  It is well-known that intuitionistic logics can be formalized by means of
Brouwerian semilattices, i.e. relatively pseudocomplemented semilattices. Then
the logical connective implication is considered to be the relative
pseudocomplement and conjunction is the semilattice operation meet. If the
Brouwerian semilattice has a bottom element 0 then the relative
pseudocomplement with respect to 0 is called the pseudocomplement and it is
considered as the connective negation in this logic. Our idea is to consider an
arbitrary meet-semilattice with 0 satisfying only the Ascending Chain
Condition, which is trivially satisfied in finite semilattices, and introduce
the connective negation x^0 as the set of all maximal elements z satisfying
x^z=0 and the connective implication x->y as the set of all maximal elements z
satisfying x^z\le y. Such a negation and implication is "unsharp" since it
assigns to one entry x or to two entries x and y belonging to the semilattice,
respectively, a subset instead of an element of the semilattice. Surprisingly,
this kind of negation and implication, respectively, still shares a number of
properties of these connectives in intuitionistic logic, in particular the
derivation rule Modus Ponens. Moreover, unsharp negation and unsharp
implication can be characterized by means of five, respectively seven simple
axioms. Several examples are presented. The concepts of a deductive system and
of a filter are introduced as well as the congruence determined by such a
filter. We finally describe certain relationships between these concepts.
