x exists, means, there is some confirmable property that x has.
(E!x =df EF(Fx)).
If it is true that x thinks then x exists.
We cannot deny our existence, because the process of denying requires existence.
If there is anything that x does or says then that x must exist.
Division is a mathematical operation between numbers or functions of numbers.
Separating objects into parts is not division.
That we can simulate the operation of division applied to pies etc, has restrictive meaning.
What would (pie)/pi mean? I don't think there is a knife sharp enough...
Originally Posted by Owen Holden
How is it that there is: language, logic or mathematics, if there are no minds.
2+2=4, is only a scrible, if there cannot be an interpretation of the symbols.
There cannot be knowledge of: truth, fact, or existence, if there is nobody to understand, can...
Originally Posted by Owen Holden
[] means it is logically necessary that. <> means it is logically possible that.
These symbols have been a part of standard modal logic since 1918, C. I. Lewis, A Survey of Symbolic Logic. ...where have you been?
[] means it is logically necessary that. <> means it is logically possible that.
These symbols have been a part of standard modal logic since 1918, C. I. Lewis, A Survey of Symbolic Logic. ...where have you been?
It is empirically true that '(the number of planets)=9', not logically...
If classical logic is inconsistent then so is classical mathematics.
A. x=y -> (Fx <-> Fy). This is an axiom of first order logic and it is a theorem of second order logic. (Leibnitz's Law)
A is a theorem of Principia Mathematica, *13.15.
1. x=y -> ([](x=x) <-> [](x=y)).
therefore...
x is unique means, there is one and only one thing that x is.
'The' in the particular, in the singular, is the meaning of 'unique'.
The definite article 'the' refers to that one and only x.
The x such that Fx, is that (unique) x which satisfies Fx.
That there is only one x which...
Reality is all that is the case.
What is not the case is not part of reality.
Even dreams are partof reality, even if the 'objects' in that dream are not.
"A point has zero volume, zero area, zero length, and cardinality 1."
All geometric concepts have no physical qualities.
Points do not have physical qualities at all.
"The phrase "X doesn't have volume" means that the concept of volume is inapplicable to X."
Wrong again.
X has zero...
I don't care at all about your re-defining of words that are clear and distinct.
You sound like a theist who wants to define classical words as they please??
Why do you think I should be concerned about your special definition of object etc.
If you want to talk about point-objects, then...
"Originally Posted by Owen Holden
What is the size of a point?
Surely there are no physical objects of 'one' dimention.
Points are (mathematical) geometrical concepts.
How does a physical object have size if it has no dimention??
Dimentions are mathematical concepts not physical...
What is the size of a point?
Surely there are no physical objects of 'one' dimention.
Points are (mathematical) geometrical concepts.
How does a physical object have size if it has no dimention??
Dimentions are mathematical concepts not physical concepts.
Originally Posted by sd01g
It seems to me that it is impossible to measure something of zero volume
Your insistence that all mathematical truths have (interpretation) meaning with respect to a particular application (physics) is false.
It is clear to me that there are no physical...
You said "0/0 is undefined because the definition of division says so."
Presumably then, your defiition of 0/0 says that 0/0 is undefined??
What exactly does your definition of division say??
Quote:
If we define division between the numbers x and y as: x/y =df (the z: x=y*z),
then...
Wrong. Self-referring statements are not always a problem. See: Quine, New Foundations,
Quine Mathematical Logic.
V=df {x:x=x}.
ie. (V e V) is not a problem for Quine.
Russell's theory of types does exactly that.
That there are undecidable propositions within a logic that can deal...
It makes no sense to say 0/0 is defined as undefinable.
If we define division between the numbers x and y as: x/y =df (the z: x=y*z),
then neither 0/0 or x/0 are unique.
That is to say neither 0/0 nor 1/0 exist, even though they are defined!
There is no unique number that 0/0 is.
There...
There is no sense to the concept "the truth".
By what decision method, can you decide 'the truth'?
Truth is not unique, it is dependent on the system used.
Watch out for the dictator "Herkl", If he does not agree with your remarks he will 'lock the whole thread', Heil Herkl!
Of course...
Extending the number system from complex numbers, (a+bi), to 4-D
hypercomplex numbers, (a+bi+cj+dk), leads to a multiplication
table such as;
(A) i^2=j^2=-1, ij=ji=k, k^2=+1, ik=ki=-j, jk=kj=-i.
Sir W. Hamilton introduced 'quaternions' by presenting the
multiplication table;
(B)...
For hypercomplex numbers (a+bi+cj+dk), k^2=+1.
For quaternions (a+bi+cj+dk), k^2=-1.
Which is it? Obviously the k of hypercomplex numbers is different from the k of quaternions.
Don't you see the inconsistency?
For me k^2=+1 and kHk=-1, i.e. H is the Hamilton product...
Of course there are zero divisors! They are numbers that are specifically defined.
(i-j) is a zero divisor because there is a non-zero number, (i+j), which when multiplied by (i-j), becomes 0.
"(As an extra hint, Forbenius proved what you're trying to do is impossible in 1877, I just found...
Extensions of complex numbers are available for 2^n dimentions.
For example:
1. (a+bi+cj+dk) = ((a+bi)+(c+di)j)
where: i<>j, i^2=j^2=-1, ij=ji=k, ik=ki=-j, jk=kj=-i, k^2=+1.
Unlike quaternions, these hypercomplex numbers are:
commutative and associative wrt addition and...
A set of connectives is truth functionally complete if any formula can be expressed using only the set of connectives. (given p, q, r, etc.)
For example:
{nor} is complete.
~p =df (p nor p)
p v q =df ~(p nor q)
p -> q =df ~p v q
p & q =df ~(~p v ~q)
(p nand q) =df ~(p & q)
p <-> q...
Certainty is an illusion of decidability.
Truth is relative to the system that decides it.
Truth is that which can or has been shown to be the case.
Absolute truth, certainty, is not attainable, because..
there is no system that can determine all truths.
All I know is that I know...
1. {nor}
2. {nand}
3. {<-, True}
4. {->, False}
5. {~, v}
6. {~, &}
7. {~, ->}
8. {~, <-}
These sets of connectives are complete in the sense that all of the other propositional connectives can be defined by them.
{~, <->} and {~, xor} are not complete.
I don't agree.
Logic and mathematics are mental phenomena, i.e. there is no logic or mathematics without mind. Indeed, there are no languages either.
There cannot be any timeless things!
Concepts are dependent on mind.
Mind is dependent on brain.
Brain is dependent on physical things...
C0nfused:
Are mathematics just an invention, a creation of humans that helps them in their everyday life, or are they actually connected to nature, and are part of it that we just happened to discover?
We discover the consequences of our inventions.
That logic/mathematics can be applied...
drunkenfool: So... Are there propositions?
Yes. Propositions are declarative sentences which are either true or false.
That is, they are meaningful statements.
I don't agree.
E!x =df EF(Fx).
~E!(the x:~(x=x)), is a theorem.
All objects described by contradicory predications do not exist!
They have no properties at all.
~E!x <-> ~EF(Fx).
Yes.
That which is equal to (1/0) does not exist.
1/x =df (the y: x*y=1), includes the definition of (1/0).
If x=0 then 1/0 = (the y:0*y=1)
But, there is no y such that: y*0=1.
That is, Ay(y*0=0).
Therefore, (the y:y*0=1) does not exist.
Yes, inevitably.
We must make sense out of the expression EyEx(x/0 = y).
Is it tautologous or is it contradictory?
i.e. is (1/0) a number?, or, is (0/0) a number?, or, is (2/0) a number? etc..
It seems contradictory to me.
I have devised a truth table method of dealing with monadic predicate logic and propositional logic within the same method...
From: Owen Holden (oorionus@yahoo.com)
Subject: Truth tables for monadic predicate logic
View: Complete Thread (3 articles)
Original Format
Newsgroups...
honestrosewater:
So the true premises I have are
1. Some T(houghts and memories) are not Q(ualia). (an O statement)
2. Some T are O(bservables). (I)
3. All Q are O. (A)
"Some T are O" and "Some O are T" are equivalent (they have equivalent Venn diagrams), yes? So I also have
4. Some...
honestrosewater:
So the true premises I have are
1. Some T(houghts and memories) are not Q(ualia). (an O statement)
2. Some T are O(bservables). (I)
3. All Q are O. (A)
"Some T are O" and "Some O are T" are equivalent (they have equivalent Venn diagrams), yes? So I also have
4. Some O...
I don't agree.
The null set exists by axiom, therefore it cannot be 'nothing' in any sense.
{} e {{}}, is a theorem which proves that {} is something and that it is not nothing.
"You're making it much more difficult than it needs to be."
Then why don't you prove that (1/0 = 1/0) is true !?
No, it is not acceptable when x is a described object.
If G(x) means x=x then ~(G(x)) means ~(x=x).
That is F(x) <-> ~(G(x)), is not a substitution instance of G(x),
in G(x)...
Quote:
Originally Posted by Owen Holden
No it does not prove ~~(1/0 =1/0), because it does not hold.
Yes, 8. ~(G(1/0)), of your post 2 is correct.
Incorrect.
Your proof (~P) is faulty.
D1. 1/z =df (the x: 1=x*z & ~(z=0))
D2. G(the x: Fx) =df Ey(Ax(x=y <-> Fx) & Gy)
You...
No it does not prove ~~(1/0 =1/0), because it does not hold.
D1. 1/z =df (the x: x*z=1 & ~(z=0))
1/0 =df (the x: x*0=1 & ~(0=0)).
(1/0) is defined by the description (the x: x*0=1 & ~(0=0)).
Both (x*0=1) and ~(0=0) are contradictory, because Ax(x*0=0) and (0=0) and ~(0=1) are...
No you cannot. If you replace G(x) with ~(G(x))
"D2. G(the x: Hx) =df Ey(Ax(x=y <-> Hx) & Gy)"
~(G(the x:Hx)) <-> ~Ey(Ax(x=y <-> Hx) & Gy).
Which is true and not false. i.e. we do not get ~(~(G(1/0)) as a theorem.
(1/0=1/0) is false!
Both you and AKG are wrong.
Extending the number system from complex numbers, (a+bi), to 4-D
hypercomplex numbers, (a+bi+cj+dk), leads to a multiplication
table such as:
(A) i^2=j^2=-1, ij=ji=k, k^2=+1, ik=ki=-j, jk=kj=-i.
Note that these hypercomplex numbers are commutative and have elementary functions.
We...