Syllogistic
Unity

Proving
the Equivalency of All
Syllogisms
Using Object Logic
Armahedi Mahzar

© 2011
Foreword
Logic is the science of thinking as it is
discovered by Aristotle. In his treatise of
syllogism he used alphabets...
Part One:
Logic Algebra of Objects
In this part the Boolean
algebra is made pictorial by
Replacing letters with
colored ob...
LOGICAL NOTATION
Two Interpretations
of Kauffman Box Algebra

Kauffman Box algebra is a rewriting
of the Spencer-Brown “Laws of Form”
Algeb...
FUNDAMENTAL LAWS
OF LOGIC
LAWS OF NEGATION
NOT TRUE = FALSE
NOT FALSE = TRUE
LAWS OF CONJUNCTION
TRUE AND TRUE = TRUE
TRUE...
Basic Box Arithmetic
LAW OF
NEGATION

LAW OF
CONJUNCTION

From this Box Arithmetic we can build a
logic algebra discovered...
Axiom of the
Logic Box Algebra
The single Axiom for Logical
Box Algebra is Huntington
tautology
The Meaning of the
Axiom:
Reductio ad Absurdum
The Huntington
Axiom box
diagram is

The diagram can be
read as
Red is True...
Rules of Inference
Rule of Substitution
any variable can be
replaced by a function of
other variables
Rule of Replacement
...
Agebraic Identities
(logical tautologies) are
theorems
Law of
Absorption
Law of
Negation
Law of
Contradiction
Law of
(De)i...
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)...
Part Two :
Syllogism

In this part we will
reformulate syllogism in a
boolean formula which is
drawn as picture of
enclosi...
Syllogism as an
Implication
“IF p AND q THEN r”
represented by

p, q and r are fundamental
propositions
p and q are premis...
Aristotle Fundamental
Propositions
Facts of Syllogism
Every Valid Syllogism is a
Tautology
Leibnitz proved that there
are only 24 Valid
Syllogisms
We will us...
The names of the valid
syllogisms are

Using symmetric properties
and Boolean Identity , we
have only to prove just the
Ba...
BARBARA
syllogism
Syllogism Barbara =
[[b[c]][a[b]]a[c]]
Proof of the validity of
Barbara Syllogism
(All Red is Green & All Green is Blue
is Blue)=TRUE

=

=

deiteration

All Red...
Part 3 :
Syllogistic Unity

In this part we will prove
the unity of valid syllogisms
by using its permutational
symmetry, ...
STEP 1: Barbara Triad
Barbara,
Baroco and
Bocardo are
equivalent to
each other. All
can be
represented
by single box
diagr...
STEP 2:
Celarent Zodiac
The twelve
syllogisms are
equivalent to each
other. All can be
represented by a
single box diagram...
STEP 3:
Celaront Triad
Celaront,
Cesaro and
Darapti are
equivalent to
each other. All
can be
represented
by single
diagram...
STEP 4:
Barbari Hexad
Barbari, Camestros,
Felapton,
Bramantip, Calemos
and Fesapo are
equivalent to each
other. All can be...
Step 5:
Syllogistic Equivalence

Barbara = Celarent
by substituting
with

Celarent = Barbari
by replacing
with

Celarent =...
24
valid
syllogisms
Conclusion:
Syllogistic Unity
Due to
all the members of the Barbara
triad, Celarent zodiac, Barbari
hexad and Celaront tri...
Afterword
The fact of syllogistic unity is
anticipated by Christine LaddFranklin who had shown that all
valid syllogisms c...
References
Aristotle :
Non-Mathematical Verbal Logic

http://classics.mit.edu/Aristotle/prior.1.i.html

George Boole:
Alge...
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Transcript of "Syllogistic unity"

  1. 1. Syllogistic Unity Proving the Equivalency of All Syllogisms Using Object Logic Armahedi Mahzar © 2011
  2. 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. 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. 4. LOGICAL NOTATION
  5. 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. 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. 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. 8. Axiom of the Logic Box Algebra The single Axiom for Logical Box Algebra is Huntington tautology
  9. 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. 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. 11. Agebraic Identities (logical tautologies) are theorems Law of Absorption Law of Negation Law of Contradiction Law of (De)iteration
  12. 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. 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. 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. 15. Aristotle Fundamental Propositions
  16. 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. 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. 18. BARBARA syllogism Syllogism Barbara = [[b[c]][a[b]]a[c]]
  19. 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. 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. 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. 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. 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. 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. 25. Step 5: Syllogistic Equivalence Barbara = Celarent by substituting with Celarent = Barbari by replacing with Celarent = Celaront by replacing with
  26. 26. 24 valid syllogisms
  27. 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. 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. 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

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