Multiple quantification

Algebraic representation of relational propositions

To represent proposition types developed in the last post algebraically, I'll introduce numerical indexes, in somewhat similar way as it is done in SETL algebra.

In standard types of categorical propositions indexes are not needed, as quantification ranges over same individuals -- in universal propositions over all individuals in the universe of discourse (UD) and in particular propositions over some individuals in given UD. Therefore indexes are assumed to be the same in the range of given expression, e.g. All S is P, represented algebraically as [-S+P], may be seen as having default index 1: [-S1+P1]; meaning that predicates S and P are applied to same individuals synchronously. It is explicit in universal disjunctive reading of the same expression: Everything is either not S or it is P, ie. S and P are applied both to the same every thing. When we introduce indexes, we can start asynchronous quantification. Using different indexes for S and P, as in [-S1+P2], we split the individuals we are quantifying over, and interpretation shifts from different predicates applying to same individuals, to different predicates applying to (possibly) different individuals. Voila, we have multiple quantification! Well, at least first elements of it. Comparing different readings of the formula without indexes with readings of indexed formula we can better recognize the shift:

Categorical: All S is P --- For all S there is some P
Disjunctive: Everything is either not S or it is P --- Everything is either not S, or there is some P for it
Hypothetical: If something is S, then it is P --- If something is S, then there is some P for it

Venn diagram for this expression is similar to that of non-indexed version, ie. for universal affirmative. To indicate that we are quantifying over different individuals, we may use different colors for circles:










Or, if we draw 3D diagrams, we can indicate different individuals by placing them on different layers:






This interpretation presumes that for each S there is at least one separate P, ie. if there is exactly one S, then there is at least one P; if there are two S-s, then there are at least two P-s, etc. In other words, P-s can be distributed between S-s; there are at least as many P-s as there are S-s.

Here are the readings of all eight basic types in indexed forms:

[-S1+P2] For every S there is some P.
[-S1-P2] For no S there is any P.
[+S1-P2] For only S there is some P.
[+S1+P2] For only S there is not some P.
(-S1+P2) For not only S there is some P.
(-S1-P2) For not only S there is not some P.
(+S1-P2) For some S there is no P.
(+S1+P2) For some S there is some P.

As distribution patterns for these formulas are similar to non-indexed versions, these formulas can be seamlessly included into syllogisms, eg.:

[-S+M]+[-M1+P2]=[-S1+P2]
All S is M and for every M there is some P, therefore for every S there is some P.

[+S1-M2]+[+M+P]=[+S1+P2]
For only S there is some M and only M is not P, therefore for only S there is not some P.

[-S1-M2]+(+M2+P3)=(-S1+P3)
For no S there is any M and for some M there is some P, therefore for not only S there is some P.

These readings are purely existential, in sense that existence of (potentially) different individuals is somehow mutually dependent, without determination of the specific nature of the dependency.

We can take "have/has" as typical relation of dependency in contrast to "are/is" in basic types of categorical propositions. In this case we have very simple move from "is" to "has" with simultaneous shift to multiple quantification:

All S is P --- [-S1+(H12+P2)] All S has P
No S is P --- [-S1-(H12+P2)] No S has P
Only S is P --- [+S1-(H12+P2)] Only S has P
Only S is not P --- [+S1+(H12+P2)] Only S has not P
Not only S is P --- (-S1+(H12+P2)) Not only S has P
Not only S is not P --- (-S1-(H12+P2)) Not only S has not P
Some S is not P --- (+S1-(H12+P2)) Some S has not P
Some S is P --- (+S1+(H12+P2)) Some S has P

There is an important difference between purely existential expression without determined relation and expression in which relation is determined -- in latter distribution patterns are not necessarily similar to those in basic categorical types. This can be seen by comparing No S has P to For no S there is any P. Latter may be rendered as Either there is no S or there is no P, regardless of relationship considered, ie. individuals belonging to S and those belonging to P do not exist simultaneously. The former instead is denial of certain relationship between individuals belonging to S and P respectively, and it may be rendered as For no S there is any P such that the first has the second. Algebraically, when we open parentheses inside the formula, it will become [-S1-H12-P2], read as eg. Either there is no S, or it does not have anything, or what is has is not P.

ADDENDUM

Oh, I forgot to present the types from last post. Here they are:
(Somewhat hurriedly composed)

Everything is itself: [1]
Everybody is himself: [-B1+1]
Something is itself: (1)
Somebody is himself: (+B1+1)
Everything is something: [(1)] ??
Everything is somehow related to itself: [-1+R11]
Everything is somehow related to something: [-1+(R12+2)]
Something is somehow related to something: (+1+R12+2)