re presenting your point (whatever it may be) in such a format.
