Proving the equivalence of all valid syllogism using the pictorial object logic

1. Syllogistic
Unity
Proving
the Equivalency of All
Syllogisms
Using Object Logic
Armahedi Mahzar
© 2011

2. Foreword
Logic is the science of thinking as it is
discovered by Aristotle. In his treatise of
syllogism he used alphabets to represent
concept in his verbal logic. George Boole
created an algebra of logic by representing
logical operations with mathematical symbols
besides letters as variables. These
symbolizations is still linear literal.
Charles Sanders Peirce rewrote boolean
algebra in a planar pictorial symbols by using
pictures as the symbols of logic, but he still
used alphabets as the symbols of variables.
The pictorial symbolization is also used by
George Spencer-Brown having a half of a box,
which he called cross, to replace the ovals of
Peirce
Louis Kauffman replaced the Brownian cross
with a complete box in his pictorial Box
Algebra of logic.
In the following slides we will make the Box
Algebra more pictorial, by replacing letters
with colored objects to get an Object Logic.
Finally, we will use the Object Logic to prove
the astounding fact of Syllogistic Unity.

3. Part One:
Logic Algebra of Objects
In this part the Boolean
algebra is made pictorial by
Replacing letters with
colored objects
Replacing mathematical
symbols with boxes
configuration

4. LOGICAL NOTATION

5. Two Interpretations
of Kauffman Box Algebra
Kauffman Box algebra is a rewriting
of the Spencer-Brown “Laws of Form”
Algebra
But it can also be interpreted as
rewriting of the Existential Graph
Algebra of Peirce
The following presentation follows
Peircean interpretation with colored
marbles as variables

6. FUNDAMENTAL LAWS
OF LOGIC
LAWS OF NEGATION
NOT TRUE = FALSE
NOT FALSE = TRUE
LAWS OF CONJUNCTION
TRUE AND TRUE = TRUE
TRUE AND FALSE = FALSE
FALSE AND TRUE = FALSE
FALSE AND FALSE = FALSE

7. Basic Box Arithmetic
LAW OF
NEGATION
LAW OF
CONJUNCTION
From this Box Arithmetic we can build a
logic algebra discovered by George Boole.
Alfred North Whitehead and Bertrand
Russel derived the whole Boolean Algebra
on five axioms. George Spencer-Brown
reduced the axiom into just two axiom in
his Laws of Form Primary Algebra. Louis
Kaufman reduced the axioms to just one in
his Box Algebra.

8. Axiom of the
Logic Box Algebra
The single Axiom for Logical
Box Algebra is Huntington
tautology

9. The Meaning of the
Axiom:
Reductio ad Absurdum
The Huntington
Axiom box
diagram is
The diagram can be
read as
Red is True
if and only if
Not Red implies
Blue and
Not Red implies
Not Blue
which is equivalent
to
Red is True
if only if
Not Red implies
a Contradiction
the Reductio ad
Absurdum
principle

10. Rules of Inference
Rule of Substitution
any variable can be
replaced by a function of
other variables
Rule of Replacement
a function of variables can
be replaced by another
equivalent function of the
same variables
Using these rules we can
derive all Boolean tautologies,
some of them is in the
following page.

11. Agebraic Identities
(logical tautologies) are
theorems
Law of
Absorption
Law of
Negation
Law of
Contradiction
Law of
(De)iteration

12. Implication in BOX
algebra
Logical Proposition
IF p THEN q = TRUE
NOT p OR q = TRUE
p AND NOT q = FALSE
NOT (p AND NOT q)=
TRUE
In the NAND box
algebra notation it
is represented by
In Boolean
Notation
(p
q) =1
p’ + q = 1
p x q’ = 0
(p x q’ )’ = 1

13. Part Two :
Syllogism
In this part we will
reformulate syllogism in a
boolean formula which is
drawn as picture of
enclosing boxes containing
colored objects that
represents concepts.

14. Syllogism as an
Implication
“IF p AND q THEN r”
represented by
p, q and r are fundamental
propositions
p and q are premises
r is conclusion

15. Aristotle Fundamental
Propositions

16. Facts of Syllogism
Every Valid Syllogism is a
Tautology
Leibnitz proved that there
are only 24 Valid
Syllogisms
We will use the NAND
interpreted box algebra of
Kauffman to prove
The syllogistic unity: all
valid syllogisms is equivalent
to each other

17. The names of the valid
syllogisms are
Using symmetric properties
and Boolean Identity , we
have only to prove just the
Barbara syllogism validity.

18. BARBARA
syllogism
Syllogism Barbara =
[[b[c]][a[b]]a[c]]

19. Proof of the validity of
Barbara Syllogism
(All Red is Green & All Green is Blue
is Blue)=TRUE
=
=
deiteration
All Red
=
=
absorption
contradiction
negation

20. Part 3 :
Syllogistic Unity
In this part we will prove
the unity of valid syllogisms
by using its permutational
symmetry, the algebraic
substitution and the
equivalency of different
algebraic expressions

21. STEP 1: Barbara Triad
Barbara,
Baroco and
Bocardo are
equivalent to
each other. All
can be
represented
by single box
diagram
Barbara
Amp Asm
Asp
Baroco
Apm Osm
Osp
Bocardo
Omp Ams
Osp

22. STEP 2:
Celarent Zodiac
The twelve
syllogisms are
equivalent to each
other. All can be
represented by a
single box diagram
Camestres: Arg Egb
Camenes : Arg Ebg
Celarent : Egb Arg
Cesare
: Ebg Arg
Ebr
Ebr
Erb
Erb
Datisi
Darii
Disamis
Diramis
: Arg Ibr
: Arg Irb
: Ibr Arg
: Irb Arg
Ibg
Ibg
Igb
Igb
Ferio
Ferison
Festino
Fresison
: Egb Irb
: Ebg Irb
: Egb Ibr
: Ebg Ibr
Org
Org
Org
Org

23. STEP 3:
Celaront Triad
Celaront,
Cesaro and
Darapti are
equivalent to
each other. All
can be
represented
by single
diagram
Celaront
Emp Asm
Osp
Cesaro
Epm Asm
Osp
Darapti
Amp Ams
Isp

24. STEP 4:
Barbari Hexad
Barbari, Camestros,
Felapton,
Bramantip, Calemos
and Fesapo are
equivalent to each
other. All can be
represented by
single box diagram
Barbari
Amp Asm
Isp
Camestros
Apm Esm
Osp
Felapton
Emp Ams
Osp
Bramantip
Apm Ams
Isp
Calemos
Apm Ems
Osp
Fesapo
Epm Ams
Osp

25. Step 5:
Syllogistic Equivalence
Barbara = Celarent
by substituting
with
Celarent = Barbari
by replacing
with
Celarent = Celaront
by replacing
with

26. 24
valid
syllogisms

27. Conclusion:
Syllogistic Unity
Due to
all the members of the Barbara
triad, Celarent zodiac, Barbari
hexad and Celaront triad are
equivalent to each other, and
the equivalency of BarbaraBarbari-Celarent-Celaront,
all of the 24 syllogism is a
member of a single equivalent
class: the union of the four
classes.
This fact can be called as the
Syllogistic Unity

28. Afterword
The fact of syllogistic unity is
anticipated by Christine LaddFranklin who had shown that all
valid syllogisms can be derived
from her particular antilogism
formula:
In fact the formula is just one
of the 24 valid antilogisms
which are equivalent to each
other, from each of them we
can also derive all valid
syllogism.

29. References
Aristotle :
Non-Mathematical Verbal Logic
http://classics.mit.edu/Aristotle/prior.1.i.html
George Boole:
Algebraic Symbolic Logic (Algebra of Logic)
http://www.freeinfosociety.com/media/pdf/4708.pdf
Charles Sanders Peirce:
Algebraic Graphical Logic (Existential Graph)
http://www.jfsowa.com/peirce/ms514.htm
George Spencer-Brown:
Algebraic Graphical Logic (Laws of Form)
http://www.4shared.com/document/bBAP7ovO/G-spencer-Brown-Lawsof-Form-1.html
Louis Kauffman:
Algebraic Pictorial Logic (Box Algebra)
http://www.math.uic.edu/~kauffman/Arithmetic.htm