# Definition: Satisfaction Relation, Model, Tautology, Contradiction

Let $$L\subseteq (\Sigma^*,\cdot)$$ be a formal language with strings $s\in L$ formed according to a syntax and let inside a domain of discourse $U,$ a semantics $I(U,L)$, and the valuation $[[]]_I$ be given.

### Tautology

We say that the interpretation $I(U,L)$ satisfies models, is a model of) an interpretable string $s\in L$, denoted by $$I\models s,$$ if and only if the corresponding valuation is true, i.e.$$[[s]]_I=1.$$ If $I\models s$ for all possible interpretations $I$, then we write $\models s$ and say that $s$ is valid. Alternatively, we call $s$ a tautology.

We say that the interpretation $I(U,L)$ does not satisfy does not model) an interpretable string $s\in L$, denoted by $$I\not{\models} s,$$ if and only if the corresponding valuation is false, i.e.$$[[s]]_I=0.$$ If $I\not {\models} s$ for all possible interpretations $I$, then we write $\not{\models} s$ and say that $s$ is invalid. Alternatively, we call $s$ a contradiction.

Chapters: 1 2 3
Definitions: 4 5 6
Examples: 7
Explanations: 8
Lemmas: 9 10 11 12 13 14 15
Motivations: 16 17
Proofs: 18 19 20 21 22 23 24 25 26 27

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