XPost: sci.logic

In Montague Semantics the bivalance principle
is expressed by the type "prop". You find that in
the intial definition of Montague:

2 Intensional Logic
By ME_a is understood rhe set of meaningful expresions of type a
1. Every variable and constant of type a is ME_a.

https://www.cs.rhul.ac.uk/~zhaohui/montague73.pdf

If I am not mistaken then Montague simply uses
the letter "t" for the type "prop".

Mild Shock schrieb:

Of course you can restrict yourself to
only so called "decidable" sentences A,

i.e. sentences A where:

True(L,A) v True(L,~A)

But this doesn't mean that all sentences
are decidable, if the language allows for
example at least one propositional variables p,

then you have aleady an example of an
undecidable sentences, you even don't
need anything Gödel, Russell, or who knows

what, all you need is bivalence, which was
already postualated by Aristoteles.

Principle of bivalence
https://en.wikipedia.org/wiki/Principle_of_bivalence

if you assume that a propostional variable
is "variably", meaning it can take different truth
values depending on different possible worlds,

or state of affairs, or valuations, or how ever
you want to call it. Then a propositional variable
is the prime example of an undecided sentence.

Mild Shock schrieb:

Thats a little bit odd to abolish incompletness.
Take p, an arbitrary propositional variable.
Its neither the case that:

True(L,p)

Nor is ihe case that:

True(L,~p)

Because there are always at least two possible worlds.
One possible world where p is false, making True(L,p)
impossible, and one possible world where p is true,

making True(L,~p) impossible.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.logic

Thats a little bit odd to abolish incompletness.
Take p, an arbitrary propositional variable.
Its neither the case that:

True(L,p)

Nor is ihe case that:

True(L,~p)

Because there are always at least two possible worlds.
One possible world where p is false, making True(L,p)
impossible, and one possible world where p is true,

making True(L,~p) impossible.

olcott schrieb:

On 7/23/2024 7:02 AM, Mild Shock wrote:

Little bit odd reference to mathematical logic for 2024.

olcott schrieb:

Curry, Harkell B. 1977. Foundations of Mathematical Logic. Page:45

https://www.liarparadox.org/Haskell_Curry_45.pdf

*It sustains this idea*

L is the language of a formal mathematical system.
x is an expression of that language.

When we understand that True(L,x) means that there is a finite
sequence of truth preserving operations in L from the semantic
meaning of x to x in L, then mathematical incompleteness is abolished.

~True(L,x) ∧ ~True(L,~x)
means that x is not a truth-bearer in L.
It does not mean that L is incomplete

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.logic

Of course you can restrict yourself to
only so called "decidable" sentences A,

i.e. sentences A where:

True(L,A) v True(L,~A)

But this doesn't mean that all sentences
are decidable, if the language allows for
example at least one propositional variables p,

then you have aleady an example of an
undecidable sentences, you even don't
need anything Gödel, Russell, or who knows

what, all you need is bivalence, which was
already postualated by Aristoteles.

Principle of bivalence
https://en.wikipedia.org/wiki/Principle_of_bivalence

if you assume that a propostional variable
is "variably", meaning it can take different truth
values depending on different possible worlds,

or state of affairs, or valuations, or how ever
you want to call it. Then a propositional variable
is the prime example of an undecided sentence.

Mild Shock schrieb:

Thats a little bit odd to abolish incompletness.
Take p, an arbitrary propositional variable.
Its neither the case that:

True(L,p)

Nor is ihe case that:

True(L,~p)

Because there are always at least two possible worlds.
One possible world where p is false, making True(L,p)
impossible, and one possible world where p is true,

making True(L,~p) impossible.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.logic

But truth bearer has another meaning.
The more correct terminology is anyway
truth maker, you have to shift away the

focus from the formula and think it is
a truth bearer, this is anyway wrong,
since you have two additional parameters
your "True" and your language "L".

So all that we see here in expression such as:

[~] True(L, [~] A)

Is truth making, and not truth bearing.
In recent years truth making has received
some attention, there are interesting papers
concerning truth makers. And it has

even a SEP article:

Truthmakers
https://plato.stanford.edu/entries/truthmakers/

A world of truthmakers? https://philipp.philosophie.ch/handouts/2005-5-5-truthmakers.pdf

olcott schrieb:

The key difference is that we no long use the misnomer
"undecidable" sentence and instead call it for what it
really is an expression that is not a truth bearer, or
proposition in L.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.logic

This view of a logic is extremply powerful.
For example we can already define a property
of a logic. For example we could say a logic

L is consistent, if it doesn't explode, i.e.
if it doesn't prove anything, i.e. if there
exists a sentences with is not in the logic:

L consistent :<=> exists A (A e S & ~(A e L))

Or take page 18 of the BLACK BOOK by
Chagrov & Zakharyaschev, Modal Logic - 1997 https://global.oup.com/academic/product/modal-logic-9780198537793

L has disjunction property :<=>
(A v B e L <=> A e L v B e L)

Theorem: Classical logic does not have disjunction property

Proof: Classical logic has LEM, i.e. p v ~p e L,
but it is neither the case p e L nor ~p e L.

Q.E.D.


Mild Shock schrieb:

But obviously sometimes sentences are
decidable, and sometimes not. Since
this depends on "True" and "L".

Actually modern logic does it much simpler,
you don't need to prescribe or explain what
a "True" and "L" does, in that you repeat

nonsense like for example:

A truth maker is any sequence of truth preserving operations
that links an expression x of language L to its semantic meaning
in language L. The lack of such a connection in L to x or ~x
means that x is not a truth-bearer in L.

Its much much easier to define a "logic".
You just take a language of sentences S.
And define a "logic" L as a subset of S.

You can imagine that L was defined as follows:

L := { A e S | True(L, A) }

But this is not necessarely the case how L is
conceived, or how L comes into being.

So a logic L is just a set of sentences. You
don't need the notion truth maker or truth bearer
at all, all you need to say you have some L ⊆ S.

You can then study such L's. For example:
- classical logic
- intuitionistic logic
- etc..

olcott schrieb:

On 7/24/2024 3:33 PM, Mild Shock wrote:

But truth bearer has another meaning.
The more correct terminology is anyway
truth maker, you have to shift away the

focus from the formula and think it is
a truth bearer, this is anyway wrong,
since you have two additional parameters
your "True" and your language "L".

So all that we see here in expression such as:

[~] True(L, [~] A)

Is truth making, and not truth bearing.
In recent years truth making has received
some attention, there are interesting papers
concerning truth makers. And it has

even a SEP article:

Truthmakers
https://plato.stanford.edu/entries/truthmakers/

A world of truthmakers?
https://philipp.philosophie.ch/handouts/2005-5-5-truthmakers.pdf

olcott schrieb:

The key difference is that we no long use the misnomer
"undecidable" sentence and instead call it for what it
really is an expression that is not a truth bearer, or
proposition in L.

A truth-bearer is any expression of language that can
be true or false. Self-contradictory expressions are not
truth bearers.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.logic

But obviously sometimes sentences are
decidable, and sometimes not. Since
this depends on "True" and "L".

Actually modern logic does it much simpler,
you don't need to prescribe or explain what
a "True" and "L" does, in that you repeat

nonsense like for example:

A truth maker is any sequence of truth preserving operations
that links an expression x of language L to its semantic meaning
in language L. The lack of such a connection in L to x or ~x
means that x is not a truth-bearer in L.

Its much much easier to define a "logic".
You just take a language of sentences S.
And define a "logic" L as a subset of S.

You can imagine that L was defined as follows:

L := { A e S | True(L, A) }

But this is not necessarely the case how L is
conceived, or how L comes into being.

So a logic L is just a set of sentences. You
don't need the notion truth maker or truth bearer
at all, all you need to say you have some L ⊆ S.

You can then study such L's. For example:
- classical logic
- intuitionistic logic
- etc..

olcott schrieb:

On 7/24/2024 3:33 PM, Mild Shock wrote:

But truth bearer has another meaning.
The more correct terminology is anyway
truth maker, you have to shift away the

focus from the formula and think it is
a truth bearer, this is anyway wrong,
since you have two additional parameters
your "True" and your language "L".

So all that we see here in expression such as:

[~] True(L, [~] A)

Is truth making, and not truth bearing.
In recent years truth making has received
some attention, there are interesting papers
concerning truth makers. And it has

even a SEP article:

Truthmakers
https://plato.stanford.edu/entries/truthmakers/

A world of truthmakers?
https://philipp.philosophie.ch/handouts/2005-5-5-truthmakers.pdf

olcott schrieb:

The key difference is that we no long use the misnomer
"undecidable" sentence and instead call it for what it
really is an expression that is not a truth bearer, or
proposition in L.

A truth-bearer is any expression of language that can
be true or false. Self-contradictory expressions are not
truth bearers.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)