This time, let us have a confirming the reason why we need the Law of Excluded Middle, daringly with formalism's view, of course it is just my idea and in terms of my daily view, rather I advocate intuitionism's view, but this time I dare to dealing with its issue through formaism's postion.

"socrates will die".
For this proposition, the rule of exclusion,

"Socrates either dies or does not die".
It is a rule that the proposition P ∨ ¬P holds.

What we prepare for the sentences with "socrates will die" and 
"Socrates either dies or does not die" must be caused from that we merely suggest only uncertain, but sometime or another day must resulting or concluding thing's presence and to it,how we should define so formally, that thing could never be error idea.

First, according to upper shpwing example sentences, toward one proposition, we can have only two posibilities or probable consequences at outputting phase. 
Nevertheless, at this premise, very typical one shortage exist. 

One of it could be e.g. tomorrow's weather suggesting proposition case could have the reality of deduced truth that if it is sunny day, to its affirmation which could be hit, this case could never have issue, because it could satisfiy set proposition, but another case at not hit, as proved wrong, that range is extremely larger than hit sase. Of course necessarily usually we can get the presumption that almost cases are not so completely sunny or a little cloudy tasty, but actually even the case that rain, never yet, either even stormy or snowy or so could be possible, not so presumably but just possible, in other words this dichotomy means one so certainly  focused subject and another so vaguely other all possibilities as uneven dealing are co-exisitng, then intirinsicalt that method from the beginning as premise setting, perfectly only expedient omission is seriously prepared, then if we restrictly deal with it so perfectly logically, we should take this sase, at one so certain hitting or missing proposition and other very larger range many propositions, then if former is set as one proposition S, S on all possibilities or probably presumable condition. If we articulate it with formula, as defining all other possibilities as set except one hit element Ss,next could be set so necessarily.

                                                       
                                                        S on Ss∨Ss ⊃ S

Here we could deleberate must head for that truly articulated dechotomy could be composed or not, and if that could be possible what we should define as truly genuine dechomy.

To some extent, if we restrictly define absolutely refined or genuine dechotomy, that must be nothing but only which could be sunny or rain, or cloudy or at the matter of fact celarly defined item only could be applicable, we could say so.

In other words, it must not to S or not S, S∨ ¬S, 
                                                rather it should be 

                                                             
                                                                      S ∨ ∀(¬S)

Necessaarily, this means at ∀(¬S), any kind of specifically described weather condition, e.g., rain, heavy rain or downpour( as stormy), or hail, snow, or typhoon, or hurricane.

Eventually, we could conclude that our annoyable this excluded midle's dilemma must be caused from that my suggesting very rough not specifying only expediently defining two items as latter denial to former given proposition. 

Then, necesarily, we should apply next formula, so far.

                                               

                                            S ∨ ∀(¬S) = S ∨ ¬S(∀', ∀", ∀'", ....)

At the matter of fact, for the present, I've got only that conclusion, then, until next chance to argue about this subject, I)ll address with more precise descripting around this subject as propositional question.

(to be continued)

Mar. 27th.   2021