Lambek calculus with one division and one primitive type permitting empty antecedents / S. L. Kuznetsov. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2009. № 2. P.  [Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 76-79].

The following assertion is proved: a deduction rule given by a scheme is admissible in the Lambek calculus with one division \mathrm{L}^*(\backslash) permitting empty antecedents if and only if it is admissible in the fragment of \mathrm{L}^*(\backslash) with one primitive type \mathrm{L}^*(\backslash; p_1). To do that, a type substitution is used which reduces the derivability in \mathrm{L}^*(\backslash) to the derivability in \mathrm{L}^*(\backslash;p_1).

Key words: Lambek calculus, admissible rules, proof nets.