In a comment, Brandon asked about how to represent in my version of logic algebra ordinary propositional formulas among modal formulas. This question has bothered me for some time and I have thought of some possibilities to introduce needed distinctions. As it stands now, we can interpret brackets and parentheses either with quantification or without it, while retaining in both cases their disjunctive versus conjunctive character. This makes it impossible to explicitly specify whether we have quantification or not. But to introduce new special symbols moves us away from the unity of notation between different levels of logic, which I consider a feature with great value, and natural to this kind of algebraic representation. As I noted in my response to Brandon, at present I prefer to introduce the distinction between quantificational and non-quantificational contexts with minimal altering of existing symbols -- quantificational contexts will be bolded and non-quantificational contexts will be plain.
Thus, we can differentiate between universal, particular and singular categorical propositions, and between propositional and modal formulas, using for all purposes the same syntax:
| Categorical formulas: | |||
| [-S+P] Every S is P Everything is either not S or is P | [-S-P] No S is P Everything is either not S or not P | [+S-P] Only S is P Everything is either S or not P | [+S+P] Only S is not P Everything is either S or P |
| [-S+P] This is either not S or is P | [-S-P] This is either not S or not P | [+S-P] This is either S or not P | [+S+P] This is either S or P |
| ([-S+P]) Something is either not S or P | ([-S-P]) Something is either not S or not P | ([+S-P]) Something is either S or not P | ([+S+P]) Something is either S or P |
| [(-S+P)] Everything is not S but B | [(-S-P)] Everything is neither S nor P | [(+S-P)] Everything is S but not P | [(+S+P)] Everything is S and P |
| (-S+P) This is not S but P | (-S-P) This is neither S nor P | (+S-P) This S is not P This is S but not P | (+S+P) This S is P This is S and P |
| (-S+P) Not only S is P Something is not S but is P | (-S-P) Not only S is not P Something is neither S nor P | (+S-P) Some S is not P Something is S but not P | (+S+P) Some S is P Something is S and P |
| Predicate/modal formulas: | |||
| [-p+q] Necessarily, if p, then q | [-p-q] Necessarily, if p, then not q | [+p-q] Necessarily, only if p, q | [+p+q] Necessarily, either p or q |
| [-p+q] If p, then q | [-p-q] If p, then not q | [+p-q] Only if p, q | [+p+q] Either p or q |
| ([-p+q]) Possibly, if p, then q | ([-p-q]) Possibly, if p, then not q | ([+p-q]) Possibly, only if p, q | ([+p+q]) Possibly, either p or q |
| [(-p+q)] Necessarily, not p but q | [(-p-q)] Necessarily, neither p nor q | [(+p-q)] Necessarily, p but not q | [(+p+q)] Necessarily, both p and q |
| (-p+q) Not p but q | (-p-q) Neither p nor q | (+p-q) p but not q | (+p+q) Both p and q |
| (-p+q) Possibly, not p but q | (-p-q) Possibly, neither p nor q | (+p-q) Possibly, p but not q | (+p+q) Possibly, both p and q |
With this alteration of symbols, Brandon translated axioms of different modal systems. Formulas may be simplified by dropping outer brackets, according to the principle that by default, formulas starting with '-', are disjunctive, formulas starting with '+', conjunctive (conforming to SETL algebra). But let for now the outer context be explicitly stated:
K: [-[-p+q]+[-[p]+[q]]]
D: [-[p]+(p)]
M: [-[p]+p]
4: [-[p]+[[p]]]
B: [-p+[(p)]]
5: [-(p)+[(p)]]
I add the rest from SEP article, no.8:
CD: [-(p)+[p]]
[M]: [-[p]+p]
C4: [-[[p]]+[p]]
C: [-([p])+[(p)]]
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