Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/ https://en.wikipedia.org/wiki/CycL https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the following definition of the "theory of simple types" in a footnote:
By the theory of simple types I mean the doctrine which says that the objects of thought (or, in another interpretation, the symbolic
expressions) are divided into types, namely: individuals, properties of individuals, relations between individuals, properties of such
relations, etc. (with a similar hierarchy for extensions), and that sentences of the form: " a has the property φ ", " b bears the relation
R to c ", etc. are meaningless, if a, b, c, R, φ are not of types
fitting together.
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the following
definition of the "theory of simple types" in a footnote:
By the theory of simple types I mean the doctrine which says that the
objects of thought (or, in another interpretation, the symbolic
expressions) are divided into types, namely: individuals, properties
of individuals, relations between individuals, properties of such
relations, etc. (with a similar hierarchy for extensions), and that
sentences of the form: " a has the property φ ", " b bears the
relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of
types fitting together.
That is a constraint on the language. Note that individuals of all sorts
are considered to be of the same type. For properies and relation the alternative would be that a predicate is false if any of the arguments
are of wrong type. For functions it is harder to find a reasonable value
if an argument is of wrong type.
This is of course irrelevant to the point that the resulting theory is
not formal unless both the definition of semantics and the definition of semantic logical entailment are fully formal.
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the following
definition of the "theory of simple types" in a footnote:
By the theory of simple types I mean the doctrine which says that the
objects of thought (or, in another interpretation, the symbolic
expressions) are divided into types, namely: individuals, properties
of individuals, relations between individuals, properties of such
relations, etc. (with a similar hierarchy for extensions), and that
sentences of the form: " a has the property φ ", " b bears the
relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of
types fitting together.
That is a constraint on the language. Note that individuals of all sorts
are considered to be of the same type.
For properies and relation the
alternative would be that a predicate is false if any of the arguments
are of wrong type. For functions it is harder to find a reasonable value
if an argument is of wrong type.
This is of course irrelevant to the point that the resulting theory is
not formal unless both the definition of semantics and the definition of semantic logical entailment are fully formal.
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote:
By the theory of simple types I mean the doctrine which says that the
objects of thought (or, in another interpretation, the symbolic
expressions) are divided into types, namely: individuals, properties
of individuals, relations between individuals, properties of such
relations, etc. (with a similar hierarchy for extensions), and that
sentences of the form: " a has the property φ ", " b bears the
relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of
types fitting together.
That is a constraint on the language. Note that individuals of all sorts
are considered to be of the same type. For properies and relation the
alternative would be that a predicate is false if any of the arguments
are of wrong type. For functions it is harder to find a reasonable value
if an argument is of wrong type.
This is of course irrelevant to the point that the resulting theory is
not formal unless both the definition of semantics and the definition of
semantic logical entailment are fully formal.
The body of knowledge is defined in terms of Rudolf Carnap Meaning Postulates and stored in a knowledge ontology inheritance hierarchy.
The predicate Bachelor(x) is stipulated to mean ~Married(x) where the predicate Married(x) is defined in terms of billions of other things
such as all of the details of Human(x).
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote:
By the theory of simple types I mean the doctrine which says that the
objects of thought (or, in another interpretation, the symbolic
expressions) are divided into types, namely: individuals, properties
of individuals, relations between individuals, properties of such
relations, etc. (with a similar hierarchy for extensions), and that
sentences of the form: " a has the property φ ", " b bears the
relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of
types fitting together.
That is a constraint on the language. Note that individuals of all sorts
are considered to be of the same type.
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
For properies and relation the
alternative would be that a predicate is false if any of the arguments
are of wrong type. For functions it is harder to find a reasonable value
if an argument is of wrong type.
(General_Knowledge ⊨ x) means True(x)
(General_Knowledge ⊨ ~x) means False(x)
~True(x) & ~False(x) means x is not an element of General_Knowledge
This is of course irrelevant to the point that the resulting theory is
not formal unless both the definition of semantics and the definition of
semantic logical entailment are fully formal.
(General_Knowledge ⊨ x) means True(x)
(General_Knowledge ⊨ ~x) means False(x)
~True(x) & ~False(x) means x is not an element of General_Knowledge
On 26/11/2025 15:54, olcott wrote:
(General_Knowledge ⊨ x) means True(x)
(General_Knowledge ⊨ ~x) means False(x)
~True(x) & ~False(x) means x is not an element of General_Knowledge
Eh? You made it sound like General_Knowledge was the system, rather than
a model, but there you have it as a model.
"animals" <are> "living things" is stipulated.
On 2025-11-26, olcott <polcott333@gmail.com> wrote:
"animals" <are> "living things" is stipulated.
So a dead rabbit isn't an animal?
On 2025-11-26, olcott <polcott333@gmail.com> wrote:
"animals" <are> "living things" is stipulated.
So a dead rabbit isn't an animal?
Pure genius!
On 2025-11-26, Kaz Kylheku <643-408-1753@kylheku.com> wrote:
On 2025-11-26, olcott <polcott333@gmail.com> wrote:
"animals" <are> "living things" is stipulated.
So a dead rabbit isn't an animal?
How about Mickey Mouse? Living thing or not? Animal or not?
On 11/26/2025 3:49 PM, Kaz Kylheku wrote:
On 2025-11-26, Kaz Kylheku <643-408-1753@kylheku.com> wrote:
On 2025-11-26, olcott <polcott333@gmail.com> wrote:
"animals" <are> "living things" is stipulated.
So a dead rabbit isn't an animal?
How about Mickey Mouse? Living thing or not? Animal or not?
{Fictional character} is mutually exclusive with
{animal} in the type hierarchy.
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote:
By the theory of simple types I mean the doctrine which says that the
objects of thought (or, in another interpretation, the symbolic
expressions) are divided into types, namely: individuals, properties
of individuals, relations between individuals, properties of such
relations, etc. (with a similar hierarchy for extensions), and that
sentences of the form: " a has the property φ ", " b bears the
relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of
types fitting together.
That is a constraint on the language. Note that individuals of all sorts
are considered to be of the same type. For properies and relation the
alternative would be that a predicate is false if any of the arguments
are of wrong type. For functions it is harder to find a reasonable value
if an argument is of wrong type.
This is of course irrelevant to the point that the resulting theory is
not formal unless both the definition of semantics and the definition of
semantic logical entailment are fully formal.
The body of knowledge is defined in terms of Rudolf Carnap Meaning Postulates and stored in a knowledge ontology inheritance hierarchy.
The predicate Bachelor(x) is stipulated to mean ~Married(x) where the predicate Married(x) is defined in terms of billions of other things
such as all of the details of Human(x).
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote:
By the theory of simple types I mean the doctrine which says that the
objects of thought (or, in another interpretation, the symbolic
expressions) are divided into types, namely: individuals, properties
of individuals, relations between individuals, properties of such
relations, etc. (with a similar hierarchy for extensions), and that
sentences of the form: " a has the property φ ", " b bears the
relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of
types fitting together.
That is a constraint on the language. Note that individuals of all sorts
are considered to be of the same type.
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
olcott kirjoitti 26.11.2025 klo 17.39:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote:
By the theory of simple types I mean the doctrine which says that
the objects of thought (or, in another interpretation, the symbolic
expressions) are divided into types, namely: individuals, properties
of individuals, relations between individuals, properties of such
relations, etc. (with a similar hierarchy for extensions), and that
sentences of the form: " a has the property φ ", " b bears the
relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of >>>> types fitting together.
That is a constraint on the language. Note that individuals of all sorts >>> are considered to be of the same type. For properies and relation the
alternative would be that a predicate is false if any of the arguments
are of wrong type. For functions it is harder to find a reasonable value >>> if an argument is of wrong type.
This is of course irrelevant to the point that the resulting theory is
not formal unless both the definition of semantics and the definition of >>> semantic logical entailment are fully formal.
The body of knowledge is defined in terms of Rudolf Carnap Meaning
Postulates and stored in a knowledge ontology inheritance hierarchy.
The predicate Bachelor(x) is stipulated to mean ~Married(x) where the
predicate Married(x) is defined in terms of billions of other things
such as all of the details of Human(x).
That, too, is irrelevant to the point that the resulting theory is not
formal unless both the definition of semantics and the definition of
semantic logical entailment are fully formal.
olcott kirjoitti 26.11.2025 klo 17.54:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote:
By the theory of simple types I mean the doctrine which says that
the objects of thought (or, in another interpretation, the symbolic
expressions) are divided into types, namely: individuals, properties
of individuals, relations between individuals, properties of such
relations, etc. (with a similar hierarchy for extensions), and that
sentences of the form: " a has the property φ ", " b bears the
relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of >>>> types fitting together.
That is a constraint on the language. Note that individuals of all sorts >>> are considered to be of the same type.
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
In the type system Gödel called minimal all of those would be
individuals and therefore of the same type.
On 11/27/2025 1:56 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.39:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote:
By the theory of simple types I mean the doctrine which says that
the objects of thought (or, in another interpretation, the symbolic >>>>> expressions) are divided into types, namely: individuals,
properties of individuals, relations between individuals,
properties of such relations, etc. (with a similar hierarchy for
extensions), and that sentences of the form: " a has the property φ >>>>> ", " b bears the relation R to c ", etc. are meaningless, if a, b,
c, R, φ are not of types fitting together.
That is a constraint on the language. Note that individuals of all
sorts
are considered to be of the same type. For properies and relation the
alternative would be that a predicate is false if any of the arguments >>>> are of wrong type. For functions it is harder to find a reasonable
value
if an argument is of wrong type.
This is of course irrelevant to the point that the resulting theory is >>>> not formal unless both the definition of semantics and the
definition of
semantic logical entailment are fully formal.
The body of knowledge is defined in terms of Rudolf Carnap Meaning
Postulates and stored in a knowledge ontology inheritance hierarchy.
The predicate Bachelor(x) is stipulated to mean ~Married(x) where the
predicate Married(x) is defined in terms of billions of other things
such as all of the details of Human(x).
That, too, is irrelevant to the point that the resulting theory is not
formal unless both the definition of semantics and the definition of
semantic logical entailment are fully formal.
In Olcott's Minimal Type Theory Rudolf Carnap Meaning
Postulates directly encode semantic meaning in the syntax.
The meaningless finite string "Bachelor" is defined as
a semantic predicate through other already defined terms
∀x (Bachelor(x) := (Male(x) ∧ Human(x) ∧ ~Married(x)))
Adapted by Olcott from Rudolf Carnap Meaning postulates.
And encoded in the syntax of Olcott's Minimal Type Theory https://philarchive.org/archive/PETMTT-4v2
On 11/27/2025 2:00 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.54:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of
semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote:
By the theory of simple types I mean the doctrine which says that
the objects of thought (or, in another interpretation, the symbolic >>>>> expressions) are divided into types, namely: individuals,
properties of individuals, relations between individuals,
properties of such relations, etc. (with a similar hierarchy for
extensions), and that sentences of the form: " a has the property φ >>>>> ", " b bears the relation R to c ", etc. are meaningless, if a, b,
c, R, φ are not of types fitting together.
That is a constraint on the language. Note that individuals of all
sorts
are considered to be of the same type.
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
In the type system Gödel called minimal all of those would be
individuals and therefore of the same type.
Then Gödel would be wrong.
olcott kirjoitti 27.11.2025 klo 17.31:
On 11/27/2025 1:56 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.39:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of >>>>>>> semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote: >>>>>>
By the theory of simple types I mean the doctrine which says that >>>>>> the objects of thought (or, in another interpretation, the
symbolic expressions) are divided into types, namely: individuals, >>>>>> properties of individuals, relations between individuals,
properties of such relations, etc. (with a similar hierarchy for
extensions), and that sentences of the form: " a has the property >>>>>> φ ", " b bears the relation R to c ", etc. are meaningless, if a, >>>>>> b, c, R, φ are not of types fitting together.
That is a constraint on the language. Note that individuals of all
sorts
are considered to be of the same type. For properies and relation the >>>>> alternative would be that a predicate is false if any of the arguments >>>>> are of wrong type. For functions it is harder to find a reasonable
value
if an argument is of wrong type.
This is of course irrelevant to the point that the resulting theory is >>>>> not formal unless both the definition of semantics and the
definition of
semantic logical entailment are fully formal.
The body of knowledge is defined in terms of Rudolf Carnap Meaning
Postulates and stored in a knowledge ontology inheritance hierarchy.
The predicate Bachelor(x) is stipulated to mean ~Married(x) where
the predicate Married(x) is defined in terms of billions of other
things such as all of the details of Human(x).
That, too, is irrelevant to the point that the resulting theory is not
formal unless both the definition of semantics and the definition of
semantic logical entailment are fully formal.
In Olcott's Minimal Type Theory Rudolf Carnap Meaning
Postulates directly encode semantic meaning in the syntax.
if the encoding is not fully formally specified the theory is not
formal.
The meaningless finite string "Bachelor" is defined as
a semantic predicate through other already defined terms
∀x (Bachelor(x) := (Male(x) ∧ Human(x) ∧ ~Married(x)))
Adapted by Olcott from Rudolf Carnap Meaning postulates.
And encoded in the syntax of Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2
That page only tells how to define a sentence in terms of other
sentences. As it does not permit any arguments on the left side of :=
the expression ∀x (Bachelor(x) := (Male(x) ∧ Human(x) ∧ ~Married(x))) is syntactically invalid.
olcott kirjoitti 27.11.2025 klo 17.43:
On 11/27/2025 2:00 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.54:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of >>>>>>> semantics and the definition of semantic logical entailment are
fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote: >>>>>>
By the theory of simple types I mean the doctrine which says that >>>>>> the objects of thought (or, in another interpretation, the
symbolic expressions) are divided into types, namely: individuals, >>>>>> properties of individuals, relations between individuals,
properties of such relations, etc. (with a similar hierarchy for
extensions), and that sentences of the form: " a has the property >>>>>> φ ", " b bears the relation R to c ", etc. are meaningless, if a, >>>>>> b, c, R, φ are not of types fitting together.
That is a constraint on the language. Note that individuals of all
sorts
are considered to be of the same type.
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
In the type system Gödel called minimal all of those would be
individuals and therefore of the same type.
Then Gödel would be wrong.
No, what he said was perfectly true about what the words meant
at the time. Your preferences may differ but there is no right
or wrong in matters of taste.
On 11/28/2025 2:58 AM, Mikko wrote:
olcott kirjoitti 27.11.2025 klo 17.31:
On 11/27/2025 1:56 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.39:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:That is a constraint on the language. Note that individuals of all >>>>>> sorts
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating >>>>>>>>> semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of >>>>>>>> semantics and the definition of semantic logical entailment are >>>>>>>> fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote: >>>>>>>
By the theory of simple types I mean the doctrine which says that >>>>>>> the objects of thought (or, in another interpretation, the
symbolic expressions) are divided into types, namely:
individuals, properties of individuals, relations between
individuals, properties of such relations, etc. (with a similar >>>>>>> hierarchy for extensions), and that sentences of the form: " a
has the property φ ", " b bears the relation R to c ", etc. are >>>>>>> meaningless, if a, b, c, R, φ are not of types fitting together. >>>>>>
are considered to be of the same type. For properies and relation the >>>>>> alternative would be that a predicate is false if any of the
arguments
are of wrong type. For functions it is harder to find a reasonable >>>>>> value
if an argument is of wrong type.
This is of course irrelevant to the point that the resulting
theory is
not formal unless both the definition of semantics and the
definition of
semantic logical entailment are fully formal.
The body of knowledge is defined in terms of Rudolf Carnap Meaning
Postulates and stored in a knowledge ontology inheritance hierarchy. >>>>>
The predicate Bachelor(x) is stipulated to mean ~Married(x) where
the predicate Married(x) is defined in terms of billions of other
things such as all of the details of Human(x).
That, too, is irrelevant to the point that the resulting theory is not >>>> formal unless both the definition of semantics and the definition of
semantic logical entailment are fully formal.
In Olcott's Minimal Type Theory Rudolf Carnap Meaning
Postulates directly encode semantic meaning in the syntax.
if the encoding is not fully formally specified the theory is not
formal.
The meaningless finite string "Bachelor" is defined as
a semantic predicate through other already defined terms
∀x (Bachelor(x) := (Male(x) ∧ Human(x) ∧ ~Married(x)))
Adapted by Olcott from Rudolf Carnap Meaning postulates.
And encoded in the syntax of Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2
That page only tells how to define a sentence in terms of other
sentences. As it does not permit any arguments on the left side of :=
the expression ∀x (Bachelor(x) := (Male(x) ∧ Human(x) ∧ ~Married(x))) >> is syntactically invalid.
∀x ∈ Human (Bachelor(x) ↔ (Male(x) ∧ Adult(x) ∧ ~Married(x)))
<sentence_7 token="FOR_ALL">
<sentence_7 token="ELEMENT_OF">
<sentence_7 token="IDENTIFIER" value="x"/>
<sentence_7 token="IDENTIFIER" value="Human"/>
</sentence_7>
<sentence_11 token="IFF">
<atomic_sentence_1 token="IDENTIFIER" value="Bachelor">
<term_2 token="IDENTIFIER" value="x"/>
</atomic_sentence_1>
<sentence_12 token="AND">
<sentence_12 token="AND">
<atomic_sentence_1 token="IDENTIFIER" value="Male">
<term_2 token="IDENTIFIER" value="x"/>
</atomic_sentence_1>
<atomic_sentence_1 token="IDENTIFIER" value="Adult">
<term_2 token="IDENTIFIER" value="x"/>
</atomic_sentence_1>
</sentence_12>
<sentence_2 token="NOT">
<atomic_sentence_1 token="IDENTIFIER" value="Married">
<term_2 token="IDENTIFIER" value="x"/>
</atomic_sentence_1>
</sentence_2>
</sentence_12>
</sentence_11>
</sentence_7>
individual means one.
a group of individuals is not one individual
----
Copyright 2025 Olcott
My 28 year goal has been to make
"true on the basis of meaning" computable.
This required establishing a new foundation
for correct reasoning.
olcott <polcott333@gmail.com> wrote:
On 11/28/2025 11:32 AM, Alan Mackenzie wrote:
[ Followup-To: set ]
In comp.theory olcott <polcott333@gmail.com> wrote:
[ .... ]
individual means one.
a group of individuals is not one individual
A group of sheep is a flock.
A group of cells is a plant or animal.
A group of stars is a galaxy.
A group of musicians is an orchestra.
Yet none of these things are individuals they are all sets.
Are you saying that an animal, say a cat, is not an individual? If so,
you are surely mistaken.
The same applies to a flock, a galaxy, or an orchestra. They all have emergent properties that the individual constituents lack.
Further examples: a newsgroup consists of posters, but its properties can
not be deduced from those of the individual posters. A motor car is a
group of components, similarly.
If you make a survey of important things in your life, most of them will
be groupings of component things. So the naive assumption that there are individuals and groups, and the two things are "of different type"
doesn't seem to be true or have relevance in normal life.
--
Copyright 2025 Olcott
My 28 year goal has been to make
"true on the basis of meaning" computable.
This required establishing a new foundation
for correct reasoning.
On 11/28/2025 12:24 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/28/2025 11:32 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
[ .... ]
individual means one.
a group of individuals is not one individual
A group of sheep is a flock.
A group of cells is a plant or animal.
A group of stars is a galaxy.
A group of musicians is an orchestra.
Yet none of these things are individuals they are all sets.
Are you saying that an animal, say a cat, is not an individual? If so,
you are surely mistaken.
Do you pay any attention at all before
you artificially contrive a baseless rebuttal ???
A group of things is equivalent to a set of things
and is never the same thing as one element of this set.
The same applies to a flock, a galaxy, or an orchestra. They all have
emergent properties that the individual constituents lack.
Further examples: a newsgroup consists of posters, but its properties can
not be deduced from those of the individual posters. A motor car is a
group of components, similarly.
If you make a survey of important things in your life, most of them will
be groupings of component things. So the naive assumption that there are
individuals and groups, and the two things are "of different type"
doesn't seem to be true or have relevance in normal life.
An element of a set is never this same set.
Have you ever heard of ZFC ???
----
Copyright 2025 Olcott
My 28 year goal has been to make
"true on the basis of meaning" computable.
This required establishing a new foundation
for correct reasoning.
[ Followup-To: set ]
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/28/2025 12:24 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/28/2025 11:32 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
[ .... ]
individual means one.
a group of individuals is not one individual
A group of sheep is a flock.
A group of cells is a plant or animal.
A group of stars is a galaxy.
A group of musicians is an orchestra.
Yet none of these things are individuals they are all sets.
Are you saying that an animal, say a cat, is not an individual? If so,
you are surely mistaken.
Do you pay any attention at all before
you artificially contrive a baseless rebuttal ???
No. Just as I haven't stopped beating my wife.
A group of things is equivalent to a set of things
and is never the same thing as one element of this set.
That may be true for some meanings of those words. But it isn't what you initially asserted, which is still in the quotes above. This was "a
group of individuals is not one individual". I think you should now
admit you were mistaken about this.
The same applies to a flock, a galaxy, or an orchestra. They all have
emergent properties that the individual constituents lack.
Further examples: a newsgroup consists of posters, but its properties can >>> not be deduced from those of the individual posters. A motor car is a
group of components, similarly.
If you make a survey of important things in your life, most of them will >>> be groupings of component things. So the naive assumption that there are >>> individuals and groups, and the two things are "of different type"
doesn't seem to be true or have relevance in normal life.
An element of a set is never this same set.
What's that got to do with anything?
Have you ever heard of ZFC ???
Of course. There's no need to be snarky.
On 11/26/2025 9:54 AM, olcott wrote:>>
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
In the type system Gödel called minimal all of those would be
individuals and therefore of the same type.
On 11/28/2025 12:24 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/28/2025 11:32 AM, Alan Mackenzie wrote:
[ Followup-To: set ]
In comp.theory olcott <polcott333@gmail.com> wrote:
[ .... ]
individual means one.
a group of individuals is not one individual
A group of sheep is a flock.
A group of cells is a plant or animal.
A group of stars is a galaxy.
A group of musicians is an orchestra.
Yet none of these things are individuals they are all sets.
Are you saying that an animal, say a cat, is not an individual? If so,
you are surely mistaken.
Do you pay any attention at all before
you artificially contrive a baseless rebuttal ???
A group of things is equivalent to a set of things
and is never the same thing as one element of this set.
The same applies to a flock, a galaxy, or an orchestra. They all have
emergent properties that the individual constituents lack.
Further examples: a newsgroup consists of posters, but its properties can
not be deduced from those of the individual posters. A motor car is a
group of components, similarly.
If you make a survey of important things in your life, most of them will
be groupings of component things. So the naive assumption that there are >> individuals and groups, and the two things are "of different type"
doesn't seem to be true or have relevance in normal life.
An element of a set is never this same set.
Have you ever heard of ZFC ???
--
Copyright 2025 Olcott
My 28 year goal has been to make
"true on the basis of meaning" computable.
This required establishing a new foundation
for correct reasoning.
On 11/28/2025 2:58 AM, Mikko wrote:
olcott kirjoitti 27.11.2025 klo 17.31:
On 11/27/2025 1:56 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.39:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:That is a constraint on the language. Note that individuals of all >>>>>> sorts
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating >>>>>>>>> semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of >>>>>>>> semantics and the definition of semantic logical entailment are >>>>>>>> fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote: >>>>>>>
By the theory of simple types I mean the doctrine which says that >>>>>>> the objects of thought (or, in another interpretation, the
symbolic expressions) are divided into types, namely:
individuals, properties of individuals, relations between
individuals, properties of such relations, etc. (with a similar >>>>>>> hierarchy for extensions), and that sentences of the form: " a
has the property φ ", " b bears the relation R to c ", etc. are >>>>>>> meaningless, if a, b, c, R, φ are not of types fitting together. >>>>>>
are considered to be of the same type. For properies and relation the >>>>>> alternative would be that a predicate is false if any of the
arguments
are of wrong type. For functions it is harder to find a reasonable >>>>>> value
if an argument is of wrong type.
This is of course irrelevant to the point that the resulting
theory is
not formal unless both the definition of semantics and the
definition of
semantic logical entailment are fully formal.
The body of knowledge is defined in terms of Rudolf Carnap Meaning
Postulates and stored in a knowledge ontology inheritance hierarchy. >>>>>
The predicate Bachelor(x) is stipulated to mean ~Married(x) where
the predicate Married(x) is defined in terms of billions of other
things such as all of the details of Human(x).
That, too, is irrelevant to the point that the resulting theory is not >>>> formal unless both the definition of semantics and the definition of
semantic logical entailment are fully formal.
In Olcott's Minimal Type Theory Rudolf Carnap Meaning
Postulates directly encode semantic meaning in the syntax.
if the encoding is not fully formally specified the theory is not
formal.
The meaningless finite string "Bachelor" is defined as
a semantic predicate through other already defined terms
∀x (Bachelor(x) := (Male(x) ∧ Human(x) ∧ ~Married(x)))
Adapted by Olcott from Rudolf Carnap Meaning postulates.
And encoded in the syntax of Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2
That page only tells how to define a sentence in terms of other
sentences. As it does not permit any arguments on the left side of :=
the expression ∀x (Bachelor(x) := (Male(x) ∧ Human(x) ∧ ~Married(x))) >> is syntactically invalid.
∀x ∈ Human (Bachelor(x) ↔ (Male(x) ∧ Adult(x) ∧ ~Married(x)))
On 11/28/2025 3:01 AM, Mikko wrote:
olcott kirjoitti 27.11.2025 klo 17.43:
On 11/27/2025 2:00 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.54:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:That is a constraint on the language. Note that individuals of all >>>>>> sorts
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating >>>>>>>>> semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of >>>>>>>> semantics and the definition of semantic logical entailment are >>>>>>>> fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the
following definition of the "theory of simple types" in a footnote: >>>>>>>
By the theory of simple types I mean the doctrine which says that >>>>>>> the objects of thought (or, in another interpretation, the
symbolic expressions) are divided into types, namely:
individuals, properties of individuals, relations between
individuals, properties of such relations, etc. (with a similar >>>>>>> hierarchy for extensions), and that sentences of the form: " a
has the property φ ", " b bears the relation R to c ", etc. are >>>>>>> meaningless, if a, b, c, R, φ are not of types fitting together. >>>>>>
are considered to be of the same type.
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
In the type system Gödel called minimal all of those would be
individuals and therefore of the same type.
Then Gödel would be wrong.
No, what he said was perfectly true about what the words meant
at the time. Your preferences may differ but there is no right
or wrong in matters of taste.
There is a correct mapping of finite strings
to the semantic meaning that they specify.
olcott <polcott333@gmail.com> wrote:
On 11/28/2025 5:08 PM, Alan Mackenzie wrote:
[ .... ]
When you broke into the middle of this context
I was still responding to the original context.
It makes me quite furious when people form
rebuttals of my work on the basis of ignoring
what I said.
You will note that I didn't do this; I took issue with your exact words
"A group of individuals is not one individual."
Disagreeing with this is fucking nuts.
"A group of individuals is not one individual."
It is, or can be, but often is. I gave at least three examples.
In other words you are trying to get away with
saying that an element of a set is exactly
one-and-the-same-thing as this same set itself.
That is fucking nuts.
No, that is a strawman of the crudest and most vulgar kind. You appear
to be unable to remember what you were replying to just a very few hours previously.
--
Copyright 2025 Olcott
My 28 year goal has been to make
"true on the basis of meaning" computable.
This required establishing a new foundation
for correct reasoning.
olcott kirjoitti 28.11.2025 klo 17.51:
On 11/28/2025 2:58 AM, Mikko wrote:
olcott kirjoitti 27.11.2025 klo 17.31:
On 11/27/2025 1:56 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.39:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:That is a constraint on the language. Note that individuals of
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness >>>>>>>>>> merely requires discarding model theory and fully integrating >>>>>>>>>> semantics directly into the syntax of the formal language. >>>>>>>>>>
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of >>>>>>>>> semantics and the definition of semantic logical entailment are >>>>>>>>> fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the >>>>>>>> following definition of the "theory of simple types" in a footnote: >>>>>>>>
By the theory of simple types I mean the doctrine which says
that the objects of thought (or, in another interpretation, the >>>>>>>> symbolic expressions) are divided into types, namely:
individuals, properties of individuals, relations between
individuals, properties of such relations, etc. (with a similar >>>>>>>> hierarchy for extensions), and that sentences of the form: " a >>>>>>>> has the property φ ", " b bears the relation R to c ", etc. are >>>>>>>> meaningless, if a, b, c, R, φ are not of types fitting together. >>>>>>>
all sorts
are considered to be of the same type. For properies and relation >>>>>>> the
alternative would be that a predicate is false if any of the
arguments
are of wrong type. For functions it is harder to find a
reasonable value
if an argument is of wrong type.
This is of course irrelevant to the point that the resulting
theory is
not formal unless both the definition of semantics and the
definition of
semantic logical entailment are fully formal.
The body of knowledge is defined in terms of Rudolf Carnap Meaning >>>>>> Postulates and stored in a knowledge ontology inheritance hierarchy. >>>>>>
The predicate Bachelor(x) is stipulated to mean ~Married(x) where >>>>>> the predicate Married(x) is defined in terms of billions of other >>>>>> things such as all of the details of Human(x).
That, too, is irrelevant to the point that the resulting theory is not >>>>> formal unless both the definition of semantics and the definition of >>>>> semantic logical entailment are fully formal.
In Olcott's Minimal Type Theory Rudolf Carnap Meaning
Postulates directly encode semantic meaning in the syntax.
if the encoding is not fully formally specified the theory is not
formal.
The meaningless finite string "Bachelor" is defined as
a semantic predicate through other already defined terms
∀x (Bachelor(x) := (Male(x) ∧ Human(x) ∧ ~Married(x)))
Adapted by Olcott from Rudolf Carnap Meaning postulates.
And encoded in the syntax of Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2
That page only tells how to define a sentence in terms of other
sentences. As it does not permit any arguments on the left side of :=
the expression ∀x (Bachelor(x) := (Male(x) ∧ Human(x) ∧ ~Married(x))) >>> is syntactically invalid.
∀x ∈ Human (Bachelor(x) ↔ (Male(x) ∧ Adult(x) ∧ ~Married(x)))
That is a different sentence. The syntax rules of
https://philarchive.org/archive/PETMTT-4v2
are different for := and =.
olcott kirjoitti 28.11.2025 klo 17.54:
On 11/28/2025 3:01 AM, Mikko wrote:
olcott kirjoitti 27.11.2025 klo 17.43:
On 11/27/2025 2:00 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.54:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:That is a constraint on the language. Note that individuals of
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness >>>>>>>>>> merely requires discarding model theory and fully integrating >>>>>>>>>> semantics directly into the syntax of the formal language. >>>>>>>>>>
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of >>>>>>>>> semantics and the definition of semantic logical entailment are >>>>>>>>> fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science)
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the >>>>>>>> following definition of the "theory of simple types" in a footnote: >>>>>>>>
By the theory of simple types I mean the doctrine which says
that the objects of thought (or, in another interpretation, the >>>>>>>> symbolic expressions) are divided into types, namely:
individuals, properties of individuals, relations between
individuals, properties of such relations, etc. (with a similar >>>>>>>> hierarchy for extensions), and that sentences of the form: " a >>>>>>>> has the property φ ", " b bears the relation R to c ", etc. are >>>>>>>> meaningless, if a, b, c, R, φ are not of types fitting together. >>>>>>>
all sorts
are considered to be of the same type.
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
In the type system Gödel called minimal all of those would be
individuals and therefore of the same type.
Then Gödel would be wrong.
No, what he said was perfectly true about what the words meant
at the time. Your preferences may differ but there is no right
or wrong in matters of taste.
There is a correct mapping of finite strings
to the semantic meaning that they specify.
Yes, amd accoprding to that mapping what Gödel said is true.
On 11/29/2025 4:20 AM, Mikko wrote:
olcott kirjoitti 28.11.2025 klo 17.54:
On 11/28/2025 3:01 AM, Mikko wrote:
olcott kirjoitti 27.11.2025 klo 17.43:
On 11/27/2025 2:00 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.54:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:That is a constraint on the language. Note that individuals of >>>>>>>> all sorts
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness >>>>>>>>>>> merely requires discarding model theory and fully integrating >>>>>>>>>>> semantics directly into the syntax of the formal language. >>>>>>>>>>>
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal >>>>>>>>>>> language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of >>>>>>>>>> semantics and the definition of semantic logical entailment are >>>>>>>>>> fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science) >>>>>>>>>
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the >>>>>>>>> following definition of the "theory of simple types" in a
footnote:
By the theory of simple types I mean the doctrine which says >>>>>>>>> that the objects of thought (or, in another interpretation, the >>>>>>>>> symbolic expressions) are divided into types, namely:
individuals, properties of individuals, relations between
individuals, properties of such relations, etc. (with a similar >>>>>>>>> hierarchy for extensions), and that sentences of the form: " a >>>>>>>>> has the property φ ", " b bears the relation R to c ", etc. are >>>>>>>>> meaningless, if a, b, c, R, φ are not of types fitting together. >>>>>>>>
are considered to be of the same type.
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
In the type system Gödel called minimal all of those would be
individuals and therefore of the same type.
Then Gödel would be wrong.
No, what he said was perfectly true about what the words meant
at the time. Your preferences may differ but there is no right
or wrong in matters of taste.
There is a correct mapping of finite strings
to the semantic meaning that they specify.
Yes, amd accoprding to that mapping what Gödel said is true.
% This sentence cannot be proven in F
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
...We are therefore confronted with a proposition which asserts its own unprovability. 15 … (Gödel 1931:40-41)
That thereby makes itself semantically unsound.
On 11/29/2025 09:57 AM, olcott wrote:
On 11/29/2025 4:20 AM, Mikko wrote:
olcott kirjoitti 28.11.2025 klo 17.54:
On 11/28/2025 3:01 AM, Mikko wrote:
olcott kirjoitti 27.11.2025 klo 17.43:
On 11/27/2025 2:00 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.54:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:That is a constraint on the language. Note that individuals of >>>>>>>>> all sorts
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness >>>>>>>>>>>> merely requires discarding model theory and fully integrating >>>>>>>>>>>> semantics directly into the syntax of the formal language. >>>>>>>>>>>>
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal >>>>>>>>>>>> language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of >>>>>>>>>>> semantics and the definition of semantic logical entailment are >>>>>>>>>>> fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science) >>>>>>>>>>
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the >>>>>>>>>> following definition of the "theory of simple types" in a
footnote:
By the theory of simple types I mean the doctrine which says >>>>>>>>>> that the objects of thought (or, in another interpretation, the >>>>>>>>>> symbolic expressions) are divided into types, namely:
individuals, properties of individuals, relations between
individuals, properties of such relations, etc. (with a similar >>>>>>>>>> hierarchy for extensions), and that sentences of the form: " a >>>>>>>>>> has the property φ ", " b bears the relation R to c ", etc. are >>>>>>>>>> meaningless, if a, b, c, R, φ are not of types fitting together. >>>>>>>>>
are considered to be of the same type.
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
In the type system Gödel called minimal all of those would be
individuals and therefore of the same type.
Then Gödel would be wrong.
No, what he said was perfectly true about what the words meant
at the time. Your preferences may differ but there is no right
or wrong in matters of taste.
There is a correct mapping of finite strings
to the semantic meaning that they specify.
Yes, amd accoprding to that mapping what Gödel said is true.
% This sentence cannot be proven in F
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 … (Gödel 1931:40-41)
That thereby makes itself semantically unsound.
OR, it proves there is the extra-ordinary, since it's only a
reading of Goedelian incompleteness.
Then, since somebody like Skolem makes for model-relativism
with regards to ordinality and cardinality, then there are
matters of book-keeping about that there's an abitrarily larger
bound than any given bound, in the un-bounded, as to why that
of course each finite language on finite inputs is decide-able,
just like any rational approximation is construct-ible (if incomplete).
The point about Montague semantics instead of Herbrand semantics
is a bad idea, since it essentially ties itself to a bounded
interpretation of interpretability itself. Similarly the
"monotonicity" and "entailment" are not well-defined in
system of "quasi-modal" logic, instead only "modal, temporal,
relevance" logic, for example about Chrysippus and Ross Anderson.
On 11/29/2025 4:17 AM, Mikko wrote:
olcott kirjoitti 28.11.2025 klo 17.51:
On 11/28/2025 2:58 AM, Mikko wrote:
olcott kirjoitti 27.11.2025 klo 17.31:
On 11/27/2025 1:56 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.39:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:That is a constraint on the language. Note that individuals of >>>>>>>> all sorts
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness >>>>>>>>>>> merely requires discarding model theory and fully integrating >>>>>>>>>>> semantics directly into the syntax of the formal language. >>>>>>>>>>>
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal >>>>>>>>>>> language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of >>>>>>>>>> semantics and the definition of semantic logical entailment are >>>>>>>>>> fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science) >>>>>>>>>
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the >>>>>>>>> following definition of the "theory of simple types" in a
footnote:
By the theory of simple types I mean the doctrine which says >>>>>>>>> that the objects of thought (or, in another interpretation, the >>>>>>>>> symbolic expressions) are divided into types, namely:
individuals, properties of individuals, relations between
individuals, properties of such relations, etc. (with a similar >>>>>>>>> hierarchy for extensions), and that sentences of the form: " a >>>>>>>>> has the property φ ", " b bears the relation R to c ", etc. are >>>>>>>>> meaningless, if a, b, c, R, φ are not of types fitting together. >>>>>>>>
are considered to be of the same type. For properies and
relation the
alternative would be that a predicate is false if any of the
arguments
are of wrong type. For functions it is harder to find a
reasonable value
if an argument is of wrong type.
This is of course irrelevant to the point that the resulting
theory is
not formal unless both the definition of semantics and the
definition of
semantic logical entailment are fully formal.
The body of knowledge is defined in terms of Rudolf Carnap
Meaning Postulates and stored in a knowledge ontology inheritance >>>>>>> hierarchy.
The predicate Bachelor(x) is stipulated to mean ~Married(x) where >>>>>>> the predicate Married(x) is defined in terms of billions of other >>>>>>> things such as all of the details of Human(x).
That, too, is irrelevant to the point that the resulting theory is >>>>>> not
formal unless both the definition of semantics and the definition of >>>>>> semantic logical entailment are fully formal.
In Olcott's Minimal Type Theory Rudolf Carnap Meaning
Postulates directly encode semantic meaning in the syntax.
if the encoding is not fully formally specified the theory is not
formal.
The meaningless finite string "Bachelor" is defined as
a semantic predicate through other already defined terms
∀x (Bachelor(x) := (Male(x) ∧ Human(x) ∧ ~Married(x)))
Adapted by Olcott from Rudolf Carnap Meaning postulates.
And encoded in the syntax of Olcott's Minimal Type Theory
https://philarchive.org/archive/PETMTT-4v2
That page only tells how to define a sentence in terms of other
sentences. As it does not permit any arguments on the left side of :=
the expression ∀x (Bachelor(x) := (Male(x) ∧ Human(x) ∧ ~Married(x)))
is syntactically invalid.
∀x ∈ Human (Bachelor(x) ↔ (Male(x) ∧ Adult(x) ∧ ~Married(x)))
That is a different sentence. The syntax rules of
https://philarchive.org/archive/PETMTT-4v2
are different for := and =.
It is equivalent. The term Bachelor(x) is still defined by
Male(x) ∧ Adult(x) ∧ ~Married(x) ∧ Human(x) thus never
circular at all as Willard Van Orman Quine insisted.
On 11/29/2025 4:20 AM, Mikko wrote:
olcott kirjoitti 28.11.2025 klo 17.54:
On 11/28/2025 3:01 AM, Mikko wrote:
olcott kirjoitti 27.11.2025 klo 17.43:
On 11/27/2025 2:00 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.54:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:That is a constraint on the language. Note that individuals of >>>>>>>> all sorts
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness >>>>>>>>>>> merely requires discarding model theory and fully integrating >>>>>>>>>>> semantics directly into the syntax of the formal language. >>>>>>>>>>>
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal >>>>>>>>>>> language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
The resulting theory is not formal unless both the definition of >>>>>>>>>> semantics and the definition of semantic logical entailment are >>>>>>>>>> fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science) >>>>>>>>>
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the >>>>>>>>> following definition of the "theory of simple types" in a
footnote:
By the theory of simple types I mean the doctrine which says >>>>>>>>> that the objects of thought (or, in another interpretation, the >>>>>>>>> symbolic expressions) are divided into types, namely:
individuals, properties of individuals, relations between
individuals, properties of such relations, etc. (with a similar >>>>>>>>> hierarchy for extensions), and that sentences of the form: " a >>>>>>>>> has the property φ ", " b bears the relation R to c ", etc. are >>>>>>>>> meaningless, if a, b, c, R, φ are not of types fitting together. >>>>>>>>
are considered to be of the same type.
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
In the type system Gödel called minimal all of those would be
individuals and therefore of the same type.
Then Gödel would be wrong.
No, what he said was perfectly true about what the words meant
at the time. Your preferences may differ but there is no right
or wrong in matters of taste.
There is a correct mapping of finite strings
to the semantic meaning that they specify.
Yes, amd accoprding to that mapping what Gödel said is true.
% This sentence cannot be proven in F
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
...We are therefore confronted with a proposition which asserts its own unprovability. 15 … (Gödel 1931:40-41)
That thereby makes itself semantically unsound.
On 11/29/2025 1:27 PM, Ross Finlayson wrote:
On 11/29/2025 09:57 AM, olcott wrote:
On 11/29/2025 4:20 AM, Mikko wrote:
olcott kirjoitti 28.11.2025 klo 17.54:
On 11/28/2025 3:01 AM, Mikko wrote:
olcott kirjoitti 27.11.2025 klo 17.43:
On 11/27/2025 2:00 AM, Mikko wrote:
olcott kirjoitti 26.11.2025 klo 17.54:
On 11/26/2025 5:37 AM, Mikko wrote:
olcott kirjoitti 25.11.2025 klo 16.21:
On 11/25/2025 3:40 AM, Mikko wrote:That is a constraint on the language. Note that individuals of >>>>>>>>>> all sorts
olcott kirjoitti 25.11.2025 klo 2.53:
Eliminating undecidability and mathematical incompleteness >>>>>>>>>>>>> merely requires discarding model theory and fully integrating >>>>>>>>>>>>> semantics directly into the syntax of the formal language. >>>>>>>>>>>>>
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal >>>>>>>>>>>>> language such as Montague Grammar or CycL of the Cyc >>>>>>>>>>>>> project can encode the semantics of anything that can >>>>>>>>>>>>> be expressed in language.
The resulting theory is not formal unless both the
definition of
semantics and the definition of semantic logical entailment are >>>>>>>>>>>> fully formal.
https://plato.stanford.edu/entries/montague-semantics/
https://en.wikipedia.org/wiki/CycL
https://en.wikipedia.org/wiki/Ontology_(information_science) >>>>>>>>>>>
*This was my original inspiration*
Kurt Gödel in his 1944 Russell's mathematical logic gave the >>>>>>>>>>> following definition of the "theory of simple types" in a >>>>>>>>>>> footnote:
By the theory of simple types I mean the doctrine which says >>>>>>>>>>> that the objects of thought (or, in another interpretation, the >>>>>>>>>>> symbolic expressions) are divided into types, namely:
individuals, properties of individuals, relations between >>>>>>>>>>> individuals, properties of such relations, etc. (with a similar >>>>>>>>>>> hierarchy for extensions), and that sentences of the form: " a >>>>>>>>>>> has the property φ ", " b bears the relation R to c ", etc. are >>>>>>>>>>> meaningless, if a, b, c, R, φ are not of types fitting together. >>>>>>>>>>
are considered to be of the same type.
An individual house, person, orange, piece of pie,
is not a group of houses, people, oranges, pieces of pie.
In the type system Gödel called minimal all of those would be >>>>>>>> individuals and therefore of the same type.
Then Gödel would be wrong.
No, what he said was perfectly true about what the words meant
at the time. Your preferences may differ but there is no right
or wrong in matters of taste.
There is a correct mapping of finite strings
to the semantic meaning that they specify.
Yes, amd accoprding to that mapping what Gödel said is true.
% This sentence cannot be proven in F
?- G = not(provable(F, G)).
G = not(provable(F, G)).
?- unify_with_occurs_check(G, not(provable(F, G))).
false.
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 … (Gödel 1931:40-41)
That thereby makes itself semantically unsound.
OR, it proves there is the extra-ordinary, since it's only a
reading of Goedelian incompleteness.
Then, since somebody like Skolem makes for model-relativism
with regards to ordinality and cardinality, then there are
matters of book-keeping about that there's an abitrarily larger
bound than any given bound, in the un-bounded, as to why that
of course each finite language on finite inputs is decide-able,
just like any rational approximation is construct-ible (if incomplete).
The point about Montague semantics instead of Herbrand semantics
is a bad idea, since it essentially ties itself to a bounded
interpretation of interpretability itself. Similarly the
"monotonicity" and "entailment" are not well-defined in
system of "quasi-modal" logic, instead only "modal, temporal,
relevance" logic, for example about Chrysippus and Ross Anderson.
The simpler easier to understand notion of Montague
Semantics is Rudolf Carnap meaning postulates that
when anchored in what is essentially a type hierarchy
can mathematically formalize any expression of
language that can ever be said and can completely
eliminate the ambiguity of the meaning of words
by using GUIDs for the placeholder of each unique
sense meaning. This seems to may "interpretation"
no longer needed.
On 11/26/2025 1:43 PM, Tristan Wibberley wrote:
On 26/11/2025 15:54, olcott wrote:
(General_Knowledge ⊨ x) means True(x)
(General_Knowledge ⊨ ~x) means False(x)
~True(x) & ~False(x) means x is not an element of General_Knowledge
Eh? You made it sound like General_Knowledge was the system, rather than
a model, but there you have it as a model.
There is no model.
It is all Rudolf Carnap Meaning Postulates
that have every single nuance of 100% of their
semantic meaning directly encoding in this formal
language arranged in a knowledge ontology
inheritance hierarchy.
On 26/11/2025 20:04, olcott wrote:
On 11/26/2025 1:43 PM, Tristan Wibberley wrote:
On 26/11/2025 15:54, olcott wrote:
(General_Knowledge ⊨ x) means True(x)
(General_Knowledge ⊨ ~x) means False(x)
~True(x) & ~False(x) means x is not an element of General_Knowledge
Eh? You made it sound like General_Knowledge was the system, rather than >>> a model, but there you have it as a model.
There is no model.
It is all Rudolf Carnap Meaning Postulates
that have every single nuance of 100% of their
semantic meaning directly encoding in this formal
language arranged in a knowledge ontology
inheritance hierarchy.
And this is the system you said of which that there has never been
anything like it?
Note that the meanings of
?- G = not(provable(F, G)).
and
?- unify_with_occurs_check(G, not(provable(F, G))).
are different. The former assigns a value to G, the latter does not.
On 30/11/2025 09:58, Mikko wrote:
Note that the meanings of
?- G = not(provable(F, G)).
and
?- unify_with_occurs_check(G, not(provable(F, G))).
are different. The former assigns a value to G, the latter does not.
For sufficiently informal definitions of "value".
And for sufficiently wrong ones too!
Eliminating undecidability and mathematical incompleteness
merely requires discarding model theory and fully integrating
semantics directly into the syntax of the formal language.
The only inference step allowed is semantic logical
entailment and this is performed syntactically. A formal
language such as Montague Grammar or CycL of the Cyc
project can encode the semantics of anything that can
be expressed in language.
| Sysop: | DaiTengu |
|---|---|
| Location: | Appleton, WI |
| Users: | 1,089 |
| Nodes: | 10 (0 / 10) |
| Uptime: | 155:08:17 |
| Calls: | 13,921 |
| Calls today: | 2 |
| Files: | 187,021 |
| D/L today: |
3,912 files (989M bytes) |
| Messages: | 2,457,192 |