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

The intuitionistic-like logic based on a poset