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A Hilbert-type axiomatic rejection $\mathbf{HAR}$ for the propositionalfragment $\mathbf{L_1}$ of Le\'{s}niewski's ontology is proposed. Also aGentzen-type axiomatic rejection $\mathbf{GAR}$ of $\mathbf{L_1}$ is proposed.Models for $\mathbf{L_1}$ are introduced. By axiomatic rejection, Ishimoto'sembedding theorem will be proved. One of our main theorems is:  \noindent \textsc{Theorem} \rm (Main Theorem) \it  $\vdash_T A \enspace \Longleftrightarrow \enspace \vdash_H \enspace A$  $\enspace \enspace \enspace \enspace \enspace \enspace \Longleftrightarrow\enspace TA \enspace \mbox{is valid in first-order predicate logic withequality}$  $\enspace \enspace \enspace \enspace \enspace \enspace \Longleftrightarrow\enspace not \dashv_H A.$  \noindent where $\vdash_T A$ means that $A$ is provable in the tableau methodof $\mathbf{L_1}$, while $\vdash_H A$ means that $A$ is provable in theHilbert-type $\mathbf{L_1}$. \rm \smallskip  In the last section, as the chracterization theorem, we shall show thetheorem which contains six equivalent statements with the cut eliminationtheorem etc.

Axiomatic Rejection for the Propositional Fragment of Le\'{s}niewski's Ontology