We consider the linear programming problem with Boolean variables (LPB) \[ \max \sum^{n}_{i=1}a_ ix_ i\quad subject\quad to\quad \sum^{n}_{i=1}b_{ij}x_ i\leq B_ j\quad (j=1,...,m) \] where \(x_ i\in \{0,1\}\), \(a_ i\geq 0\), \(B_ j\geq 0\), \(b_{ij}\geq 0\) \((i=1,...,n)\). For any j, \[ \sum^{n}_{i=1}b_{ij}x_ i-B_ j\leq 0\quad \Leftrightarrow \quad F_ j(\tilde x)=0 \] where \(\tilde x=(x_ 1,...,x_ n)\), \(F_ j:\) \(\{0,1\}^ n\to \{0,1\}\), \(F_ 0(\tilde x):=\bigvee^{m}_{j=1}F_ j(\tilde x)\), \(F_ 0\), \(\{F_ j\}\) are monotone Boolean functions. Then the weakly determined (WD) problem of LBP with a partially given set of permissible solutions can be formulated as: \[ \max \sum^{n}_{i=1}a_ ix_ i \] subject to \(M^ f_ 0\subset M_ 0^{F_ 0}\), \(M^ f_ 1\subset M_ 1^{F_ 0}\), \(x_ i\in \{0,1\}\), where \(M_ 0^{F_ 0}\), \(M_ 1^{F_ 0}\) are unknown sets of permissible and inadmissible solutions: \(M_ 0^{F_ 0}=\{\tilde x|\) \(F_ 0(\tilde x)=0\}\), \(M^ F_ 1=\{\tilde x|\) \(F_ 0(\tilde x)=1\}\). The sets \(M_ 0^{f_ 0}\) and \(M^ f_ 1\) are given in the WD-LPB model. This paper suggests a two stage solution of the WD-LPB problem. At the first stage a set of extremal non-contradictory vectors are found. These vectors (ENCV) are the solutions of the WD-LPB problem. The properties of the monotone function \(F_ 0\) is used to find the ENCV set. At the second stage a pattern recognition algorithm tests the extremal vectors to find the permissible ones among them. The sets \(M^ f_ 0\) and \(M^ f_ 1\) in the WD-LPB are used as a training information to recognize for any \(\tilde x\) the predicate \(\ll \tilde x\in M_ 0^{F_ 0}\gg\).