Re: William T. Parry gets rid of Disjunction introduction

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.

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.

> not as the nonsensical P □ 
> Q. But you claim to have gotten rid of →, so how this is to be 
> interpreted remains a mystery (and getting rid of → makes no sense since 
> → represents a specific truth table which still exists regardless of 
> whether you've assigned a symbol to it or not)
> 
>>> If you want to use this as a binary operator you'd actually need to 
>>> *define* it. You don't seem to grasp this. You can't just introduce a 
>>> new operator and expect people to know what it means.
>>>
>>
>> □ Already means necessity, it is not that hard unless
>> one makes great effort to pretend to not understand
>> what is already unequivocally clear. >
>>> Also, moving into the domain of modal logic would be an incredibly 
>>> strange thing for you to do given that in previous posts you claimed to 
>>
>> The only thing that I am using is logical necessity.
> 
> So how would you interpret 'necessity' without models?
> 

To make it easy to understand we have the above
propositional logic truth tables. They provide
the framework.

> André
> 
>>> reject the idea of models. But modal logic is *replete* with models. 
>>> Modal logic operates over a set of many models. □P means that P is 
>>> true in all accessible models of the system.
>>>
>>> André
>>>
>>
>>
> 


-- 
Copyright 2026 Olcott

My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.

The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.

My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.

(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).