* Logic


               Alphabet (Set of elements/objects)

                       Sentence Letters


                               2 place , 3 place


                               x y (tt, tf, ft, ff)


                                      And (Conjunction), Or (Disjunction),


                                      (Conditinal)If Then(t,f,tt), If and only if(t,f,f,t)



                                      Universal all (For all), 


                                      Existensial some (For a),

                                              There is at least one thing such that



               Formation Rules/Syntax/Transformations/Relations

               Universe of Discourse/Semantics/Pragmatics

                       Truth Functions

               Rules and methodes of deduction

                       Modus Ponens

                       Equivalence Thesis

                               If x then y = Either not x or y

                       Tautology - Always logicaly true statements L -true

                               (A <-> B) v (A <-> -B)

                               ~(p ^ q) <-> (~p) v (~q)

                               p V ~p

                               (p ^ q) -> p




                       Contradiction-Always logically false statements   L-false

                               A is and is not x

                               p ^ ~p

                       Logically determinate

                               a sentence that is logiclly true or false - there truth value

                               is determined by logical structure alone

                       Logically indeterminate

                               the truth value is not determined by  logical structure alone

                               but in conjunction with the way the world happens to be

                       Logically intermediate

                               False under some interpretations and true under others


                               x=>y, x implies y if there is no interpretation that makes x

                                      and y false


                               x<=> y , x is equivalent to y if there is no interpretation

                               r which x and y have opposing truth values.

                               Principle of replacement


                               Let s be a set of propositions and let r and s be propositions

                               generated by S. r and s are equivalent if r<->s is a

                               tuatology.  The equivalence of r and s is denoted by r <=>.


                               (p^q) V (~p ^ q) <=>q

                               p->q<=> ~q-> ~p

                               p v q <=> q V p


                               Equivalence is to logic what equalit is to algebra.


                       Types of Equivalence

                               Double Negation --x <=> x

                               Idempotency (x & x) <=> x, (x v x) <=> x


                                      x & y <=> y & x

                                      x v y <=> y v x

                                      x <-> y <=> y <-> x


                                      (x& y) & z <=> x & (y & z)


                                      x -> y <=> -y -> -x

1                              Distribution

                                      x & (y v z) <=> (x & y)  v (x & z)


                                      -(x & y) <=> (-x v -y)


                                      x & y -> z <=> x ->(y->z)


                                      x ->y <=> -(x & - y)

                                      x ->y <=> (-x v y)


                                      -(x <-> y) <=> (-x <-> y)

                                      (-x <-> y) <=> (x <-> -y)

                                      (x <-> y) <=> (x & y) v (-x & -y)

                                      (x <-> y)<=>(x->y) & (y->x)

                               Consequence- gneralization of implication

                                      Let D ba a set of well formed formulas

                                      D |= x , x is a consequence of D

                                      If there is no interpretation that makes each member

                                              of D true and x false.

                                      x => y  iff {x} |= y

                                      x <=> y iff {x} |=y and {y} |= x

                                       x is L-true iff { } |=x

                                      x is L-false iff { } |= -x

                                              inconsistence = L-false

                                              consistent formulas are true under at

                                              least one interpretation

                               x is L indeterminate iff neither {x } |= -x nor {-x} |= x.

                               Satisfiability - generalization of the concept of consistency

                                      A property of sets of sentences

                                      D is satisfiable if there is some interpretation that

                                              makes each of its members true.

                                      x is consistent iff { x} is satisfiable.

                                      ( x1, ....xn) is satisfiable iff (x1 & ....& xn)

                                      D is satisfialbe iff there is some x such that

                                              D is not |= X.

                               quantifier converson laws(4)

                                      (v) x <=> -(Some v) -x


                               distribution collection laws (7)

                                      (v) (x & y) <=> (v)x & (v)y

                                      (Some v) (x & y) => (Some v) x & (Some v) y


                               confinement expansion laws (10)

                                      (v) (x & y) <=> (v) x & y



                                      D is a theory if  D is nonempty and each sentence of

                                      D that is a consequence of D is a member of D.


                                       D is a theory iff D is nonempty and for each

                                       sentence s if s element L(D) and D |= s then          

                                      s element D.

                                      The sentences that are a member of a theory are

                                      referred to as its thesis and said to be accepted by

                                      the theory. Sentences who's negations are accepted

                                      are sid to be rejected by the theory. A theory is

                                      neutral with respect to those of its sentences that it

                                      neither accepts nor rejects.

                                      A formula is accepted by a theory if its universal

                                              closure is accepted.

                                      A set is said to be decidable if there is a decision

                                      procddure for determining whether any given object

                                      is a member of the set.  The set of well formed

                                      formulas and the set of tautaulogies are thus        

                                      decidable sets.

                                      Modes of theory presentation.

                                      Axomatic mode

                                      Specify a decidable set of sentences, called axioms,

                                      hen delcare the theses of the theory to be those

                                      nces that are consequence sof the set. When a set is

                                      nted axomaticcally the thesis are usually called



                                      Sematnic mode

        `                              Specify the vocabular of a theory with an    

                                      interpretaton, and then declare the thesis of the

                                      theory to be those of its sentences that are true

                                      under the interpretation.


               Let S ba  set of propositions and let r and s be propositions generted by S.

               We say that r implies s if r -> is a tautology. We write r=>s to indicate this



               disjunctive addition

                       p => ( V q)

               p ->q <=> (~p) v q

               p <-> q <=> (p ^ q) V (~p ^ ~q)


Basic logical laws


                       p v q <=> q v p     p ^ q <=> q ^ p


                       ( v  )  v  <=> v ( v )    ~ reverse ^


                       ^   v  <=> v  ^  v       ~reverse v ^

               Identity Laws

                       p v 0 <=> p  p ^ 1 <=> p

               Negation Laws

                       p ^ ~ p <=> 0    p v  ~ p <=> 1

               Idempotenty Laws

                       p v p <=> p    p ^ p <=> p

               Null Laws

                       p ^ 0 <=> 0  p V 1 <=> 1

               Absorption Laws

                       p ^ (p v q)<=> p,   p v (p ^ q) <=> p

               DeMorgans Laws

                       ~(p v q) <=> (~p) ^ (~q)       ~(p ^q) <=>(~p) v (~q)

                Involution Law

                       ~p(~p) <=> p




        A theory is :

               Consistency- if there is no sentence that it accepts and rejects

               Completness - if it is neutral w.r.t none of its sentenses

                       A complete theory is decisive if given any one of its sentences it

                       will either reject or accept the sentence.

                       If a theory is inconsistent then it is complete.


                       A set of sentence is independent if no one of its members is a

                       consequence of the other members.            

               All semantic theories are consistent and complete but not all axiomatic


               A theory is axiomatizable if ech of its theses is a consequence of some

                       decidable subset of the theory.

               A theory is said to be categorical if ti has a model and any two of its

                       models are isomorphic. If t is catagorical then t is complete.


               Well Formed Formulas

                       Recursive definition

                               1.Sentence letters are all well formed

                               2. If y and z are well formed, then so are the following

                                      -y, y&z, y v z, y -> z, y<->z

                               3. Nothing is well formed unless its being so can be

                                      established on the basis of 1 and 2.

                       Derivation tree of the construction of well formed formulas

                       argument = any finite nonempty sequence of well formed      


                               Premise > conclusion

                               Some arguments have no premise


                       If x is a sentence letter, then x is true under I if and only if I assigns

                        truth value truth to x.

                       If x = (y & z) then x is true under I iff y and z are both true under I.


                       It is always possible given a formula x and an interpretatin I, to

                       calculate the truth value of X under I.


                       An intepretation of logic of quantifiers,

                               a nonempty set, the universe or domain of the        

                               interpretation, an assignment of appropriate extensions

                               tothe extralogical vocabulary of lq.

                       Domain - natural numbes

                       's' for the number of stars is even

                       'e' for one is even



               Logical Argumentation:

                       Deductive - conclusively true

                       Inductive- not conclusively true, only probably true

                       Valid argument - impossible for its premises to be true while

                                              its conclusion is false

                                      Does not require that either the premises or the

                                      conclusion of a valid argument be true. It only

                                      requires that the conclusion be true if the premisis

                                      are true. A valid argument of all whose premisies

                                      are true is said to be sound.

                                      No argument having the form that has true premisis

                                              and a false conclusion. An argument is

                                              deductively valid if it has a validating

                                              argument form.

                       Methode of counterexample to establish invalidity.

                                      An argument does not have a validating argument


                       The methode of truth trees


Logical Relations: Countable, subset of, element of, intersection, union, product,

consistent, min, max, optimize, continous, discontiniouis, irregular, central, indentity,

universal, existential, void, combined, converse, relative product, symmetric, reflexive,

transitive, conex, orderings, partial, complete, well formed, mapping, inverse, sequences,

equivalence, congruence, abstract, isomorphic, domain of, properties of ordered pairs n-

place relation a property of ordered n-tuples, desireablity, probability/density, utility,

believability(probability, justified (deduce, induce), relative (opinion),  statistical,

normalize, linear, nonlinear, finite, stability, terminal, recursion, information value,

consistent, exhaustive, mutually exclusive, expansion,universal closure,  Deduce,

conclude, induction, deduction


then = implies=follows from=only if=if=is sufficient for=is necessary for

iff=is necessary and sufficient for=is equivalent to=



Other Relations:Fractality =Embedability =(Non-Destructive) Compressibility =

Turning Inside- Out ness =Scale Invariance =Spin Density =Information Density =

Charge Density =Sustainability =Share-Ability =Perfect Distributability =

Perfect Marketability, symmetry (a=b->b=a)..


Mathematical System

        (1) A set or a universe U

        (2) Definitions - sentences that explain the meaning of conepts that relate to the

               universe. Any term used in describing the universe itself is said to be

               undefined. All definitions are given in terms of these undefined concepts

               of objects.

        (3) Axioms - assertions about the properties of the universe and rules for creating

               and justifiying more assetions. These rules always include the system of

               logic that we have developed tothis point.

        (4)  Theorems - the additional assertions mentioned above.


        Proof - A proof of a theorem is a finite sequence of logically valid steps that

               demonstrate that the premisis of a theorem imply the conclusion.


A research mathematician might require only a few steps to prove a theorem to a

colleaque, but might take an hour to give an effective proof to a class of students. What

constitutes a proof depends on the audience.


Computer proof theory

        a, a-> b, b-> c, ..., x ->y, y->z =>z  can be proved by a computer using truth

        tables:  (a ^ (a -b) ^....^ (y->z)) -> z.

        The truth table will have 2^26 cases and at 1000 cases per second, it would take

        aproximately 64,000 seconds (18 hours) to verity the theorem.

        A similar theorem would , p1, p1 -> p2, ... p99->p100=> p!00 has 2^100 cases.


Rules of Formal Proofs

        1. A proof must end in a finite number of steps

        2. Each step must be either a premise or a proposition that is implied from

               previoius steps using any valid equivalence or implication.

        3. For a direct proof, the last step must be the conclusion of the theorem.

               For  an indirect proof, the last step must be a contradiction.

A direct proof is a proof in which the truth of the premisis of a theorem are shown to

directly imply the truth of the theorem's conclusion.



Direct proof of ~p v q, s v p, ~q => s:


Step    Proposition    Justification

1       ~p v q         Premise

2       ~q             Premise

3.      ~p             Disjunction Simplification 1, 2

4       s v p          Premise

5       s              Disjunctive simplification 3, 4


Indirect Proofs

        The method of indirect proof is based on teh equivalence P->C <=> ~P(P ^~C).

        If p +. C, then  P ^ ~C is always false; P ^ ~C is a contradiction.

        This means that a valid method of proof is to negate the conclusion of a theorem

        and add this negation to the premisis.        If a contradiction can be implied from this

        set of propositions, the proof is complete.


Definition: Propostion over the Universe. Let u be a nonempty  set. A propostion over U

is a sentence that contains a variable that can take on any value U and that has a definite

truth value as a result of any such substitution.  All of the laws of logic are valid for

propositions over a universe.  If p and q are propostions over the ingtegers, we can be

certain that p ^ q => p, because (p ^q) -> p is a tuatology and is true no matter what

values the varibles p and q are given.  If we specify p and  q to be p(n): n<=4 and q(n):

n<=8, we can also say that p implies p ^ q.


Truth Set


Definitinon: If p(n) is a proposition over U, the truth set of p(n) is Tp(n) = {a element of

U| p(a) is true}


The truth set of the propostion {1,2} ^ A = NULL taken as a proposition over the power

set of {1,2,3,4} is {NULL, {3}, {4},{3,4}}.


Definition: Tautology and Contradiction. A proposition over U is a tautology if its truth

set is U. It is a contradiction if its truth set is empty.


The truth set of compound propostions can be exressed in terms of simple propositions.


Tp^q = Tp ^ Tq

Tp v q = Tp v Tq

T~p = T(c p)

Tp<->q = (Tp ^ Tq) v (T(c p) ^ T(c p)

Tp->q = T(c p) v Tq


Equivalence: Two propositions are equivalent if p<->q is a tautology.  In terms of truth

sets, this means that p and q are equivalent if Tp = Tq.



Implication. If p and q are propostions over U, p implies q if p->q is a tautology.


Mathematical Induction: A technique for proving propostions over positive integers.

Mathematical induction reduces the proof that all of the positive integers belong to a

truth set to a finite number of steps. 


The principle of mathematical induction.  Let p(n) be a proposition over the positive

integers, then p(n) is a tautology if


(a) p(I) is true, and

(b) n >= I and p(n) => p(n+1)


Note: The truth of p(1) is called the basis for the induction proof. The premise that p(n) is

true in Statement (b) is called the induction hypothesis. The proof the p(n) implies p(n+1)

is called the induction step of the proof. 


Variations on the definitons of Mathematical Induction (Generalized)

If p(n) is a propostion over {k0, k0 + 1, k0 +2, ...}, where k0 is any integer, then p(n) is a

tautology if:


(1) p(k0) is true, and

(2) k >= k0 and p(k) => p(k +1)


The Course of Values Principle.  If p(n) is a propistion over {k0, k0+1, k0 +2, ...}, then

p(n) is a tautology if


(1) p(k0) is true, and

(2) k >= k0, p(k0), p(k0 +1), ...., p(k) => p(k +1).



An example of mathematical induction.


Consider the implication over the postive integers p(n):


        q0 -> q1, q1 -> q2, ...., qn-1 ->qn, q0 =>qn.


A proof that p(n) is a tautology follows:


        Basis: p(1) is q0->q1, q0 => q1.  This is the logical rule of detachment which we

                       know as true.  Wirte out the turth table of ((q0->q1) ^q0) -> q1 to

                       verifty this step.


        Induction: Assume that n >= 1 and p(n) is true.  We want to prove that p(n +1)

               must be true.  That is:


               q0->q1, ..., qr-1 ->qn, qn->qn +1, q0 => qn+1.


        Here is a direct proof of p(n + 1):


        Steps   Proposition(s) Justification


(1) - (n +1) q0 -> q1,,,,,,qn-1 ->qn, q0      Premises

(n +2) qn                                    (1) - (n +1), p(n)

(n +3)  qn ->qn+1                             Premise

(n +4)  qn +1                                 (n+2), (n+3),

                                              Detachment #





If p(n) is a proposition over a universe U, its turth set Tp(n) is equal to a subset of U. In

many cases such as when p(n) is an equation, we are most concerned with whether Tp(n)

=U, that is whether p(n) is a tautology.  Since the conditions Tp(n) not equal NULL

(Existensial) and Tp(n) =U (Universal ) ar so often an issue we have a special system of

notation for them.


Existensial Quantifier


If p(n) is a proposition over U with Tp(n) not equal 0, we commonly say there exists an n

in U such that p(n) (is true). We abbreviate this sentence, with  the symbols (E n) u

(p(n)), E is called the existential quatifier.


Universal Quantifier


If p(n) is a proposition over U with Tp(n) = U, we commonly  say for all n in U, p(n) is

(is true)".   We abbreviate this proposition with the symbols (An)u(p(n)). A is termed the

universal quantifier.


(Ax) (Ar(x) -> B(x)) is true = for all x such that Ar(x) = x lives in air, B(x), x is a bird.


Negation of a Quantified Proposition

~(Ax) (Ar(x) -> B(x)) <=> (Ex) (~(Ar(x) -> B(x))) <=> (E x) ((Ar(x) -> B(x)) .


The negation of a universially quantified proposition is an existentially quantified

propostion.    When you negate an existensially quantified propostion, you obtain a

universially quantified propostion. Symbolically, ~((A n) u (p(n)) <=> (E n) u (~p(n)),

and ~((E n) u (p(n))) <=> (A n)u(~p(n)).


Multiple Quantifiers


p(x,y) : x^2 -y^2 = (x+y) (x-y) is a tautology over the set of all pairs of real numbers

because it is true for each pair (x,y) in R x R. The asserttion that p(x,y) is a tuatology

could be quantified as (Ax)R ((Ay)R (p(x,y))) or (Ay) R ((Ax) R (p(x,y)).


Key Concepts In Proof


1. All theorems in mathematics can be expressed in "If P then C" (P=>C) format, or in

"C1 iffi C2" fromat.  The latter is equivalent to "If C1 then C2 and if C2 then C1."


2.If P then C, P is a premise (or hypothesis) and C is the conclusion.  It is important to

realize that theorem makes a statement that is dependent on the premise being true.


3.There are two basic methods for proving P => C:

        (a) Direct: Assume P is true and prove C is true; and

        (b) Indirect (proof by contradiction): Assume P is true C is false

               and prove that this leads to a contradiction.

4. The methode of proof for "iff" theorems is found in the law (P <->C) <=> ((P->C) ^ (C

->P)). Hence to prove an "If and only if" statment one must porve an "if ..then ..."

statment and its converse.


Propositional Connectives and Truth Values as Mappings


The propositional connectives thus far intorduced require two input sentences (p,q) into a

third sentence r = f(p,q).  x v y =   r = v(x,y). Since each sentence can be attributed the

truth value 0 or 1, it is convenient to allow X to denote a universal set of sentences and

then define the truth function T: X->{0,1), where T(p) =1 if the proposition p is true and

T(p) = 0 if the propositon if false. Consequently, as a mapping, f: X x X -> X, defined by

f(p,q) = r.  The disjunctive operation, p v q, r = v (p,q), is a function of two variables. The

truth value of the composite statement r can be found by knowing the truth values of the

sentences p and q and by knowing precisely which function f is being employed.   The

truth table is a function denoted by F. Commuting diagram:


                       X x X >  f > X .T > {0,1}<F<{0,1} x {0,1} < T x T< X x X


The meaning of the diagram is this: starting with a pair of sentences (p,q) in X x X

processing given truth values, the same value will be obtained by either traversing the

diagram across the top and then down, or by going down first and then traversing the

diagram horizontally along the bottom.  In sum the diagram states that:


T(f(p,q)) = F(T(p), T(q)).












Sets: Set, Finite Set, Cardinatily, Subset, Equality(=), Intersection( ^), Union(U),

Disjoint(A ^ B=Null) (no elements in common), Universe, Compliment, Symmetric

Difference (In a and b but not in both), Venn Diagram,


Cartesian Product AXB = set of all possible ordereed pairs whose first component come

from a, and whose second component comes from b. , Power Sets P(A) - If A is any set,

the power set of A is the set of all subsets of a, including the empty set and A itself.

Summation Notation and Generalizations.


Permutations: (Subclass of the rule of the cartesian product)

Partitions of sets = Let A be a set. A partition of A is any set  of one or more nonempty

subsets or blocks A1, A2, ... of A  such that A1 U A2 ... and subsets Ai are mutually

dischoint Ai ^Aj = Null for i not equal to j. Let A = {a, b, c, d} then  3 partition of a are:

{{a},{b}, {c,d}}, 2. {{a,b}, {c, d}}, 3. {{a}, {b}, {c},{d}}.

How to partition a carton of 24 cans, 4 six packs, 3 8 packs, 2 twelve packs, in all cases

the sum of all packs must be 24, and a can must be in one and only one pack.


Basic Law of Addition= #A=#A1+#A2+....#An.  The sophomore computer science

majors where told they must take one (and only one) of the following courses. Calculus,

Data Structures, or Compiler Construction, in a given semester.  The numbers in each,

course, respectively, for sophomore C. S. majors, were 75, 60, 55. How many sophomore

C.S. majors are there? 


A= the set of all sophomore .c.s majors, A1=set of all c.s. majors who took Calculus.

A2=set of all c.s. majors who took Calculus.   A3=set of all sophmores C.S. majors who

took Compiler, Construcion.


Since all sophmore C. S. majors must take at least one of the courses, the number we

want is : #A=#A(A1 U A2 U A3) = #A1+#A2+#A3 minus all duplications.


#A=#(A1U A2 U A3)

        = #A1 +# A2 + #A3 - (duplications - triplications)

        =#A1+#A2+A3- duplications + triplications

        =#A1+#A2+#A3 - #(A1 ^ A2)-#(A1^A3) - #(A2 ^A2) + #(A1^A2^A3)

= 75+60+55-25-12-15+10 = 148.


Combinations: (Another subclass of the rule of products)

A = {a,b,c,d}?

Order is important in permutations.

        There are P(4,3) = 4!/(4-3)!=24.

Order is not important in combinations.

        How many ways can we simply list, or choose, three letters from the set A={a,

        b,c,d}, all three elment subsets of A: The notation of choosing 3 elements from 4

is (4 3) =(C)ombinatorics(4,3) (4 choose 3), or the number of combinations for 4 objects

taken 3 at a time:


               (n k) =C(n:k)  (read "n choose k") is the number of combinatinos of n

               objects taken k at a time.


Relation between permutation and combination problems: (Used in Probability)


        If A is any finite set of n elementsm, the number of k-element subsets of A is:

         (n k) =  n!/(n-k)!k!,  where 0<=k<=n. 


The binomial theorem gives us a procedure for expanding (x+y)^n where n is a postive

integer.  The coefficients of the expansion of (x + y)^n can be expressed compactly in the

form of the number (n k)- the binomial coefficient.]


The binomial theorem. (x + y)^n = (n 0) x^n + (n 1)x^n-1 y ^1 + .....+ ....= Sum (N, K=0)

(n k) x^n-k y^k.



Set Relations:

        0 e B = an object is a member of set b

        a  i  b =a is included in b

        If a I b and b i a, then a = b

        a  s  b = a is a subset of b

        a + b = union of a or  b

        a x b = intersection of a and b

        a - b = relative complement

        Order n tuple set of < o1, o2, ...>

        n-ary relation of a set of objects


Combinatorics (Art of Counting)

Rule of products - If n operations must be performed, and the number of operations for

each operations is p1, p2, .. and pn, with each p independent of various choices, then the

n operations can be performed p1p2..;pnways.


The number of elements in the poser set of A, P(A) is #P(A)=2^#A.


Permutation= N!=N Factorial 3 Elements (a,b,c), then the number of permutiation is 3! =


Permutation of k elments taken from a set of n elements: P(n:k)= N!/(n-k)!.

Of (8)eight people who want to be president, .vp, treasurur (3), P(8,3)= 8!/(8-3)!=336

ways of choosing officers.






        Basic model 1:

               [opposition]>[chaos of  elements & relations & transformations-     

               falseness, ambiguity, uncertainty/most information=improbable       

               distributions]>[order(organization<positive feedback)>

               Transformations(Evaluation of information within the system>

               Stability of the system (limits of spectrum, levels of stability<negative

               feedback)>Equilibrium of the system (Max. entropy (most probable

               distributions=less information), certitude(Logic Deductive)>

               permanence of the system /unity(more than sum of parts/order out of

               instability/synergy/syncronicities(Overall system)/wholeness



        Basic Model II:


               preexistence of any sytem>all imaginatinos possible>

               chaos>opposition (+,-, ~)>all universes possible/impossible>chaos[chaos of  elements &

               relations & transformations-falseness, ambiguity, uncertainty/most

               information=improbable distributions]>order(organization<positive feedback)>anaolog

               spectrum>Logic (True, False, Neutral)>Duality(Bipolar,Binary), N-ary relations N-

               Polarities>A universe in general > a universe of type

               x(objects(properties),relations,meanings of a unvierse)>a real universe(genesis,

               history(past, present, future, parrallel universes,dimensions of a universe(12(12(12)))>

                       Precreation>possible universes>possible ideas>possible sounds,vibrarions,energy

                       votexes, genes, memes>possilbe notation systems>symbols>alphabets


               Transformations(Evaluation of information within the system>

               Stability of the system (limits of spectrum, levels of stability<negative

               feedback)>Equilibrium of the system (Max. entropy (most probable

               distributions=less information), certitude(Logic Deductive)>

               permanence of the system /unity(more than sum of parts/order out of

               instability/synergy/syncronicities(Overall system)/wholeness


        List of all ideas of a universe:


               Precreation>possible universes>possible ideas>possible sounds,vibrarions,energy

                       votexes, genes, memes>possilbe notation


                       places),wordsds>languages(syntax, semantics,

                       pragmantics)>sciences,knowledge, facts,social systems and   

                       arts>time frames(seasons, fesitvals, biological clocks, day,week, month,

                               year, generation, centruy, age, epoch,)>artifacts(buildings,  


                       worlds>solar systems>galaxies>supergalaxies>unvierses>




        Binary, Bipolar relations:

               +postive/true  -negative/false


               good           evil

               male           female

               yin            yang

               true           false

               white          black



        Trinary, Tripolar relations:

               +positive, neutral, -negative

               thesis, synthesis, antitheis


        N-ary, N-Polar relations:

               Spectrum/Analog/Numbers Theory

                -infinity ............1............to infinity