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I have a question.

Let X be some set. Let $v$ be valuation.

I saw definition which says that $X \models \beta$ means that for any valuation $v$: if $v(X)\subseteq\{1\}$, then $v(\beta)=1$.

Question 1. Does $()\subseteq \{1\}$ mean at the same time that "all premisses in X are true or X is empty set"?

Question 2. If so, what does really $X\not\models\beta$ says?

Option (1): "there exists $v$ such that $v(X)\subseteq \{1\}$ and $v(\beta)=0$". If so, does it mean that at the same time: "all premisses are true and $\beta$ is true or $X=\emptyset$ and $\beta$ is true"?

or

Option (2): "there exists $v$ such that $v(X)=1$ and $v(\beta)=0$". If so, should I read it "all premisses are true and $\beta$ is false?

Which option is proper? Or maybe both options are wrong? The question basically is: what does $X\not \models \beta$ mean and how to write it by using valuation?

Please, help me understand.

pyrka123
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1 Answers1

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  1. $X$ is a set of formulas. Thus, to say that $v(X) \subseteq \{ 1 \}$ means that all formulas in $X$ a satisfied (evaluated to $1$, i.e. True) by $v$.

More customarily: $v \vDash X$.

  1. When $\beta$ is a formula, to say that $X \nvDash \beta$ means that $\beta$ is not logical consequence of the set $X$ of premises, i.e. that there is some valuation $v$ such that $v \vDash X$ and $v \nvDash \beta$, i.e. such that $v$ satisfies all formulas in $X$ and falsifies $\beta$.

Thus, option (2) is the correct one.

When $X$ is the empty set. we have that $\emptyset \vDash \beta$ means that $\beta$ is a tautology, i.e. a formula that is true in every interpretation. We usually write it: $\vDash \beta$.

Thus, to write $\nvDash \beta$ means: it is not true that $\vDash \beta$, i.e. it means that $\beta$ is not a tautology.

Mauro ALLEGRANZA
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