  We re-examine d=4 hidden-variables-models for a system of two spin-$1/2$
particles in view of the concrete model of Hardy, who analyzed the criterion of
entanglement without referring to inequality. The basis of our analysis is the
linearity of the probability measure related to the Born probability
interpretation, which excludes non-contextual hidden-variables models in $d\geq
3$. To be specific, we note the inconsistency of the non-contextual
hidden-variables model in $d=4$ with the linearity of the quantum mechanical
probability measure in the sense $\langle\psi|{\bf a}\cdot {\bf
\sigma}\otimes{\bf b}\cdot {\bf \sigma}|\psi\rangle+\langle\psi|{\bf a}\cdot
{\bf \sigma}\otimes{\bf b}^{\prime}\cdot {\bf
\sigma}|\psi\rangle=\langle\psi|{\bf a}\cdot {\bf \sigma}\otimes ({\bf b}+{\bf
b}^{\prime})\cdot {\bf \sigma}|\psi\rangle$ for non-collinear ${\bf b}$ and
${\bf b}^{\prime}$. It is then shown that Hardy's model in $d=4$ does not lead
to a unique mathematical expression in the demonstration of the discrepancy of
local realism (hidden-variables model) with entanglement and thus his proof is
incomplete. We identify the origin of this non-uniqueness with the
non-uniqueness of translating quantum mechanical expressions into expressions
in hidden-variables models, which results from the failure of the above
linearity of the probability measure. In contrast, if the linearity of the
probability measure is strictly imposed, which is tantamount to asking that the
non-contextual hidden-variables model in $d=4$ gives the CHSH inequality
$|\langle B\rangle|\leq 2$ uniquely, it is shown that the hidden-variables
model can describe only separable quantum mechanical states; this conclusion is
in perfect agreement with the so-called Gisin's theorem which states that
$|\langle B\rangle|\leq 2$ implies separable states.
