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# Exactly what is “Truth”?

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There remains my question about what makes abstract relationships (like 2 + 2 = 4) true?
They are "true" in a given formal system if they are one of the axioms or follow from them by means of the inference rules.

In [imath]\mathbb{Z}_3[/imath] it is true that 2 + 2 = 1.

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They are "true" in a given formal system if they are one of the axioms or follow from them by means of the inference rules.

In [imath]\mathbb{Z}_3[/imath] it is true that 2 + 2 = 1.

That is true, but sounds to me no more than what I had already said:

...these may be considered true by definition, because we define the symbols "2" and "4" and the relationship "2 + 2 = 4" to be true.

I'm not sure that it answers:

...but the fact that humans can construct different expressions for the same truth suggests, to me, that there is an underlying truth that both are descriptions of. If so, where do such truths come from, and what makes them true?

The answer that I would, tentatively, put forwards is that abstract relationships are true because they are implicit in physical reality. E.g. If you have four apples and divide them into two equal lots, you have two apples in each lot. Hence 2 + 2 = 4 is true because physical reality makes it so.

However, it seems to me that such empirical derivation cannot ensure that it is always true. Merely that it is observed to generally be true. Thus there could be a circumstance in which 2 + 2 != 4, we just have not met it yet!

Does that make sense?

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Rather than consider physical reality an empirical basis for mathematical asserts, I say that certain things can be described by an appropriate construct. Counting apples is well described by the usual arithmetic but predicting the position of a turnstyle isn't, when the number of clicks for a complete round is exceeded. Mapping [imath]\mathbb{Z}[/imath] onto the [imath]\mathbb{Z}_3[/imath] of my previous example describes a turnstyle having a total of 3 positions. Rigid rotations of arbitrary angle, around a fixed axis, can be described by SO(2) or SU(1). The constructs are independent of reality, we just find what kind of construct some given thing is best described by.

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I disagree. This suggests that if there is no evidence, then a truthful statement cannot be made.

No. If there is no evidence, there is lack of information to deterrmine truth or falsity.

So the statement:

is true, provided that your definition of reality includes both physical reality and beliefs.

Yes it does. Beliefs are reality. Everything is in reality. We exist in reality. Everything we know is reality.

There remains my question about what makes abstract relationships (like 2 + 2 = 4) true?

Definitions make them true. Relation 2+2=4 is not abstract; it's concrete.

At the beginning of the baseball season, Joe Girardi, the manager of the Yankees baseball club, wore number 27 to signify that Yankees will win 27th baseball title. Yankees had 26 at the beginning of the season. If you add one more to 26, by definition, we call it twenty seven. We call that evidence, by definition, 27. We can call it "aoubafiua," but if gave such random names to all quantities it would unnecessarily make our life difficult; twenty seven is a more organized title for reference purposes. numerical descriptions are even easier, i.e. 27, 28, 29, 30.

It's all based on evidence. If you pluck one needle of grass, we say you have a unit, one leaf, or 1. If you pluck another one, you will have more than 1. You will have a 1, and another 1. That one and another one, we call two by definition. If you pluck another one, you will have more than 2, and that we call 3 by definition.

So 1 + 1 + 1 = 3 is concrete, based on evidence, and now defined as such so that we don;t have to go through the thinking and defining process again.

Now, since we now that the definition holds true for leafs of grass, we trust that it will work with distances, with molecules, with speeds. When we put our definitions to test, against evidence, they hold true.

Because our definitions are consistent with evidence in many applications, then we accept them as universally true.

But whatever the statement is, it must be consistent with evidence to be true. Otherwise, it is either false, or indeterminable.

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No. If there is no evidence, there is lack of information to determine truth or falsity.

I agree. Furthermore, in the absence of evidence, any statement about a "fact" is neither true nor false, its status is "unknown". However the "fact", as a thing-in-itself, is either true or false.

An example: There is now evidence that there is water on the surface of the moon. So we now know that it is true. But before there was any evidence, the validity of the statement "there is water on the moon" was unknown.

However, the water did not suddenly appear just because we found out about it. It was already there. So it was true that there was water on the moon before we had the evidence that enabled us to know that it was true.

I.e. You appear to be conflating the truth of a fact with the truth of a statement about that fact.

Definitions make them true. Relation 2+2=4 is not abstract; it's concrete.

You appear to be making two conflicting statements here:

Firstly, you appear to be claiming that abstract truths are true by definition. That is one possible solution. The problem with that solution is, surely 2 + 2 = 4 was true millions of years before there were men around to define it to be so?

Then, I guess, by saying that "2+2=4 is not abstract; it's concrete" you mean that it is empirically known to be true. That is a different solution:

It's all based on evidence. If you pluck one needle of grass, we say you have a unit, one leaf, or 1. If you pluck another one, you will have more than 1. You will have a 1, and another 1. That one and another one, we call two by definition. If you pluck another one, you will have more than 2, and that we call 3 by definition.

So 1 + 1 + 1 = 3 is concrete, based on evidence, and now defined as such so that we don;t have to go through the thinking and defining process again.

I like this idea, but it has the flaw that, no matter how many blades of grass we pick, we only know that 1 + 1 + 1 = 3 for those needles of grass. We can assume that this is always true, but we cannot show that it is so. We could be wrong.

Now, since we now that the definition holds true for leafs of grass, we trust that it will work with distances, with molecules, with speeds. When we put our definitions to test, against evidence, they hold true. Because our definitions are consistent with evidence in many applications, then we accept them as universally true.

As I've pointed out above, there is a difference between something being true in every case that we have tested, and asserting that it is universally true. Asserting that something is universally true does not make it so. Take, for example, the lottery. You try every week, but you always loose. So do your friends, so does everyone you know. You come to the conclusion that the lottery is a con, because it cannot be won. The evidence proves that. That is, until someone does win.

But whatever the statement is, it must be consistent with evidence to be true. Otherwise, it is either false, or indeterminable.

Agreed, but that relates to the truth of statements, not abstract entities.

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Beliefs are reality. Everything is in reality. We exist in reality. Everything we know is reality.

I disagree with all of this except for "We exist in reality." While it is true that people have beliefs, having beliefs does not make those beliefs consistent with reality. Beliefs are one's interpretation of reality not necessarily representations of reality, even though the believer believes them to be so.

Not everything we know is in reality. I know about fire breathing dragons. They do not exist in reality. Just because we can imagine something in refined detail does not make it real or true. But I would agree that it is true that we can imagine the existance of fire breathing dragons. So while dragons are not representations of truth or reality, imagination is.

.....But whatever the statement is, it must be consistent with evidence to be true. Otherwise, it is either false, or indeterminable.

I think this is a more accurate statement than the ones above, but still open to interpretation. There was a time when the observable evidence clearly demonstrated that all celestial objects revolved around the Earth which was fixed at the center of the universe. As such, this was believed to be true by a vast majority of the population at the time. Was it any less false then than it is now? I say no. While the evidence, both observable and from scriptural accounts, was interpreted to be in support of the geocentric model of the universe, the truth is that the evidence was being misinterpreted (or maybe even manipulated considering the influence of the church).

So while I would agree that evidence is an essential element in understanding what is the truth, it should also be understood evidence is being interpreted, and today's conclusive evidence may be tomorrow's discarded notion, and the truth remains elusive.

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Everything is in reality. Reality is always what it is. It is what it is.

Only statements can be true or false.

If humans had no communication, everything would still be in reality. It would be what it is. There would be no truth or falsity. Only when you introduce communication, to describe reality, we have truth and lies. Only statements about reality can be true or false. Reality itself just is.

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Everything is in reality. Reality is always what it is. It is what it is.

Only statements can be true or false.

Good, we are getting to the nub of it. You suggest that only statements can be true or false. I suggest that a type of abstract entities, abstract relationships, are also true.

There are two types of abstract entities:

a) The numbers 1, 2, 3 , 4 etc. These are neither true nor false, they merely are.

:eek_big: The relationships between them. E.g. 2 + 2 = 4. These are true.

So "2 + 2 = 4" is a statement. What is it about? It's about an abstract relationship between the numbers 2 and 4. So "2 + 2 = 4" is not ONLY a statement, its also an abstract relationship 2 + 2 = 4. Note: I put the statement in double quotes to distinguish them.

So there are two (2 + 2 = 4)'s in this analysis:

a) The statement "2 + 2 = 4" which is about an abstract entity.

:doh: The abstract entity 2 + 2 = 4 itself.

Both are believed to be true:

The statement "2 + 2 = 4" is true to the extent that it reflects the abstract entity. This is evidenced empirically, as you have suggested, but the evidence does not make it true. It is the relationship between the statement and the abstract entity it relates to, and the truth of that abstract entity, that makes it true. If 2 + 2 = 4 was not true as an abstract entity, "2 + 2 = 4" would not be true as a statement. The empirical evidence merely allows us to know that it is true.

So what makes the abstract relationship 2 + 2 = 4 true? That is the question I was asking...

I'll also throw in the question "can abstract relationships be false?". I think not. 2 + 2 = 4 is an abstract relationship, it is meaningful. 2 + 2 = 5 is false, but I'd suggest that there is no such abstract relationship. It's just numbers I've arranged into an equation. So perhaps it is false because there is no such abstract relationship?

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While it is true that people have beliefs, having beliefs does not make those beliefs consistent with reality. Beliefs are one's interpretation of reality not necessarily representations of reality, even though the believer believes them to be so.

I suspect that you are using the term "reality" to mean "physical reality". As such I would agree with your statement. However, there is a sense in which beliefs are a part of reality. As are ideas and observations. I don't mean to imply that they exist physically, but they do exist. Otherwise, how could we discuss them?

I know about fire breathing dragons. They do not exist in reality. Just because we can imagine something in refined detail does not make it real or true. But I would agree that it is true that we can imagine the existence of fire breathing dragons. So while dragons are not representations of truth or reality, imagination is.

I think that you are agreeing that concepts (which you call "imagination") exist independently of the physical reality of the entity they describe. Otherwise, what would make the statement "fire breathing dragons do not exist" true? If there was no concept "fire breathing dragon", then the statement "fire breathing dragons do not exist" would be meaningless.

There was a time when the observable evidence clearly demonstrated that all celestial objects revolved around the Earth which was fixed at the center of the universe. As such, this was believed to be true by a vast majority of the population at the time. Was it any less false then than it is now? I say no.

Agreed.

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Beliefs are reality. Everything is in reality. We exist in reality. Everything we know is reality.

I disagree with all of this except for "We exist in reality." While it is true that people have beliefs, having beliefs does not make those beliefs consistent with reality.

I, er... believe he meant that beliefs are reality in the sense that the fact that someone believes something is reality. This is distinct from what one believs being reality. If John believes Sue secretly loves him and doesn't show it cuz she's got hubbie and kids, even if what he believes is false it is still a fact of reality that John believes it.

.....But whatever the statement is, it must be consistent with evidence to be true. Otherwise, it is either false, or indeterminable.

I think this is a more accurate statement than the ones above, but still open to interpretation. There was a time when the observable evidence clearly demonstrated that all celestial objects revolved around the Earth which was fixed at the center of the universe. As such, this was believed to be true by a vast majority of the population at the time. Was it any less false then than it is now? I say no. While the evidence, both observable and from scriptural accounts, was interpreted to be in support of the geocentric model of the universe, the truth is that the evidence was being misinterpreted (or maybe even manipulated considering the influence of the church).

An ounce of formal logic and A implies B isn't the same as B implies A.

The implicant, or antecedent, clause can also be called "sufficient condition" and the consequent can be called "necessary condition" and must means necessity. The meaning was not consistent implies true; it was consistent is necessary for being true.

P. S. I meant to say the evidence was not being manipulated and it wasn't really the influence of the church. The influence was Aristotle and the evidence was the "obvious fact" that planet Earth is not moving. I mean, we're sitting right on it! Surely we'd be aware if it were moving!

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There are two types of abstract entities:

a) The numbers 1, 2, 3 , 4 etc. These are neither true nor false, they merely are.

:eek_big: The relationships between them. E.g. 2 + 2 = 4. These are true.

...

So there are two (2 + 2 = 4)'s in this analysis:

a) The statement "2 + 2 = 4" which is about an abstract entity.

:doh: The abstract entity 2 + 2 = 4 itself.

You have a contradiction between first a and b, and the second a and b. In the first, you claim that entities just are, which I said before and agree with; and statements are true or false, which I also said and agree with. In the second you depart and say that now the entity is 2+2=4 itself is true. I can not agree with that. An entity just is, as said before.

Both are believed to be true:

Only statement can be true. The entity just exists.

So "2 + 2 = 4" is a statement. What is it about? It's about an abstract relationship between the numbers 2 and 4.

It is a "generalized" statement that holds true universally through repeated empirical observation. As an entity it just exists in imagination, and as a statement it holds true generally as it relates to any evidence.

So what makes the abstract relationship 2 + 2 = 4 true? That is the question I was asking...

Again, the definitions make it true. Let's talk about definitions.

When born, we first learn definitions. This is a "cap." The shape we perceive through sensory mechanisms that serves a specific function is defined in alphabetical and lingual manner as a "cap." (letters are defined as certain lingual sounds.) This is a "bottle." This is a "liquid." This is a "juice." This is an "apple." We have definitions. Definitions help us communicate about reality in an organized manner.

Similarly, we define 1 as a unit of something, bounded by wherever our sensory perceptions draws distinction. (Our sensory perception tells us that a table is bounded by air, and air is not part of the unit of table. Our perception draws a distinction. A table is one unit within its bounds.)

Then, we define 2 as one more than 1. We define four as two more than two. The statement is true by definition and it is consistent with evidence in all applications as far as we have tested it.

(It is consistent because, when we perceive one table, and another, and another, then we perceive more than one. When we perceive one more than one, then by definition we perceive two. Then three. Then four, and so on by definition.)

The number entity or an equation, as a part of our imagination or thinking, or as written somewhere, just is; it exists neither true nor false; it just is. But the statement is true as it says something about reality.

I'll also throw in the question "can abstract relationships be false?". I think not. 2 + 2 = 4 is an abstract relationship, it is meaningful. 2 + 2 = 5 is false, but I'd suggest that there is no such abstract relationship. It's just numbers I've arranged into an equation. So perhaps it is false because there is no such abstract relationship?

Sure. An equation can exist and be false. 2+2=5 exists, right here, in reality. But, as a statement it is false by definition; because we define 5 as 1 more than 4, and we define 4 as 2 + 2.

2+2=5 is not consistent with evidence, and is false.

The fact that definitions are embedded in our understanding of 2+2=5 statement is of no moment. Definitions are embedded in all communications. When you read the dictionary, which is a compilation of defintions, it will not tell you that a definition of something consists of definition of this and definition of that. The dictionary just tells you what the definition is without mentioning definitions. It is assumed and understood that all is defined; that you understand it. All I have to tell you is 2+2=5, and I do not have to define 2 or 5 for you. Simlarly, if a chair is defined as a furniture designed to seat one person, the dictionary will not tell you: a chair is defined as what is defined as furniture to what is defined as designed to what is defined to seat to what is defined as one to what is defined as person. It assumed that you uinderstand that all is defined and you uinderstand the definitions.

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You have a contradiction between first a and b, and the second a and b. In the first, you claim that entities just are, which I said before and agree with; and statements are true or false, which I also said and agree with. In the second you depart and say that now the entity is 2+2=4 itself is true. I can not agree with that. An entity just is, as said before.

No there is not a contradiction. You omitted the following from your quote:

I suggest that a type of abstract entities, abstract relationships, are also true.

This made clear that I regard abstract relationships as a type of abstract entity. Therefore it is legitimate to use the term "abstract entity" provided that I make it clear what type of abstract entity I was referring to (an abstract relationship). If that confused you, then I apologize. If it helps, please substitute the word "relationship" for "entity" in the second set of a's and b's. I.e.

So there are two (2 + 2 = 4)'s in this analysis:

a) The statement "2 + 2 = 4" which is about an abstract relationship.

B) The abstract relationship 2 + 2 = 4 itself.

I hope that clarifies my meaning.

Only statement can be true. The entity just exists.

See above. I was referring to abstract relationships, which are a type of abstract entity that can be true. If you accept that abstract relationship can be true, then your statement "Only statement can be true" is false. If, on the other hand, you do not accept that abstract relationships (like 2 + 2 = 4) can be true (in and of themselves), please explain your logic:

a) Are you denying that abstract relationships exist in and of themselves, separately from statements about them? Or...

B) Are you denying that abstract relationships (although they exist) can be true? Or...

c) Are you saying that the truth of a statement about an abstract relationship makes the abstract relationship true? Or what???

So what makes the abstract relationship 2 + 2 = 4 true? That is the question I was asking...

Again, the definitions make it true...

So here you appear to acknowledge that abstract relationships can be true? So why do you say:

Only statement can be true.

This seems to be inconsistent.

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Jedaisoul, you are free to disagree and I respect that. I think we've come to an impass at the fundamental level: the definition of abstract entity.

If abstract entity is an entity, and if entities just are, then an abstract entity just is.

If abstract entity is an entity, and if entities can not be true or false, then abstract entity can not be true or false.

The entity, 2+2=5 for example, does not speak, does not state anything. The statement is in your head. You interpret it according to definitions.

In communication it goes like this: INTENDED STATEMENT BY SOURCE -> ENTITY ->RECEIVED STATEMENTBY A TARGET.

Source generates entity to convey a statement. When I speak to you I generate sound, sound propagates through air, sound reaches your sensors, you interpret the sound.

The sound is an entity; it is not a statement. The statement exists only in my head as I interpret it, and in your head as you perceive it.

If I speak Lithuanian to you, and if you do not understand Lithuanian and I do, then the sound is an entity in air. But the statement in my head carries one meaning, and in your head a totally different meaning, even though the entity is the same: the Lithuanin sound. The Statement is only in your head, regardless of the entity. The entity just is, and only the statement can be true or false.

Same with abstract entitites and relationships. When I generate 2+2=4 to convey some meaning, the 2+2=4 just exists, but the statement is created in my and your head; the meaning, the definitions. The entity just is, but the meanings can be true or false.

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Jedaisoul, you are free to disagree and I respect that. I think we've come to an impass at the fundamental level: the definition of abstract entity.

I agree that we have reached an impasse. You are, of course, entitled to your views, but I'm disappointed that you have not answered my questions.

I've come to the conclusion that you are denying the existence of abstract relationships as things in themselves that are not dependent on a statement of them to make them real. At least that seems to be what you are saying. There certainly seem to be unresolved inconsistencies in what you've said, but as you are not willing to answer questions, nor comment on my reply #56, there is nothing more to say.

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you are denying the existence of abstract relationships as things in themselves that are not dependent on a statement of them to make them real. .

No I do not. I specifically agreed that your relationship 2+2=4 is an entity in itself. I just do not care how you characterize relationships; as abstrtact or concrete. The only pertinent question is whether you are considering the relationship as (1) an entity itself, or (2) a statement.

The abstract relationship, as an entity just is, and as a statement in your head or my head can be true or false.

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No I do not. I specifically agreed that your relationship 2+2=4 is an entity in itself. I just do not care how you characterize relationships; as abstrtact or concrete. The only pertinent question is whether you are considering the relationship as (1) an entity itself, or (2) a statement.

The abstract relationship, as an entity just is, and as a statement in your head or my head can be true or false.

Thank you for clarifying your meaning. I can't see how "2 + 2 = 4" as a statement is capable of being true or false, but 2 + 2 = 4 as an abstract relationship is not. However I accept that that is your view, and you are entitled to your view. 'Nuf said.

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What makes a relationship into a statement is your perception and brain. You create it in your head.

A sound in air just is. But once you perceive it in your ears, and interpret it according to your definitions in your head, the relationship of sound waves becomes a statement in your head. That is the only place where a relationship exists as a statement; otherwise it is just an entity outside of your head.

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