Course / training:

Method for Logical Analysis


Principles of Formal logic.



Combinatory Explosion in Logical Systems





Combinatory Explosion in Logical Systems



Introduction:

the emergence of a System of Logic.



Judgment makes use of information.


Our judgments and estimations are in many ways like our other reactions and decisions.
They are based on information we have available - at conscious as well as subconscious levels.

Information and difference.


About the concept of 'information' a lot of different views and understands exist. To avoid misunderstandings, and grasp its essential meaning, it's useful to first look at its most characteristic property. This becomes apparant when we try to conceive of a situation where she is totally absent.
Clearly, without any information there is only utter chaos, senseless 'noise', total vagueness, a completely non-knowing.
Information only comes into play with the perception of any difference. As soon as a distinction can be made - eg a difference between on/off, in/out, true/false, etc. - some order emerges. Only then reasoning becomes possible, and logic applies. Every amount of information thus implies at least one difference.

Information and ordering.


Any difference on its turn implies at least two 'things', phenomena or states in an area in reality. Thus, based on distinctions, combinations of things can be be considered.
Between these things simultaneously exists at least one order, ie the one that necessarily follows from their difference as perceived.
Moreover, in order to make any sense to us, information in general can not purely consist of loose data. We view the information in a certain cohesion. This implies that it allows for some ordering to be distinguished.
Viewed in reverse, every ordered state, or structure, represents in itself a certain amount of information.

Logical relations.


Given a random collection of elements, we may take a look at the logical relations that are possible between those elements. The logical relations refer to the range of states or values which the elements can take seperately, as well as through their mutual derivation relations.

Information and reasoning.


By combining data, we may obtain more complex forms of information.
Naturally, we do this by means of our thinking.
Every train of thought, and in fact every process of information processing, has the general form of a reasoning, that is to say:
Reasoning:
A number of input data are combined, and next certain data are derived from the combination.
The ways in which those combinations can be made, and the values that these combinations can take, are determined by the laws of logic.
Of course these properties certainly apply to judgments and estimations. They are to be considered as reasonings, in that they operate on certain information, and produce information thereafter.

Laws of logic.


The logical laws only apply to the relations between data, ie the combinations and derivations, and not to the individual data (such as direct observations and feelings). They also apply independently of the content and the nature of the data, which includes possible variations in subject, domain, problem, purpose, application, scope, etc..

Levels of logical complexity.


Each form of reasoning, or argument, consists of a combination of one or more distinct logical relations.
Orderings, and therefore arguments, are possible in every imaginable form, but also in every unthinkable form: they are virtually unlimited in possible variation, complexity and size. As will be show below, in a few steps this already reaches far beyond the limits of the imagination and comprehension of people, and even beyond the capacities of calculation and data storage of physical and even theoretical computers of any conceivable size.
Fortunenately, all of these possible forms can be sorted out en judged with help of the laws of logic. Therefore, understanding the laws of logic is indispensable for every judgment being meaningful and reliable. For the optimal use of logic, a clear understanding of the minimal levels of logical complexity and their proportions is indispensable.

Logical possibilities in information.


In this overview we'll have a look at possible logical relations given an arbitrairy sets of units (or items).
This concerns the problem of combinatory explosion.
Below some rules are given about quantication of combinatory explosion in propositional logic and in predicate logic.

System of Logic.


S!

: a logical system ('apparatus', calculus).

S!

PPL :

S!

is a system in propositional logic (

PPL

) (or higher).

S!

PDL-I :

S!

is a system in predicate logic (

PDL-I

), first-order logic (

FOL

) (or higher).
SEM!(

S!

) : the semantics, a set of ordering rules, of

S!

.

L!

: a formal system (language system).

L!

PPL :

L!

is a language/system in propositional logic (or higher).

L!

PDL-I :

L!

is a language/system in predicate logic, first-order logic,

FOL

, (or higher).
SYN!(

L!

) : the syntaxis, or grammer, a set of ordering rules, of

L!

.
WFF*(

L!

)) : the set of well-formed statements (formulas) of

L!

.

1.  Starting parameters.



1.1.  Objects.


Applicable in

PPL

and further.

D

* : (referential) domain or population, set of elements d[i];   with (i

=

1, .. d).
d : domain- or population-size; total number of objects, domain-elements ('things', phenomena, items, variables) d[i] within

D

*.

D

*

=

{d[1], .. d[i], .. d[d] }.
d

=

|

D

*

|

.

Example.


With two items (d

=

2 ), the set

D

·d may consist of the following elements (objects), represented by proposition symbols and stated in arbitrary order:
{ (d

=

2 ) (

D

·d

=

{' A' ,' B'} ) }.
Eventually, the domain may be empty. That would make the inference system

S!

[s1] however extremely minimal, if not futile.
Some examples of statements in such a 'minimal' system, stated in a formal language:
{ (d

=

0 ) (

D

·d

=

{} ) : ({}

=

(v) {} ); (({})

$

=

(v)

$

0 ); ({}

=

(r)

$

0 ); etc.}.
Likewise, the domain may consist of only one element. But then the inference systems

S!

[s1] also remains very simple.
Some examples of statements in such a 'primitive' system, stated in a formal language:
{ (d

=

1 ) (

D

*

=

{d[1]} ) : ((d[1] )

$

=

(v)

$

1 ); ((d[1] )

$

=

(v)

$

0 ); etc.}.

Range.


When the number of objects is less then one, any reasoning becomes meaningless.
On the other hand, when it is infinite, an inconceivable amount of reasoning concerning the domain becomes practically undecidable.
For a domain which is manageable, the following applies:
{ (d  

=

|

D

*(

mgb

)

|

); (1

d

<

0 ) }.

1.2.  Values.


Generally applicative to objects.

V

* : value system or 'value palette', set of values v[j];   waarbij (j

=

1, ..v).
v : total number of values, state values, or object values, (valences); e.g. truth values, v[j] in

V

*.

V

*

=

{v[1], .. v[j], .. v[v] }.
v

=

|

V

*

|

.

Example.


With two values (v

=

2 ) the set

V

·v may consist of the following elements (values), represented by value constants and stated in arbitrary order:
{ (v

=

2 ) (

V

·v

=

{0 ,1 } ) }.

Range.


When the number of values is less then two, any assignment of value becomes meaningless, and thus any attempt to meaningful reasoning becomes impossible.
On the other hand, when this number is infinite, almost any reasoning concerning the domain becomes practically undecidable.
For a value set which is sensible and manageable, the following applies:
{ (v  

=

|

V

*(

mgb

)

|

); (2

v

<

0 ) }.
In

PDL

some further parameters come into play.
(2a) p : total number of predicate-variables (attributes, predicate names); including identity, '='.
(2b) r : (maximum) number of argument-places, or arity, for each predicate name.
(We may eventually use for simplicity and security the maximum over all predicate names).
(2c) n : total number of elements (individuals, objects) in the referential domain (the population).
The (maximum) number of items a in

PDL

is a derivate of the latter three.
{p

a

(p *MAX(1,(r *n)) }.
In other words, when for a

PDL

system we have sufficient information about the parameters p, r and n, we can calculate a, and may reason further following the rules for a

PPL

system.

2.  Combinatory possibilities.



2.1.  Semantic expansion.



2.1.1.  Elementary object states.


Object states constitute pairs or tupels (the Cartesisch product) from the v values and d elements.
They reflect the domain at an observational level.
At a semantical level these are truth statements with respect to separate objects.
In logical language these are e.g. literals, ground instances, or 'witnesses'.
They resemble samples from a population.

H

·(v,d) : The set of all possible unique object states.
h : The total number of possible unique object states.

Example.


With two truth values (v

=

2, binary system) and two items (d

=

2) the set

H

·(v,d) may consist of the following elements (object states), represented by proposition symbols and stated in arbitrary order:
(

H

·(v

=

2,d

=

2)

=

  {' A' ,'A' ,' B' ,'B' } ).

Size.


{ v, d

|

(h ((h

=

|

H

·(v,d)

|

);
(h  

=

(

|

V

·v

|

,

|

D

·d

|

);

=

v *d   ) )h )d, v }.

In a binary system.


Under (v

=

2 ) applies:

H

·(v,d) is just as large as the doubling of set

D

·d.
Eg., under (v

=

2 );
from (d

=

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10,

..

} );
follows (h

=

{ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,

..

} ).

Range.


The number h remains liniar (polynomial) in d.
(2

h

<

0 ).

Complexity class.


The set

H

·(v,d) remains within the class of countable infinite sets (denumerable sets).
Thus it can be searched algorithmically - with a singletape Turing machine - in linear polynomial computational time (

P-TIME

).
(

H

·(v,d)

POLY

(d**1 );

TIME

(d );

P-TIME

).

2.1.2.  Domain states.


Domain states consist of conjunct combinations of all objects with their specific values, ie various object states.
They reflect the domain in a purely descriptive way.
At a semantical level these are truth statements with respect to the state of the entire domain.
They are similar to the cells (categories of variance) in a so-called contingency table (cross tabulation, of 'crosstab'), which forms the basis of numerous statistical measures for the comparision of variances, in particular Chi-square2), and variants or derivates of the latter, such as correlation coefficient, regression coefficient, Student's t, F, Fisher z, etc..

B

·(v,d) : The set of all possible unique domain states.
b : The total number of possible unique domain states.

Example.


With two truth values (v

=

2, binary system) and two items (d

=

2) the set

B

·(v,d) may consist of the following elements (domain states), represented by proposition symbols and stated in arbitrary order:
(

B

·(v

=

2,d

=

2)

=

{'( A B)' ,'( A B)' ,'(A B)' ,'(A B)' } ).

Size.


The number b equals the number of repetition variantions, or, sequence variations with replacement i.e. repetition, with size (length) d from v elements.
{ v, d

|

(b (b

=

|

B

·(v,d)

|

;
(b  

=

(d1 := 1,

..

d )
v;  

=

v **d )b )d, v }.
This number determines the length of the digital truth-value patterns of the logical relations.
It is equal to the number of rows in the truth-values table.

In a binary system.


Under (v

=

2 ) applies:

B

·(v,d) is just as large as the set of all possible subsets - the power set - of

D

·d. I.e.:
(v

=

2) (b

=

|

B

·(2,d)

|

;

=

|

P

**d

|

).
Eg., under (v

=

2 );
from (d

=

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10,

..

} );
follows (b

=

{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1 024,

..

} ).

Range.


The number b remains exponential in d.
(2

b

<

0 **0 );

Complexity class.


The set

B

·(v,d) remains within the class of uncountable infinite sets.
Thus it can be searched algorthmically in exponential computational time (

EXP-TIME

).
(

B

·(v,d)

EXP-TIME

(d ) ).

2.1.3.  Logical relations.


Logical relations reflect the domain at an analytical level.
At a semantical level they constitute conditional truth statements with respect to the entire domain or parts of it.
In logical languages these are e.g. truth-value patterns, formulas, propositions, theorems, and the like.
They correspond to the columns in the truth-values table.

T

·(v,d) : The set of all possible unique logical relations.
t : The total number of possible unique logical relations.

Example.


With two truth values (v

=

2, binary system) and two items (d

=

2), the set

T

·(v,d) may consist of the following elements (logical relations), represented by proposition symbols and stated in arbitrary order:

T

·(v

=

2,d

=

2)

=


{' T' ,' F'
,' A' ,' B' ,'A' ,'B'
,'( A B)' ,'( A B)' ,'(A B)' ,'(A B)'
,'( A B)' ,'( A B)' ,'(A B)' ,'(A B)'
,'( A B)' ,'( A

#

B)' } ).

Size.


The number t equals the numer of order variations or permutations with repetition with size (length) b from v elements.
{ v, d, b

|

(t ((t

=

|

T

·(v,d)

|

);
(t  

=

(b1 := 1,

..

b )
v;  

=

|

B

·(v,b)

|

;

=

v **

|

B

·(v,d)

|

;

=

v **(v **d) ) )t )b, d, v }.

In a binary system.


Under (v

=

2 ) applies:

T

·(v,d) is just as large as the power set of the power set of

D

·d.
(v

=

2) (t

=

|

T

·(2,d)

|

;

=

|

P

**b

|

;

=

|

P

**|

P

**d

|

|

).
Eg., under (v

=

2 );
from (d

=

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10,

..

} );
follows (t

=

{4, 16, 256, 65 536, 4 294 967 296, 1.844674407371 *(10**19), 3.402823669209 *(10**38), 1.157920892373 *(10**77), 1.340780792994 *(10**154), 1.797693134862 *(10**308),

..

} ).

Range.


The number t remains hyper-exponential in d.
(2

t

<

0 **(0 **0 ) );

Complexity class.


The set

T

·(v,d) can be searched in hyper-exponential computational time (

2-EXP-TIME

).
(

T

·(v,d)

2-EXP-TIME

(d ) ).

2.1.4.  The Truth-values table.


Given a domain

D

·d and value set

V

·v, the possible immediate logical relations-between-relations are completely defined through the so-called truth-values table.
This is construed on basis of systematic value assignment (validation) of the objects by a simple standarized algorithm. The objects each in turn go through the range of values, through so-called 'nested' cycles (loops), thus getting their unique, ordered truth-value patterns. Next, all other ordered value combinations are filled in. In this way a closed, coherent table arises of all possible sequences of (truth) values given parameters (d,v).
The value patterns are simular to statements with at least one proverb, or clause, in other words, 'sentences'.
In a binary system, they are expressed by binary numbers. Each of them has length of b value constants. These are simular to characters or symbols in written language.
The length b equals the amount of information in standard units: bits.
Furthermore, the table represents, with perfect garantees, all possible elementary logical relations together with their definite, immediate mutual logical relations.

Size.


The total number of cells in the truth-values table, tw, naturally becomes even larger than t, being the product of the number of domain states (rows) b, and the number of domain-state relations (columns) t:
{ v, d, b, t

|

(tw ((tw

=

|

T

·(v,d)w

|

);
(tw
 

=

|

(

T

*(v,d),

B

*(v,d) )

|

;  

=

(

|

T

*(v,d)

|

*

|

B

*(v,d)

|

);  

=

(

|

B

*(v,b)

|

*

|

B

*(v,d)

|

);
 

=

v **(v **d ) *v **d;  

=

v **((v **d) +d ) ) )tw )t, b, d, v }.

In a binary system.


Under (v

=

2 ) applies:
{ (v

=

2 ) ( tw  

=

|

T

·(2,d)w

|

;  

=

2 **((2 **d) +d ) ) }.
E.g., under (v

=

2 );
from (d

=

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10,

..

} );
follows (tw

=

{8, 64, 2 018, 1 048 576, 137 438 953 472, 1.18059162071744 *(10**21), 4.35561429658752 *(10**40), 2.96427748447488 *(10**79), 6.86479766012928 *(10**156), 1.840837770098688 *(10**311),

..

};
respectively in bytes (64-bits): {0.125, 1, 32, 16384, 2 147 483 648, 1.844674407371 ·(10**19), 6.805647338418 ·(10**38), 4.631683569492 ·(10**77), 1.072624634395 ·(10**155), 2.876309015779 ·(10**309),

..

} );

2.1.5.  Arguments.


Or reasonings, inference schemas.
In setting up an argument we usually don't make use of all possible logical relations about the subject in question, but a certain selection, or subset, of

T

·(v,d).
Arguments constitute of unique combinations of logical relations (i.e. without doubles).
At a semantic level these are sets of conditional truth-value statements which we regard in conjunction.
They reflect the domain at a discursive level.
In logical languages these are sets of truth-value patterns, formulas, propositions, theoremes, and the like: so called 'theories'.
For a solid analysis we have to take all possible selections into account.

U

·(v,d) : The set of all possible unique arguments.
u : The total number of possible unique arguments.

Example.


In an extremely simple, so to say 'primitive' logical system, with two truth values (v

=

2, binary system) and only one item (d

=

1), the set

U

·(v,d) may consist of the following subsets (arguments), represented here as combinations of logical relations stated in terms of propositiesymbols, and in arbitrary order:

U

·(v

=

2,d

=

1)

=


{ {}
,{' T'}
,{' T' ,' F'} {' T' ,' A'} {' T' ,'A'}
,{' T' ,' F' ,' A'} ,{' T' ,'F' ,'A'} ,{' T' ,' A' ,'A'}
,{' T' ,' F' ,' A' ,'A'}
,{' F'}
,{' F' ,' A'} ,{' F' ,'A'}
,{' F' ,' A' ,'A'}
,{' A'}
,{' A' ,'A'}
,{A'} }.

Size.


The number u equals the sum of all possible unique unsorted selections (without internal repetition) from

T

·(v,d) - i.e. of the binomial coefficients of t above the length (number of logical relations) t1 of each subset of

T

·v,d.
{ v, d, t

|

(u ((u

=

|

U

·(v,d)

|

);
(u1 ((U*[u1]

U

·(v,d) ); ((u[u1]

=

|

U*[u1]

|

)   (1

u[u1]

t )
(t1 ((u[u1]

=

t1 ) ((U*[u1] U·t1 ) (U·t1

U

·(v,d) ) ) )t1 ) ) )u1 );
(u
 

=

(u1 := 1,

..

u )
u[u1];
 

=

(t1 := 1,

..

t )
(

|

U·t1

|

);
 

=

(t1 := 1,

..

t )

binomial

(t, t1 );
 

=

|

T

·(v,t)

|

;

=

2 **

|

T

·(v,d)

|

;

=

2 **(v **(v **d)) ) )u )t, d, v }.

In a binary system.


Under (v

=

2 ) applies:

U

·(v,d) is just as large as the power set of the power set of the power set of

D

·d.
(v

=

2 ) (u

=

|

U

·(2,d)

|

;

=

|

P

**t

|

;

=

|

P

**|

P

**b

|

|

;

=

|

P

**|

P

**|

P

**d

|

|

|

).
Eg., under (v

=

2 );
from (d

=

{1, 2, 3,

..

} );
follows (u

=

{16, 65 536, 1.157 920 892 373 ·(10**77),

..

} ).

Range.


The number u remains ultra-exponential in d.
(2

u

<

2 **(0 **(0 **0 ) ) );

Complexity class.


The set

U

·(v,d) can be searched in ultra-exponential computational time (

3-EXP-TIME

).
(

U

·(v,d)

3-EXP-TIME

(d ) ).

2.2.  Syntactic expansion.


When we want to record semantic structures outside our own thinking, or communicate them to others, we'll have to represent them into an empirical form, with a tempero-spatial structure, for example into linear structure of some language.

2.2.1.  Derivations.


Or inference patterns.
In order to 'translate' arguments into syntactic structure, we'll first have to chose for a certain order in time. This requires linearisation.
At the level of semantics the order of thinking steps is completely irrelevant for their logical content and implications. Therefore, this order is at the level of syntax - at least in logic - totally arbitrary. She may be chosen at will for reasons of 'taste', convenience, convention, association or esthetics - or, in a more rational way, for the sake of cognitive ergonomics, dan wel efficiency of algorithm i.e. systematics of demonstration or refutation. Because of this arbitrary nataure - and the fact that most people build their reasonings rather haphazardly - in practical instances we may encounter all possible variations in sorting order.
Derivations are unique order variations, or permutations, or sequenties, without doubles, of each unique combination of logical relations (likewise without doubles).
They are similar to records in a sequential data base.

Size.


The total number of possible unique derivations y becomes the sum of all possible unique sorted selections (wihhout internal repetition) from

T

·(v,d); i.e. of the faculty of the size u[u1] of each subset of

U

·v,d.
{ v, d, t, u

|

(y ((y

=

|

Y

·(v,d)

|

);
(y1 ((Y*[y1]

Y

·(v,d) ) ((y[y1]

=

|

Y*[y1]

|

)   (1

y[y1]

t )
(t1 ((y[y1]

=

t1 ) ((Y*[y1] Y·t1 ) (Y·t1

Y

·(v,d) ) ) )t1 ) ) )y1 );
(y
 

=

(y1 :=1,

..

y )
y[y1];
 

=

(u1 :=1,

..

u )
(u[u1]

!

);
 

=

(t1 := 1,

..

t )
(

|

U·t1

|

*

(t1

!

) );
 

=

(t1 := 1,

..

t )
(

|

Y·t1

|

);
 

=

(t1 :=1,

..

t )
(

binomial

(t, t1 )

*

(t1

!

) );
 

=

(t1 :=1,

..

t )
(t

*

(t -1 )

..

*

(t -t1 +1 ) ) ) )y )u, t, d, v }.

In a binary system.


Eg., under (v

=

2 );
from (d

=

{1, 2,

..

} );
follows (y

=

{64, 56 874 039 553 216,

..

} ).

Obviously, with a fairly commonplace number of items d, the number y will already become of rather bizarre proportions - at least by human standards, but also in terms of mechanical computability. Nevertheless, the complete set of those possible sorting orders

Y

·(v,d), constitutes the actual range in which we create our everyday 'trains of thought', whereby we surely have full freedom of arranging steps of thought in any order we wish.
At the other hand, it is also the search space in which we preceive the reasonings of others, and try to capture their 'sense', or logical validity in a more precise way.

Complexity class.


The set

Y

·(v,d) weliswaar shows an enormous 'explosion' in size relative to the set

U

·(v,d), but this increase is not exponential, and therefore

Y

·(v,d) remains within the same complexity class.

2.2.2.  Reasoning links ('thinking steps').


These are the distinct elements within the inference patterns.
They are similar to data fields, cells or 'addresses' in a data record.

Size.


The total number of reasoning links e becomes the sum of the product of the number of possible derivations y[y1] and the length (number of logical relations) t1 of each subset of

U

·v,d.
{ v, d, t, u, y

|

(e ((e

=

|

E

·(v,d)

|

);
(e1 ((E[e1]

E

·(v,d) ); ((E*[e1]

=

{E[e1] } ) ((e[e1]

=

|

E*[e1]

|

) (e[e1]

=

1 );
(t1 y1 ((y[y1]

=

t1 ) ((y[y1]

=

(e[e1] *t1 ) ) ((E[e1] Y*[y1] )
((

|

E·t1

|

=

(

|

Y·t1

|

*t1 ) ) ((E[e1] E·t1 ) (E·t1

E

·(v,d) ) ) ) ) ) )y1, t1 ) ) ) )e1 );
(e
 

=

(e1 :=1,

..

e )
e[e1];
 

=

(t1 := 1,

..

t )
(

|

U·t1

|

*

(t1

!

)

*

t1 );
 

=

(t1 := 1,

..

t )
(

|

Y·t1

|

*

t1 );
 

=

(t1 := 1,

..

t )
(

|

E·t1

|

);
 

=

(t1 :=1,

..

t )
(

binomial

(t, t1 )

*

(t1

!

)

*

t1 );
 

=

(t1 :=1,

..

t )
(t

*

(t -1 )

..

*

(t -t1 +1 )

*

t1 ) ) )e )y, u, t, d, v }.

In a binary system.


Eg., under (v

=

2 );
from (d

=

{1, 2,

..

} );
follows (e

=

{196, 853 110 593 298 256,

..

} ).

2.2.3.  Reasoning links, encoded.


This concerns the amount of information which is minimally required to represent the links within the derivations.
The most 'parsimonious' solution will follow a rigorously unambiguous grid of coding, that is in terms of (truth) value-patterns, in other words binary numbers. As said before, each of these has a length of b value constants, being equal to the amount of information in standard units: bits.

Size.


The total number of coded reasoning links g becomes the sum of the product of the number of possible reasoning links e[e1] and the length (number of value constants) b under parameters {v,d}.
{ v, d, b, t, u, y, e

|

(g ((g

=

|

G

·(v,d)

|

);
(g1 ((G[g1]

G

·(v,d) ); ((G*[g1]  

=

{G[g1] } ) ((g[g1]

=

|

G*[g1]

|

) (g[g1]

=

1 );
(t1 y1 ((y[y1]

=

t1 ) (e1 ((E*[e1] E·t1 ) ((G[g1] E*[e1] )
((

|

G·(t1,b)

|

=

(

|

E·t1

|

*b ) ) ((

|

G·(t1,b)

|

=

(

|

Y·t1

|

*

t1

*

b ) );
((G*[g1] G·(t1,b) ) (G·(t1,b)

G

·(v,d) ) ) ) ) ) )e1 )y1, t1 ) ) ) )g1 );
(g
 

=

(g1 :=1,

..

g )
g[g1];
 

=

(t1 := 1,

..

t )
(

|

U·t1

|

*

(t1

!

)

*

t1

*

b );
 

=

(t1 := 1,

..

t )
(

|

Y·t1

|

*

t1

*

b );
 

=

(t1 := 1,

..

t )
(

|

E·t1

|

*

b );
 

=

(t1 := 1,

..

t )
(

|

G·(t1,b)

|

);
 

=

(t1 :=1,

..

t )
(

binomial

(t, t1 )

*

(t1

!

)

*

t1

*

b );
 

=

(t1 :=1,

..

t )
(t

*

(t -1 )

..

*

(t -t1 +1 )

*

t1

*

b ); ) )g )e, y, u, t, b, d, v }.

In a binary system.


Eg., under (v

=

2 );
from (d

=

{1, 2,

..

} );
follows (g

=

{392, 3 412 442 373 193 024,

..

};
respectively in bytes (64-bits): {6 125, 53 319 412 081 141 (ca. 53 Terabyte),

..

} ).
In other words, to write down all possible unique logical derivations, in terms of their reasoning links, extensionally, in standard binary code, without any syntactical duplicates or other (i.e. semantical) redundancy, in a binary value system, for just two objects, we already need a 64-bits storage capacity of at least about 53 Terabyte.

3.  Interpretation.


A crucial issue in daily life is: how can we assert the value of an arbitrary amount of information? This may apply in terms of, for instance, usefulness, credibility, plausibility, reliability, .. and, in the end, veracity. This actually means that we want to be able to reduce information back to some real properties of the domain to which she is refering. In other words, we're looking for a sensible, preferably an optimal interpretation.

3.1.  Syntactic reduction.


The information which we come across is often 'wrapped up', or encoded, in a certain empirical form, such as speech sounds, writing signs, or other symbols.

3.1.1.  Language signs to reasoning links.


The first step in finding out information content is decoding the data in accordance with the specific coding system, the language in which she is represented. This first means recognizing words and sentences.
For the purpose of this overview we assume the utterly simple case of a strictly unambiguous, artificial system of binary coding. We may then for instance trace back the bits from the set

G

 (v,d) to reasoning links of the set

E

 (v,d).

3.1.2.  Reasoning links to derivations.


The second step is recognizing derivations in the preceding set reasoning links. For this we make use of syntactical rules, the grammar. With these we can easily group the reasoning links of the set

E

 (v,d) into various derivations of the set

Y

·(v,d).

3.1.3.  Derivations into arguments.


The third step is eliminating the order 'effects' of the reasoning steps within the derivations. As we have seen above, this order is syntactically totally arbitrary.
Thus we can simply chose for one single, general, fully univocal and consistent sorting principle, with which all order variations vanishes. A principle that is often sufficient, is for instance sorting for length respectively alphabet.
The result of this is that we reduce the derivations of the set

Y

·(v,d) to arguments of the set

U

·(v,d).

3.2.  Semantical reduction.


How can we assess the logical 'status', i.e. the semantic value, of an arbirary argument? This means we want to be able to judge a certian subset of

U

·(v,d) at its truth valuee, in particular its logical validity.

3.2.1.  Consistent arguments.


What we like to know at least is whether arguments are to be taken 'seriously' at all, that is to say, qualify for an assessment of logical validity. Therefore we reasonably first take a look at arguments that, in principle, can be 'made true', or in other words, are satisfiable. That means that at least they are free from internal conflicts, in other words, they are consistent.
We must then take into account the collection of all the largest possible consistent subsets of

U

·(v.a).
There are four different variants of this, depending on the stage in the selection procedure that we can apply.
All of these variants, of course, we will view again without (syntactic) duplications within or between the subsets.

3.2.1.1.  Consistent Arguments: without immediate contradictions.


These are subsets of

U

·(v,d) without proven falsehood (falsum) or statements which are complementary.
The set of all possible unique consistent arguments, say

U

·(v,d)(

C

)
, thus consists of a (semantically weaker) selection of subsets from

U

·(v,d).

Example.


With two truth values (v

=

2, binary system) and two items (d

=

2) the set

U

·(v,d) (

C

)
may consist of the following subsets (consistent arguments), each with elements (logical relations) represented by proposition symbols and stated in arbitrary order:

U

·(v

=

2,d

=

2)
(

consis

)
 

=


{ {

$

(1111)}
,{

$

(1100)} ,{

$

(1010)} ,{

$

(0011)} ,{

$

(0101)}
,{

$

(1000)} ,{

$

(0100)} ,{

$

(0010)} ,{

$

(0001)}
,{

$

(1110)} ,{

$

(1101)} ,{

$

(1011)} ,{

$

(0111)}
,{

$

(1001)} ,{

$

(0110)} }.
The syntactical expressions (formulas) usually applied thereby for the associated logical relations are, respectively:

U

·(v

=

2,d

=

1)
(

C

)
 

=


{ {' T'}
,{' A'} ,{' B'} ,{'A'} ,{'B'}
,{'( A B)'} ,{'( A B)'} ,{'(A B)'} ,{'(A B)'}
,{'( A B)'} ,{'( A B)'} ,{'(A B)'} ,{'(A B)'}
,{'( A B)'} ,{'( A

#

B)'} }.

Size.


The size of this set is, with slightly larger basic parameters, rather hard to compute. Here follows is an approximation formula:
{ v, d, t, u

|

(u(

C

)
((u(

C

)

=

|

U

·(v,d)(

C

)

|

);
(u1 ((U*[u1]

U

·(v,d) ); ((U*[u1] ) (CONSIS(U*[u1] );
((U*[u1](

C

)

:=

U*[u1] ) (U*[u1](

C

)

U

·(v,d)(

C

)
)
(u[u1](

C

)

=

|

U*[u1](

C

)

|

)   (1

u[u1](

C

)

(t /2 ) ) ) ) )u1 );
(

U

·(v,d)(

C

)

:=

( U*[u1](

C

)
) );
(u(

C

)
 

=

(u1 := 1,

..

u )
(U*[u1](

C

)

U

·(v,d)(

C

)
)

|

u[u1] ) );
(u(

C

)
 

<

(t1 :=1,

..

(t /2 ) )
(

binomial

((t /2 ), t1 )

*

b ) ) )u )t, d, v }.
{ (k1

U

·(v,d) (

consis,max

)

:

  size =.. [ingewikkeld]
) }.

In a binary system.


Eg., under (v

=

2 );
from (d

=

{1, 2,

..

} );
follows (u (

consis,max

)

=

{5, 941,

..

} ).

3.2.1.2.  Consistent arguments of (1), without semantic redundancy within arguments.


This set is identical to the preceding one, but now after (exhaustive) paraphrasing reduction within each subset.
In effect, all logically 'weaker' elements within the subsets disappear, without loss of information . As a result, many 'double' subsets appear, which of course may be deleted, likewise without loss of information.
The set of all possible unique minimal consistent arguments, say

U

·(v,d)(

Cm

)
, thus consists of a (semantically equivalent) compression of

U

·(v,d)(

C

)
- through paraphrasing and then doublure reductions of its subsets.

Size.


Interestingly, the size of this set is equal to that of the set

T

·(v,d), in which each logical relation takes up precisely one separate subset, with exception of the one element falsum.
{ v, d, t, u(

C

)

|

(u(

Cm

)
((u(

Cm

)

=

|

U

·(v,d)(

Cm

)

|

);
(u(

Cr

)
((u(

Cr

)

=

|

U

·(v,d)(

Cr

)

|

);
(u1(

C

)
((U*[u1](

C

)

U

·(v,d)(

C

)
);
(u1(

Cr

)
((U*[u1](

Cr

)

:=

parf-reduc

(U*[u1](

C

)
) (U*[u1](

Cr

)

U

·(v,d)(

Cr

)
) )u1(

Cr

)
) )u1(

C

)
) )u(

Cr

)
);
(

U

·(v,d)(

Cr

)

:=

( (u1(

C

)
:= 1,

..

u(

C

)
)
(u1(

Cr

)
(u1(

Cr

)

=

u1(

C

)
)u1(

Cr

)

|

U*[u1](

Cr

)
) ) );
((

U

·(v,d)(

Cm

)

:=

doub-reduc

(

U

·(v,d)(

Cr

)
);
(u1(

Cm

)
((U*[u1](

Cm

)

U

·(v,d)(

Cm

)
);
((u[u1](

Cm

)

=

|

U*[u1](

Cm

)

|

)   (u[u1](

Cm

)

=

1 ) ) )u1(

Cm

)
) );
(u(

Cm

)
 

=

(U*[u1](

Cm

)

U

·(v,d)(

Cm

)
)

|

u[u1](

Cm

)
;
 

=

(t -1 ) ) )u(

Cm

)
)u(

C

),
t, d, v }.

In a binary system.


Eg., under (v

=

2 );
from (d

=

{1, 2, 3,

..

} );
follows (u (

consis,max;mnm

)

=

{3, 15, 255,

..

} ).

3.2.1.3.  Consistent arguments of (2), without semantic redundancy between arguments.


Next we consider the arguments of

U

·(v,d)(

Cm

)
that may each, being a domain state from

B

 (v,d), directly constitute an image of the associated domain.
As we saw before, each of these consists of d object states

H

 (v,d): for each object precisely one.
The paraphrase reduced set of all possible unique minimal consistent arguments, say

U

·(v,d)(

Cmr

)
, thus consists of a selection of subsets of

U

·(v,d)(

Cm

)
.

Size.


The size of

U

·(v,d)(

Cmr

)
is of course equal to b.
{ v, d, b, u(

Cm

)

|

(u(

Cmr

)
((u(

Cmr

)

=

|

U

·(v,d)(

Cmr

)

|

);
((

U

·(v,d)(

Cmr

)

:=

parf-reduc

(

U

·(v,d)(

Cm

)
) );
(u1(

Cmr

)
((U*[u1](

Cmr

)

U

·(v,d)(

Cmr

)
)
((u[u1](

Cmr

)

=

|

U*[u1](

Cmr

)

|

)   (u[u1](

Cmr

)

=

d ) ) )u1(

Cmr

)
);
(

U

·(v,d)(

Cmr

)

:=

( U*[u1](

Cmr

)
) );
(u(

Cmr

)
 

=

(U*[u1](

Cmr

)

U

·(v,d)(

Cmr

)
)

|

u[u1];
 

=

b ) ) )u(

Cmr

)
)u(

Cm

),
b, d, v }.

Example.


The previous version of the 'minimal' set,

U

·(v

=

2,d

=

1)
(

Cm

)
,   after this operation:

U

·(v

=

2,d

=

1)
(

Cmr

)
 

=

{ {

$

(10)} ,{

$

(01)} };
The syntactical expressions (formulas) usually applied thereby for the associated logical relations are, respectively:

U

·(v

=

2,d

=

1)
(

Cmr

)
 

=

{ {' A'} ,{'A'} }.
We can view the situation also with a slightly larger set.

Example.


With two truth values (v

=

2, binary system) and two items (d

=

2) the set

U

·(v,d)(

Cmr

)
may consist of the following subsets with elements (truth-value patterns), in arbitrary order:

U

·(v,d)(

Cmr

)
 

=

{ {

$

(1000) } ,{

$

(0100) } ,{

$

(0010) } ,{

$

(0001) } }.
The syntactical expressions (formula's) usually applied thereby for the associated logical relations are, respectively:

U

·(v,d)(

Cmr

)
 

=

{ {'( A B)' } ,{'( A B)' } ,{'(A B)' } ,{'(A B)' } }.

In a binary system.


Eg., under (v

=

2 );
from (d

=

{1, 2, 3,

..

} );
follows (u (

consis,max;mnm,reduc

)

=

{2, 4, 8,

..

} ).

3.2.1.4.  Consistent arguments of (3), with all derivable consistent reasoning links.


These are the so-called 'maximally consistent' subsets (Not to be confused with the set under (1)).
For this type of (sub)sets precise criteria are defined.
(1) It only contains arguments which are compatible (consistent) with the other elements within the (sub)set.
(2) Each semantically different element that will be added to the (sub)set will lead to contradiction with one ore more elements present, thus inconsistency of the (sub)set.
The set of all possible unique 'maximally consistent' arguments, say

U

·(v,d)(

mC

)
, consists of all subsets of

U

·(v,d)(

Cmr

)
that are maximally extended while still staying consistent.
The result thus consists of a selection from

U

·(v,d)(

C

)
of those (unique, consistent) subsets of maximal size that are (naturally) still internally consistent, and also mutually consistent.
Each subset of the set

U

·(v,d) (

mC

)
belongs to the class, or type, of so called Hintikka sets.
(The reverse doesn't necessarily apply).

Size.


The size of

U

·(v,d)(

mC

)
equals half of t.
{ v, d, b, t, u(

C

)

|

(u(

mC

)
((u(

mC

)

=

|

U

·(v,d)(

mC

)

|

);
(u1(

C

)
((U*[u1](

C

)

U

·(v,d)(

C

)
); (((u[u1](

C

)

=

|

U*[u1](

C

)

|

) (1

u[u1](

C

)

(t /2 ) ) );
((u[u1](

C

)

=

(t /2 ) )
(u1(

mC

)
((U*[u1](

mC

)

:=

U*[u1](

C

)
) (U*[u1](

mC

)

U

·(v,d)(

mC

)
) )u1(

mC

)
);
(

U

·(v,d)(

mC

)

:=

( U*[u1](

C

)
) );
(u(

mC

)
 

=

|

(u1(

mC

)
:= 1,

..

u(

mC

)
)
(U*[u1](

mC

)

U

·(v,d)(

mC

)
) u[u1];
 

=

b ) ) )u(

mC

)
)u(

C

),
t, d, v }.

In a binary system.


Eg., under (v

=

2 );
from (d

=

{1, 2, 3,

..

} );
follows (u(

mC

)

=

2, 8, 128, 32 768,

..

} ).

Example.


The previous version of the 'minimal' set,

U

·(v

=

2,d

=

1)
(

consis

)
  after this operation:

U

·(v

=

2,d

=

1)
(

mC

)
 

=

{ {

$

(11), ,

$

(10)} ,{

$

(11), ,

$

(01)} };
The syntactical expressions (formula's) usually applied thereby for the associated logical relations are, respectively:

U

·(v

=

2,d

=

1)
(

mC

)
 

=

{ {' T' ,' A'} ,{' T' ,'A'} }.
We can view the situation also with a slightly larger set.

Example.


With two truth values (v

=

2, binary system) and two items (d

=

2) the set

U

·(v,d) (

mC

)
may consist of the following elements (truth value patterns), here stated in arbitrary order, and in usually applied syntactical formula forms:

U

·(v

=

2,d

=

2)
(

mC

)
 

=


{ {

$

(1000) ,

$

(1100) ,

$

(1010) ,

$

(1001) ,

$

(1110) ,

$

(1101) ,

$

(1011)

$

(1111) }
,{

$

(0100) ,

$

(1100) ,

$

(0101) ,

$

(0110) ,

$

(1110) ,

$

(1101) ,

$

(0111)

$

(1111) }
,{

$

(0010) ,

$

(0011) ,

$

(1010) ,

$

(0110) ,

$

(1110) ,

$

(1011) ,

$

(0111)

$

(1111) }
,{

$

(0001) ,

$

(0011) ,

$

(0101) ,

$

(1001) ,

$

(1101) ,

$

(1011) ,

$

(0111)

$

(1111) } }.
The syntactical expressions (formulas) usually applied thereby for the associated logical relations are, respectively:

U

·(v

=

2,d

=

2)
(

mC

)
 

=


{ {'( A B)' ,' A' ,' B' ,'( A B)' ,'( A B)' ,'(A B)' ,'( A B)' ,' T' }
,{'( A B)' ,' A' ,'B' ,'( A

#

B)' ,'( A B)' ,'( A B)' ,'(A B)' ,' T' }
,{'(A B)' ,'A' ,' B' ,'( A

#

B)' ,'( A B)' ,'(A B)' ,'(A B)' ,' T' }
,{'(A B)' ,'A' ,'B' ,'( A B)' ,'( A B)' ,'(A B)' ,'(A B)' ,' T' } }.

3.2.2.  True arguments.


At last we examine which of the consistent arguments may be true.
For such an so-called evaluation there are two possibilities.

3.2.2.1.  Validity - independent of domain.


Logically valid is in any case one or more of the consistent arguments of the (maximal) collection of possible minimal consistent arguments,

U

·(v,d) (

Cm

)
.
The set of all possible unique logically valid arguments, say

U

·(v,d) (

Cmd

)
, thus in all cases consists of one complex element: the disjunction of all arguments from that set, each of which as said consisting of one logical relation.

Size.


{ v, d, u(

Cm

)

|

(u(

Cmd

)
((u(

Cmd

)

=

|

U

·(v,d)(

Cmd

)

|

);
(n ((n

=

u(

Cm

)
) ((n

=

(t -1 ) )
(

U

·(v,d)(

Cmd

)
 

:=

{U*[1](

Cm

)
U*[2](

Cm

)

..

U*[n](

Cm

)
};
 

:=

( (u1(

Cm

)
:= 1,

..

u(

Cm

)
)
(U*[u1](

Cm

)

U

·(v,d)(

Cm

)
)u1(

Cm

)

|

U*[u1](

Cm

)
) ) ) )n );
(u(

Cmd

)
 

=

1 ) )u(

Cmd

)
)u(

Cm

)
, d, v }.

3.2.2.2.  Truth - dependent of domain.


Only when we apply the arguments to a specific domain, will clarify whether one or more domain states of

B

·(v,d) apply, or so to say 'are the case'. By this, paraphrasing reduction takes place of each of the yet 'open' options from the disjunction of consistent arguments

U

·(v,d)(

Cmd

)
(through transferent equivalence reduction). Because this disjunction tegelijk contains all possibilities of satisfiable arguments, this operation boils down to exhausting degressive reduction.
The final result of this is the set of all possible unique true arguments, zeg

U

·(v,d)(

rdu

)
. This consists in all possible cases of one element: 'truth' (verum, that is value '

$

1').
With this, we lose by the way every possiblity to derive any further information, in other words, all logical power. This last step therefore brings us at the end of the entire cycle of expansion and reduction of a logical system within the range of its value scale.

Size.


{ v, d, u(

Cmr

)

|

(u(

rdu

)
((u(

rdu

)

=

|

U

·(v,d)(

rdu

)

|

);
(u1(

Cmr

)
((U*[u1](

Cmr

)

U

·(v,d)(

Cmr

)
) ((U*[u1](

Cmr

)
)

$

=

1 ) ((U*[u1](

Cmr

)

$

1 )
((

U

·(v,d)(

rdu

)

=

U*[u1](

Cmr

)
) (

U

·(v,d)(

rdu

)

$

1 ) ) ) ) )u1(

Cmr

)
);
(u(

rdu

)
 

=

1 ) )u(

rdu

)
)u(

Cmr

)
, d, v }.

C.P. van der Velde © 2014, 2018.