On algebra of logic 2

Brandon's thoughts on my version of logic algebra lead me to think anew about propositional version of it. And I came to conclusion that in addition to thinking of the propositional version of it as simple transformation where term letters are replaced by proposition letters, we can interpret it also as modal propositional logic. And it reveals amazing unity between different "layers" of logic.

As said, we can move to ordinary propositional logic (in singleton universe as Brandon calls it) by simply replacing terms by propositions. Then we should interpret parentheses simply as conjunction and brackets as disjunction. But this in not fully equivalent translation of quantified term logic. Rather, it is equivalent to term logic with single individual.

But if we transfer quantification also, then we have modal propositional logic. Parentheses, as usual, set up particular context, which should be read as In some case... Brackets, as usual, set up universal context, read as In every case... (Ordinary propositional logic instead presumes the preamble It is the case that...)

[-p+q] In every case (/necessarily), if p is true, q is also
[-p-q] In every case (/necessarily), if p is true, q is not
[+p-q] In every case (/necessarily), if p is true, q may be true
[+p+q] In every case (/necessarily), if p is true, q may be not true
(-p+q) In some case (/possibly), p is not true, but q is
(-p-q) In some case (/possibly), neither p nor q is true
(+p-q) In some case (/possibly), p is true but not q
(+p+q) In some case (/possibly), both p and q are true

([+p+q]) In some case (/possibly), either p or q is true
[(+p+q)] In every case (/necessarily), both p and q are true
...

Further, [p] is read as Necessarily p, ie. In every case p is true
[-p] Necessarily -p; p is impossible; In every case p is not true
(p) Possibly p; In some case p is true
(-p) Possibly -p; In some case p is not true

In singleton universe, or monotonic propositional logic, (p) is read as just p is true, or It is the case that p (contextually), and it is the same as [p], read as It is the case that p (universally), with all contextual references (places, times, persons...) replaced with absolute references.