# What is the either or logical error

A certain proposal is rejected. A certain fact is not given. The question of truth related to a certain assertion is answered negatively. We need the negation of a statement when we face a certain error or lie. Lots *Signal words for the negation *are available: different, excepted, excluded, except, objection, objection, false, false, false report, error, free from, contrast, opposite, lied, just as little, illusion, misleading, erroneous, error, no, by no means, by no means Lie, by no means, no, not, nothing, never, never, nobody, never, nowhere, nowhere, without, deception, invalid, impossible, untrue, unreal, prohibition, supposed, neither / nor, refuted, contradiction, probably hardly. Sometimes the negative signal word is missing in a negation.

A: A certain rectangular door is 2 meters high.

a *concrete negation *from A: The relevant dimension is 2.15 m.*definition*: The "abstract negation" of the usual statement A is that statement which A negates without any additional information being given. - For the abstract negation of A one writes briefly: (not A). It applies *heuristic rule*: The smaller the logical content of the *ordinary statement *A is, the greater the logical content of (not A) (see below the De Morgan theorems, the four negative clauses of predicate logic, the double negation theorem and the contraposition clause with application to a chain).

A: On 08/11/1999 there was a total solar eclipse on planet Earth. (true)

(not A) as a short answer: No.

(not A): On 08/11/1999 there was no total solar eclipse on planet earth.

(untrue)*De Morgan's sentences*: For all common statements A and B, the following two statements of the apply *equivalence*:

(2) [not (A or B)] is equivalent to [(not A) and (not B)]

The logical content of statements A and B is not important here. Findings (1) and (2) are therefore *general*. Consequently owns *the abstract negation of a conjunction* the logical status of an adjunction and *the abstract negation of an adjunction* the logical status of a conjunction.*definition*: With the exception of the abstract negation of A, all statements that contradict the usual statement A are “concrete negations” of statement A. - One can negate the predicate, the subject, the accusative object, the dative object, a certain property , the indication of possession, the spatial reference, the temporal reference, the means, the purpose and the number. By comparing a certain concrete negation NA with the statement A, it becomes clear in which detail the contradiction lies. The abstract negation (not A) follows from any concrete negation of the statement A, but not vice versa. The logical content of (not A) is therefore smaller than the logical content of any concrete negation of proposition A. If proposition B is an implication of proposition A, which is not equivalent to A, then (not B) is a concrete negation of A.

A: Anton Huber is in Augsburg on November 23, 2004 with a Mercedes 200 E (official

License plate: A - K 1993) driven into the standing Opel Astra by Maria Schmid.

*concrete negations of A*:

_{1}: In the accident in question, Anton Huber was not the driver of the first car,

*rather*Max Huber.

N

_{2}: The accident in question did not happen on November 23, 2004,

*rather*on October 23, 2004.

N

_{3}: The accident in question did not happen in Augsburg,

*rather*in Ulm.

N

_{4}: In the accident in question, the first car was not a Mercedes 200 E,

*rather*a VW Passat.

N

_{5}: The registration number of the first car was not A-K 1993,

*rather*A-KM 1993.

N

_{6}: The second car involved in the accident in question was not an Opel Astra,

*rather*a Ford model.

N

_{7}: The second car was not stationary at the time of the accident in question,

*rather*rolled forward.

N

_{8}: At the time of the accident in question, Maria Schmid was not the owner of the second car,

*rather*the spouse.

N

_{9}: The driver of the first car is not on the second,

*rather*Maria Schmid is on the standing position when reversing

Mercedes drove up.

N

_{10}: Anton Huber did not have a car accident on November 23, 2004.andBoth the Mercedes 200 E with the registration number A - K 1993 and the Opel Astra from Maria Schmid were driven without accidents until they were scrapped. - This is the one

*total mismatch of the statement A and the concrete negation N*

_{10}.

*the abstract negation of A*:

(not A) is equivalent to the adjunction

(N

_{1 }orN

_{2}orN

_{3}orN

_{4}orN

_{5 }orN

_{6}orN

_{7}orN

_{8}orN

_{9})

*omitting "but ...".*

According to De Morgan's theorems, the abstract negation of A has the logical status of an adjunction. Because the statement A can be represented as a conjunction. The purpose of the accident report form is to clearly and completely document the accident in question as a conjunction:

*conjunction*

[(1) In the accident in question, Anton Huber was the driver of the first car.

and (2) The accident in question happened on November 23, 2004.

and (3) The accident in question happened in Augsburg.

and (4) In the accident in question, the first car was a Mercedes 200 E.

and (5) The registration number of the first car was A - K 1993.

and (6) The second car involved in the accident in question was an Opel Astra.

and (7) The second car was at a standstill at the time of the accident in question.

and (8) Maria Schmid was the owner of the second car at the time of the accident in question.

and (9) The driver of the first car ran into the second car in the accident in question.].

When applying De Morgan's theorems, one must note that the statement N_{1} "In the accident in question, Anton Huber was not the driver of the first car." Is a concrete negation of statement A, but *the abstract negation of *(1) "In the accident in question, Anton Huber was the driver of the first car."

The abstract negation or a certain concrete negation of the usual statement A is very often ambiguous. As *Adjunct *or finding a *Contravalence* the negation in this case is mostly a weak statement. In this case, by specifying a certain possibility, the negation becomes unambiguous and a strong statement. By adding “but ...”, the abstract negation becomes a concrete negation and the abstract negation is reinforced with a reason. Because A holds for all ordinary statements *the sentence about negation*: From any concrete negation of the ordinary proposition A, its abstract negation follows. - According to the proposition about ascertaining truths, the relevant proposition is equivalent to

A: On June 30, 2004, Kathrin Müller was 16 years old.

(not A): Kathrin Müller was not 16 years old on June 30, 2004.

a weak statement.

A concrete negation of A is N1: On June 30, 2004, Kathrin Müller was 18 years old.

_{1}is a strong statement.

(not n_{1}): On June 30, 2004, Kathrin Müller was not 18 years old.

_{1}) is a weak statement.

_{1}follows (not A).

(2) From A it follows (not N

_{1}).

*The principle of double negation*: The ordinary proposition A is equivalent to the abstract negation of (not A).

From this axiom of logic follows a *Criterion for abstract negation*: The abstract negation of the usual statement A can be recognized by the fact that the statement A is equivalent to the abstract negation of (not A).

A: In each rectangle the diagonals are the same length.

B: There is a rectangle where the diagonals are not the same length.

Is the statement B the abstract negation of A?

(not B): There is no rectangle where the diagonals are not the same length.

The statement B is the abstract negation of A.

*The contraposition clause*: The conclusion “From A follows B.” is equivalent to “From (not B) follows (not A).", Where A and B are arbitrary ordinary statements. - In the example for the indirect proof (see below) the contraposition theorem is applied to the conclusions [From (4) follows (5).] And [From (8) follows (9)]: (From “The integer a is not divisible by 2. "follows" The square number a^{2} is not divisible by 2. ".) is equivalent to (From" The square number a^{2 }is divisible by 2. "It follows that" The whole number a is divisible by 2. ") - *Sixth application of logic in science*: To one *Fallacy *to uncover, one can use the contraposition theorem.

A: Fritz Meier has three million euros in the lottery when it was played on December 15, 2007

won.

B: Fritz Meier placed a tip for the lottery draw on December 15, 2007.

(not B): Fritz Meier did not give a tip for the lottery draw on December 15, 2007.

(not A): Fritz Meier did not have three million euros in the draw on December 15, 2007

Won the lottery.

The conclusions [from A follows B.] and [from (not B) follows (not A).] Are equivalent, *but both invalid*. Fritz Meier could have benefited from another person's tip as part of a tip community.

When using the contraposition theorem you have to look carefully, otherwise you will fall on someone else *Fallacy* in: From “The implication (C => D) is given.” does not follow “The implication [(not C) => (not D)] is given.”. According to the contraposition clause, “[(not C) => (not D)] is given.” Is equivalent to “(D => C) is given.”. Both conclusions are invalid if statements C and D are not equivalent. Because the reversal of the deduction direction is only allowed if the statements C and D are equivalent. Therefore everyone needs to be usable *criteria* are based on a general equivalence.

The congruence theorem sss yields a *Criterion for the congruence of triangles*:

The congruence of two triangles can be recognized by

that they match in the three side dimensions.

D: The triangle ABC and the triangle FGH are congruent. (relational property)

E: The triangles ABC and FGH correspond in the three side dimensions. (Criteria)

The statements D and E are equivalent for all pairs of triangles (*general equivalence*).

1st case: Both triangles have the side dimensions: 5.7 cm, 6.5 cm and 8.0 cm. The criterion E is fulfilled. So the triangles in question are congruent.

2nd case: The triangle ABC has the side dimensions: 5.7 cm, 6.5 cm and 8.0 cm. The triangle KLM has the side dimensions: 5.7 cm, 7.2 cm and 8.0 cm. The criterion E is not met. So the triangles in question are not congruent.*The theorem about the negation of equivalent statements*: The statement that the statement D is equivalent to E is equivalent to "The abstract negation of D is equivalent to (not E).". - With the help of the criterion for equivalence and the contraposition theorem can be proven.

D: The triangle ABC and the triangle FGH are congruent.

E: The triangles ABC and FGH correspond in the three side dimensions.

(not D): The triangle ABC and the triangle FGH are not congruent.

(not E): The triangles ABC and FGH do not match in at least one of the side dimensions.

*The principle of excluded contradiction*: From “The usual statements A and B contradict each other” follows (A is untrue. Or B is untrue.). - The *ordinary statement *A and the statement (not A) cannot even both be untrue. Because either the statement A is untrue or (not A). In contrast, there is the possibility that the usual statement A and a certain concrete negation N_{A. }both are untrue. The *abstract negation *a certain guess also has the logical status of a guess. *A concrete negation *a certain presumption leads to a clarification of the question of truth if it is demonstrably true (a rebuttal of the presumption). For all common statements A, the statements (1) to (6) apply:

(1) “A is true.” Is equivalent to “(not A) is untrue.”.

(2) “A is untrue.” Is equivalent to “(not A) is true.”.

(3) From “A is true.” Follows “N_{A. } is untrue. "

(4) From “N_{A.} is true. ”follows“ A is untrue. ”.

(5) From “N_{A.} is untrue. ”does not follow“ A is true. ”.

(6) “A is untrue.” Does not follow “N_{A.} is true.".*The theorem about the indirect proof *is a *Axiom of logic*: The statement “A is true.” Is equivalent to “(not A) is untrue.”, Where A is any ordinary statement. - An indirect proof is a proof of truth: the abstract negation of a certain theorem is refuted. The abstract negation of the relevant theorem proves in most cases not only to be untrue but also to be *adversarial*. This rebuttal means that the proposition in question is true.

**Example of a**

*indirect evidence*The truth of theorem A "There is no rational number q,

which is the equation x

^{2 }= 2 fulfilled. "Should be proven.

Proof using nine conclusions and four true universal laws,

by using the chain-shot theorem, the contraposition clause

and the theorem on conjunctions in logical content

and by refuting the statement (not A):

(1): There is a rational number q which satisfies the equation x^{2} = 2 fulfilled.

From (1) follows (2): The number q has the normal representation: q = a: b, where a and b are certain integers that are relatively prime, and b is greater than zero.

has a clear normal representation. "is true by definition.)

From [(1) and (2)] it follows (3): a^{2} : b^{2} = 2 | • b^{2}

(3) is equivalent to (4): a^{2 }= 2 b^{2}

From (4) follows (5): The whole number a is divisible by 2.

integer a is not divisible by 2. => The square number a

^{2}is not divisible by 2.)

From [(4) and (5)] it follows (6): (2 n)^{2 } = 2 b^{2} , where n is an integer.

(6) is equivalent to (7): 4 n^{2} = 2 b^{2} | : 2

(7) is equivalent to (8): b^{2} = 2 n^{2 }

From (8) follows (9): The whole number b is divisible by 2.

integer b is not divisible by 2. => The square number b

^{2}is not divisible by 2.)

From [(5) and (9)] it follows (10): The numbers a and b have a common factor, namely two.

The implications of (not A) (2) and (10) contradict each other. The statement (not A) is therefore adversarial. It follows that (not A) is untrue. So theorem A is true. - It is wrongly claimed that the proof idea in the above example is in Euclid's famous work "The Elements". Rather, the relevant indirect proof, which Aristotle has already mentioned several times, has been inserted subsequently from a presumably older work (compare: Euclid, The Elements, translated from the Greek according to Heiberg's text by Clemens Thaer, Darmstadt 1973, 5th edition, p. 462).

Predicate logic (here: “predicate” = property) deals with universal and existential clauses. Existence clauses, which have the logical status of a factual assertion, refer either to a part of the universe (local existence clause) or to the whole universe (universal existence clause). Analytical existence statements refer either to a finite set (local existence statement) or to an infinite set (universal existence statement). An analogous classification is also possible for universal sentences. *The four negative sentences of predicate logic* are valid by definition:

(1) [not (All x have the property Z.)] is equivalent to “There is an x which does not have the property Z”.

(2) [not (All x do not have the property Z. ”] Is equivalent to“ There is an x which has the property Z ”.

(3) [not (There is an x which has the property Z.)] is equivalent to “All x do not have the property Z.”.

(4) [not (There is an x that does not have the property Z.)] is equivalent to “All x have the property Z.”.

These four statements of equivalence also apply to universal laws and universal existential propositions. So owns *the abstract negation of a universal law* (universal universal proposition) the logical status of a universal existential proposition. And *the abstract negation of a universal existence proposition* has the logical status of a universal law.

A: There are planets with a ring system.

The proposition A is a universal existence proposition. This is about the property Z of planets, that they have a ring system. The abstract negation (not A) is untrue, but has the logical status of a universal law. “In the whole universe there is no planet with a ring system.” Is equivalent to “Every planet is without a ring system.” This universal law is refuted by the counterexample Saturn. Incidentally, the exploration of our planetary system with the help of space probes has shown that the other three gas planets Jupiter, Uranus and Neptune also have rings.

Manfred Brill

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