Deep scientific ideas have at least one distinctive property: they can be applied both by philosophers in abstract fundamental debates and by engineers in concrete practical applications. Mathematical approaches to modeling of vagueness also possess this property. Problems connected with vagueness have been discussed at the beginning of XXth century by philosophers, logicians and mathematicians in developing foundations of mathematics leading to clarification of logical semantics and establishing of mathematical logic and set theory. Those investigations led also to big step in the history of logic: introduction of three-valued logic. In the second half of XXth century some mathematical theories based on vagueness idea and suitable for modeling vague concepts were introduced, including fuzzy set theory proposed by Lotfi Zadeh in 1965 [16] and rough set theory proposed by Zdziss¸aw Pawlak in 1982 [4] having many practical applications in various areas from engineering and computer science such as control theory, data mining, machine learning, knowledge discovery, artificial intelligence. Concepts in classical philosophy and in mathematics are not vague. Classical theory of concepts requires that definition of concept C hast to provide exact rules of the following form: if object x belongs to concept C, then x possess properties P1, P2, . . . , Pn; if object x possess properties P1, P2, . . . , Pn, then x belongs to concept C.
a belongs to fuzzy set X what corresponds to vagueness of concepts. In particular if μX (a) = 0, then object a does not belong to fuzzy set X and if μX (a) = 1, then object a belongs to fuzzy set X in full degree what corresponds to classical notion of belonging to a set.
In rough set theory concepts are represented by subsets of a given space U and objects are represented by granules, some collections of objects, in classical rough set theory these granules are equivalence classes of some equivalence relation R on U . In rough set theory sets are represented and analyzed by two operators: lower and upper approximations, denoted respectively by R∗, R∗ and defined for set X ⊆ U as follows: R∗(X) = [{Y ∈ U/R : Y ⊆ X}
R∗(X) = [{Y ∈ U/R : Y ∩ X 6= ∅}.
Set X ⊆ U is rough iff R∗(X) 6= R∗(X). With every set X ⊆ U there are associated three sets called regions: positive region
P OS(X) := R∗(X), negative region
N EG(X) := U \ R∗(X) = R∗(X ) = [{E ∈ U/R : E ∩ X = ∅} ′ where X′ = U \ X and boundary region
BN D(X) := R∗(X) \ R∗(X) = U \ (P OS(X) ∪ N EG(X)).
Set X ⊆ U is rough iff BN D(X) 6= ∅. In the case of any set X ⊆ U positive region of X can be interpreted as a set of objects from U which surely belongs to X, negative region can be interpreted as a set of objects from U which surely do not belong to X, whereas boundary region can be interpreted as a set of objects from U which possibly belong to X.
One can note that both approaches to modelling vagueness are some generalizations in which crisp sets are particular cases: in fuzzy set theory fuzzy set Xof objects from U is crisp iff for all a ∈ U either μX (a) = 1 or μX (a) = 0; in rough set theory set X ⊆ U is crisp iff BN D(X) = ∅. Both theories are also essentially connected with Łukasiewicz’s ideas.
The main object of the paper is to present and compare fuzzy sets and rough
sets approaches to vagueness and uncertainty modelling and analysis, in particular we
will discuss representation of vague concepts in both theories. We will also present
Łukasiewicz’s arithmetization of propositional calculus semantics and Łukasiewicz’s
involvement in discussion on meaning and logical values of propositions about future
which led to introduction of the third logical value and to proposing three-valued logic.
Our comparison of fuzzy set theory and rough set theory approaches to vagueness
modelling will be made with respect to a characterization of vagueness proposed in
contemporary philosophy. This characterization includes a second order vagueness condition
which, roughly speaking, requires that a boundary of a concept cannot be a crisp set.
We will conclude the paper with presenting and discussion solutions to that problem
in the rough set approach to vagueness modelling including that proposed by Andrzej
Skowron in [