  The aim of the present paper is to show that the concept of intuitionistic
logic based on a Heyting algebra can be generalized in such a way that it is
formalized by means of a bounded poset. In this case it is not assumed that the
poset is relatively pseudocomplemented. The considered logical connectives
negation, implication or even conjunction are not operations in this poset but
so-called operators since they assign to given entries not necessarily an
element of the poset as a result but a subset of mutually incomparable elements
with maximal possible truth values. We show that these operators for negation
and implication can be characterized by several simple conditions formulated in
the language of posets together with the operator of taking the lower cone.
Moreover, our implication and conjunction form an adjoint pair. We call these
connectives "unsharp" or "inexact" in accordance with the existing literature.
We also introduce the concept of a deductive system of a bounded poset with
implication and prove that it induces an equivalence relation satisfying a
certain substitution property with respect to implication. Moreover, the
restriction of this equivalence on the base set is uniquely determined by its
kernel, i.e. the class containing the top element.
