Reconciling Schrödinger's Cat with the Principle of Explosion

That's not what I am saying. I'm saying that the truth value of a mathematical proposition can only be derived from a set of unproven axioms. The concept of mathematical truth is meaningless outside of an axiomatic.

It seems to me that you're wrong. Firstly, I don't know why you're focusing so much on mathematics and mathematical propositions. You should be focusing on all propositions, mathematical and not. Secondly, many propositions about everyday life, such as "It is sunny outside" and "The decorative fruit is in the bowl on the chest" have truth values that do not depend on some set of axioms.

To me, it is not the axioms that give meaning, it is the undefined terms and concepts. An axiomatic truth is true even when there is no axiom that asserts it.

You're not thinking here. You're trying too hard to avoid thinking.

I'm just saying that A) all natural language statements are ambiguous and thus their "true or false" value depends inter alia on how they are interpreted; and B) even in mathematical language that is (or tries to be) devoid of ambiguity, the truth of a statement depends on the axioms one arbitrarily postulates as "true" at the begining of the demonstration.

IOW, there's no such thing as an absolutely true statement (true in all cases and forever, however interpreted or framed axiomatically). It all depends on a frame of reference or another.

That's not so hard to get, is it?

Take the word "fruit" that you used in one of your examples. It may seem reasonnably clear to you but it is in fact ambiguous. In the biological sense of the word, a tomato, a walnut and a peapod are "fruits", aka they contain seeds and developed from the ovary of a flower. In common English, they are not fruits. On the other hand, strawberries may be considered as fruits in common English but they are not fruits in the biological sense, as they grow out of an inflorescence (a group of flowers) and not from the ovary of a single flower...

And I don't even know what a "decorative fruit" is. Does it mean an actual (common English language) fruit meant for decoration only, or a piece of plastic in the shape and color of a fruit (ie something that looks like a fruit but is not a fruit)?

I'm curious as to what evidence there is to support that claim.



I just disagreed with you on that point in my previous reply to you. To me, the truth of a statement does not depend on the axioms; it depends on the undefined terms and concepts.



There are necessarily true statements, such as "5 + 5 = 10" and "A triangle has three sides," that are true in all cases and forever.



It means the latter.

Check any word in any dictionnary, and you will find several different meanings. It follows that all words are ambiguous. Like the word "fruit" in my example above.

The "undefined terms and concepts" you speak of need to be posited axiomatically. Any geometry starts with axioms such as "there exist spaces".

The statement "5 + 5 = 10" is only true in base-10 numbering. In base-6, 5 + 5 = 14.

There's no way out of it. All truths are relative to a reference frame arbitrarily chosen. It's basically the same idea as gallilean or special relativity, applied to language.

(Not to say that all frames are equally useful in all circumstances. That's where it becomes interesting.)

Just as a matter of interest...do you think you are getting anywhere with this poster ?

Your well stated points are elementary ones well known as problems to logicians. Indeed some of them were the reason Wittgenstein 'dumped' his earlier Tractatus (based on logical propositions) and turned instead to the adage 'meaning is usage' (implying that semantics and hence 'truth' is always contextual).

Articles on the so called 'explosion' point out that following Russell's Paradox, logicians have tried to developed alternative 'logics' to delimit it, so the attempts to it to relate to QM, which is well known to be 'counter intuitive' is somewhat superfluous.

The ultimate issue is obviously the meaning of the word 'truth' hence my suggestion that he reads up on 'theories of truth' .Until he recognises that, I suggest you are being strung along.

My mother would have said that Rock Hudson was one.

I'm getting some way towards sharpening my own thoughts on the topic, I think. What he does with this material is none of my business. You can bring a horse to water etc.

Well, that explains why you are having trouble understanding Quantum Mechanics, doesn't it.

The Copenhagen Interpretation implies 2 = 3.

Schrödinger himself intended "Schrödinger's Cat is simultaneously alive and not alive" to be a contradiction, so your defense that such a proposition is not a contradiction is foolish.

As long as he doesn't sing, I guess...

No, the undefined terms and concepts do not need to be expressed axiomatically in order for them to have meaning. In my view, which is how I have been taught and which I believe is how high school geometry in the United States is commonly taught, the undefined terms and concepts are stipulated before the axioms and postulates are presented. Geometry (2004) by Ron Larson, Laurie Boswell, and Lee Stiff, my most favorite book, takes that approach.



That's an interesting example. There does seem to be some sort of domain of discourse we use, whether implicitly or explicitly, that limits the possible interpretations of statements. Your example suggests "the statement '5 + 5 = 10' is unnecessary," but I firmly believe "5 + 5 = 10" is necessary. It appears you and I are presupposing different domains of discourse for the statement "5 + 5 = 10."

It also seems self-contradictory to talk about "base-10" and "base-6" numbering, because the very names of those numbering systems is dependent on base-10 numbering. There may be a flaw to uncover in your example.

I really don't think you should justify cannibalism that way. You said that kids are tasty with salt and pepper... and that they provide nutrition. But that doesn't mean that you can't stick with meat and vegetables.

And on a side note, elephants don't fly even if you think they have wings.

The existence of a space, of straight lines, etc have to be affirmed, established through axioms. Otherwise, if there is no such thing as a space, there can be no geometry.

What do your books say about axioms? I can't believe they ignore the concept altogether.


That would be what I call a frame of reference.

I'm not going to dwell on numbering systems. I learnt base numbering in second grade, at age 7, and it always seemed quite logical to me... I suggest you read another school book about it.

Undefined terms and concepts, such as point, line, and a point lying on a line, do not have to be affirmed and established through axioms. As I know from personal experience, their meanings can be stipulated without axioms. In stipulating the meanings of the undefined terms and concepts without axioms, a person just has to know what the undefined terms and concepts mean. There are some things a person just has to know. A person must be intellectually gifted by God, biology, genetics, or something in order to properly know the meanings of the undefined terms and concepts.

My books do not ignore the concept of axioms. Some of my books, such as the aforementioned Geometry, present axioms after the meanings of the undefined terms and concepts are stipulated. At that time, the axioms can be presented using the already known undefined terms and concepts. From my own personal experience, which has been influenced by my books, an axiom is a statement that is taken to be true, but is not formally accompanied by proof.

Okay. Why do you think they are not accompanied by proof?

Axioms aren't formally accompanied by proof by definition of axiom.

Why do mathematicians need them, you think? Those guys usually HATE unproven statements.

Mathematicians need them because they need a place to begin their intellectual pursuit.

Exactly. A place to begin. Why and how do you think mathematicians chose a particular set of axioms rather than another?