experchange > comp.theory

olcott (03-07-20, 07:54 PM)
Modal Logic
◊P ↔ ¬□¬P // Possibly(P) ↔ ¬Necessarily(¬P)
□P ↔ ¬◊¬P // Necessarily(P) ↔ ¬Possibly(¬P)
∀P (P ∈ Analytical_Knowledge(P) ↔ □P)

A universal truth predicate that refutes the Tarski Undefinability
Theorem can be easily derived simply as membership in the set of
Analytical_Knowledge.

∀P (□P ≡ Sound_Deductive_Proof(P) ≡ Theoerem(P))
Assumes Haskell Curry definition of axiom (see linked paper).

Copyright 2020 Pete Olcott



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André G. Isaak (03-07-20, 08:12 PM)
On 2020-03-07 10:54, olcott wrote:
> Modal Logic
> ◊P ↔ ¬□¬P    //  Possibly(P) ↔ ¬Necessarily(¬P)
> □P ↔ ¬◊¬P    //  Necessarily(P) ↔ ¬Possibly(¬P)
> ∀P (P ∈ Analytical_Knowledge(P) ↔ □P)
> A universal truth predicate that refutes the Tarski Undefinability
> Theorem can be easily derived simply as membership in the set of
> Analytical_Knowledge.
> ∀P (□P ≡ Sound_Deductive_Proof(P) ≡ Theoerem(P))
> Assumes Haskell Curry definition of axiom (see linked paper).


Do you actually know any modal logic? If not, you probably shouldn't be
using it.

You understand that □P does not necessarily imply □□P, right? So if □P
is true but □□P is false, what consequences does that have on the
theoremhood of P?

André
olcott (03-07-20, 08:53 PM)
On 3/7/2020 12:12 PM, André G. Isaak wrote:
> On 2020-03-07 10:54, olcott wrote:
> Do you actually know any modal logic? If not, you probably shouldn't be
> using it.
> You understand that □P does not necessarily imply □□P, right? So if □P
> is true but □□P is false, what consequences does that have on the
> theoremhood of P?
> André


Can you provide a concrete example of that?

P ≡ "Cats are not dogs"
(□P ≡ Sound_Deductive_Proof(P) Theoerem(P))

So are you saying that the necessary statement: "Cats are not dogs"
is not necessarily necessary?

Please give me an example where "Cats are not dogs" is not necessarily
necessary and do not commit any fallacy of equivocation error of
changing the set of semantic properties assigned to the terms cat and dog.

When-so-ever two elements of the universal set have different sets of
properties we can know with complete certainty that they are not the
same element.

André G. Isaak (03-08-20, 07:26 AM)
On 2020-03-07 11:53, olcott wrote:
> On 3/7/2020 12:12 PM, André G. Isaak wrote:
> Can you provide a concrete example of that?
> P ≡ "Cats are not dogs"
> (□P ≡ Sound_Deductive_Proof(P) Theoerem(P))
> So are you saying that the necessary statement: "Cats are not dogs"


"necessary" has no model-independent meaning. What makes this a
necessary statement as opposed to a contingent one?

> is not necessarily necessary?
> Please give me an example where "Cats are not dogs" is not necessarily
> necessary and do not commit any fallacy of equivocation error of
> changing the set of semantic properties assigned to the terms cat and dog.


I asked if you had ever actually studied modal logic. The answer,
apparently, is no.

□ and ◊ are simply quantifiers. They have no interpretation independent
of a model, and you don't provide a model. Whether "Cats are not dogs"
is true, necessarily true, necessarily necessarily true, etc. depends
entirely on the model you are using.

André
olcott (03-08-20, 08:23 AM)
On 3/7/2020 11:26 PM, André G. Isaak wrote:
> On 2020-03-07 11:53, olcott wrote:
> "necessary" has no model-independent meaning.


So you don't necessarily have to keep breathing to remain alive?

> What makes this a
> necessary statement as opposed to a contingent one? Semantic logical entailment.


> I asked if you had ever actually studied modal logic. The answer,
> apparently, is no.


The only thing that I know ab out modal logic is this
>>>> ◊P ↔ ¬□¬P // Possibly(P) ↔ ¬Necessarily(¬P)
>>>> □P ↔ ¬◊¬P // Necessarily(P) ↔ ¬Possibly(¬P)


The above precisely define the English words in terms of each other.
This utterly eliminates the error of
equating impossible with "I really don't brelieve it"

The only expressions of language that are impossibly true are direct
contradictions.

> □ and ◊ are simply quantifiers. They have no interpretation independent
> of a model, and you don't provide a model. Whether "Cats are not dogs"
> is true, necessarily true, necessarily necessarily true, etc. depends
> entirely on the model you are using.
> André


So in other words you believe that cats may actually be dogs some of the
time?
André G. Isaak (03-08-20, 09:14 AM)
On 2020-03-07 23:23, olcott wrote:
> On 3/7/2020 11:26 PM, André G. Isaak wrote:
> So you don't necessarily have to keep breathing to remain alive?
> Semantic logical entailment.
> The only thing that I know ab out modal logic is this


Then you really know virtually nothing whatsoever about modal logic. You
should therefore stop using it until you actually know what it is.

Though I doubt this will benefit you, here is an incredibly brief
explanation:

A model in modal logic consists of:

(A) A set of possible worlds, each of which can be thought of as a model
in the non-modal sense. Exactly what is meant by 'possible' is entirely
dependent on what the model is being used for.
(B) An accessibility relation which defines which worlds are 'visible'
to one another.

□P means that P is true in the world under consideration and in all
worlds visible to that world.

◊P means that P is true in at least one world visible to the world under
consideration.

□□P means that P is true in the world under consideration, in all worlds
visible to that world, and in all the worlds visible to those worlds
including ones which might not be visible to the world under consideration.

And so forth.

Unless you explain which possible worlds are under consideration and the
accessibility between those worlds, □ (universal modal quantifier) and ◊
(existential modal quantifier) don't mean anything.

> The above precisely define the English words in terms of each other.
> This utterly eliminates the error of
> equating impossible with "I really don't brelieve it"
> The only expressions of language that are impossibly true are direct
> contradictions.


Wait a minute -- so 'impossibly false' is a type of false, but
'impossibly true' is also a type of false? You really need better
terminology.

> So in other words you believe that cats may actually be dogs some of the
> time?


Again, that depends on the model under consideration. What worlds are
being considered? alternate evolutionary histories? things which can be
imagined? worlds corresponding to our desires?

André
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