On algebra of logic 1

Contexts

Brackets and parentheses play important role in this version of logic algebra. Brackets are used to set up universal context, parentheses to set up particular context. With this interpretation [P] means Everything is P and (P) means Something is P. Accordingly [-P] means Everything is non-P, which is equivalent to -(P) -- Nothing is P. Also (-P) means Something is non-P, which is equivalent to -[P] -- Not everything is P.

How to represent singular propositions? In SETL singulars are treated as having "wild quantity" and are marked with ±, signifying that they may be treated as universals as well as particulars. Singulars are peculiar indeed as they may be seen as predicates belonging to single individual. But it seems to me we don't need any special quantity-marker for singulars. It is the question of semantics, not of special logical treatment. When used in universal context, singular term should be read "hypothetically" (as universals are also). Eg. with T for Tim [-'T'+P] should be read If there is Tim, he is P, or Either Tim is not (there) or he is P. But ('T'+P) is read just Tim is P, ie. Something is Tim and he is P. Again, ['T'-P] is read Only Tim is P, or Either something is Tim or it is not P. Singularity is marked here by single quotes, signifying name, and by dropping '+' from before it, but logical treatment does not differ from that of universals.

Nested Contexts

Contexts may be nested. Universal contexts may contain other universal contexts as well as particular contexts, and vice versa, particular contexts may contain universal and other particular contexts. Eg. All S is P and Q is expressed as [-S+(P+Q)]. As universal context does not by itself instantiate any individuals, (P+Q) should not be read here as positing some P which is Q. Read in universal context it says If there is any S, then it is something that is P and Q. If there is no S, there need not be any P that is Q. Hence as well [(P+Q)] does not posit any P that is Q. It says, If there is anything, then it is something, that is P and Q. As already said, in case of single letter [P] expresses Everything is P, or If there is anything, then it is (something that is) P. Same is expressed by [(P)], where parentheses are spelled out -- Everything is something, that is P, or If there is anything, then it is something, that is P.

By adding extra parentheses we can particularize any expression in universal context. Eg. [((P))] means Everything is something, some of which is P. Similarly in case of complex expressions, eg. [((P+Q))] means Everything is something, some of which is P that is Q. From here it is easy to see that ((P+Q)) expresses There is something, some of which is P that is Q. And to express There is something, some of which is P, I'll use ((P)). When used inside conjunction, as in (S+(P+Q)), which is read Some S is something that is P and Q, there are several ways to proceed. We can abstract from some S by just dropping S, and say ((P+Q)), which means There is something, some of which is P and Q. We can also rise to all S: [-S+((P+Q))] All S is something, some of which is P and Q. But we can also replace S with (P+Q), saying Something is P and Q, for something is S, and it is P and Q, therefore something is P and Q, or Some P is Q (equivalent to conjunction elimination).

Universal context nested in particular context expresses idea, that there is something, that is wholly ... (whatever is contained in nested universal context). Eg. (+S+[P+Q]) expresses, that Some S is either P or Q, or There is something, that is S and either P or Q. Similarly ([P+Q]) says that Something is either P or Q, or There is something, all of which is either P or Q. What about ([P])? This is just the other way of expressing (P) Something is (wholly) P. Hence, any depth of nesting of alternating universal and particular contexts for single letter may be reduced to the outermost context: [([([(P)])])] reduces to [P], but in case of complex expressions both innermost and outermost contexts should be retained.

But what happens when we double universal context, eg. [[P]]? It seems there is no difference between [P] All is (some) P and [[P]] All is only P. But when we use it in both sides of disjunction, the difference in meaning becomes important. [[P]+[Q]] All is only P or only Q is clearly different from [P+Q] All is P or Q. Compare this to [(P)+(Q)] All is some P or some Q, which seems to be equivalent to All is P or Q. In last cases disjunction is applied to every individual, but in case of [[P]+[Q]] to the whole. [P+Q] is disjunction of predicates, [[P]+[Q]] is disjunction of propositions. We may follow this in case of Boole's example:
(1) Every inhabitant (of an island) is either European or Asiatic:
[-I+[E+A]]=[-I+E+A]
Everyone is either European or Asiatic: [E+A] (Only Europeans are not Asiatics)
(2) Every inhabitant is European or every inhabitant is Asiatic:
[[-I+E]+[-I+A]]=[-I+[E]+[A]]
Either everyone is European or everyone is Asiatic: [[E]+[A]]

In (1), as said, disjunction of predicates is applied to every individual in given domain, but in (2) predicates are applied to the whole domain at once. We may call it double universalisation in contrast to double particularization as in ((P)).