Exactly what is “Truth”?

[jedaisoul]

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.

Yes, I know you believe that. You are entitled to your beliefs, but that does not make them credible. An abstract relationship is not a sound in the air. But that is what you seem to believe, and if so, I really do not want to discuss this any further.

[watcher]

Your mind is clouded by the fact that you believe that what you know what you are talking about here is “real”. I can show quite clearly (via logic; except that you have to understand mathematics) how you have come to convince yourself of that delusion and that it is indeed a delusion.

 

mathematics doesn't tell us if something is real or unreal. it has no power to tell us if we are delusional or not. this is the point of jedi. but it does tell us if we we are being precise or just approximating. the physical meaning of our equations are an entirely different fields of study. math does not directly tell us what exists out there. our consciousness does.

[jedaisoul]

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

I still largely disagree with lawcat, but I'm indebted to the vigorous discussion we've had for helping me reach further conclusions about abstract entities and relationships. So my position is changing, and I'd like to express it here...

 

I had seen abstract relationships as something that existed in and of themselves, separate from physical reality, and from our ideas about them. So, for example, Pythagoras's theorem existed an abstract relationship from the start of our universe, and Pythagoras merely discovered it. This led to questions like "where do abstract relationships come from?" and "what makes abstract relationships true?", which I was unable to answer satisfactorily.

 

I've come to the conclusion that abstract relationships are nothing more than ideas in out minds. What has existed from time immemorial is the behaviour of physical entities from which we abstract the entities and relationships. So abstract entities and relationships are human constructs. Hence, I now say that the physical relationships expressed in Pythagoras' theorem have existed since time immemorial, but Pythagoras (or whoever) abstracted that relationship as an idea.

 

However, I still distinguish abstract entities and relationships (as ideas) from statements about them. So I maintain that statements are not the only things that can be true. Abstract relationships can be true. However, as they are human constructs, they can also be false. What makes an abstract relationship true or false? How well it encapsulates the physical behaviour it is abstracted from. Hence, empirical evidence is important in allowing us to know whether an abstract relationship is true, but it does not make it true.

 

So, my thanks again to lawcat for his stimulating input to the thread.

[HydrogenBond]

Empirical evidence does not make something entirely true. Contained with the correlation is a margin of error, where it is not true. Empirical can head toward the truth, but by definition can never reach the whole truth, until there is no margin of error. But with statistics this would imply a probability of 1.0, which is only a theoretical limit, implying the whole truth can never be reached this way.

 

Rather than alter the approach for truth, by staying at probability=1, we added a philosophy for reality that says reality, at the deepest level, is based on uncertainty. If that is true, then the margin of error for any correlation, is a truth onto its own, since it follows this other truth. So now the correlation is composed to two intertwined truths making the entire relationship true, any which way.

 

For example, swine flu, like bird flu, west nile virus, etc., have a correlation that contains truth within a margin of error. In the reality of chaos and uncertainty, the margin of error is a truth onto its own. The last two examples never really panned out. But we can attribute that to chaos, messing with the original composite truth, proving that chaos does indeed control the universe.

[lawcat]

my thanks again to lawcat for his stimulating input to the thread.

 

You're welcome. I enjoy good debate as well. It helps sharpen the argument, and see the shortcomings of own position. So, I thank you as well.

[Qfwfq]

If you like to sharpen the argument, and see the shortcomings of your positions I suggest looking up the story of Euclid's fifth. This is the story of the transition from the ancient to the modern notion of what mathematics is. It is what separated the concept of "true or false" in mathematics from that of reality, opening up oceans of possible constructs which are explored regardless of whether each one be a description of the real world. Meantime:

 

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.

Actually the matter is a bit more than just giving a meaning to the symbols. What you are saying here is just that one could e. g. use the symbol 4 to stand for what we usually call 9 and maybe also use the symbol 2 for what we usually call 5 and this obviously makes a difference, one could also change the meanings of the + and = symbols. But it is actually more than this, consider the following example:

 

Let's use the symbol = for equality and + for addition; it is commutative and associative. Let's use the usual symbol 0 for the neutral element of + so that e. g. 2 + 0 = 2 and likewise for the other symbols in lieu of 2. This of course also goes for 1 = 0 + 1 and let's use the usual symbol 1 as being the element which "follows" 0 and the usual symbol 2 for the element that follows 1 (2 = 1 + 1) and 3 for the next one (3 = 2 + 1). We also choose the usual 4 = 3 + 1 and 5 = 4 + 1. Some equalities on these symbols, so far defined, are already consequentially true:

 

2 + 2 = 4

3 + 2 = 5

 

For instance (using associativity too): 2 + 2 = 2 + (1 + 1) = (2 + 1) + 1 = 3 + 1 = 4. But which other statements are already provable? Obviously, we haven't enough axioms to say what symbols are the result of 3 + 3, or of 4 + 3 or of 5 + 5. One may easily see it as only a matter of choosing further symbols, but how about the following further axiom: 5 + 5 = 3, would this be inconsistent with the choices so far made above? Certainly it can't be proven true, but can it be proven false? Can 4 + 4 = 2 be proven false, could it also be chosen as a further axiom?

[lawcat]

Can 4 + 4 = 2 be proven false, could it also be chosen as a further axiom?

 

AS you can see, based on your previous definitions, 4+4=2 is obviously false ((2+2)+(2+2) =/= 2, because left side is more than 2).

 

You can make 4+4=2 axiomatic, but you must revise previous definitions of 4, or of 2, if you want to make it consistent with previous findings.

 

Or, you must define 4+4 to mean something totally different, and = and 2 to mean something total different. For example, you can define 4+4 to mean "two fermions", and = to mean "can not occupy," and 2 to mean "common position at same time."

 

Then, 4+4=2 would mean: two fermions can not occupy common position at same time.

 

The entity you've created, 4+4=2, would just be neither true nor false, but the statement in our head, two fermions can not occupy common position at same time, would be subject to truth or falsity based on evidence.

[Qfwfq]

Uhm, ahem! Having asked if this would be inconsistent with the choices so far made, you obviously must count out revising previous definitions of 4, or of 2, or anything I had defined.

 

AS you can see, based on your previous definitions, 4+4=2 is obviously false ((2+2)+(2+2) =/= 2, because left side is more than 2).

I had not defined a symbol such as > or as < (IOW any notion of ordering).

 

I had loosely used the word follows but implicitly defined it as "A follows B" meaning B = A + 1 and, at that point, the word becomes superfluous: 1 = 0 + 1 was consequent to what had already been stated apart from choosing the first symbol after +, = and 0. This I had done with the sole aim of avoiding a slight conceptual hitch, but it could be done away with. Summing up the definitions/axioms:

 

1. = ---> equality
   
2. + ---> (binary internal operator which is) associative and commutative
   
3. 0 ---> neutral element of + meaning that for a generic element A: 0 + A = A
   
4. 1 ---> next chosen element symbol (1 = 0 + 1 is true by above and the only other relation positable before choosing further symbols would be 1 + 1 = 0)
   
5. 2 ---> next chosen element symbol, defined by positing: 2 = 1 + 1
   
6. 3 --> as above, positing: 3 = 2 + 1
   
7. 4 --> as above, positing: 4 = 3 + 1
   
8. 5 --> as above, positing: 5 = 4 + 1
   

 

Forgive me another formal imperfection of yesterday: I was tacitly infering a couple of things but should have said them explicitly: We are defining these symbols as distinct, in the sense that the = relation holds between each of the above symbols and itself but not one of them and another (different one). Putting it briefly = is not just equivalence but actual equality. We also posit that distinct elements + 1 give distinct elements; by associativity is rests consequent also, for any C (as well as for 1), that A + C = B + C cannot hold unless A = B. Having begged this pardon, I sum up yesterday's questions a bit more clearly:

 

Can the truth or falsity of 5 + 5 = 3 be determined?

Can the truth or falsity of 4 + 4 = 2 be determined?

If neither can be determined, can they be both posited as two further axioms?

 

As a hint, I'll answer a simpler, single question independent of the above two: Can we posit 4 + 2 = 3 as a further axiom? The answer is no, we may determine it is false as follows:

 

4 + 2 = 3 + 1 + 2 = 3 + 3

4 + 2 = 3 (by the proposed new axiom)

3 + 3 = 3 (transitive property of equality)

0 + 3 = 3 (0 being defined as additive neutral)

3 + 3 = 0 + 3 (transitivity of = again)

3 = 0 (clarifications above, about distinct elements and associativity of +)

 

As you may see, without having defined any ordering relation, we can show that an axiom 4 + 2 = 3 would be inconsistent. What about the other two I proposed?

[lawcat]

I think we can. I just want to reassert that definitions are necessary.

 

The 5+5=3 becomes,

 

3+3+3+1 = 3,

 

Then, 3+3+3+1 = 0+3,

 

then, 3+3+1 = 0,

 

if 0+A = A, and if A = 3+3+1, then

 

3+3+1 = 0 + 3+3+1, or

 

0+3+3+1 = 3+3+1, but here

 

0+3+3+1 = 0, axiom is false, by definition.

___

 

Now, to go back to your rejection of ordering:

The definition 0+ A = A sets up ordering because

If 0 + A = A, then for any B such that B =/= 0 (neutral, not more nor less): A + B =/= A.

This creates ordering. Based on 0 +A = A, it is not wrong to use more or less.

It goes like this:

 

1 =/= 0

 

1 + 1 =/= 0; from axiom 1 + 1 = 0, by definition 1 + 1 = 2, the axiom becomes 2 = 0, but by definition 0 + 2 = 2 which is inconsistent with 0 + 2= 0. So axiom 1 + 1 = 0 is false, and 1 + 1 =/= 0. So on and so forth for all, for all A + B such that B =/=0:A + B =/=A. The operator + tells ust that there is B more than just A on the left side for all B =/= 0; and from definitions 1, or 2, or 3, or 4 , . . . =/= 0.

[Qfwfq]

Good effort! :P

 

Of course definitions are necessary; the issue was with empirical observation of reality. You correctly conclude:

then, 3+3+1 = 0,

 

0+3+3+1 = 3+3+1, but here

 

0+3+3+1 = 0

but you don't specify by which definition the axiom is false. Do my definitions and axioms of yesterday supply a manner of equating 3 + 3 + 1 with any of the symbols distinct from 0?

 

As for the ordering, I'm quite fine with your application of modus tollens to reject 1 + 1 = 0 and I'm fine with:

So on and so forth for all, for all A + B such that B =/=0:A + B =/=A.

but I don't conclude there being an ordering from it. You make an assumption in:

The operator + tells ust that there is B more than just A on the left side for all B =/= 0; and from definitions 1, or 2, or 3, or 4 , . . . =/= 0.

based on the usual notion of addition. Without this assumption, rejecting A + B = A doesn't distinguish between "more than" an "less than". Neither does it reject A + B = 0 unless A and B are such that the definitions can equate their "sum" with a distinct symbol.

 

Hint: Peano's axioms (which define the natural numbers or counting numbers) include the statement that there is one, and only one, element that is not equal to any element + 1 which we could term "the initial element". This unique element is 0 and of course 3 + 3 + 1 would be inconsistent with it, if we added that further axiom. Lo and behold, it also makes the natural numbers ordered by the + operation!

 

But without adding axioms to yesterday's ones, either of the two I proposed could be added... but not both. 4 + 4 = 2 would imply 4 + 2 = 0, equivalent to 3 + 3 = 0 hence inconsistent with 3 + 3 + 1 = 0 because 3 + 3 + 1 = 3 + 3 can't be. So the two proposed axioms are 5 + 1 = 0, which terminates defining new elements, and 5 + 2 = 0 which allows choosing one more symbol for 5 + 1 but none further. We could define 187 symbols, from 0 to 186, and then posit 186 + 1 = 0. With any of these choices, given any A and calling B = A + 1 there is an element C such that A = B + C; however there is no distinction between "positive and negative" elements and + does not imply an ordering.

[lawcat]

You love to drive me crazy. :P

 

Good effort! ;)

you don't specify by which definition the axiom is false. Do my definitions and axioms of yesterday supply a manner of equating 3 + 3 + 1 with any of the symbols distinct from 0?

Axiom is false by 0 + A = A. A is any element. By axiom derivation 3 + 3 + 1 = 0 is either true or false, but it equals an element A. If A = 0, then the axiom should become 0 + 0 = 0 which is true by definition 0 + A = A. if not, then the axiom 3+3+1=0 is false because 3 + 3 + 1 = A such that A =/= 0. Aditional Proof can be obtained by defining 9 as 8+1, and 8 as 7+1, and 7 as 6+1, and 6 as 5+1. But if you limit your univers to 6 elements as you have done above, then obviously you can not prove anything above 5 unles we specifically define that For any element A and element B, there is element C such that A + B = C. This you have not done, but I presumed. If my presumption is correct then 3+3+1 becomes 3+4 which is A + B, and there must be a C such that A + B = C.

 

but I don't conclude there being an ordering from it. . . . Lo and behold, it also makes the natural numbers ordered by the + operation! . . . there is no distinction between "positive and negative" elements and + does not imply an ordering.

 

I must disagree fundamentally here. The definitions tell us that 1+1=2, 2+1=3, 3+1=4, 4+1=5, ... The +1 creates ordering, so that we can say qualitatevely for each element B, such that B = A + 1, that something more, additional, extra is required to complete B from A becaue B = A is lacking by definitions, and that extra is +1 always.

[Qfwfq]

You love to drive me crazy. :cheer:

Of course I do! :cheer:

:P

Axiom is false by 0 + A = A. A is any element. By axiom derivation 3 + 3 + 1 = 0 is either true or false, but it equals an element A. If A = 0, then the axiom should become 0 + 0 = 0 which is true by definition 0 + A = A. if not, then the axiom 3+3+1=0 is false because 3 + 3 + 1 = A such that A =/= 0.

This simply means that it implies A = 0 if you call 3 + 3 + 1 by the name of A. There's no reason A must differ from 0, negating A = 0 is simply inconsistent with positing A = 3 + 3 + 1 and 3 + 3 + 1 = 0.

 

Aditional Proof can be obtained by defining 9 as 8+1, and 8 as 7+1, and 7 as 6+1, and 6 as 5+1.

This isn't additional proof, it's additional definitions! Without them, nothing is in contradiction of 3 + 3 + 1 = 0.

 

But if you limit your univers to 6 elements as you have done above, then obviously you can not prove anything above 5 unles we specifically define that For any element A and element B, there is element C such that A + B = C. This you have not done, but I presumed. If my presumption is correct then 3+3+1 becomes 3+4 which is A + B, and there must be a C such that A + B = C.

Slow down, hold it man! It ain't the universe, heck, it's just one of infinitely many possible constructs! I thought you were a lawyer, not The Judge...;)

 

More serious, this doesn't show that C can't be any of the already defined elements, though there may be limitations on which one it can and can't be.

 

I must disagree fundamentally here. The definitions tell us that 1+1=2, 2+1=3, 3+1=4, 4+1=5, ... The +1 creates ordering, so that we can say qualitatevely for each element B, such that B = A + 1, that something more, additional, extra is required to complete B from A becaue B = A is lacking by definitions, and that extra is +1 always.

How about something that completes A from B instead? Consider my previous post where I say that there could be some C such that B + C = A, given B = A + 1, if one hasn't chosen any axiom to the effect of there being the initial element.

 

In the integers there is no initial element, there is a distinction between positive and negative ones and the above C is -1. If instead we make the choice of 3 + 3 + 1 = 0 then the above C is 5 + 1.

 

B = A + 1 = (B + C) + 1 = B + (5 + 1) + 1 = B + (3 + 3) + 1 = B + (3 + 3 + 1) = B + 0 = B

 

No inconsistency with A = B + 5 + 1.

[lawcat]

This is where my knowledge stops, and your explanation begins. And if so, probably in a separate thread, becuase I am unsure how this ties into previous truth discussion.

[lawcat]

This simply means that it implies A = 0 if you call 3 + 3 + 1 by the name of A. There's no reason A must differ from 0, negating A = 0 is simply inconsistent with positing A = 3 + 3 + 1 and 3 + 3 + 1 = 0.

3+3+1 must equal some A for the proof of the axiom. The axiom predicts that 3+3+1 =0, and we must test whether A = 0, or A =/=0, for the proof. That is the end of that discussion.

 

 

 

How about something that completes A from B instead? Consider my previous post where I say that there could be some C such that B + C = A, given B = A + 1, if one hasn't chosen any axiom to the effect of there being the initial element.

 

In the integers there is no initial element, there is a distinction between positive and negative ones and the above C is -1. If instead we make the choice of 3 + 3 + 1 = 0 then the above C is 5 + 1.

 

B = A + 1 = (B + C) + 1 = B + (5 + 1) + 1 = B + (3 + 3) + 1 = B + (3 + 3 + 1) = B + 0 = B

 

No inconsistency with A = B + 5 + 1.

 

I honestly do not know whay you would use circular argument; plug the axiom into proof.

 

First, you posit an axiom 3+3+1=0. Then you define B+C=A, and B=A+1. Then you plug your axiom into defintions as if it was already proven true, to derive B: if A=0 in axiom, then B=0+1, and if 0+B=B, then B=1. IF B=1, and A=0, then 3+3+1=0 becomes 1+(5+1) = 0. So, from B+C=A, then C = 5+1.

 

Then with that C =5+1 you go on to prove that is true. Circular!

 

Axiom: 3+3+1=0

Definitions: B+C=A, and B=A+1

From previous definitions: 3+1=4

 

Then, axiom: 3+4=0

 

If B=3, and C=4, and A=0, for axiom B+C=A,

then, from B=A+1 --> axiom leads to 3=0+1 or 0+1=3

 

But, from defintion 0+A=A --> 0+1=1

 

If 0+1 = 0+1, then 1 = 3, which is false by definition:

 

1 = 1 + 1 + 1

0 + 1 = 1 + 1 + 1

0 = 1 + 1

0=2

0=0+2 or 0+2=0

but by defintion, 0 + 2 = 2

 

The axiom is false.

 

To tie this to the Truth discusion: WE ALWAYS check statements against definitions, but we don't plug unproven statements blindly into definitions. Definitions are your evidence. Statements are measured for truth and falsity on whether they are consistent with evidence, and evidence for abstract relationships are definitions.

[Qfwfq]

I am unsure how this ties into previous truth discussion.

It is the question about the meaning of truth in mathematical/logical propositions. I agree we have got a bit lost in the details but I'm showing how empirical evidence does not close the case; as well as defining symbols one needs axioms. This is the take in modern mathematics and logic.

 

Euclid wrote about lines, points &c. without defining them because he meant what folks usually called by these words (well, the Greek ones of course) and he was tacitly assuming the surface to be flat as well as indefinitely extended. He was never satisfied with having to posit what is known as his fifth axiom and thought it to be logically redundant but could never prove it being consequential to the other ones, the same discomfort afflicted mathematicians and logicians down through those long, long centuries until quite recently, when a few folks realized there are mutually exclusive alternative choices which are fully consistent with the other axioms; you can posit one or another of them being true. This is history. The change in perspective opened up entire worlds (and not only geometries).

Foundations of Mathematics -- from Wolfram MathWorld

Geometry -- from Wolfram MathWorld

 

The moral of the fable: Empirical evidence does not determine the logical truth; observing a given real thing determines which construct it is best described by. Are you counting apples? Use the applecountin' numbers, usually called "the natural numbers". If the thing is a clicking turnstyle with N positions then you just replace the third Peano axiom with N = 0 (except it isn't called N within the construct, so the axiom is just 1 + the previous number = 0) and you go ringin' 'roud the rosies with a pocket full of posies. Alternatively this construct (called a cyclic group of order N) can be defined by the classes of integers which have the same remainder for division by N.

Cyclic Group -- from Wolfram MathWorld

Number Theory -- from Wolfram MathWorld

 

3+3+1 must equal some A for the proof of the axiom.

You don't prove axioms, you choose them. So long as they aren't inconsistent with each other your construct can be fine.

 

I honestly do not know whay you would use circular argument; plug the axiom into proof.

I wasn't proving the axiom I was showing how C works. It is simpler if we define:

6 = 5 + 1

so that we can use it to write the solution for C as 6 instead of clumsy 5 + 1. My axiom was:

3+3+1=0

Then I defined B=A+1, required C be such that B+C=A and showed that C = 6 and "tested" it by adding it to B and geting A. Since the value of C is determined by the requirement you can't choose it arbitrarily:

If B=3, and C=4, and A=0, for axiom B+C=A,

then, from B=A+1 --> 3=0+1 or 0+1=3

 

But, from defintion 0+A=A --> 0+1=0

 

The axiom is false.

Of course it's the choice C = 4 that's inconsistent with what I had done IOW my requirement that adding C to any number give the previous one, under the 6 + 1 = 0 axiom. Can you see? Inconsistency occurs only when you let extraneous assumptions slip in. You're highly accustomed to apple counting, you're unaccustomed to cyclic additive groups. The proposition 5 + 5 = 3 is false in applecountin' numbers, it is true in the cyclic group of order 7. OTOH in applecountin' numbers, 5 + 5 = 10 is true and so it is in any cyclic group of order greater than 10; in smaller ones it is false.

 

How about truth in logic, more in general? Consider the liar's paradox:

 

This proposition is false.

 

Is the above proposition false? Is it true? What can be said about it? What do you think about that black beast called self-referentiality and about Tarski's theory of truth?

Tarski's Truth Definitions (Stanford Encyclopedia of Philosophy)

Tarski, A. "The Semantic Conception of Truth and the Foundations of Semantics." Philos. Phenomenol. Res. 4, 341-376, 1944.

Tarski, A. "Der Wahrheitsbegriff in den formalisierten Sprachen." Studia Philos. 1, 261-405, 1936.

 

Statements are measured for truth and falsity on whether they are consistent with evidence, and evidence for abstract relationships are definitions.

definitions and axioms. Evidence tells us how well axioms fit some real thing but, of course, that doesn't concern abstract (logical-mathematical) statements.

[lawcat]

You don't prove axioms, you choose them. So long as they aren't inconsistent with each other your construct can be fine.

 

Axiom = self evident statement that follows from definitions. (I followed your naming of 3+3+1 as axiom for communication purposes here. But your axiom 3+3+1=0 is not an axiom. It is not self evident. It is an unproven statement. it is not an axiom.)

 

I wasn't proving the axiom I was showing how C works. It is simpler if we define:

6 = 5 + 1

so that we can use it to write the solution for C as 6 instead of clumsy 5 + 1. My axiom was:

3+3+1=0

Then I defined B=A+1, required C be such that B+C=A and showed that C = 6 and "tested" it by adding it to B and geting A. Since the value of C is determined by the requirement you can't choose it arbitrarily:Of course it's the choice C = 4 that's inconsistent with what I had done IOW my requirement that adding C to any number give the previous one, under the 6 + 1 = 0 axiom. Can you see? Inconsistency occurs only when you let extraneous assumptions slip in. You're highly accustomed to apple counting, you're unaccustomed to cyclic additive groups. The proposition 5 + 5 = 3 is false in applecountin' numbers, it is true in the cyclic group of order 7. OTOH in applecountin' numbers, 5 + 5 = 10 is true and so it is in any cyclic group of order greater than 10; in smaller ones it is false.

 

This has nothing to do with apple counting. I simply followed your definitions. Your axiom is not an axiom. It needs to be proven consistent with definitions if it is to be used. But before you proved it, you merged it with definitions and ended up proving circular argument. You do not do that. You must keep the two separate and compare your statment 3+3+1=0 with definitions for consistency.

 

definitions and axioms.

 

Ok, you can include axioms too. But lets be careful that when we do, we are only including that which is self evident. Statements are measured for truth and falsity on whether they are consistent with evidence, and evidence for abstract relationships are definitions and axioms.

 

Evidence tells us how well axioms fit some real thing but, of course, that doesn't concern abstract (logical-mathematical) statements.

Yes, Axioms are self evident from existing evidence. For abstract statements, definitions are evidence, and axioms are self evident from definitions (evidence).

 

Your distinction of apples, or cyclic counting, goes to the issue of evidence (definitions). What is evidence?

Evidence is a reliable perception of relevant subject matter. Then, the question becomes whether, for your inquiry, apples are a relevant subject matter. If not, you need to confine yourself to evidence (definitions) that are relevant. "Relevant evidence" means evidence having any tendency to make the existence of any fact that is of consequence to the determination of the proof more probable or less probable than it would be without the evidence.

[Qfwfq]

Axiom = self evident statement that follows from definitions.

Seems this is the whole source of misunderstanding. B)

 

Although I know the word originally meant a proposition taken to be self evidently true, I'm in the habit of today's mathematicians, and many others, who use it synonymously with postulate and absolutely nothing past lack of inconsistency is required. I apologize for the misunderstanding ensuing from my lexical habit, I guess lawyers use the term as the philosopher of old.

 

Your axiom is not an axiom. It needs to be proven consistent with definitions if it is to be used.

This is better, but folks typically don't directly prove them consistent because the only way is to find no inconsistency, theorem after theorem; indeed the lack of these is the only definition of consisency in an axiomatic (or formal) system. How do you accomplish that before having used the friggin' t'ings?

 

But before you proved it, you merged it with definitions and ended up proving circular argument. You do not do that. You must keep the two separate and compare your statment 3+3+1=0 with definitions for consistency.

What I did was not the first thing but test the second. What you call circularity is just the matching up, things tallied, no inconsistency found and you would find none if you checked all possibilities. So 6 is the inverse of 1, 5 is the inverse of 2, + induces no ordering on these seven elements. The neutral is its own inverse, in any group.

 

Ok, you can include axioms too. But lets be careful that when we do, we are only including that which is self evident. Statements are measured for truth and falsity on whether they are consistent with evidence, and evidence for abstract relationships are definitions and axioms.

Generous, thanks. :) But, considering what I bolded and that you mean it with statements including the axioms, isn't there a whiff of circularity?

 

Yes, Axioms are self evident from existing evidence. For abstract statements, definitions are evidence, and axioms are self evident from definitions (evidence).

So, in the end, your meaning of "evidence" is not reality but the definitions? Egash, maybe not, ooooooh....

What is evidence?

Evidence is a reliable perception of relevant subject matter. Then, the question becomes whether, for your inquiry, apples are a relevant subject matter. If not, you need to confine yourself to evidence (definitions) that are relevant. "Relevant evidence" means evidence having any tendency to make the existence of any fact that is of consequence to the determination of the proof more probable or less probable than it would be without the evidence.

So it appears that your definitions are constrained to the same reality and therefore your evidence is reality. Is that what you understand for abstract statements?

 

With = being the usual notion of equality, let there be one element Mary such that, for any element A it holds that Mary § A = A and be there another element Susan.

Be it that, with A distinct from B, A § Susan is distinct from B § Susan.

Define Cynthia = Susan § Susan.

Define Nadia = Cynthia § Susan.

Define Deborah = Nadia § Susan.

 

What constraint is there, how many fine ladies may/must we define before we (might) posit Mary = the last one of them § Susan? Is the notion of a sewing circle necessary as evidence for this to be possible? Is it sufficient to know we can (somehow) find enough elements? Are the definitions really sufficient "evidence" to decide which element the ring is closed after?