## 2 Fundamentals
### Contradictions  
In this paper, we differentiate between the term contradiction--which occurs when a statement is in opposition either with itself or an established fact--and the term contradictory, which we will explain below. In the literature on Requirements Engineering, many definitions of contradictions can be found, see Section 3_Related Work_. To get the most generically valid and scientifically accepted definition, we base our theory on the logical philosophy of [PERSON]. The foundation of his logic--also known as term logic, traditional logic, or formal logic--developed in his work Metaphysics is the law of non-contradiction (LNC) [6]. There, he argues that it is impossible that the same thing belongs and does not belong simultaneously in an identical way to the same object [7]. \"The doctrine of the square of opposition [as seen in Figure 2; note by the author] originated with [PERSON] in the fourth century BC and has occurred in logic texts ever since. Although severely criticized in recent decades, it is still regularly referred to\" [8] and will hence serve as a basis for our purposes.  
Figure 1: Formulation and building blocks.  
The relations can be described as follows:  
* \"Every S is P\" and \"Some S is not P\" are contradictories.
* \"No S is P\" and \"Some S is P\" are contradictories.
* \"Every S is P\" and \"No S is P\" are contraries.
* \"Some S is P\" and \"Some S is not P\" are subcontraries.
* \"Some S is P\" is a subaltern of \"Every S is P\".
* \"Some S is not P\" is a subaltern of \"No S is P\".  
Therefore, we have four main oppositions: contradictories, contraries, subcontraries, and subalterns.  
1. Contradictory opposites, e.g., \"he is sick\"/\"he is not sick\", are mutually exhaustive and mutually inconsistent. This means, that one statement must be true and the other false or vice versa. They cannot both be true or false at the same time.
2. Contrary opposites, e.g., \"it is black\"/\"it is white\", are also mutually inconsistent, but not exhaustive. While they cannot both be true, they can both be false.
3. Subcontraries, e.g., \"you can--If you want to--call in sick\"/you can--If you want to--not call in sick\" are mutually consistent. While they can simultaneously be true at the same time, they cannot simultaneously be false at the same time.
4. The statement \"some people are sick\" is the subaltern of \"everybody is sick\", while the latter is the superaltern of the former. If the superaltern is true, the subaltern must also be true and if the subaltern is false, the superaltern must also be false.  
By these definitions, the four central kinds of opposition--contradictory, contrariety, subcontrariety, and subaltern--are mutually inconsistent.  
In addition to the contradictions considered so far, there are other types. [PERSON] differentiates between the Aristotelian LNC-contradictions, dialectic contradictions, and antinomies [9].  
Dialectic contradictions are comparable to antagonisms or so-called,,conflict of goals\".  
For example:  
* The vehicle must have high performance
* The vehicle must have low consumption  
They don't stand in a mathematical/logical conflict but are incompatible in practice.  
Antinomies denote conceptual or propositional structures in which the truth value oscillates. A famous example is: [PERSON] says, \"[PERSON] speaks the truth,\" and [PERSON] says, \"[PERSON] lies.\" They are often confused with self-contradictions [9].  
In this paper, we will tackle the LNC conflicts, except for the subcontraries. As they can be valid simultaneously, subcontraries are not contradictions that need to be resolved for the RE work.  
Figure 2: Square of opposition.