Blogicum

Opposite Terms

2010-10-09T18:01:00.008+03:00

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

2010-10-01T21:05:00.004+03:00

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

2009-01-10T23:53:00.003+02:00

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

2009-01-06T14:24:00.007+02:00

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›‹›〉 ?
〈‹‹››‹›〉 = 〈‹›〉 = 〈‹‹››‹›〉 ?
〈‹‹››〉 = 〈〉 = 〈‹›‹›〉 ?

Peirce and Bricken accommodated

2009-01-03T10:40:00.008+02:00

(Corrected 4.01.09)

Both Peirce's alpha graphs and Bricken's boundary logic may be seamlessly accommodated into my logic algebra. For this I'll adopt following conventions into my notation:

1) Letters g and f signify any propositional formula (or graph).
2) Negation of a formula is expressed by enclosing this formula between ‹ and ›.
3) Appearance of a formula in odd depth of negation is marked by enclosing this formula between ≺ and ≻; in even depth (incl 0) -- between ≼ and ≽; in any depth (incl 0) -- between ⋘ and ⋙.
4) Contents of the negated formula are joined by conjunction or disjunction depending on immediate context of the negated formula, i.e. [‹g f›] = ‹[g f]› and (‹g f›) = ‹(g f)›.
5) When formula is true independently of the context, context markers may be dropped.
6) Default context (Blank sheet) is conjunctive (i.e. Peircean).

Now we can demonstrate equivalences of three systems:

Peirce α	Bricken BL	Tom
Basic formulas
g	g	g
‹g›	‹g›	‹g›
gf	‹‹g›‹f››	(gf) ⇔ [‹‹g›‹f››]
‹‹g›‹f››	gf	(‹‹g›‹f››) ⇔ [gf]
‹g‹f››	‹g›f	(‹g‹f››) ⇔ [‹g›f]
Transformations
⇔ ‹‹›‹›› ⇔ ‹‹›› ⇔ T	‹›‹› ⇔ ‹› ⇔ T	(‹‹›‹››) ⇔ (‹‹››) ⇔() ⇔ ○ ⇔ ⇔ [‹›] ⇔ [‹›‹›]
‹›‹› ⇔ ‹› ⇔ F	⇔ ‹‹›‹›› ⇔ ‹‹›› ⇔ F	[‹‹›‹››] ⇔ [‹‹››] ⇔ [] ⇔ □ ⇔ (‹›) ⇔ (‹›‹›)
‹›g ⇔ ‹›	‹‹›g› ⇔	(‹›g) ⇔ □ ⇔ [‹‹›g›]
‹‹›g› ⇔	‹›g ⇔ ‹›	(‹‹›g›) ⇔ ○ ⇔ [‹›g]
‹‹g›› ⇔ g	‹‹g›› ⇔ g	‹‹g›› ⇔ g
≼gf≽ ⇒ ≼g≽	≺gf≻ ⇒ ≺g≻	(≼gf≽) ⇒ (≼g≽) [≺gf≻] ⇒ [≺g≻]
≺g≻ ⇒ ≺gf≻	≼g≽ ⇒ ≼gf≽	(≺g≻) ⇒ (≺gf≻) [≼g≽] ⇒ [≼gf≽]
g⋘f⋙ ⇔ g⋘gf⋙	g⋘f⋙ ⇔ g⋘gf⋙	g⋘f⋙ ⇔ g⋘gf⋙

Bricken's boundary logic

2009-01-02T12:58:00.004+02:00

William Bricken's boundary logic (BL) (see also) is interesting dual to Peirce's existential graphs. In his system blank sheet represents FALSE (Peirce: TRUE) and composition of structures represents disjunction (Peirce: conjunction). Accordingly basic logical operations are represented as follows:

‹void› -- FALSE
() -- TRUE
(p) -- NOT p
p q -- p OR q
((p) (q)) -- p AND q
(p) q -- IF p THEN q

Basic transformation rules:

()() = () -- calling [T v T = T]
(()) = ‹void› -- crossing [~T = F]
(() p) = ‹void› -- occlusion [~(T v p) = F & ~p = F]
() p = () -- dominion [T v p = T]
((p)) = p -- involution (~(~p) = p)
p {q p} = p {q} -- pervasion (from any depth) [p v ~(... q v p...) = p v ~(...q...)]

As parentheses operate mostly as negation, they may be replaced by ‹›:
‹p› -- NOT p
‹p q› -- NOT (p OR q)
‹‹p›‹q›› -- NOT (NOT p OR NOT q) = p AND q
‹p› q -- NOT p OR q = IF p THEN q

Hence, it is similar to operation in my notation inside brackets, where negation applies to disjunction:
‹p q› -- [‹p q›] = [-[p+q]] = [(-p-q)]
‹‹p›‹q›› -- [‹‹p›‹q››] = [-[-p-q]] = [(+p+q)]
‹p› q -- [‹p› q] = [-p+q]

Compendium of Peirce's Existential Graphs

2009-01-01T18:05:00.021+02:00

Based on Zalamea. See also Zeman, Sowa and Dau.

p, q, f -- propositional letters
P, Q, F -- predicate letters

Alpha

Peirce's modified notation	My (modified) notation
pq -- conjunction	(pq)
‹p› -- negation	‹p›
‹p‹q›› -- implication	[‹p›q]
‹‹p›‹q›› -- disjunction	[pq]

A1 Erasure: Any evenly enclosed graph may be erased.
A2 Insertion: Any graph may be inserted in any oddly enclosed region.
A3 Iteration: Any graph may be iterated (i.e. repeated) in a strict region of that graph.
A4 Deiteration: Any graph whose occurrence could result from iteration may be deiterated (i.e. erased).
A5 Double cut: A double cut may be inserted or erased around any graph in any region.

A1 (ER): pq ⇒ p                                       (pq) ⇒ p
A2 (IN): ‹p› ⇒ ‹pq›                                 ‹p› ⇒ ‹pq›
A3, A4 (IT and DI): p‹q› ⇔ p‹pq›         (p‹q›) ⇔ (p‹pq›)
A5 (DC): p ⇔ ‹‹p››                                 p ⇔ ‹‹p››

Beta

LI -- Line of Identity

B1 Erasure: Any evenly enclosed portion of an LI may be erased.
B2 Insertion: Any portions of LI's may be joined in an oddly enclosed region.
B3 Continuous iteration: Any LI may be extended towards strict regions. Any LI may branch in its region.
B4 Continuous deiteration: Any LI may be retracted towards regions with lesser cuts.

B1: P---Q ⇒ P- -Q                       (PQ) ⇒ (P)(Q)
B2: ‹P- -Q› ⇒ ‹P---Q›            ‹(P)(Q)› ⇒ ‹(PQ)›
B3, B4: P--- ‹q› ⇔ P-‹- q›    (P)‹q› ⇔ (P‹q›)?

Some X is R to every Y ⇒ To every Y some X is R.
-‹-‹-R2-›-› ⇒ ‹‹-R2-›-›            (X1[‹Y2›R12]) ⇒ [‹Y2›(X1R12)]?

Gamma

{f} -- possibly not f                                          (‹f›)
‹{f}› -- not possibly not f = necessarily f     ‹(‹f›)› = [f]
{‹f›} -- possibly not not f = possibly f          (‹‹f››) = (f)

‹{f}› ⇒ ‹‹f›› ⇒ f ⇒ ‹‹f›› ⇒ {‹f›}

G1: In an even area, any Alpha cut may be half-erased to become a Gamma cut.
‹› ⇒ {}                                                                ‹› ⇒ (‹›)
G2: In an odd area, any Gamma cut may be half-completed to become an Alpha cut.
‹{}› ⇒ ‹‹››                                                      ‹(‹›)› ⇒ ‹‹››

Boundary mathematics and algebra of sets

2008-12-22T18:40:00.013+02:00

(Corrected 23.12.08)

Brandon brought to my attention an interesting comparison between boundary mathematics and algebra of logic as developed in this blog. However I didn't quite follow his equivalences between the two systems. Of course this might be caused by the fact that I met with the subject of boundary mathematics first time in his blog. Nevertheless, the subject provoked me to think about similarities and differences between the two.

Jeffrey James, in his thesis "A Calculus of Number Based on Spatial Forms", bases his system on "making distinctions out of the void". The basic element is a single boundary in the void -- unit, represented by o, or generally instance, represented by (X). The other boundaries are "black hole", represented by □, or generally abstract, represented by [X]; and inverse, represented by △ in empty case or generally by <X>. Only instance has actually a spatial interpretation as boundary. Boundaries can be collected and nested to form complex structures.

In following I will interpret [(X)] ≗ X as set X and formulate set algebra, taking lead from Wikipedia article. Translation into propositional formulas should be pretty straightforward by replacing set letters with propositional letters. I believe it may be regarded as extension of Jeffrey's notation, or as combination of his boundary mathematics with my algebra of logic. It's quite new to me, so there may be lot of things that need to be corrected.

Union of sets A and B: [AB]
Intersection of sets A and B: AB
Complement of set A (to universal set): <A>
Complement of set A relative to set B: <A>B
Difference of set A from B: A
Symmetric difference of sets A and B: [(<A>B)(A)]
Cartesian product of sets A and B: ([A][B])
Inclusion of set A in set B: [<A>B]

Commutativity: [AB] ≗ [BA]; AB ≗ BA
Associativity: [A[BC]] ≗ [[AB]C] ≗ [ABC]; A(BC) ≗ (AB)C ≗ ABC
Distributivity: [A(BC)] ≗ [AB][AC]; A[BC] ≗ [(AB)(AC)]

Null (or empty) set: □
Universal set: [o]?

Identity laws: [A□] ≗ A; A[o] ≗ A
Complement laws: [A<A>] ≗ [o]; A<A> ≗ □

Idempotent laws: [AA] ≗ A; AA ≗ A
Domination laws: [A[o]] ≗ [o]; A□ ≗ □
Absorption laws: [A(AB)] ≗ A; A[AB] ≗ A

Proof of idempotent laws:
[AA] ≗ [AA][o] ≗ [AA][A<A>] ≗ [A(A<A>)] ≗ [A□] ≗ A
AA ≗ [(AA)□] ≗ [(AA)(A<A>)] ≗ A[A<A>] ≗ A[o] ≗ A

DeMorgan's laws: <[AB]> ≗ <A>; <AB> ≗ [<A>]
Double complement or involution: <<A>> ≗ A
Complement laws of universal and empty sets: <[o]> ≗ □; <□> ≗ [o]

Uniqueness of complements: [<([AB]=[o] AB=□)> B=<A>] ≗ [<[AB]<(AB)>> B=<A>]

Reflexivity: [<A>A]
Antisymmetry: [<A>B][A] ≗ A=B
Transitivity: [<[<A>B][C]>[<A>C]]

If A, B and C are subsets of S:
Least element and greatest element: [<□>A]? and [<A>S]
Joins: [<A>[AB]]; [<[<A>C][C]>[<[AB]>C]]
Meets: [<AB>A]; [<[<C>A][<C>B]>[<C>(AB)]]

[<A>B] ≗ AB=A ≗ [AB]=B ≗ A=□ ≗ [<><A>]

Relative complements:
C<AB> ≗ [(C<A>)(C)]
C<[AB]> ≗ C<A>
C<B<A>> ≗ [(AC)(C)]
[(B<A>)]C ≗ [([(BC)]<A>)] ≗ [(B[(C<A>)])]
[(B<A>)C] ≗ [BC]<A<C>>
A<A> ≗ □
□<A> ≗ □
A<□> ≗ A
B<A> ≗ <A>B
<B<A>> ≗ [A]
[o]<A> ≗ <A>
A<[o]> ≗ □

De Morgan's types 3

2008-01-28T08:10:00.000+02:00

In a comment to Brandon's post about subalternation and existential import I said:

"difference between traditional and modern understanding of the relation between A and I propositions is in a sense similar to the difference between deduction and implication. In deduction we assume premises and are able to deduce conclusion, in implication we don't assume antecedent."

In De Morgan's spicular notation, modified as noted in my previous post (ie. '.' marks particularity), this idea is easy to express. Difference between modern and traditional interpretations of A proposition is "a matter of dot":

S.))P -- traditional interpretation
S))P -- modern interpretation

Dot makes explicit the enthymetical assumption that there are S-s we are talking about. It doesn't matter whether they are real or nominal only.

De Morgan's types 2

2008-01-27T18:40:00.000+02:00

I'll try now to write down syllogisms contained in De Morgan's zodiac in modified spicular notation, where dot does not signify negation but particularity. I'll present each universal syllogism together with it's opposing particular syllogisms.

X))Y))Z=X))Z
Y((X(.(Z=Y(.(Z
X(.(Z((Y=X(.(Y

Z((Y((X=Z((X
Y))Z).)X=Y).)X
Z).)X))Y=Z).)Y

X))Y)(Z=X)(Z
Y((X(.)Z=Y(.)Z
X(.)Z)(Y=X(.(Y

Z)(Y((X=Z)(X
Y)(Z(.)X=Y).)X
Z(.)X))Y=Z(.)Y

X)(Y()Z=X))Z
Y)(X(.(Z=Y).(Z
X(.(Z()Y=X(.)Y

Z()Y)(X=Z((X
Y()Z).)X=Y(.)X
Z).)X)(Y=Z).(Y

X()Y))Z=X()Z
Y()X).(Z=Y(.(Z
X).(Z((Y=X).(Y

Z((Y()X=Z()X
Y))Z).(X=Y).(X
Z).(X()Y=Z).)Y

Now, this was a piece of cake. Iconicity of the spicular notation is a great help.

De Morgan's categorical types

2008-01-25T22:21:00.000+02:00

In a comment to the last post, Brandon asked what would be the equivalent of De Morgan's "zodiac" (section 48 of the "Syllabus of a Proposed System of Logic") in my notation?

Here it goes:
1. Let's enumerate symbols in zodiac with clock numerals, so that 12 is in the top, 3 on right, 6 in bottom, 9 on left.
1 )( 2 () 3 )) 4 ).) 5 ).) 6 ).( 7 () 8 )( 9 (.) 10 ).) 11 ).) 12 ))

Corresponding formulas in my notation (for coherence I'll use De Morgan's letters):
A. Read clockwise
1 (-X-Y) 2 (+X+Y) 3 [-X+Y] 4 (-X+Y) 5 (-X+Y) 6 [-X-Y]
7 (+X+Y) 8 (-X-Y) 9 [+X+Y] 10 (-X+Y) 11 (-X+Y) 12 [-X+Y]

B. Read counterclockwise (ie. assymmetric formulas change)
1 (-X-Y) 2 (+X+Y) 3 [+X-Y] 4 (+X-Y) 5 (+X-Y) 6 [-X-Y]
7 (+X+Y) 8 (-X-Y) 9 [+X+Y] 10 (+X-Y) 11 (+X-Y) 12 [+X-Y]

Universal syllogisms are composed of universal premises at 12, 3, 6 and 9, read either clockwise or counterclockwise:

a1) 12[-X+Y]+3[-Y+Z]=[-X+Z]
a2) 3[+Z-Y]+12[+Y-X]=[+Z-X]
b1) 3[-X+Y]+6[-Y-Z]=[-X-Z]
b2) 6[-Z-Y]+3[+Y-X]=[-Z-X]
c1) 6[-X-Y]+9[+Y+Z]=[-X+Z]
c2) 9[+Z+Y]+6[-Y-X]=[+Z-X]
d1) 9[+X+Y]+12[-Y+Z]=[+X+Z]
d2) 12[+Z-Y]+9[+Y+X]=[+Z+X]

For every universal syllogism there are two opposed syllogisms, constructed of one universal premise retained from universal syllogism and of a particular premise adjacent to it in zodiac but "external", which is contradictory to the conclusion of the corresponding universal syllogism. Premises are read in opposite direction to the universal syllogism they oppose, and retained universal premise is converted. I understand that adjacent "external" formulas in comparison to eg. 12 and 3 are correspondingly 11 and 4. Hence:

a11) 12[+Y-X]+11(+X-Z)=(+Y-Z)
a12) 4(+X-Z)+3[+Z-Y]=(+X-Y)
a21) 3[-Y+Z]+4(-Z+X)=(-Y+X)
a22) 11(-Z+X)+12[-X+Y]=(-Z+Y)
b11) 3[+Y-X]+2(+X+Z)=(+Y+Z)
b12) 7(+X+Z)+6[-Z-Y]=(+X-Y)
b21) 6[-Y-Z]+7(+Z+X)=(-Y+X)
b22) 2(+Z+X)+3[-X+Y]=(+Z+Y)
c11) 6[-Y-X]+5(+X-Z)=(-Y-Z)
c12) 10(+X-Z)+9[+Z+Y]=(+X+Y)
c21) 9[+Y+Z]+10(-Z+X)=(+Y+X)
c22) 5(-Z+X)+6[-X-Y]=(-Z-Y)
d11) 9[+Y+X]+8(-X-Z)=(+Y-Z)
d12) 1(-X-Z)+12[+Z-Y]=(-X-Y)
d21) 12[-Y+Z]+1(-Z-X)=(-Y-X)
d22) 8(-Z-X)+9[+X+Y]=(-Z+Y)

Wow! That was not easy but it works. Was that a memory-device?

Missing types 7

2008-01-23T22:29:00.000+02:00

To be uneducated is a great misery. Today I discovered De Morgan's eight types of categorical propositions (in "Syllabus of Proposed System of Logic"), which match my types, even if he seems not to have recognized the same natural language expressions for new types. His notation is interesting:

Universal propositions:
[-X+Y]: X))Y All Xs are some Ys
[-X-Y]: X).(Y All Xs are not (all) Ys
[+X+Y]: X(.)Y Everything is either some X or some Y (or both)
[+X-Y]: X((Y Some Xs are all Ys

Particular propositions:
(+X-Y): X(.(Y Some Xs are not (all) Ys
(+X+Y): X()Y Some Xs are some Ys
(-X-Y): X)(Y Some things are not either (all) Xs nor (all) Ys
(-X+Y): X).)Y All Xs are not some Ys

'X)' and '(X' is read '(all) X'
'X(' and ')X' is read 'some X'
'.' is negative copula
'X' is positive term, 'x' negative.

For me, this notation would be clearer, if dot would always mark particular proposition. Then the meaning of symbols would be almost iconically given. E.g.:

Universal propositions:
X))Y All Xs are only Ys
X)(Y All Xs are not any Ys
X()Y Only Xs are not only Ys
X((Y Only Xs are any Ys

Particular propositions:
X(.(Y Some Xs are not any Ys
X(.)Y Some Xs are some Ys
X).(Y Some things are neither any Xs nor any Ys
X).)Y All Xs are not some Ys

He calls this notation spicular, borrowing the name from Sir Hamilton, who characterized it as "horrent with mysterious spiculae."

On algebra of logic 4

2008-01-18T23:31:00.001+02:00

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)]]

On algebra of logic 3

2008-01-09T00:14:00.001+02:00

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
...

On algebra of logic 2

2008-01-06T14:22:00.000+02:00

Brandon's thoughts on my version of logic algebra lead me to think anew about propositional version of it. And I came to conclusion that in addition to thinking of the propositional version of it as simple transformation where term letters are replaced by proposition letters, we can interpret it also as modal propositional logic. And it reveals amazing unity between different "layers" of logic.

As said, we can move to ordinary propositional logic (in singleton universe as Brandon calls it) by simply replacing terms by propositions. Then we should interpret parentheses simply as conjunction and brackets as disjunction. But this in not fully equivalent translation of quantified term logic. Rather, it is equivalent to term logic with single individual.

But if we transfer quantification also, then we have modal propositional logic. Parentheses, as usual, set up particular context, which should be read as In some case... Brackets, as usual, set up universal context, read as In every case... (Ordinary propositional logic instead presumes the preamble It is the case that...)

[-p+q] In every case (/necessarily), if p is true, q is also
[-p-q] In every case (/necessarily), if p is true, q is not
[+p-q] In every case (/necessarily), if p is true, q may be true
[+p+q] In every case (/necessarily), if p is true, q may be not true
(-p+q) In some case (/possibly), p is not true, but q is
(-p-q) In some case (/possibly), neither p nor q is true
(+p-q) In some case (/possibly), p is true but not q
(+p+q) In some case (/possibly), both p and q are true

([+p+q]) In some case (/possibly), either p or q is true
[(+p+q)] In every case (/necessarily), both p and q are true
...

Further, [p] is read as Necessarily p, ie. In every case p is true
[-p] Necessarily -p; p is impossible; In every case p is not true
(p) Possibly p; In some case p is true
(-p) Possibly -p; In some case p is not true

In singleton universe, or monotonic propositional logic, (p) is read as just p is true, or It is the case that p (contextually), and it is the same as [p], read as It is the case that p (universally), with all contextual references (places, times, persons...) replaced with absolute references.

Axiom

2008-01-02T10:02:00.000+02:00

It cannot be that there is nothing
Nothing is unthinkable
Even if some local setting
Empty is (as it may be)
It itself is something that
Excludes the global nullity
Therefore
Something is

On algebra of logic 1

2007-12-26T16:05:00.001+02:00

Contexts

Brackets and parentheses play important role in this version of logic algebra. Brackets are used to set up universal context, parentheses to set up particular context. With this interpretation [P] means Everything is P and (P) means Something is P. Accordingly [-P] means Everything is non-P, which is equivalent to -(P) -- Nothing is P. Also (-P) means Something is non-P, which is equivalent to -[P] -- Not everything is P.

How to represent singular propositions? In SETL singulars are treated as having "wild quantity" and are marked with ±, signifying that they may be treated as universals as well as particulars. Singulars are peculiar indeed as they may be seen as predicates belonging to single individual. But it seems to me we don't need any special quantity-marker for singulars. It is the question of semantics, not of special logical treatment. When used in universal context, singular term should be read "hypothetically" (as universals are also). Eg. with T for Tim [-'T'+P] should be read If there is Tim, he is P, or Either Tim is not (there) or he is P. But ('T'+P) is read just Tim is P, ie. Something is Tim and he is P. Again, ['T'-P] is read Only Tim is P, or Either something is Tim or it is not P. Singularity is marked here by single quotes, signifying name, and by dropping '+' from before it, but logical treatment does not differ from that of universals.

Nested Contexts

Contexts may be nested. Universal contexts may contain other universal contexts as well as particular contexts, and vice versa, particular contexts may contain universal and other particular contexts. Eg. All S is P and Q is expressed as [-S+(P+Q)]. As universal context does not by itself instantiate any individuals, (P+Q) should not be read here as positing some P which is Q. Read in universal context it says If there is any S, then it is something that is P and Q. If there is no S, there need not be any P that is Q. Hence as well [(P+Q)] does not posit any P that is Q. It says, If there is anything, then it is something, that is P and Q. As already said, in case of single letter [P] expresses Everything is P, or If there is anything, then it is (something that is) P. Same is expressed by [(P)], where parentheses are spelled out -- Everything is something, that is P, or If there is anything, then it is something, that is P.

By adding extra parentheses we can particularize any expression in universal context. Eg. [((P))] means Everything is something, some of which is P. Similarly in case of complex expressions, eg. [((P+Q))] means Everything is something, some of which is P that is Q. From here it is easy to see that ((P+Q)) expresses There is something, some of which is P that is Q. And to express There is something, some of which is P, I'll use ((P)). When used inside conjunction, as in (S+(P+Q)), which is read Some S is something that is P and Q, there are several ways to proceed. We can abstract from some S by just dropping S, and say ((P+Q)), which means There is something, some of which is P and Q. We can also rise to all S: [-S+((P+Q))] All S is something, some of which is P and Q. But we can also replace S with (P+Q), saying Something is P and Q, for something is S, and it is P and Q, therefore something is P and Q, or Some P is Q (equivalent to conjunction elimination).

Universal context nested in particular context expresses idea, that there is something, that is wholly ... (whatever is contained in nested universal context). Eg. (+S+[P+Q]) expresses, that Some S is either P or Q, or There is something, that is S and either P or Q. Similarly ([P+Q]) says that Something is either P or Q, or There is something, all of which is either P or Q. What about ([P])? This is just the other way of expressing (P) Something is (wholly) P. Hence, any depth of nesting of alternating universal and particular contexts for single letter may be reduced to the outermost context: [([([(P)])])] reduces to [P], but in case of complex expressions both innermost and outermost contexts should be retained.

But what happens when we double universal context, eg. [[P]]? It seems there is no difference between [P] All is (some) P and [[P]] All is only P. But when we use it in both sides of disjunction, the difference in meaning becomes important. [[P]+[Q]] All is only P or only Q is clearly different from [P+Q] All is P or Q. Compare this to [(P)+(Q)] All is some P or some Q, which seems to be equivalent to All is P or Q. In last cases disjunction is applied to every individual, but in case of [[P]+[Q]] to the whole. [P+Q] is disjunction of predicates, [[P]+[Q]] is disjunction of propositions. We may follow this in case of Boole's example:
(1) Every inhabitant (of an island) is either European or Asiatic:
[-I+[E+A]]=[-I+E+A]
Everyone is either European or Asiatic: [E+A] (Only Europeans are not Asiatics)
(2) Every inhabitant is European or every inhabitant is Asiatic:
[[-I+E]+[-I+A]]=[-I+[E]+[A]]
Either everyone is European or everyone is Asiatic: [[E]+[A]]

In (1), as said, disjunction of predicates is applied to every individual in given domain, but in (2) predicates are applied to the whole domain at once. We may call it double universalisation in contrast to double particularization as in ((P)).

Multiple quantification

2007-12-26T14:03:00.000+02:00

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)

Relational categorical propositions

2007-12-16T10:33:00.000+02:00

I know Sommers and Englebretsen and others have treated these matters, but I'd like to think it over myself.

There are three reasons why term logic is regarded inadequate:

1. Inability to treat reflexive inferences, eg.
(a) Everybody killed himself
(b) Cato killed Cato
(c) Someone killed Cato

2. Inability to treat propositions with multiple quantification, eg.
(d) All horses are animals
(e) All horses' heads are animals' heads

3. Inability to treat transitions between term logic and propositional logic, eg.
(f) Every inhabitant of the island is either Asiatic or European
(g) Either every inhabitant of the island is Asiatic or s/he is European

Relational problems

First and second problem belong both to relational difficulties. Normal logical form for (a) is
Everybody is someone who killed himself
or, in more general form
Everybody is someone who is somehow related to himself
and in existential form
Everybody is himself
or in its most general form
Everything is itself
which might be called the axiom of identity.

This axiom can be weakened in several ways:
One way to weaken it is to go back by one step and to consider only one class of beings as in
Everybody is himself
or
Every blog is itself...
for, as everything is itself, everything in every class of things is also. This is the way of constricting the subject.

The other way is to loosen the subject as in
Something is itself
for, as everything is itself and there is something, something is itself. Constricting and loosening of the subject may be taken together:
Somebody is himself.

Third way is to loosen the predicate as in
Everything is something
for, as it is itself, it is something. This is strong form of traditional categorical proposition.

Fourth way is to loosen the copula, ie. to generalize the way how subject is related to predicate:
Everything is somehow related to itself
for, as it is itself, it is somehow related to itself.

All this may be taken together in strong form as in
Everything is somehow related to something
which might be called the axiom of relatedness, or in weak form as in
Something is somehow related to something
which may be considered a general form of relational categorical proposition in contrast to simple categorical proposition in form
Something is something.

From these considerations we can resolve the first difficulty:

Everybody killed himself, (assertion)
therefore Cato killed himself. (weakening)
Everybody is himself, (axiom)
therefore Cato is himself. (weakening)
Cato killed himself,
and Cato is himself,
therefore Cato killed Cato.
Cato killed Cato,
and Cato is somebody,
therefore somebody killed Cato
(and Cato killed somebody).

Spe Salvi

2007-12-12T22:00:00.000+02:00

Pope Benedict wrote about eternal life:

12. [...] To imagine ourselves outside the temporality that imprisons us and in some way to sense that eternity is not an unending succession of days in the calendar, but something more like the supreme moment of satisfaction, in which totality embraces us and we embrace totality—this we can only attempt. It would be like plunging into the ocean of infinite love, a moment in which time—the before and after—no longer exists. We can only attempt to grasp the idea that such a moment is life in the full sense, a plunging ever anew into the vastness of being, in which we are simply overwhelmed with joy. This is how Jesus expresses it in Saint John's Gospel: “I will see you again and your hearts will rejoice, and no one will take your joy from you” (16:22). We must think along these lines if we want to understand the object of Christian hope, to understand what it is that our faith, our being with Christ, leads us to expect.

This would be fulfillment (and therefore ceasing) of all hope indeed -- the fullness of being, where God is all in all.

What would Hell be in these terms? It would also be ceasing of all hope, the fullness or realizing that you have Failed. It is, one might say, fulfailment of all hope.

Missing types 6

2007-12-10T08:59:00.000+02:00

Finally I reach to parasyllogisms with mixed universal and particular premises, seen traditionally as invalid. Here we go:

Particular minor and universal major

Not only S is M, which alone is not P,
therefore not only S is something, that alone is not P.

Not only S is M, which alone is P,
therefore not only S is something, that alone is P.

Not only S is not M, all of which is P,
therefore not only S is something, that alone is not P.

Not only S is not M, none of which is P,
therefore not only S is something, that alone is P.

Some S is M, which alone is not P,
therefore some S is something, that alone is not P.

Some S is M, which alone is P,
therefore some S is something, that alone is P.

Some S is not M, all of which is P,
therefore some S is something, that alone is not P.

Some S is not M, none of which is P,
therefore some S is something, that alone is P.

Before giving it in tabular form, let's introduce the relevant algebraic forms. First, notice the unintuitive reading of the expression [+S+P] -- Only S is not P or S alone is not P. But this is consistent with universal disjunctive reading Everything is either S or P. When we want to express in particular context that Some S is something, that alone is not P, we have to write (+S+[[P]]). This is equivalent with Only P is not something, some of which is S -- [+P+((S))]. To see that they are equivalent, let's draw their Venn diagram. Unknown something will be marked with X:

Some S is something, that alone is not P (+S+[[P]])

Only P is not something, some of which is S [+P+((S))]

Also notice, that as X may represent M as well as -M, premises with same extremes give equivalent results, whether middle term in both premises is +M or -M.

And here is the table with algebraic expressions:

Table 1	[-M+P]	[-M-P]	[+M+P]	[+M-P]
(-S+M)	(-S+P)	(-S-P)	(-S+[[P]])	(-S+[[-P]])
(-S-M)	(-S+[[P]])	(-S+[[-P]])	(-S+P)	(-S-P)
(+S+M)	(+S+P)	(+S-P)	(+S+[[P]])	(+S+[[-P]])
(+S-M)	(+S+[[P]])	(+S+[[-P]])	(+S+P)	(+S-P)

Universal minor and particular major

All S is M, some of which is P,
therefore all S is something, some of which is P.

All S is M, some of which is not P,
therefore all S is something, some of which is not P.

No S is M, and not only Mis P,
therefore all S is something, some of which is P.

No S is M, and not only M is not P,
therefore all S is something, some of which is not P.

Only S is not M, some of which is P,
therefore only S is not something, some of which is P.

Only S is not M, some of which is not P,
therefore only S is not something, some of which is not P.

Only S is M, but not only M is P,
therefore only S is not something, some of which is P.

Only S is M, but not only M is not P,
therefore only S is not something, some of which is not P.

Table 2	(-M+P)	(-M-P)	(+M+P)	(+M-P)
[-S+M]	(-S+P)	(-S-P)	[-S+((P))]	[-S+((-P))]
[-S-M]	[-S+((P))]	[-S+((-P))]	(-S+P)	(-S-P)
[+S+M]	(+S+P)	(+S-P)	[+S+((P))]	[+S+((-P))]
[+S-M]	[+S+((P))]	[+S+((-P))]	(+S+P)	(+S-P)

Equivalencies between tables

As said before, tables 1 and 2 are reducible to each other, if we reinterpret upper row in Table 1 as containing minor premises, and left column as containing major premises. Letters S and P have to be reinterpreted accordingly. I'll give the equivalences between two tables as references from reinterpreted Table 1 cells to equivalent cells of Table 2, with numbering of rows and columns referring to data section only, without headers:

Table 1'	[-M+S]	[-M-S]	[+M+S]	[+M-S]
(-P+M)	R4C4	R2C4	R3C4	R1C4
(-P-M)	R4C2	R2C2	R3C2	R1C2
(+P+M)	R4C3	R2C3	R3C3	R1C3
(+P-M)	R4C1	R2C1	R3C1	R1C1

Bolded references are to cells in their own respective position, colored references to paired cells of their own color.

Missing types 5

2007-12-06T22:57:00.001+02:00

What about the rest of combinations of the premises? To say it briefly, all of these do yield definite conclusions, but none of these are expressible with our basic types. But this is not a reason to reject them and say that we cannot reason about them or that they do not contain any useful information for further reasoning. To separate these from "normal" syllogisms I'll call them parasyllogisms. Let's have a closer look at them. I'll start with parasyllogisms where both premises are particular.

Not only S is M, and not only M is P, (-S+M)+(-M+P)
therefore not only S is not some P. (-S)≠(P)
Not only S is something, that is not alone P.
There is a predicate, that not only S do not share with some P.

Not only S is M, and not only M is not P, (-S+M)+(-M-P)
therefore not only S is not not only P. (-S)≠(-P)
Not only S is something, that is not alone not P.
There is a predicate, that not only S do not share with not only P.

Not only S is M, and some M is P, (-S+M)+(+M+P)
therefore not only S has something common with some P. ((-S)+(P))
Not only S is something, some of which is P.
There is a predicate, that not only S share with some P.

Not only S is M, and some M is not P, (-S+M)+(+M-P)
therefore not only S has something common with not only P. ((-S)+(-P))
Not only S is something, some of which is not P.
There is a predicate, that not only S share with not only P.

---------

Not only S is not M, and not only M is P, (-S-M)+(-M+P)
therefore not only S has something common with some P. ((-S)+(P))
Not only S is not something, that is not alone P.
There is a predicate that not only S share with some P.

Not only S is not M, and not only S is not P, (-S-M)+(-M-P)
therefore not only S has something common with not only P. ((-S)+(-P))
Not only S is not something, that is not alone not P.
There is a predicate that not only S share with not only P.

Not only S is not M, some of which is P, (-S-M)+(+M+P)
therefore not only S is not some P. (-S)≠(P)
Not only S is not something, some of which is P.
There is a predicate that not only S do not share with some P.

Not only S is not M, some of which is not P, (-S-M)+(+M-P)
therefore not only S is not not only P. (-S)≠(-P)
Not only S is not something, some of which is not P.
There is a predicate that not only S do not share with not only P.

---------

Some S is M, and not only M is P, (+S+M)+(-M+P)
therefore some S is not some P. (S)≠(P)
Some S is something, that is not alone P.
There is a predicate that some S do not share with some P.

Some S is M, and not only M is not P, (+S+M)+(-M-P)
therefore some S is not not only P. (S)≠(-P)
Some S is something, that is not alone not P.
There is a predicate that some S do not share with not only P.

Some S is M, and some M is P, (+S+M)+(+M+P)
therefore some S has something common with some P. ((S)+(P))
Some S is something, some of which is P.
There is a predicate that some S share with some P.

Some S is M, but some M is not P, (+S+M)+(+M-P)
therefore some S has something common with not only P. ((S)+(-P))
Some S is something, that is not only P.
There is a predicate that some S share with not only P.

-------------

Some S is not M, and not only M is P, (+S-M)+(-M+P)
therefore some S has something common with some P. ((S)+(P))
Some S is not something that is not all P.
There is a predicate that some S share with some P.

Some S is not M, and not only M is not P, (+S-M)+(-M-P)
therefore some S has something common with not only P. ((S)+(-P))
Some S is not something, that is not all not P.
There is a predicate that some S share with not only P.

Some S is not M, some of which is P, (+S-M)+(+M+P)
therefore some S is not some P. (S)≠(P)
Some S is not something, some of which is P.
There is a predicate that some S do not share with some P.

Some S is not M, some of which is not P, (+S-M)+(+M-P)
therefore some S is not not only P. (S)≠(-P)
Some S is not something, some of which is not P.
There is a predicate that some S do not share with not only P.

-------------------

All this is taken together in the following table:

	(-M+P)	(-M-P)	(+M+P)	(+M-P)
(-S+M)	(-S)≠(P)	(-S)≠(-P)	((-S)+(P))	((-S)+(-P))
(-S-M)	((-S)+(P))	((-S)+(-P))	(-S)≠(P)	(-S)≠(-P)
(+S+M)	(S)≠(P)	(S)≠(-P)	((S)+(P))	((S)+(-P))
(+S-M)	((S)+(P))	((S)+(-P))	(S)≠(P)	(S)≠(-P)

Each parasyllogism with both premises particular affirms either the existence of a predicate that extreme terms share or the existence of a predicate that they do not share. Speaking in predicate logic language, we leave the safe waters of first order logic and sail into less known and limitless ocean of second order logic.

Missing types 4

2007-11-29T11:19:00.000+02:00

Let's look now at syllogisms with combined universal and particular premises. Here are some syllogisms in first figure with particular minor premise and universal major:

Not only S is M, all of which is P, therefore not only S is P.
Not only S is M, none of which is P, therefore not only S is not P.
Not only S is not M, but only M is not P, therefore not only S is P.
Not only S is not M, but only M is P, therefore not only S is not P.
Some S is M, all of which is P, therefore some M is P.
Some S is M, none of which is P, therefore some S is not P.
Some S is not M, but only M is not P, therefore some S is P.
Some S is not M, but only M is P, therefore some S is not P.

Only fifth and sixth are traditionally recognized (Darii and Ferio) in first figure. And here are corresponding algebraic formulas:

	[-M+P]	[-M-P]	[+M+P]	[+M-P]
(-S+M)	(-S+P)	(-S-P)
(-S-M)			(-S+P)	(-S-P)
(+S+M)	(+S+P)	(+S-P)
(+S-M)			(+S+P)	(+S-P)

What will happen if we use universal minor and particular major, which use is ruled out in traditional syllogistics?

All S is M, but not only M is P, therefore not only S is P.
All S is M, but not only M is not P, therefore not only S is not P.
No S is M, some of which is P, therefore not only S is P.
No S is M, some of which is not P, therefore not only S is not P.
Only S is not M, but not only M is P, therefore some S is P.
Only S is not M, but not only M is not P, therefore some S is not P.
Only S is M, some of which is P, therefore some S is P.
Only S is M, some of which is not P, therefore some S is not P.

None of these are recognized in traditional syllogistics as belonging to first figure, but some are found in other figures with converted or inverted premises. Here is the table with formulas:

	(-M+P)	(-M-P)	(+M+P)	(+M-P)
[-S+M]	(-S+P)	(-S-P)
[-S-M]			(-S+P)	(-S-P)
[+S+M]	(+S+P)	(+S-P)
[+S-M]			(+S+P)	(+S-P)

Remembering that terms in formulas may be switched we have again the complete catalog of syllogisms with mixed premises. Traditionally recognized syllogisms are following (colored):

table 1, row 4, column 2 -- Darii, Datisi
table 1, row 4, column 3 -- Ferio, Festino, Ferison, Fresison
table 1, row 5, column 5 -- Baroco
table 2, row 5, column 4 -- Disamis, Dimaris
table 2, row 5, column 5 -- Bocardo

We could reduce two tables into single, in which case top row would be interpreted twice -- once as major, and once as minor premise, and left column accordingly as minor or major. In second case Ss and Ps should be reinterpreted in all expressions, premises as well as conclusions. Subjects in conclusions will thus be on second position. In this case Disamis and Dimaris join Darii and Datisi, but Bocardo will not join Baroco, but will take the position in row 2, column 2 (green). Thus we are left with eight mixed syllogism types, four of which have been dealt with in traditional syllogistics.

Missing types 3

2007-11-28T16:06:00.000+02:00

Influences on Syllogistics

Categorical syllogisms are essentially conjunctions of two categorical propositions sharing one term, which relates the other two terms in definite manner, as expressed in conclusion. I'll put minor premise before major, as it seems more natural way of sequencing premises. It is also historically justified by example of Aristotle himself. Although he set major premise before minor, it seems he did so because terms were in different order. So, his perfect syllogism reads as A belongs to all B, which belongs to all C, therefore A belongs to all C. He saw conclusion of such exposition as immediately and self-evidently true. Psychological self-evidence was reason for laying this inference the cornerstone of his deductive system. Because normal term-sequence in our logical forms is different, sequence of propositions should also be different, if it is to have similar psychological force. Traditionally recognized syllogisms in first figure may thus be expressed as following:

All S is M, all of which is P, therefore all S is P. (Barbara)

All S is M, none of which is P, therefore no S is P. (Celarent)

Some S is M, all of which is P, therefore some S is P. (Darii)

Some S is M, none of which is P, therefore some S is not P. (Ferio)

Let's add now our new forms to the first figure:

No S is M, but only M is P, therefore no S is P.

No S is M, but only M is not P, therefore all S is P.

First surprise! We have valid syllogisms yielding universal conclusions in first figure with universal negative as minor premise. Further:

Only S is M, and only M is P, therefore only S is P.

Only S is M, and only M is not P, therefore only S is not P.

Only S is not M, all of which is P, therefore only S is not P.

Only S is not M, none of which is P, therefore only S is P.

Now we have 8 universal syllogisms in first figure, with two conclusions of each universal type! What about other combinations of universal premises? Let's see:

All S is M, and only M is P, therefore, presuming not all is M,
not only S is not P (or: something is neither S nor P).

All S is M, and only M is not P, therefore, presuming not all is M,
not only S is P (or: something is not S, but is P).

No S is M, all of which is P,
therefore, presuming there is some M, not only S is P.

No S is M, none of which is P,
therefore, presuming there is some M, not only S is not P.

Only S is M, all of which is P,
therefore, presuming there is some M, some S is P.

Only S is M, none of which is P,
therefore, presuming there is some M, some S is not P.

Only S is not M, and only M is P,
therefore, presuming not all is M, some S is not P.

Only S is not M, and only M is not P,
therefore, presuming not all is M, some S is P.

Next surprise! All combinations of universal propositions yield conclusions! 8 universal and 8 particular conclusions, two for each type. Half of these presume some existential claims about middle term, i.e. that there is some M (when middle term is distributed in both premises) or that not everything is M (there is something, which is not M -- when middle term is distributed in neither). These existential claims refer to the domain of discourse and are not satisfied only if M comprehends the whole domain or is internally contradictory.

Let's now use SETL expressions expanded by contextual markers to see the whole algebraic picture at once. I'll put major premises in top row and minor premises in left column:

	[-M+P]	[-M-P]	[+M-P]	[+M+P]
[-S+M]	[-S+P]	[-S-P]	(-S-P)¹	(-S+P)¹
[-S-M]	(-S+P)²	(-S-P)²	[-S-P]	[-S+P]
[+S-M]	(+S+P)²	(+S-P)²	[+S-P]	[+S+P]
[+S+M]	[+S+P]	[+S-P]	(+S-P)¹	(+S+P)¹

¹ Presupposition: Not all is M
² Presupposition: There is some M

We see here some interesting (or disturbing?) results:
1) Two negations give conclusion.
2) Premises with middle in both undistributed give conclusions.
3) As terms in all expressions may be switched (together with their operators), resulting in equivalent expressions (converses or inverses), the table actually comprises all syllogisms with universal premises in all figures -- colored expressions are those traditionally recognized. In first row are Barbara and Celarent, which is equivalent to Cesare (with major premise converted). In second row is Camestres (with inverted major), and the same is also Camenes (with major inverted and minor converted). In third row are Darapti and Felapton (with minor inverted in both), latter is also Fesapo (with major converted and minor inverted), and then Bramantip (with both major and minor inverted) in green, for conclusion was chopped into particular form because universal form was not recognized.

Missing types 2

2007-11-27T17:06:00.000+02:00

Christening new types

So, there are four additional types of categorical propositions, introduced in my previous post, two universal and two particular. How should we call these? I propose to dub Only S is P "universal exclusive", for it excludes P from belonging to anything besides S, and Only S is not P "universal inclusive", for it subsumes everything not belonging to S under P. Similarly Not only S is not P would be called "particular exclusive", for it excludes some indefinite particular from both S and P, and Not only S is P "particular inclusive". Latter name is a bit obscure, although we could justify it by pointing out that some indefinite particular (-S) is subsumed under P. But as it is equivalent with Some P is not S one might object that it excludes some P from S. As I can't find a better name, I'll stick with this, as it completes a nice symmetry. Thus eight types, each paired with its contradictory, are:

Universal affirmative ----- Particular negative
Universal negative ----- Particular affirmative
Universal exclusive ----- Particular inclusive
Universal inclusive ----- Particular exclusive

Venn diagrams, distribution and algebra

In following, dashed lines mark undistributed terms and continuous lines distributed terms.

Universal exclusive: Only S is P

In Sommers-Englebretsen term logic (SETL) algebraic notation:
-(-S)-P

This becomes a bit clumsy, and I would prefer the following notation:
[+S-P]

Universal inclusive: Only S is not P

SETL:

-(-S)-(-P)

My notation:

[+S+P]

Particular inclusive: Not only S is P

SETL:

+(-S)+P

My notation:

(-S+P)

Particular exclusive: Not only S is not P

SETL:

+(-S)-P

My notation:

(-S-P)

To make the SETL algebra easily workable with new types, I branch off from it by adding "environment markers" to its set of symbols. Brackets '[' and ']' mark universal context and parentheses '(' and ')' mark particular context. In SETL these contexts were marked by - and + as quantity-markers respectively. This reading can still be retained as default interpretation in unmarked contexts. Default connector will be disjunction in universal and conjunction in particular context. Thus, in algebraic notation all types are as following:

Universal affirmative: [-S+P] -- Everything is either not S or P
Universal negative: [-S-P] -- Everything is either not S or not P
Universal exclusive: [+S-P] -- Everything is either S or not P
Universal inclusive: [+S+P] -- Everything is either S or P
Particular affirmative: (+S+P) -- Something is both S and P
Particular negative: (+S-P) -- Something is S but not P
Particular exclusive: (-S-P) -- Something is neither S nor P
Particular inclusive: (-S+P) -- Something is not S but is P

When we use context markers, terms in all expressions may change position together with their +/- marker without changing the meaning of the expression. This becomes important when we introduce our new types into syllogisms. All nice features of SETL still work, with additional note that negation of an expression will change its context also:

-[-S+P] = (+S-P)
-[-S-P] = (+S+P)
-[+S-P] = (-S+P)
-[+S+P] = (-S-P)