Distilling down farther, it would appear that there are two types of truth, universal and individual. Your theory of correspondence appears to work well with the universal version.

It probably works well on the individual version too as long as the "facts" are placed in the perspective of the individual.

In these two examples the first is universal and the set of the second is the "universe" of the individual's total experience and understanding.

Yes. I can see how a mapping of opinions when given the same facts would overlap where shared opinion regarding what is true exists, but would not overlap where the divergence in pre-existing belief created a subsequent divergence in understanding. I suppose that I'm wondering what "the set of truth" would describe/include, or if attempting to describe such a set is even necessary to acquire the above results? It seems like "the set of truth" above is meant as the set of individual 'truth' and by my lights, it is no different than the "set of belief/opinion".



Again, to have a set of "the truth" is very ambiguous usage of the term truth. I do not think that that is a helpful step in understanding what truth is. It seems the removed set above would be where the facts directly correspond to shared opinion, based upon shared understanding and/or interpretation of the facts.



I think that attempting to do this very well may prove fatal to the project of becoming aware of what truth is. While false beliefs do exist, thought/belief has everything to do with how truth is engaged by the mind. A careful analysis of thought/belief formation instantiates truth in a universally applicable, and extant manner. Behavioral observation confirms. This situates truth outside of human conceptions. We can get into that later.



If by "not subject to human interpretation" you mean that the objective facts do not change accordingly, I'm in agreement for reasons already given. Facts are the case.



As far as I can tell, this seems to be a result of the methodology being employed. I mean, it is a logical consequence of how the pursuit began.



Belief is insufficient for truth - false beliefs exist. This is what grounds my objection to the equation of belief and truth that is necessary in order to hold individual truth; ie. "his truth", "her truth", "your truth", and "my truth". Here, if by "the truth" you mean what really happened between the two of them, then that would be equating "the truth" with the way things were, the actual state of affairs, the events that took place. It would be equating truth to objective reality.

Edited to correct an important oversight... change sufficient in the beginning of the last paragraph to insufficient. Oooops.

A little blurb on set theory ... just for the record.

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on ZF set theory, the notion of class is informal, whereas other set theories, such as NBG set theory, axiomatize the notion of "class", e.g., as entities that are not members of another entity.

Every set is a class, no matter which foundation is chosen. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.

Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.

In this case you pick a group of people who have some trait in common such as those who believe the Earth is flat and they become a class or "set". Other groups who have differerent "qualities" go into different sets and then the sets are compared for overlap, equality, etc.

This post was not directed at anyone in particular but was requested earlier by the OP.

The "solution set" is the class of ideas where facts do not correspond with the "truth". This may seem counter intuitive but the set where they do correspond (form equality) is the trivial answer.

So ... when is the truth inconsistent with the facts? Examples?

In quantum physics the location of the electron is given by a "cloud" diagram that covers a large area compared to the size of the particle.

When seeking the fact "Where is the electron in this atom?" you may get the following answers.

Physics student "It's location is uncertain but it is somewhere in that described area".

Engineer "It's everywhere in the described area at once."

Mathematician "We don't know for certain."

Each speaks the truth in his perspective. Can one fact spawn different truths? Can a specific question have different answers?

Some people absolutely do not classify a blow job as "sex" or "sexual relations." They consider it more like a massage with a happy ending.

Therefore, when Bill said he did not have sexual relations with that woman, he was telling the truth according to his idea of what sexual relations is. (Sexual relations is an act that could result in a pregnancy.)

I prefer a stricter definition - replace "could" with "does".

It isn't sex unless conception actually occurs.

I think that there is no difference between individual and general truth, only (some or many) individuals are not aware of it.

Something seems off metal...



How are these two statements compatible while retaining coherency?

What do we call the largest set?

A set is group of objects, mathematical or otherwise. In the case of truth an individual would be a set of one. The universal set would be the set of everyone.

A class is a group of sets (which may be individuals) who share some common characteristic, i.e., a class action suit, people who don't eat pie, etc. If you take that concept it to the extreme, a class can be a set of one. True, but essentially meaningless for differentiation unless part of a subset in use.

One could conceive a class of numbers who define the circumference of a circle by the radius. The class would include the set which consists of only the number pi. In this case the class equals the set.

The largest set is the class of anything.

The smallest set is the class of nothing ... generally known as the empty set.

Pertaining to this thread several specific sets or classes of humans and concepts have been discussed.

All humans.
All facts.
All opinion.
All truth.
All falsehoods.
All beliefs.
All understood concepts.
All misunderstood concepts.

Obviously the sets can be broken infinitely and classed infinitely but as the sets or classes are defined, logic would find mathematical relationships.

The set of humans would equal the set of humans with opinions if all humans were defined to have opinions. Some do not.

The trick is in the definitions as the set of truth is inextricably linked with a class of humans. The set of facts is not linked to humanity in it's definition. Some human truths would only fall into a set of one human. By contrast, the truth of quantum physics, for example, fall outside human understanding and follow whatever physical and mathematical laws that apply ... understood properly by humans or not.

An interesting class of sets would be the truths that do not depend upon human understanding and those who only depend on human understanding.

So, we are back to the theory of correspondence. If you want to prove the one to one correspondence of fact and truth, search for examples where they do not fit the theory. If the set is empty, you have your proof.

Give me a bit, metal. Thank you for the recent explanations.

If every set is a class, then "a class is a collection of sets" means a class is a collection of classes. "A class that is not a set is called a proper class" would mean that proper classes do not exist because if "every class is a collection of sets" then there are no classes that are not a set. If there are no classes which are not a set, then proper classes cannot exist.

You see my problem here?

We would have to do away with the notion of proper classes in order to remain coherent.

As a matter of definitions and contexts:

The quoted text is explicit about the definition of class being context dependent. No actual definition of set is given, but that definition might also vary in different contexts. I don't assume that all of these statements are intended to apply within the same framework. Specifically, (my understanding is that) proper classes are not used in ZF.

As a matter of pure logic:

Based on the statements made, and avoiding the assumption that the explanations given are complete definitions, it is possible for a collection of sets and to not be a set if that collection fails to have a quality required of sets. (Such as the ability to be included in another set).

If the purpose is to delineate distinctions between groupings based upon common denominators, we must be able to set these things out. While I understand that some sets are constituted by an overlap of smaller sets, and that other sets may constitute all smaller sets of any given kind, I do not understand how the two prior descriptions are compatible with one another.

In short... NBG sounds coherent. ZF does not.

I'm looking forward to putting this method to use, but the guidelines for our this must be coherent, lest we'll find ourselves violating the law of non-contradiction. I think it would be interesting to pursue, identify, and describe an axiomatic set, according to NBG set theory. Perhaps even set out the contingencies that make such a pursuit possible.

I think you are stumbling upon the original "set of all sets paradox".

Here is an explanation from Wiki.

Early paradoxes: the set of all sets
Main article: Russell's paradox

In 1897 the Italian mathematician Cesare Burali-Forti discovered that there is no set containing all ordinal numbers. As every ordinal number is defined by a set of smaller ordinal numbers, the well-ordered set of all ordinal numbers should define an ordinal number Ω which does not belong to the set. On the other hand, Ω must belong to the set of all ordinal numbers. Therefore, the set of all ordinal numbers cannot exist.

By the end of the 19th century Cantor was aware of the non-existence of the set of all cardinal numbers and the set of all ordinal numbers. In letters to David Hilbert and Richard Dedekind he wrote about inconsistent sets, the elements of which cannot be thought of as being all together, and he used this result to prove that every consistent set has a cardinal number.

After all this, the version of the "set of all sets" paradox conceived by Bertrand Russell in 1903 led to a serious crisis in set theory. Russell recognized that the statement x = x is true for every set, and thus the set of all sets is defined by {x | x = x}. In 1906 he constructed several paradox sets, the most famous of which is the set of all sets which do not contain themselves. Russell himself explained this abstract idea by means of some very concrete pictures. One example, known as the Barber paradox, states: The male barber who shaves all and only men who don't shave themselves has to shave himself only if he does not shave himself.

There are close similarities between Russell's paradox in set theory and the Grelling–Nelson paradox, which demonstrates a paradox in natural language.

Note the part where x = x is true of all sets. thus Facts are facts!
This concept separates individual sets (like facts) from complex sets such as the combination of truth and the human understanding of truth.

Again, examples are used to illustrate the concepts which do not fit the set. Do you follow the consistency of my posts? As I stated earlier, I tend to be brief whenever possible. I do not think the concepts in this thread are all that abstract. It needs only the search for the set of concepts that do not justify the correspondence of fact and truth.

Indeed metal, Russell's paradox does exactly what I've set out. It is a direct result of those two aformentioned statements being incompatible with one another. The one is the negation of the other, and vice-versa. Russell also recognized that X=X is meaningless.

Yes, metal I recognize the consistency of your thinking wrt the methodology being employed(set theory). The brevity is fine as well. It tends to keep things more in focus, and I can appreciate that as well. The last statement here puzzles me. Could you confirm whether or not I'm grasping the scope here...

There was a QM question posed earlier wrt to location of the electron within the cloud. Was that put forth as an example which satisfies the criterion of the search in the above quote, on your view? I mean, do you think that that is an example which does not justify the correspondence of fact and truth?

I used the QM example to illustrate several different concepts.

Firstly, the QM example is a physical-mathematical concept that falls outside of human opinion. There are obvious individuals who would argue against the meaning or reality of QM, but the assumption here is that the math and physics are consistent, understandable, and stand alone whether humans exist or not. Belief doesn't matter.

That said, a comparison of facts about QM to truths as understood by the mind of an intelligent human versed in the math and physics of QM still results in different comprehension of what an electron cloud does, means, represents, and illustrates. They have different truths, depending upon perspective.

The physics student knows the mass and charge of the electron and the fact that the position and/or momentum cannot be determined beyond given parameters but he/she knows it is there somewhere.

The engineer knows the "reality" of the electron is how it acts in the real world and doesn't care where it is, only what it does. He knows that, functionally, the electron acts like it occupies every possible location therefore it is located at every possible location with regards to matter-matter interaction. The fact that is has only one mass and one charge is irrelevant to the truth.

The mathematician seeks the answer of "Where is the electron mathematically?" The laws of uncertainty simply do not allow the exact location beyond known parameters leaving the answer, "I don't know for certain." as the truth.

The truths to each are disparate and have nothing to do with beliefs.

When dealing with humans, facts, and beliefs ... the divergence from corresponding sets occurs so rapidly as to become meaningless or simply a number of sets equal to the number of different beliefs which is essentially infinity.

Another similar example might be the set of five string theories that existed in the eighties and nineties before M theory. If you consider just those who studied and understood string theory, each of the five theories were mathematically consistent and gave an understood "truth" about sub-atomic physics with the exception that none of the five string theories worked with each other mathematically... a very bad truth indeed.

Along came M theory with it's single added dimension of freedom and instantly all the five string theories became simplified, worked mathematically, and appeared to be just different aspects of the same thing as viewed from from a different perspective.

In the universe of string theory was M theory the truth? Were the original five theories the truth, but not the whole truth? Can truths evolve into deeper truths with increased human understanding? Is an absolute, well understood, fully documented human truth still true if it changes over time?

I would disagree with the notion that the math stands alone whether humans exist or not, but we need not get into that. I would agree that the math works and is understandable and consistent. That's all we need to agree upon for our purposes here.



It seems that you're employing the term "truth" in place of belief or conclusions. I mean, do they have different truths, or do they come to different conclusions, regarding what they believe to be the case, based upon the same facts, depending upon their perspective? Truth is not determined by perspective, rather it is necessarily presupposed within it. It seems to be both, more accurate and useful, for us to say that they have come to different conclusions based upon their perspective. The difference here being critical to understanding the role that truth presupposition plays in our thinking. If truth depends upon perspective, then we have subordinated truth.



Ok.



Ok - up until the last claim. What are you referring to there as "the truth"? Does the following convey what you mean...

The fact that it has only one mass and one charge is irrelevant to the fact that it acts like it occupies every possible location.



I think that if we are more disciplined in our use of the term "truth", we can avoid the problems that arise from employing "the truth" to represent things other than truth(perspective, fact, certainty). If "I don't know for certain" is a true statement, it is not true of the electron. Rather, it is true of the mathematician. In other words, it corresponds to the mathematician's state of mental affairs, not to the objective state of electron affairs.



While I have a genuine desire to employ set theory we must not subordinate truth in the process. The above is simply not true metal. You've called personal conclusions, facts, and perspectives "the truth". Conclusions are always arrived at from first believing the premisses are true. Perspectives wholly depend upon belief. Fact, on the other hand, does not. Thus, truth corresponds to fact/reality... not opinion, perspective, and conclusions. Those things may or may not be true, depending upon whether or not they correspond to fact/reality.



True, but they all converge upon truth-presupposition. Namely, the presupposition of 'loose' truth/reality correspondence that is necessary in order to form thought/belief. That is the largest set - it includes everything humanly thought, believed, and or known.



Purely mathematical problems require purely mathematical solutions. M theory is not "the truth". When we say "the whole truth" what does that really mean? Such ambiguous terminological usage is entirely context dependent, and when a little unpacking is done, it often becomes apparent that the term is being used to mean something else.

I've never seen "an absolute, well understood, fully documented human truth". Are you talking about a bit of human knowledge?