Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and X in
L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between premises
P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and X in
L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between premises
P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and X in
L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between
premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
I am not sure what aspect of logic would be changed by this system that
is why I opened up this discussion.
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and X in
L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between
premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
I am not sure what aspect of logic would be changed by this system that
is why I opened up this discussion.
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and X
in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between
premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese
L can be any natural or formal language as long as it has the above interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
I am not sure what aspect of logic would be changed by this system that
is why I opened up this discussion.
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and X
in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between
premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese
L can be any natural or formal language as long as it has the above interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
I am not sure what aspect of logic would be changed by this system that
is why I opened up this discussion.
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property >>>> of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and X
in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between
premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese
L can be any natural or formal language as long as it has the above
interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
I am not sure what aspect of logic would be changed by this system that
is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to every
element of the entire body of analytical truth, whether it be facts
about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property >>>>> of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and X
in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between
premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>
L can be any natural or formal language as long as it has the above
interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
I am not sure what aspect of logic would be changed by this system that >>>> is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to every
element of the entire body of analytical truth, whether it be facts
about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property >>>>> of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and X
in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between
premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>
L can be any natural or formal language as long as it has the above
interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
I am not sure what aspect of logic would be changed by this system that >>>> is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to every
element of the entire body of analytical truth, whether it be facts
about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
On 3/19/2023 10:52 AM, olcott wrote:
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the
property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and
X in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between
premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>>
L can be any natural or formal language as long as it has the above
interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges from >>>>> correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
I am not sure what aspect of logic would be changed by this system
that
is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to every
element of the entire body of analytical truth, whether it be facts
about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
The predicate True(L,X) is provided to explicitly contradict Tarski's conclusion that no such predicate can possibly exist.
True(L,X) means that there is a semantic connection from expressions of language L that are stipulated to be true to X.
On 3/19/2023 10:52 AM, olcott wrote:
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the
property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and
X in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between
premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>>
L can be any natural or formal language as long as it has the above
interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges from >>>>> correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
I am not sure what aspect of logic would be changed by this system
that
is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to every
element of the entire body of analytical truth, whether it be facts
about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
The predicate True(L,X) is provided to explicitly contradict Tarski's conclusion that no such predicate can possibly exist.
True(L,X) means that there is a semantic connection from expressions of language L that are stipulated to be true to X.
On 3/19/2023 12:30 PM, olcott wrote:
On 3/19/2023 10:52 AM, olcott wrote:
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the
property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and >>>>>>> X in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between
premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>>>
L can be any natural or formal language as long as it has the above
interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges from >>>>>> correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
I am not sure what aspect of logic would be changed by this system >>>>>> that
is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to every
element of the entire body of analytical truth, whether it be facts
about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
The predicate True(L,X) is provided to explicitly contradict Tarski's
conclusion that no such predicate can possibly exist.
True(L,X) means that there is a semantic connection from expressions
of language L that are stipulated to be true to X.
expressions of language L that are stipulated to be true correspond to Haskell Curry elementary theorems of T.
Let T be such a theory. Then the elementary statements which belong to T
we shall call the elementary theorems of T; we also say that these
elementary statements are true for T. Thus, given T, an elementary
theorem is an elementary statement which is true. https://www.liarparadox.org/Haskell_Curry_45.pdf
expressions of language L that are stipulated to be true also correspond
to the basic facts of natural language such as cats are animals thus
cats are not ten story office buildings.
That it is true in F that G is unprovable in F requires a semantic
connection from elementary theorems of F to G in F, otherwise G is
untrue in F.
On 3/19/2023 12:30 PM, olcott wrote:
On 3/19/2023 10:52 AM, olcott wrote:
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the
property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
True(L,X) means that a semantic connection exists between (a) and >>>>>>> X in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between
premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>>>
L can be any natural or formal language as long as it has the above
interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges from >>>>>> correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
I am not sure what aspect of logic would be changed by this system >>>>>> that
is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to every
element of the entire body of analytical truth, whether it be facts
about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
The predicate True(L,X) is provided to explicitly contradict Tarski's
conclusion that no such predicate can possibly exist.
True(L,X) means that there is a semantic connection from expressions
of language L that are stipulated to be true to X.
expressions of language L that are stipulated to be true correspond to Haskell Curry elementary theorems of T.
Let T be such a theory. Then the elementary statements which belong to T
we shall call the elementary theorems of T; we also say that these
elementary statements are true for T. Thus, given T, an elementary
theorem is an elementary statement which is true. https://www.liarparadox.org/Haskell_Curry_45.pdf
expressions of language L that are stipulated to be true also correspond
to the basic facts of natural language such as cats are animals thus
cats are not ten story office buildings.
That it is true in F that G is unprovable in F requires a semantic
connection from elementary theorems of F to G in F, otherwise G is
untrue in F.
On 3/19/2023 12:55 PM, olcott wrote:
On 3/19/2023 12:30 PM, olcott wrote:
On 3/19/2023 10:52 AM, olcott wrote:
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary >>>>>>>> consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the
property
of Boolean true.
(b) Some expressions of language L are a semantically necessary >>>>>>>> consequence of others.
True(L,X) means that a semantic connection exists between (a)
and X in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between >>>>>>>> premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>>>>
L can be any natural or formal language as long as it has the
above interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its
operations on the basis of this system or such a system diverges >>>>>>> from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this
foundation. https://en.wikipedia.org/wiki/Principle_of_explosion >>>>>>>
I am not sure what aspect of logic would be changed by this
system that
is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to every >>>>> element of the entire body of analytical truth, whether it be facts
about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>>
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
The predicate True(L,X) is provided to explicitly contradict Tarski's
conclusion that no such predicate can possibly exist.
True(L,X) means that there is a semantic connection from expressions
of language L that are stipulated to be true to X.
expressions of language L that are stipulated to be true correspond to
Haskell Curry elementary theorems of T.
Let T be such a theory. Then the elementary statements which belong to T
we shall call the elementary theorems of T; we also say that these
elementary statements are true for T. Thus, given T, an elementary
theorem is an elementary statement which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
expressions of language L that are stipulated to be true also correspond
to the basic facts of natural language such as cats are animals thus
cats are not ten story office buildings.
That it is true in F that G is unprovable in F requires a semantic
connection from elementary theorems of F to G in F, otherwise G is
untrue in F.
If a semantic connection exists then this semantic connection can be specified syntactically.
If G states that there is no syntactic connection from the elementary theorems of F to G in F, then G is stating that G is untrue in F.
On 3/19/2023 1:22 PM, olcott wrote:
On 3/19/2023 12:55 PM, olcott wrote:
On 3/19/2023 12:30 PM, olcott wrote:
On 3/19/2023 10:52 AM, olcott wrote:
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary >>>>>>>>> consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the >>>>>>>>> property
of Boolean true.
(b) Some expressions of language L are a semantically necessary >>>>>>>>> consequence of others.
True(L,X) means that a semantic connection exists between (a) >>>>>>>>> and X in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between >>>>>>>>> premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese
L can be any natural or formal language as long as it has the
above interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its >>>>>>>> operations on the basis of this system or such a system diverges >>>>>>>> from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this >>>>>>>> foundation. https://en.wikipedia.org/wiki/Principle_of_explosion >>>>>>>>
I am not sure what aspect of logic would be changed by this
system that
is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to every >>>>>> element of the entire body of analytical truth, whether it be facts >>>>>> about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>>>
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
The predicate True(L,X) is provided to explicitly contradict
Tarski's conclusion that no such predicate can possibly exist.
True(L,X) means that there is a semantic connection from expressions
of language L that are stipulated to be true to X.
expressions of language L that are stipulated to be true correspond to
Haskell Curry elementary theorems of T.
Let T be such a theory. Then the elementary statements which belong to T >>> we shall call the elementary theorems of T; we also say that these
elementary statements are true for T. Thus, given T, an elementary
theorem is an elementary statement which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
expressions of language L that are stipulated to be true also correspond >>> to the basic facts of natural language such as cats are animals thus
cats are not ten story office buildings.
That it is true in F that G is unprovable in F requires a semantic
connection from elementary theorems of F to G in F, otherwise G is
untrue in F.
If a semantic connection exists then this semantic connection can be
specified syntactically.
If G states that there is no syntactic connection from the elementary
theorems of F to G in F, then G is stating that G is untrue in F.
The conventional proof that G is true is not a proof that G is true in
F, it is a proof that G is true in meta-F.
LP = "This sentence is not true"
is not true because LP is not a truth bearer
This sentence is not true: "This sentence is not true"
is true because LP is not a truth bearer.
G = "this sentence cannot be proven in F"
cannot be proven in F because G has a vacuous truth object.
?- G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Proves that G has a vacuous truth object.
This sentence cannot be proven: "this sentence cannot be proven in F"
is true because G has a vacuous truth object.
TT = "This sentenced is true"
is not true because TT has a vacuous truth object.
This sentence is true.
What is it true about?
It is true about being true.
What is it true about being true about?
It is true about being true about being true...
On 3/19/2023 1:22 PM, olcott wrote:
On 3/19/2023 12:55 PM, olcott wrote:
On 3/19/2023 12:30 PM, olcott wrote:
On 3/19/2023 10:52 AM, olcott wrote:
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary >>>>>>>>> consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the >>>>>>>>> property
of Boolean true.
(b) Some expressions of language L are a semantically necessary >>>>>>>>> consequence of others.
True(L,X) means that a semantic connection exists between (a) >>>>>>>>> and X in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between >>>>>>>>> premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese
L can be any natural or formal language as long as it has the
above interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its >>>>>>>> operations on the basis of this system or such a system diverges >>>>>>>> from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this >>>>>>>> foundation. https://en.wikipedia.org/wiki/Principle_of_explosion >>>>>>>>
I am not sure what aspect of logic would be changed by this
system that
is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to every >>>>>> element of the entire body of analytical truth, whether it be facts >>>>>> about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>>>
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
The predicate True(L,X) is provided to explicitly contradict
Tarski's conclusion that no such predicate can possibly exist.
True(L,X) means that there is a semantic connection from expressions
of language L that are stipulated to be true to X.
expressions of language L that are stipulated to be true correspond to
Haskell Curry elementary theorems of T.
Let T be such a theory. Then the elementary statements which belong to T >>> we shall call the elementary theorems of T; we also say that these
elementary statements are true for T. Thus, given T, an elementary
theorem is an elementary statement which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
expressions of language L that are stipulated to be true also correspond >>> to the basic facts of natural language such as cats are animals thus
cats are not ten story office buildings.
That it is true in F that G is unprovable in F requires a semantic
connection from elementary theorems of F to G in F, otherwise G is
untrue in F.
If a semantic connection exists then this semantic connection can be
specified syntactically.
If G states that there is no syntactic connection from the elementary
theorems of F to G in F, then G is stating that G is untrue in F.
The conventional proof that G is true is not a proof that G is true in
F, it is a proof that G is true in meta-F.
LP = "This sentence is not true"
is not true because LP is not a truth bearer
This sentence is not true: "This sentence is not true"
is true because LP is not a truth bearer.
G = "this sentence cannot be proven in F"
cannot be proven in F because G has a vacuous truth object.
?- G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Proves that G has a vacuous truth object.
This sentence cannot be proven: "this sentence cannot be proven in F"
is true because G has a vacuous truth object.
TT = "This sentenced is true"
is not true because TT has a vacuous truth object.
This sentence is true.
What is it true about?
It is true about being true.
What is it true about being true about?
It is true about being true about being true...
On 3/19/2023 2:21 PM, olcott wrote:
On 3/19/2023 1:22 PM, olcott wrote:
On 3/19/2023 12:55 PM, olcott wrote:
On 3/19/2023 12:30 PM, olcott wrote:
On 3/19/2023 10:52 AM, olcott wrote:
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary >>>>>>>>>> consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the >>>>>>>>>> property
of Boolean true.
(b) Some expressions of language L are a semantically necessary >>>>>>>>>> consequence of others.
True(L,X) means that a semantic connection exists between (a) >>>>>>>>>> and X in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists
between premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from >>>>>>>>>> cheese
L can be any natural or formal language as long as it has the
above interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its >>>>>>>>> operations on the basis of this system or such a system
diverges from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this >>>>>>>>> foundation. https://en.wikipedia.org/wiki/Principle_of_explosion >>>>>>>>>
I am not sure what aspect of logic would be changed by this
system that
is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to
every
element of the entire body of analytical truth, whether it be facts >>>>>>> about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>>>>
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
The predicate True(L,X) is provided to explicitly contradict
Tarski's conclusion that no such predicate can possibly exist.
True(L,X) means that there is a semantic connection from
expressions of language L that are stipulated to be true to X.
expressions of language L that are stipulated to be true correspond to >>>> Haskell Curry elementary theorems of T.
Let T be such a theory. Then the elementary statements which belong
to T
we shall call the elementary theorems of T; we also say that these
elementary statements are true for T. Thus, given T, an elementary
theorem is an elementary statement which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
expressions of language L that are stipulated to be true also
correspond
to the basic facts of natural language such as cats are animals thus
cats are not ten story office buildings.
That it is true in F that G is unprovable in F requires a semantic
connection from elementary theorems of F to G in F, otherwise G is
untrue in F.
If a semantic connection exists then this semantic connection can be
specified syntactically.
If G states that there is no syntactic connection from the elementary
theorems of F to G in F, then G is stating that G is untrue in F.
The conventional proof that G is true is not a proof that G is true in
F, it is a proof that G is true in meta-F.
LP = "This sentence is not true"
is not true because LP is not a truth bearer
This sentence is not true: "This sentence is not true"
is true because LP is not a truth bearer.
G = "this sentence cannot be proven in F"
cannot be proven in F because G has a vacuous truth object.
?- G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Proves that G has a vacuous truth object.
This sentence cannot be proven: "this sentence cannot be proven in F"
is true because G has a vacuous truth object.
G has a vacuous truth object
G has a vacuous truth object
G has a vacuous truth object
G has a vacuous truth object
TT = "This sentenced is true"
is not true because TT has a vacuous truth object.
This sentence is true.
What is it true about?
It is true about being true.
What is it true about being true about?
It is true about being true about being true...
On 3/19/2023 2:21 PM, olcott wrote:
On 3/19/2023 1:22 PM, olcott wrote:
On 3/19/2023 12:55 PM, olcott wrote:
On 3/19/2023 12:30 PM, olcott wrote:
On 3/19/2023 10:52 AM, olcott wrote:
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary >>>>>>>>>> consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the >>>>>>>>>> property
of Boolean true.
(b) Some expressions of language L are a semantically necessary >>>>>>>>>> consequence of others.
True(L,X) means that a semantic connection exists between (a) >>>>>>>>>> and X in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists
between premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from >>>>>>>>>> cheese
L can be any natural or formal language as long as it has the
above interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its >>>>>>>>> operations on the basis of this system or such a system
diverges from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this >>>>>>>>> foundation. https://en.wikipedia.org/wiki/Principle_of_explosion >>>>>>>>>
I am not sure what aspect of logic would be changed by this
system that
is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to
every
element of the entire body of analytical truth, whether it be facts >>>>>>> about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>>>>
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
The predicate True(L,X) is provided to explicitly contradict
Tarski's conclusion that no such predicate can possibly exist.
True(L,X) means that there is a semantic connection from
expressions of language L that are stipulated to be true to X.
expressions of language L that are stipulated to be true correspond to >>>> Haskell Curry elementary theorems of T.
Let T be such a theory. Then the elementary statements which belong
to T
we shall call the elementary theorems of T; we also say that these
elementary statements are true for T. Thus, given T, an elementary
theorem is an elementary statement which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
expressions of language L that are stipulated to be true also
correspond
to the basic facts of natural language such as cats are animals thus
cats are not ten story office buildings.
That it is true in F that G is unprovable in F requires a semantic
connection from elementary theorems of F to G in F, otherwise G is
untrue in F.
If a semantic connection exists then this semantic connection can be
specified syntactically.
If G states that there is no syntactic connection from the elementary
theorems of F to G in F, then G is stating that G is untrue in F.
The conventional proof that G is true is not a proof that G is true in
F, it is a proof that G is true in meta-F.
LP = "This sentence is not true"
is not true because LP is not a truth bearer
This sentence is not true: "This sentence is not true"
is true because LP is not a truth bearer.
G = "this sentence cannot be proven in F"
cannot be proven in F because G has a vacuous truth object.
?- G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Proves that G has a vacuous truth object.
We are therefore confronted with a proposition which asserts its own unprovability. (Gödel 1931:39-41)
Thus Gödel's G is simplified to this:
G = ¬(F ⊢ G)
Translated into Prolog like this:
?- G = not(provable(F, G)).
Found to be incorrect by this:
?- unify_with_occurs_check(G, not(provable(F, G))).
false
Because the Prolog G has an “uninstantiated subterm of itself” we can know that unification will fail because it specifies “some kind of
infinite structure.”The quotes come from: (Clocksin and Mellish 2003:255)
So G is unprovable in F because G is incorrect, thus not because F is incomplete.
This sentence cannot be proven: "this sentence cannot be proven in F"
is true because G has a vacuous truth object.
TT = "This sentenced is true"
is not true because TT has a vacuous truth object.
This sentence is true.
What is it true about?
It is true about being true.
What is it true about being true about?
It is true about being true about being true...
On 3/19/2023 1:22 PM, olcott wrote:
On 3/19/2023 12:55 PM, olcott wrote:
On 3/19/2023 12:30 PM, olcott wrote:
On 3/19/2023 10:52 AM, olcott wrote:
On 3/19/2023 10:45 AM, olcott wrote:
On 3/19/2023 1:00 AM, olcott wrote:
On 3/18/2023 7:32 PM, olcott wrote:
On 3/18/2023 6:17 PM, olcott wrote:
Just like with syllogisms conclusions a semantically necessary >>>>>>>>> consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the >>>>>>>>> property
of Boolean true.
(b) Some expressions of language L are a semantically necessary >>>>>>>>> consequence of others.
True(L,X) means that a semantic connection exists between (a) >>>>>>>>> and X in L. *Axiom(P) ⊨□ X*
Provable(L,P,X) means that a semantic connection exists between >>>>>>>>> premises P and X in L. *P ⊨□ X*
The Moon is made from green cheese ⊨□ The Moon is made from cheese
L can be any natural or formal language as long as it has the
above interfaces.
When this is called the foundation of correct reasoning
*and indeed is the actual foundation of correct reasoning*
that means that every system of logic either derives all of its >>>>>>>> operations on the basis of this system or such a system diverges >>>>>>>> from
correct reasoning into incorrect thus erroneous reasoning.
The kludge of the principle of explosion is eradicated by this >>>>>>>> foundation. https://en.wikipedia.org/wiki/Principle_of_explosion >>>>>>>>
I am not sure what aspect of logic would be changed by this
system that
is why I opened up this discussion.
The foundation of correct reasoning is just that and applies to every >>>>>> element of the entire body of analytical truth, whether it be facts >>>>>> about the world or mathematical relationships.
(A & ~A) ⊨□ FALSE
FALSE ⊨□ FALSE
TRUE ⊨□ TRUE
A & B ⊨□ A
A & B ⊨□ B
The Moon is made from green cheese ⊨□ The Moon is made from cheese >>>>>>
False(X) ⊨□ True(~X)
X = "this sentence is not true"
(~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
The predicate True(L,X) is provided to explicitly contradict
Tarski's conclusion that no such predicate can possibly exist.
True(L,X) means that there is a semantic connection from expressions
of language L that are stipulated to be true to X.
expressions of language L that are stipulated to be true correspond to
Haskell Curry elementary theorems of T.
Let T be such a theory. Then the elementary statements which belong to T >>> we shall call the elementary theorems of T; we also say that these
elementary statements are true for T. Thus, given T, an elementary
theorem is an elementary statement which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
expressions of language L that are stipulated to be true also correspond >>> to the basic facts of natural language such as cats are animals thus
cats are not ten story office buildings.
That it is true in F that G is unprovable in F requires a semantic
connection from elementary theorems of F to G in F, otherwise G is
untrue in F.
If a semantic connection exists then this semantic connection can be
specified syntactically.
If G states that there is no syntactic connection from the elementary
theorems of F to G in F, then G is stating that G is untrue in F.
The conventional proof that G is true is not a proof that G is true in
F, it is a proof that G is true in meta-F.
LP = "This sentence is not true"
is not true because LP is not a truth bearer
This sentence is not true: "This sentence is not true"
is true because LP is not a truth bearer.
G = "this sentence cannot be proven in F"
cannot be proven in F because G has a vacuous truth object.
?- G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
Proves that G has a vacuous truth object.
This sentence cannot be proven: "this sentence cannot be proven in F"
is true because G has a vacuous truth object.
TT = "This sentenced is true"
is not true because TT has a vacuous truth object.
This sentence is true.
What is it true about?
It is true about being true.
What is it true about being true about?
It is true about being true about being true...
The following foundation of correct reasoning simultaneously gets rid of Gödel Incompleteness Tarski Undefinability the principle of explosion
and every other divergence from correct reasoning that is allowed by
logic systems since the syllogism.
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
P is a subset of expressions of language L
T is a subset of (a)
Provable(P,X) means P ⊨□ X
True(T,X) means X ∈ (a) or T ⊨□ X
False(T,X) means T ⊨□ ~X
The above system only applies to the analytic side of the analytic
synthetic distinction which includes all of math and logic yet excludes expressions of language that can only be verified as true with input
from the sense organs.
The following foundation of correct reasoning simultaneously gets rid of Gödel Incompleteness Tarski Undefinability the principle of explosion
and every other divergence from correct reasoning that is allowed by
logic systems since the syllogism.
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
P is a subset of expressions of language L
T is a subset of (a)
Provable(P,X) means P ⊨□ X
True(T,X) means X ∈ (a) or T ⊨□ X
False(T,X) means T ⊨□ ~X
The above system only applies to the analytic side of the analytic
synthetic distinction which includes all of math and logic yet excludes expressions of language that can only be verified as true with input
from the sense organs.
On 4/1/2023 4:14 PM, olcott wrote:
The following foundation of correct reasoning simultaneously gets rid of
Gödel Incompleteness Tarski Undefinability the principle of explosion
and every other divergence from correct reasoning that is allowed by
logic systems since the syllogism.
Just like with syllogisms conclusions a semantically necessary
consequence of their premises
Semantic Necessity operator: ⊨□
(a) Some expressions of language L are stipulated to have the property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
P is a subset of expressions of language L
T is a subset of (a)
Provable(P,X) means P ⊨□ X
True(T,X) means X ∈ (a) or T ⊨□ X
False(T,X) means T ⊨□ ~X
The above system only applies to the analytic side of the analytic
synthetic distinction which includes all of math and logic yet excludes
expressions of language that can only be verified as true with input
from the sense organs.
When this is called the foundation of correct reasoning it is called
that because this is the actual way that analytic truth really works.
When any existing formal system violates the way that analytic truth
really works then it is incorrect in the absolute sense because all
formal systems and mathematics are instances of analytic truth.
When other formal systems are requited to inherit True(x), Provable(x)
and False(x) from this foundation then only their divergence is
abolished.
Semantic relations that are specified syntactically are still allowed
and long as they are semantically coherent.
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