Uncertainty about the definition of entail > Definition > We say that a set of propositions (the "premises") $Γ$ entails a proposition $φ$ (the "conclusion") if for every truth assignment $A$, if $A(φ)$=1 when we have $A(ψ)$=1 for all $\psi$ in $Γ$. > > Note: if a truth assignment assigns "false" (i.e. 0) to a premise in $Γ$, it doesn't matter what it assigns to $φ$; this truth assignment still satisfies the definition. The note below makes me confuse. Does it mean if there exists $A(ψ)$=0 for $\psi$ in $Γ$, $Γ$ entails $φ$ still hold?

The definition must be rephrased as :

> there are **no** truth assignment $A$ such that : $A(\psi)=1$ for every $\psi \in \Gamma$ and $A(\varphi)=0$.

Thus, what happens with a truth assignment $A'$ that assigns "false" (i.e. $0$) to some premise $\psi \in \Gamma$ ?

It doesn't matter what they "do" to $\varphi$, because we are only concerned with the truth assignements that "satisfy" (i.e. assign $1$) to **all** $\psi \in \Gamma$.

* * *

You must read the definition as a "recipe" to test for _entailment_ :

_(i)_ consider a truth assignement $A$ : are all formulae $\psi$ in $\Gamma$ _true_ for $A$ ?

* If NO, skip it (i.e. skip $A$).

* If YES, then check if $\varphi$ is true.




_(ii)_ if $\varphi$ is true for $A$, then take a new truth assignment $A'$ and go to step _(i)_.

_(iii)_ if $\varphi$ is false for $A$, then stop : $\varphi$ is **not** entailed by $\Gamma$.