Syllogistic Unity
Proving the Equivalency of All Syllogisms Using Object Logic Armahedi Mahzar
Š 2011
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.
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
LOGICAL NOTATION
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
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
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.
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 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
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.
Agebraic Identities (logical tautologies) are theorems Law of Absorption Law of Negation Law of Contradiction Law of (De)iteration
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
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.
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
Aristotle Fundamental Propositions
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
The names of the valid syllogisms are
Using symmetric properties and Boolean Identity , we have only to prove just the Barbara syllogism validity.
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 All Red is Blue)=TRUE
=
=
deiteration
=
=
absorption
contradiction
negation
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
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
STEP 2: Celarent Zodiac
The twelve syllogisms are equivalent to each other. All can be represented by a single box diagram
Camestres: Arg Egb Ebr Camenes : Arg Ebg Ebr Celarent : Egb Arg Erb Cesare : Ebg Arg Erb
Datisi Darii Disamis Diramis
: Arg Ibr Ibg : Arg Irb Ibg : Ibr Arg Igb : Irb Arg Igb
Ferio Ferison Festino Fresison
: Egb Irb Org : Ebg Irb Org : Egb Ibr Org : Ebg Ibr Org
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
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
Step 5: Syllogistic Equivalence
Barbara = Celarent by substituting with
Celarent = Barbari by replacing with
Celarent = Celaront by replacing with
24 valid syllogisms
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
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.
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