Logic without A <-> (A <-> t) ?

[Jan Burse]

Dear All

Is there a logic that does not have the following identity?

A <-> (A <-> t)

I think minimal logic has it, we can recast it as follows:

A -> (A -> (B -> B))
A -> ((B -> B) -> A)
(A -> (B -> B)) -> (((C -> C) -> A) -> A)

Which are all derivable. But I wonder whether there is
some logic that breaks in one of these or in the original
identify.

Bye

[Jan Burse]

Jan Burse schrieb:

> Is there a logic that does not have the following identity?
>
> A <-> (A <-> t)

Ok, can even pose the simpler question.

Is there a logic that does not have
the following identity?

A <-> (t -> A)

In minimal logic the following is derivable:

A -> ((B -> B) -> A)

(((B -> B) -> A) -> A

In which logic not?

Bye

[Daryl McCullough]

Jan Burse says...

>Ok, can even pose the simpler question.
>Is there a logic that does not have
>the following identity?
>
> A <-> (t -> A)
>
>In minimal logic the following is derivable:
>
> A -> ((B -> B) -> A)
> (((B -> B) -> A) -> A
>
>In which logic not?

I believe that in relevance logic, you cannot
derive B -> A unless B is relevant to the conclusion
A. So even if A is true, you cannot derive B -> A
in general.

--
Daryl McCullough
Ithaca, NY

[Jan Burse]

Daryl McCullough schrieb:

> Jan Burse says...
>
>> Ok, can even pose the simpler question.
>> Is there a logic that does not have
>> the following identity?
>>
>> A <-> (t -> A)
>>

> I believe that in relevance logic, you cannot

> derive B -> A unless B is relevant to the conclusion
> A. So even if A is true, you cannot derive B -> A
> in general.

What if I use A->A for B. Why can I not
derive (A->A)->A?

(A->A is normally a good replacement for t)

Bye