Re: William T. Parry gets rid of Disjunction introduction

On 2026-07-04 14:58, olcott wrote:
> On 7/4/2026 11:59 AM, André G. Isaak wrote:
>> On 2026-07-04 10:44, olcott wrote:
>>> On 7/4/2026 10:08 AM, André G. Isaak wrote:
>>>> On 2026-07-04 07:21, olcott wrote:
>>>>> On 7/4/2026 1:47 AM, Mikko wrote:
>>>>>> On 03/07/2026 18:04, olcott wrote:
>>>>
>>>>>>> P ⇒ Q
>>>>>>> P → Q
>>>>>>> P ⊃ Q
>>>>>>>
>>>>>>> are all abolished and replaced with the binary
>>>>>>> form of logical necessity: P □ Q
>>>>
>>>> There is no 'binary form of logical necessity.
>>>
>>> That is why I just created one.
>>
>> But you didn't define it.
>>
>>>> □ is a unary operator and an expression like P □ Q makes absolutely 
>>>> no sense.
>>>>
>>>
>>> Q is a necessary consequence of P makes perfect sense
>>> and corrects a fundamental error in the definition of
>>> valid deductive inference.
>>
>> That would normally be written as □(P → Q), 
> 
> That does not perfectly preserve the complete
> necessity between P and Q.

What on earth does it mean for there to be a "complete necessity between 
P and Q". Necessity applies to propositions. It doesn't hold *between* 
things.

□(P → Q) means that Q is necessarily implied by P. If you mean something 
other than that you're really going to have to clarify what you mean.

> P □ Q   P → Q
> 0 ? 0   0 1 0
> 0 ? 1   0 1 1
> 1 0 0   1 0 0
> 1 1 1   1 1 1
> Q is a necessary consequence of P
> is the same as English If P then Q
> (a) false when P is true and Q is false
> (b) true  when P is true and Q is true
> (c) otherwise does not have a truth value.

So again your veering into the territory of three-valued logic. If a 
truth table contains any symbol other than T or F, you're dealing with a 
three or more valued logic which means you have to completely redefine 
every single logical operator before you can proceed.

And there's nothing about the above table which in any way captures the 
meaning of 'necessity' so it's entirely unclear why you want to use the 
□ symbol here. Your '□' doesn't have any relation to necessity any more 
than '→' does above.

Also note that formal logic is *not* c++. There's no such thing as 
operator overloading, so you can't take a unary operator and use it for 
some ill-defined binary operation as well. You need a new symbol since □ 
is already taken.

P □ Q makes as much sense as P ¬ Q or P ∀ Q

>> So how would you interpret 'necessity' without models?

I note you didn't answer this. The notion of necessity in modal logic is 
intrinsically tied to model theory.

How exactly are you defining 'necessity' if you're not making use of models?

André

-- 
To email remove 'invalid' & replace 'gm' with well known Google mail 
service.