Re: William T. Parry gets rid of Disjunction introduction

On 2026-07-04 15:36, olcott wrote:
> On 7/4/2026 4:29 PM, André G. Isaak wrote:
>> 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.
>>
> 
> The English if P then Q only actually tells you
> P(true) then necessarily Q(true)
> Q(false) then necessarily P(false)
> IT DOES NOT TELL YOU MORE THAN THIS AND
> IT IS STUPID MISTAKE TO ASSUME OTHERWISE
> THE WAY THAT IMPLICATION STUPIDLY DOES.

You're introducing the word 'necessarily' here without any attempt to 
explain what is meant by this (by you).

What is the difference between

a) Q is a necessary consequence of P.
b) Q is a consequence of P

Give an example where b holds true but where a is false. If you can't do 
that, then your use of 'necessary' is completely meaningless verbiage.

>> 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
>>
> 
> Do you know what propositional logic is?
> then that is one way to avoid models.

Propositional calculus uses models. It's also extremely limited.

André

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