* 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