Logical Evidence

A room of all n-digital logical relations becomes transformed into the resulting room of logical evidence with two-gradated dimensions in order to portray the texture of all including possible logical relations symmetrically.

Elaboration of the room of logical evidence

The 4-dimensional room of logical evidence contains 16 logical relations symmetrically – here ablated to the two dimensions of presentation.

Contemplation

The most resulting logical relations are more or less trivial besides the implication and the exclusion.

Implication

The implication becomes associated with a logical - on falsification tested - hypothesis.

X	Y	X Þ Y	*The hypothesis,* on X follows Y, tested on falsification, is
F	F	T	not falsified
F	T	T	not falsified
T	F	F	Falsified
T	T	T	not falsified

Exclusion

The exclusion becomes associated with a logical - on verification tested - hypothesis.

X	Y	X ≠> Y	*The hypothesis,* Y doesn’t follow X, tested on verification, is
F	F	F	not verified
F	T	T	Verified
T	F	F	not verified
T	T	F	not verified
REMARK: The exclusion-operator is implemented by the author and is not commonly accepted in sciences!

In this context, the hypothesis, Y doesn’t follow X, is coincidentally to the hypothesis, on NOT X follows Y.

Categorisation

The 16 possible relations in 4 groups.

Table of Logical Evidence
Totality		Conditionality
T T T T	TRUE	T F T T	Implication
F F F F	FALSE	F T F F	NOT Implication
T F F T	Equivalent	T T F T	NOT Exclusion
F T T F	NOT Equivalent	F F T F	Exclusion
Connectivity		Variety
F T T T	NOT Conjunction	F F T T	NOT X
T F F F	Conjunction	T T F F	X
F F F T	NOT Disjunction	F T F T	NOT Y
T T T F	Disjunction	T F T F	Y
REMARK: the categories are implemented by the author!

Elaboration of a Logical Singular Hypothesis

The logical singular hypothesis, on a logical expression follows another logical expression is to decline into several hypothesises, who are to check both ways, on falsification and on verification, and the results are to match to the relations of logical evidence

.Table of Logical Declination
Hypothesises	Tests	Variety				Relation
		T	T	F	F	X
		T	F	T	F	Y
On X *follows* Y	Falsification		F			Implication
On X *follows* Y	Verification	V				Conjunction
On X *follows* NOT Y	Falsification	F				NOT Conjunction
On X *follows* NOT Y	Verification		V			NOT Implication
On NOT X *follows* Y	Falsification				F	Disjunction
On NOT X *follows* Y	Verification			V		Exclusion
On NOT X *follows* NOT Y	Falsification			F		NOT Exclusion
On NOT X *follows* NOT Y	Verification				V	NOT Disjunction
On Y *follows* X	Falsification			F		NOT Exclusion
On Y *follows* X	Verification	V				Conjunction
On Y *follows* NOT X	Falsification	F				NOT Conjunction
On Y *follows* NOT X	Verification			V		Exclusion
On NOT Y *follows* X	Falsification				F	Disjunction
On NOT Y *follows* X	Verification		V			NOT Implication
On NOT Y *follows* NOT X	Falsification		F			Implication
On NOT Y *follows* NOT X	Verification				V	NOT Disjunction

Figure of Logical Declination

Each both ways of each checked hypothesis of the logical declination of the logical singular hypothesis, on falsification and on verification, are hypothesis-specific complements and the assigned relations in the room of evidence also. In the figure, the hypothesis-depending complementary relations of logical evidence are connected by arrows from falsification-assigned relation to verification-assigned relation. The different colours are associated with the coloured groups of the table before.

Derivation of Logical Relations

The table below shows 2 given logical relations who represent the two-digital logical variety, X and Y, and 6 derived logical relations by logical declination by the logical singular hypothesis of logical conclusion, each case tested on verification. The other 8 logical relations are there logical complements.

Table of Logical Derivation

Hypothesises		Check				Matching Relations
Variety	Tests	T	T	F	F	X
Variety	Tests	T	F	T	F	Y
On X *follows* Y	Falsification		F			Implication
On X *follows* Y	Verification	V				Conjunction
On X *follows* NOT Y	Falsification	F				NOT Conjunction
On X *follows* NOT Y	Verification		V			NOT Implication
On NOT X *follows* Y	Falsification				F	Disjunction
On NOT X *follows* Y	Verification			V		Exclusion
On NOT X *follows* NOT Y	Falsification			F		NOT Exclusion
On NOT X *follows* NOT Y	Verification				V	NOT Disjunction
Conditionality	Tests	T	F	T	T	Implication
Conditionality	Tests	T	T	F	T	NOT Exclusion
On Implication *follows* NOT Exclusion	Falsification			F		NOT Exclusion
On Implication *follows* NOT Exclusion	Verification	V			V	Equivalence
Connectivity	Tests	F	F	F	T	NOT Disjunction
Connectivity	Tests	T	F	F	F	Conjunction
On NOT Disjunction follows Conjunction	Falsification				F	Disjunction
On NOT Disjunction follows Conjunction	Verification					FALSE

Figure of Logical Derivation

Elaboration of Logical Quantity and Quality

Table: Dichotomic Progression of the Theoretical Well of Hypothesises into Derived Basis-Structures of Quantity and Quality
Quantity				Well		Quality
Decimal	Singularly	Dimensional	Digital	Logical	Evidence	Dichotom. Index	Cause-Effect-Scheme
a- symmetrical	non- symmetrical	full- symmetrical	hyper- symmetrical	Well of quantity	Well of quality	Abs.: Ranges +-: Scale	The hierarchical cause-effect-scheme includes inter-dependence between X and Y
0	0	00	0000	F F F F	FALSE	-1	never… not, always…
1	1	01	0001	F F F T	NOT Disjunction	-2	, only IF none… not, only IF any…
2	2	02	0010	F F T F	Exclusion	-3	, only IF effect is uncaused… not, only IF cause or no effect…
3	3	03	0011	F F T T	NOT X	-4	CAUSE Complement
4	4	10	0100	F T F F	NOT Implication	-5	, only IF effect is missing… not, only IF effect or no cause…
5	5	11	0101	F T F T	NOT Y	-6	EFFECT Complement
6	6	12	0110	F T T F	NOT Equivalent	-7	not, only IF any equally… , only IF all differently…
7	7	13	0111	F T T T	NOT Conjunction	-8	not, only IF all… , only IF any not…
8	8	20	1000	T F F F	Conjunction	+8	, only IF all… not, only IF any not…
9	9	21	1001	T F F T	Equivalent	+7	, only IF all equally… not, only IF any differently…
10	a	22	1010	T F T F	Y	+6	EFFECT (suppositious interdependent to X)
11	b	23	1011	T F T T	Implication	+5	, only IF effect or no cause… not, only IF effect is missing…
12	c	30	1100	T T F F	X	+4	CAUSE (suppositious interdependent to Y)
13	d	31	1101	T T F T	NOT Exclusion	+3	, only IF cause or no effect… not, only IF effect is uncaused…
14	e	32	1110	T T T F	Disjunction	+2	not, only IF none… , only IF any…
15	f	33	1111	T T T T	TRUE	+1	never not… , always…
2006-09-26 © Tobias Waehneldt

Contemplation

The table shows the derivation of quantity and quality from the theoretical well of logical evidence, which is derived from the theory of hypothesises.

Derivation of Quantity

The table shows the n-digital logical relation of logical evidence as Boolean expressions (False, True),

- their transformation into the -dimensional binary-digital expressions (0, 1),

- their transformation into the number system with the base of , in this case the quaternary number system and

- their transformation into a singular number system with a base of , in that each number has its own symbol, in this case the hexadecimal number system.

Derivation of Quality

- The table shows the logical evidence, which is derived from logical declination of the singular hypothesis and their transformation into preferably simply adapted and generalized linguistic fragments of expressions of a hierarchical cause-effect-scheme respectively non-hierarchical interdependencies of influencing variables like hardly or not distinguishable causes or effects.

- The absolute value of dichotomic index describes ranges, each dichotomic scaled (prefixes) with logical qualities and their logical complements, their negations.

- Each case of the cause-effect-scheme, without the cases of logical variety, without the causes, the effects and their complements, are linguistic interpreted both ways, positive and negative and respectively their plausibility more a less indifferent but commutable.

Generalization

In order to generalize the qualitative fragments of linguistic expressions, the causes and effects are to contemplate variously exchangeable, so that the linguistic expression determines the relation of both and their assignment. In order to supports the capabilities of categorisation and operating two further table are appended.

Table: Simple Linguistic Declination of Logical Evidence 2006-09-30 © Tobias Waehneldt
… all AND none …	… any retracts/contradicts/converse …	… none resists/insists/persists …	… all equally …
… all …	… if / whether …	…whereon / how …	… none …
… any …	… because / in so far …	… although / anyhow …	… any not …
… all differently …	… any resists/insists/persists …	… none retracts/contradicts/converse …	… any OR any not …

Table: Category-Declination of Logical Evidence 2006-09-29 © Tobias Waehneldt
FALSE-Injunction	Explication	Inclusion	Identity
Conjunction	Exclusion, Inclusion, Explication, Conclusion, Disclusion, Identity, Disparity, and Injunctions are implemented by the author!		Conclusion
Disjunction			Disclusion
Disparity	Exclusion	Implication	TRUE-Injunction

Table: Operator-Declination of Logical Evidence 2006-09-29 © Tobias Waehneldt
FIN	EPC	ICS	EQV
AND	EPC, ICS, ECS, FIN, TIN are implemented by the author!		NOR
OR			NAND
XOR	ECS	IMP	TIN

Outlook

The adaptation of qualitative and quantitative logical concepts is very difficult but very helpful. It increasingly determines the communication. But doubtlessly the very complex contextual and associative components of communication transporting more than logical information. The social context and their necessities is the root of communication, to show emotions, emphasis, rivalry, identifications, ideals, to get orientation and much more. Such things don’t simply fit into a room of discrete logical relations. But the logical evidence can help to develop more differentiated forms of communication and supports the capabilities to communicate.