Aequipollentia

Universal Forms

Model for the form "Every A is r to every B" is "Every _ is _ to every _", algebraically [-_+_-_].
We have 8 sentences with this form:
1. "Every A is r to every B" [-A+r-B]
2. "Every A is r to every -B" [-A+r-(-B)]
3. "Every A is -r to every B" [-A+(-r)-B]
4. "Every A is -r to every -B" [-A+(-r)-(-B)]
5. "Every -A is r to every B" [-(-A)+r-B]
6. "Every -A is r to every -B" [-(-A)+r-(-B)]
7. "Every -A is -r to every B" [-(-A)+(-r)-B]
8. "Every -A is -r to every -B" [-(-A)+(-r)-(-B)]

Model for the form "Every A is r to only B" is "Every _ is _ to only _", algebraically [-_-_+_].
8 senteces again, arranged according to equivalencies to former sentences:
1. "Every A is -r to only -B" [-A-(-r)+(-B)]
2. "Every A is -r to only B" [-A-(-r)+B]
3. "Every A is r to only -B" [-A-r+(-B)]
4. "Every A is r to only B" [-A-r+B]
5. "Every -A is -r to only -B" [-(-A)-(-r)+(-B)]
6. "Every -A is -r to only B" [-(-A)-(-r)+B]
7. "Every -A is r to only -B" [-(-A)-r+(-B)]
8. "Every -A is r to only B" [-(-A)-r+B]

Model for the form "No A is r to any B" is "No _ is _ to any _", algebraically [-_-_-_].
We have 8 sentences with this form:
1. "No A is -r to any B" [-A-(-r)-B]
2. "No A is -r to any -B" [-A-(-r)-(-B)]
3. "No A is r to any B" [-A-r-B]
4. "No A is r to any -B" [-A-r-(-B)]
5. "No -A is -r to any B" [-(-A)-(-r)-B]
6. "No -A is -r to any -B" [-(-A)-(-r)-(-B)]
7. "No -A is r to any B" [-(-A)-r-B]
8. "No -A is r to any -B" [-(-A)-r-(-B)]

Model for the form "Every A is only to B not r" is "Every _ is only to _ not _", algebraically [-_+_+_].
8 senteces again, arranged according to equivalencies to former sentences:
1. "Every A is only to -B not r" [-A+r+(-B)]
2. "Every A is only to B not r" [-A+r+B]
3. "Every A is only to -B not -r" [-A+(-r)+(-B)]
4. "Every A is only to B not -r" [-A+(-r)+B]
5. "Every -A is only to -B not r" [-(-A)+r+(-B)]
6. "Every -A is only to B not r" [-(-A)+r+B]
7. "Every -A is only to -B not -r" [-(-A)+(-r)+(-B)]
8. "Every -A is only to B not -r" [-(-A)+(-r)+B]

Clusive Forms

Model for the form "Only A is not r to every B" is "Only _ is not _ to every _", algebraically [+_+_-_].
8 sentences, same arrangement:
1. "Only -A is not r to every B" [+(-A)+r-B]
2. "Only -A is not r to every -B" [+(-A)+r-(-B)]
3. "Only -A is not -r to every B" [+(-A)+(-r)-B]
4. "Only -A is not -r to every -B" [+(-A)+(-r)-(-B)]
5. "Only A is not r to every B" [+A+r-B]
6. "Only A is not r to every -B" [+A+r-(-B)]
7. "Only A is not -r to every B" [+A+(-r)-B]
8. "Only A is not -r to every -B" [+A+(-r)-(-B)]

Model for the form "Only A is not r to only B" is "Only _ is not _ to only _", algebraically [+_-_+_].
8 sentences, same arrangement:
1. "Only -A is not -r to only -B" [+(-A)-(-r)+(-B)]
2. "Only -A is not -r to only B" [+(-A)-(-r)+B]
3. "Only -A is not r to only -B" [+(-A)-r+(-B)]
4. "Only -A is not r to only B" [+(-A)-r+B]
5. "Only A is not -r to only -B" [+A-(-r)+(-B)]
6. "Only A is not -r to only B" [+A-(-r)+B]
7. "Only A is not r to only -B" [+A-r+(-B)]
8. "Only A is not r to only B" [+A-r+B]


Model for the form "Only A is r to any B" is "Only _ is _ to any _", algebraically [+_-_-_]. ("To every B only A is r" "Only A is not to every B not r")
8 sentences, same arrangement:
1. "Only -A is -r to any B" [+(-A)-(-r)-B]
2. "Only -A is -r to any -B" [+(-A)-(-r)-(-B)]
3. "Only -A is r to any B" [+(-A)-r-B]
4. "Only -A is r to any -B" [+(-A)-r-(-B)]
5. "Only A is -r to any B" [+A-(-r)-B]
6. "Only A is -r to any -B" [+A-(-r)-(-B)]
7. "Only A is r to any B"; "To every B only A is r" [+A-r-B]
8. "Only A is r to any -B" [+A-r-(-B)]

Model for the form "Only A is non-r to not only B" is "Only _ is not only to _ not _", algebraically [+_+_+_]. ("Only to B is not only A not r" "Only A is not only to B not r")
8 sentences, same arrangement:
1. "Only -A is (not r) to not only -B"; "Only to -B is not only -A not r" [+(-A)+r+(-B)]
2. "Only -A is (not r) to not only B"; "Only to B is not only -A not r" [+(-A)+r+B]
3. "Only -A is (not -r) to not only -B"; "Only to -B is not only -A not -r" [+(-A)+(-r)+(-B)]
4. "Only -A is (not -r) to not only B"; "Only to B is not only -A not -r" [+(-A)+(-r)+B]
5. "Only A is (not r) to not only -B"; "Only to -B is not only A not r" [+A+r+(-B)]
6. "Only A is not only to B not r"; "Only to B is not only A not r" [+A+r+B]
7. "Only A is (not -r) to not only -B"; "Only to -B is not only A not -r" [+A+(-r)+(-B)]
8. "Only A is (not -r) to not only B"; "Only to B is not only A not -r" [+A+(-r)+B]


Comparision


1. "Every A is r to every B" [-A+r-B]
"Every A is -r to only -B" [-A-(-r)+(-B)]
"No A is -r to any B" [-A-(-r)-B]
"Every A is only to -B not r"; "" [-A+r+(-B)]
"Only -A is not r to every B" [+(-A)+r-B]
"Only -A is not -r to only -B" [+(-A)-(-r)+(-B)]
"Only -A is -r to any B" [+(-A)-(-r)-B]
"Only -A is (not r) to not only -B"; "Only -A is not to only -B not r" [+(-A)+r+(-B)]

2. "Every A is r to every -B" [-A+r-(-B)]
"Every A is -r to only B" [-A-(-r)+B]
"No A is -r to any -B" [-A-(-r)-(-B)]
"Every A is only to B not r"; "To only B any A is not r" [-A+r+B]
"Only -A is not r to every -B" [+(-A)+r-(-B)]
"Only -A is not -r to only B" [+(-A)-(-r)+B]
"Only -A is -r to any -B" [+(-A)-(-r)-(-B)]
"Only -A is (not r) to not only B"; "Only to B is not only -A not r" [+(-A)+r+B]

3. "Every A is -r to every B" [-A+(-r)-B]
"Every A is r to only -B" [-A-r+(-B)]

"No A is r to any/no B" [-A-r-B]

"Every A is only to -B not -r" [-A+(-r)+(-B)]
"Only -A is not -r to every B" [+(-A)+(-r)-B]
"Only -A is not r to only -B" [+(-A)-r+(-B)]
"Only -A is r to any B" [+(-A)-r-B]
"Only -A is (not -r) to not only -B"; "Only to -B is not only -A not -r" [+(-A)+(-r)+(-B)]

4. "Every A is -r to every -B" [-A+(-r)-(-B)] 

"Every A is r to only B" [-A-r+B]

"No A is r to any -B" [-A-r-(-B)]
"Every A is only to B not -r" [-A+(-r)+B]
"Only -A is not -r to every -B" [+(-A)+(-r)-(-B)]
"Only -A is not r to only B" [+(-A)-r+B]
"Only -A is r to any -B" [+(-A)-r-(-B)]
"Only -A is (not -r) to not only B"; "Only to B is not only -A not -r" [+(-A)+(-r)+B]

5. "Every -A is r to every B" [-(-A)+r-B]
"Every -A is -r to only -B" [-(-A)-(-r)+(-B)]
"No -A is -r to any B" [-(-A)-(-r)-B]
"Every -A is only to -B not r" [-(-A)+r+(-B)]

"Only A is not r to every B" [+A+r-B]

"Only A is not -r to only -B" [+A-(-r)+(-B)]
"Only A is -r to any B" [+A-(-r)-B]
"Only A is (not r) to not only -B"; "Only to -B is not only A not r" [+A+r+(-B)]

6. "Every -A is r to every -B" [-(-A)+r-(-B)]
"Every -A is -r to only B" [-(-A)-(-r)+B]
"No -A is -r to any -B" [-(-A)-(-r)-(-B)]
"Every -A is only to B not r" [-(-A)+r+B]
"Only A is not r to every -B" [+A+r-(-B)]
"Only A is not -r to only B" [+A-(-r)+B]
"Only A is -r to any -B" [+A-(-r)-(-B)] 

"Only A is not only to B not r"; "Only to B is not only A not r" [+A+r+B]

7. "Every -A is -r to every B" [-(-A)+(-r)-B]
"Every -A is r to only -B" [-(-A)-r+(-B)]
"No -A is r to any B" [-(-A)-r-B]
"Every -A is only to -B not -r" [-(-A)+(-r)+(-B)]
"Only A is not -r to every B" [+A+(-r)-B]
"Only A is not r to only -B" [+A-r+(-B)] 

"Only A is r to any B"; "To every B only A is r" [+A-r-B]

"Only A is (not -r) to not only -B"; "Only to -B is not only A not -r" [+A+(-r)+(-B)]

8. "Every -A is -r to every -B" [-(-A)+(-r)-(-B)]
"Every -A is r to only B" [-(-A)-r+B]
"No -A is r to any -B" [-(-A)-r-(-B)]
"Every -A is only to B not -r" [-(-A)+(-r)+B]
"Only A is not -r to every -B" [+A+(-r)-(-B)]
"Only A is not r to only B" [+A-r+B]
"Only A is r to any -B" [+A-r-(-B)]
"Only A is (not -r) to not only B"; "Only to B is not only A not -r" [+A+(-r)+B]

Algebra of relations

Let a, b, c... be terms for individuals and r, s, t... be terms for specific relations. That means we shall not quantify over individuals nor over specific relations. Let A, B, C... be terms for classes of individuals and R, S, T... terms for classes of relations. We shall quantify over classes.

Simple relation is represented algebraically by
a+r+b , which is equivalent to (a+r+b)
and it may be read as "a is r to b" or "a is in relation r to b" or "a is r in relation to b".

Until it is clear that r is relation between a and b, indexes may be skipped, so
a₁+r₁₂+b₂
is implied.

This relation can be denied in several ways:

1. Relation r can be denied to exist between individuals a and b
a-r+b "a is not-r to b"
I.e. there is a, there is b, but there is not r between these.

2. It can be denied about individual a, that it is related by r to b
a-(r+b) "a is not r to b"
This is algebraically equivalent to a+[-r-b], i.e. "there is a, but either there is no b, or a is not r to it"="a is r to only -b". This is slightly more general than a-r+b

3. It can be denied about individual b, that a is r in relation to it
(-(a+r)+b) "to b, it is not a that is r in relation to it"

4. The whole sentence can be denied. This is algebraically equivalent to
-(a+r+b) "It is not the case that a is r to b" = [-a-r-b]


Relations between classes

Without quantified relations.
Contradictory pairs:

(A+r+B) "Some A is r to some B"            [-A-r-B] "Every A is -r to every B"
(A+r-B) "Some A is r to not only B"        [-A-r+B] "Every A is r to only B"
(A-r+B) "Some A is -r to some B"           [-A+r-B] "Every A is r to every B"
(A-r-B) "Some A is -r to not only B"       [-A+r+B] "Every A is -r to only B"
(-A+r+B) "Not only A is r to some B"       [A-r-B] "Only A is not -r to every B"
(-A+r-B) "Not only A is r to not only B"   [A-r+B] "Only A is not r to only B"
(-A-r+B) "Not only A is -r to some B"      [A+r-B] "Only A is not r to every B"
(-A-r-B) "Not only A is -r to not only B"  [A+r+B] "Only A is not -r to only B"

(A+[r+B]) "Some A is -r to only B"         [-A-[r+B]] "No A is -r to only B"
(A+[r-B]) "Some A is r to every B"         [-A-[r-B]] "No A is r to every B"
(A+[-r+B]) "Some A is r to only B"         [-A-[-r+B]] "No A is r to only B"
(A+[-r-B]) "Some A is -r to every B"       [-A-[-r-B]] "No A is -r to every B"
(-A+[r+B]) "Not only A is -r to only B"    [A-[r+B]] "Only A is -r to only B"
(-A+[r-B]) "Not only A is r to every B"    [A-[r-B]] "Only A is r to every B"
(-A+[-r+B]) "Not only A is r to only B"    [A-[-r+B]] "Only A is r to only B"
(-A+[-r-B]) "Not only A is -r to every B"  [A-[-r-B]] "Only A is -r to every B"

([A+r]+B) "To some B only A is -r"         [-[A+r]-B] "To no B only A is -r"
([A+r]-B) "To not only B only A is -r"     [-[A+r]+B] "To only B only A is -r"
([A-r]+B) "To some B only A is r"          [-[A-r]-B] "To no B only A is r"
([A-r]-B) "To not only B only A is r"      [-[A-r]+B] "To only B only A is r"
([-A+r]+B) "To some B every A is r"        [-[-A+r]-B] "To no B every A is r"
([-A+r]-B) "To not only B every A is r"    [-[-A+r]+B] "To only B every A is r"
([-A-r]+B) "To some B every A is -r"       [-[-A-r]-B] "To no B every A is -r"
([-A-r]-B) "To not only B every A is -r"   [-[-A-r]+B] "To only B every A is -r"

Exclusive modes

It seems my exclusive modes have found honorable forefather in Petri Hispani. See p. 272ff in his "Parvorum logicalium":

Opposite Terms

Let opposite of term A be Ā. Ie. Ā is most distant term from A in its kind (species?). White and black, good and bad, friend and enemy... Then, on condition there is distribution of terms in some dimension, there is square of opposition of terms:

 A   opposite   Ā
   com     ment
       ple
   com     ment
-Ā subopposite -A



 A |          -A
---|----------------|---
       -Ā           | Ā

It is not Greimas' semiotic square, which deals with oppositions of signs.

Back

Hei-hoo! I'm back from "Sabbatical" when I renovated my new (old) appartment and blogged in Estonian at Tabernaakel. Now, back to singulars :).

First, I'll stick to the earlier conventions of + and - and drop some innovations, eg implicative context. Instead I'll introduce singulars using same symbols, i.e. angle brackets. E.g. speaking of simple terms we have [A] "Only A", (A) "Some A" and now we have also 〈A〉 "This A". Of complex terms we have [A+B] "Only A or B", [A-B] "Only A or non-B", [-A+B] "No A that is not B", [-A-B] "No A that is B", (A+B) "A that is B", (A-B) "A that is not B", (-A+B) "Non-A that is B" and (-A-B) "Non-A that is not B". Speaking of simple existence sentences we have [A] "Only A exists" (or "Everything is A"), (A) "Some A exists" (or "Something is A") and 〈A〉 "This A exists" (or "This is A").

In last message I stayed confused about treatment of singulars. Earlier I suggested that there is no problem with treating singulars like other terms, eg "Socrates is wise" as (S+W), even if this would be read as "Some Socrates is wise". This is awkward, even if we can benefit from being able to express "Only Socrates is wise" [S-W]. The awkwardness remains even if we introduce explicitly names by using quotes ('S'+W), reading it "Someone named 'Socrates' is wise". This is improvement, because we don't let name function directly as predicate any more, but it functions now indirectly the same way: "Someone of those named 'Socrates' is wise". And ['S'-W] would make explicit the flaw we might have not noticed in previous version: "Only someone of those named 'Socrates' is wise". It is not what is meant by "Socrates is wise", where we mean certain Socrates, namely the one we are speaking about (even if you don't know for sure which one I am speaking about). But now let's use specific context for this kind of reference "This Socrates is wise" (ie. the unique one we are talking about): 〈S+W〉. This context-marker has property of being auto-dual: -〈S+W〉=〈S-W〉 ie. "It is not the case, that (this) Socrates is wise"="Socrates is not wise".

It seems we are still able to refer to Socrates as the only wise one: [〈S〉-W].

First act of intellect

Until now I have treated unemphasized contexts explicitly as applying to propositional calculus. Let's use lowercase letters for propositions and uppercase letters for terms. Thus [pq], (pq) and 〈pq〉 are interpreted accordingly as "p or q is true", "both p and q are true" and "if p is true, q is true". But unemphasized contexts appear also inside term logic categorical expressions, e.g. [-S+(P+Q)] -- All S is P and Q (or [‹S›(PQ)] or 〈S(PQ)〉). So far I haven't dealt explicitly with expressions like [SP], (SP) etc., except in case of set algebra, where uppercase letters were interpreted as sets. But can unemphasized contexts be meaningfully interpreted as outer contexts in term logic? Of course they can. As outer contexts they may be used to construct complex terms, not propositions as in case of emphasized contexts. E.g., given bindings M=man and D=dead, (MD) may be read as "man that is dead", or "dead man", without asserting propositionally that "man is dead". Latter is expressed by (MD). For an other example, consider e.g. 〈MW〉, where M=men and W=went to war, reading it as "all men that went to war", contrasted to 〈MW〉, read as "all men went to war".

In this way unemphasized contexts may be used just to select terms without asserting anything about them. As such they are just complex terms and do not have truth-value. Nevertheless, they have characteristics of consistency and inconsistency, e.g. (A‹A›) is inconsistent term while [A‹A›] is trivial term. This enables us to treat the intellect's first act explicitly.

(Hmm. What about single terms?)

Implicative context introduced

Let's introduce implicative context by 〈 and 〉.

〈pp〉 = ○

Standard equivalencies:

〈pq〉	[‹p›q]	(‹p‹q››)
〈‹p›q〉	[pq]	(‹‹p›‹q››)
〈‹p‹q››〉	[‹‹p›‹q››]	(pq)
〈‹pq›〉	[‹p‹q››]	(p‹q›)
〈‹p‹q››〉	[‹pq›]	(‹p›‹q›)
〈p‹q›〉	[‹p›‹q›]	(‹pq›)
〈‹p›‹q›〉	[p‹q›]	(‹p‹q››)
〈‹‹p›‹q››〉	[‹p‹q››]	(‹p›q)

〈pqr〉 = 〈p〈qr〉〉 = 〈(pq)r〉
〈〈pq〉r〉 = ([pr]〈qr〉)
〈pq〉 = 〈‹q›‹p›〉
〈‹p›q〉 = 〈‹q›p〉
〈‹p›p〉 = p
〈p‹p›〉 = ‹p›

〈‹‹››pq〉 = 〈pq〉 = 〈‹pq›‹›〉 = p → q
〈‹‹››pp〉 = 〈pp〉= 〈‹pp›‹›〉 = ○
〈‹‹››p〉 = 〈p〉 = 〈‹p›‹›〉 ?
〈‹‹››‹›〉 = 〈‹›〉 = 〈‹‹››‹›〉 ?
〈‹‹››〉 = 〈〉 = 〈‹›‹›〉 ?