* Logic

Language

Alphabet
(Set of elements/objects)

Sentence
Letters

Predicate/Propositions

2
place , 3 place

Operators

x
y (tt, tf, ft, ff)

Conectives

And
(Conjunction), Or (Disjunction),

Not(Negation),

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

Quantifiers

Universal
all (For all),

Existensial
some (For a),

There
is at least one thing such that

Syntacatagoromatic/Extra

Braces

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

q->(pVq)

(p^q)<->(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

Implication

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

and
y false

Equivalence

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

Commutativity

x
& y <=> y & x

x
v y <=> y v x

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

Associativity

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

Contraposition

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

1 Distribution

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

DeMorgans's

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

Importation/Exportation

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

Nameless

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

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

Biconditinals

-(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

Theories

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

theorems.

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.

Implication

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

implilcation.

disjunctive
addition

p
=> ( V q)

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

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

Basic logical laws

Commutative

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

Associative

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

Distributive

^ 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.

Independence

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

theories.

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

formulas

Premise
> conclusion

Some
arguments have no premise

Interpretation

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

form.

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.

Proof

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
#

Quantifiers

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:

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! =

3*2*1=6

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.

Derivations:

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

systems>symbols>alphabets>names(people(clans),

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,

objects)>individuals>people>familes>cities>states>nations>

worlds>solar
systems>galaxies>supergalaxies>unvierses>

Relations:

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