On algebra of logic 3

Introducing Domains

I'd like expressions of the algebra to refer to some domain. To bring this about, let's think of these formulas as containing a hidden variable referring to the domain, let's say A for actual world. (To mark domain I'll use capitalized italic letters.) Then to say Everything is P in actual world A we should write [-A+P], which might be read also as Everything either is not in actual world or is P. In more general way [P] refers by default to some domain, where everything is P. To make this explicit we write it [-D+P], where D stands for the specific domain, and brackets quantify over every individual object. In this sense D is the "top predicate" in given domain. Every predicate that refers to anything at all in given domain refers to something to which D refers. It is different from other predicates that might also refer to all individuals in given domain in that D refers to nothing outside given domain, while other predicates might refer to something in other domains also. When the domain is contextually clear, we may hide the domain letter, but when it is unclear or we are explicitly reasoning over multiple domains, we should explicate the domain, to which given expression is referring. In case of particular propositions the domain is by default marked positively, ie. (P) with domain explicated is (+D+P), meaning Something in domain D is P, or Something is in domain D and is P.

Possibility of making domains explicit broadens significantly the use of algebra. We can now express that some predicate refers to something only in given domain, eg. [+D-P], read as Something is P only in domain D, or Only in domain D is anything P. Of course we can as well express that Something is P not only in domain D -- (-D+P).

We can have empty domains -- [-D] (Everything is not in domain D) as well as -(D) (Nothing is in domain D).

We can reason about relations of domains -- eg.:
[-E-F]: Domains E and F are disjoint, having no common individuals.
[-E+F]: Domain E is included in (/is subdomain of) domain F; ie. all individuals in domain E (if it is not empty) are included in domain F.
[+E-F]: Only domain E includes domain F; ie. domain E is superdomain of F; only individuals of domain E are included in domain F.
[+E+F]: Only domain E is not in domain F; ie. any individual that is not in domain E is in domain F.
(-E-F): Something is neither in domain E nor in domain F.
(-E+F): Something that's not in domain E is in domain F.
(+E-F): Something in domain E is not in domain F.
(+E+F): Something is both in domain E and in domain F.

And we can define universal domain, including (all individuals from) all domains: [U], meaning everything is in U, or it is not at all, in any way, really or virtually, or in whatever form. U is top-domain, domain of all domains. U cannot be empty: (U).

Existential import

Every predicate is instantiated in some domain, but they can be applied in domains where they are not instantiated. Thus (1) Every A is B in D is true even if there are no A-s in D. And (2) Some A is B in D can be true only if there is some A in D. So (1) implies (2) only if there are A-s in D. This corresponds to standard modern interpretation of existential import. But even if there are no A-s in D, A has to be instantiated in some domain (real or imaginary or abstract or whatever), to be counted as predicate, ie. to be meaningful. Hence, globally it cannot be said that there are no A-s, if A is applied in any domain at all as predicate. This seems to be the position of those who insist that both universal and particular propositions have existential import (eg. J.P.N. Land, in the article "Brentano's Logical Innovations" in Edward Bruckner's Logic Museum, says: "In an ordinary proposition the subject is necessarily admitted to exist, either in the real or in some imaginary world assumed for the nonce."). And finally we may distinguish one domain, usually (but not necessarily) the actual world, in relation to which neither universal nor particular propositions need to have existential import, which seems to have been the position in logica antiqua. Consider for example Aristotle: "Take the proposition 'Homer is so-and-so', say 'a poet'; does it follow that Homer is, or does it not? The verb 'is' is here used of Homer only incidentally, the proposition being that Homer is a poet, not that he is, in the independent sense of the word." (De Interpretatione, 11). For last two positions universal propositions unconditionally imply particular ones. (See also this article in Logic Museum)


I think these three positions differ in that in logica antiqua existence was granted to real essences only but talk was meaningful about nominal essences also; in Land's position existence is broadly granted to real as well as nominal essences; but for fathers of modern logic nominal essences were largely meaningless mumblings. At least this is my impression. If domains are not explicated, then logical, physical and metaphysical existence is undistinguished and confusion results.


ADDENDUM

We can use domains with propositional logic as well.
[-E+p] In domain E proposition p is true
[-E-p+q] In domain E, if p is true, then q is true
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