Re: Gödel's Basic Logic Course at Notre Dame (Was: Analytic Truth-makers)

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.
>
>