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## 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.

| | | | | created: 2018-02-06 21:38:50 | modified: 2020-05-04 19:44:03 | by: bookofproofs