SUO: *Date 19 Apr 2002 -- Theory Query
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SUO WG Members,
Luckily, the exploration of Model Theory on the
Ontology Sublist has reached the shores of some
concrete examples just when we really need them.
Below, I have summarized the first couple of examples from the text
in a way that I think will help to illustrate the kinds of "modular"
or "phylogenetic" organization that we always find arising among the
axioms and symbols of "naturally occurring" theories and their models.
Here, the hierarchy is hereditary in such a manner that the bearings
of axioms and symbols on the species of structure evolved accumulate
along the descent down each branch. Hence, the first example should
serve to flesh out a good share of the details that we would find in
the neigborhood of the Partial Order - Lattice - Linear Order segment
of the Lindenbaum ordering in Figure 2.14 (page 95) of Sowa's KR book.
Example 1. http://suo.ieee.org/ontology/msg04023.html
o-------------------------------------------------o
| Partial Order 1.4.1 |
| |
| $L$ = {=<} |
| |
| 1. (`A`xyz)(x =< y & y =< z => x =< z) |
| 2. (`A`xy) (x =< y & y =< x => x = y) |
| 3. (`A`x) (x =< x) |
o------------------------o------------------------o
|
|
o------------------------o------------------------o
| Simple Order (Linear Order) |
| |
| $L$ = {=<} |
| |
| 4. (`A`xy) (x =< y or y =< x) |
o------------------------o------------------------o
|
|
o------------------------o------------------------o
| Dense Order |
| |
| $L$ = {=<} |
| |
| 5. (`A`xy) (x =< y & x =/= y => |
| (`E`z)(x =< z & z =/= x & |
| z =< y & z =/= y )) |
| 6. (`E`xy) (x =/= y) |
o------------------------o------------------------o
|
|
o------------------------o------------------------o
| Dense Order Without Endpoints |
| |
| $L$ = {=<} |
| |
| 7. (`A`x)(`E`y)(x =< y & x =/= y) |
| 8. (`A`x)(`E`y)(y =< x & x =/= y) |
o-------------------------------------------------o
Example 2. http://suo.ieee.org/ontology/msg04024.html
o-------------------------------------------------o
| Boolean Algebra 1.4.3 |
| |
| $L$ = {+, ·, ~, 0, 1} |
| |
| 1. Associativity of + and · |
| x + (y + z) = (x + y) + z |
| x · (y · z) = (x · y) · z |
| |
| 2. Commutativity of + and · |
| x + y = y + x |
| x · y = y · x |
| |
| 3. Idempotent Laws |
| x + x = x |
| x · x = x |
| |
| 4. Distributive Laws |
| x + (y · z) = (x + y) · (x + z) |
| x · (y + z) = (x · y) + (x · z) |
| |
| 5. Absorption Laws |
| x + (x · y) = x |
| x · (x + y) = x |
| |
| 6. De Morgan Laws |
| ~(x + y) = ~x · ~y |
| ~(x · y) = ~x + ~y |
| |
| 7. Laws of Zero and One |
| 0 =/= 1 |
| |
| x + 0 = x x + 1 = 1 |
| x · 0 = 0 x · 1 = x |
| |
| x + ~x = 1 |
| x · ~x = 0 |
| |
| 8. Law of Double Negation |
| ~~x = x |
o------------------------o------------------------o
|
|
o------------------------o------------------------o
| Boolean Algebra as Partial Order |
| |
| $L$ = {+, ·, ~, 0, 1} |
| plus defined set {=<} |
| |
| The partial order =< is defined by: |
| |
| x =< y if and only if x + y = y |
o------------------------o------------------------o
/ \
o----------------------o o----------------------o
| Atomic | | Atomless |
| Boolean Algebra | | Boolean Algebra |
| | | |
| $L$ = {+, ·, ~, 0, 1}| | $L$ = {+, ·, ~, 0, 1}|
| plus defined set {=<}| | plus defined set {=<}|
| | | |
| (`A`x) | | ~(`E`y) |
| (0 =/= x | | (0 =/= y |
| => | | & |
| (`E`y) | | (`A`z) |
| (y =< x & 0 =/= y | | (z =< y |
| & | | => |
| (`A`z) | | z = 0 or z = y |
| (z =< y | | )) |
| => | | |
| z = 0 or z = y | | |
| ))) | | |
o----------------------o o----------------------o
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