 We take as the language of Intuitionistic Linear Logic without exponentials the set \(\{\wedge , \vee , \rightarrow , * , 0, 1, \top , \bot \}\), where \(\wedge , \vee , \rightarrow, *\) are binary connectives and \(0, 1,\top , \bot\) propositional constants. We denote the logic by \(\bILL\). The axioms and rule of inference below provide a Hilbert-style axiomatization of this logic.