The following paper refers to one of the most important directions in Jan Lukasiewicz's research [1939,1951] - Aristotle's syllogistic, and more precisely, to metalogic studies of this theory. In his studies of Aristotle's syllogistic, as well as of systems of propositional calculi [1952, 1953], Lukasiewicz makes use of the notion of decidability, also termed saturation of a deductive system. He bases this notion on the notion of rejected proposition, originated by himself [1921]. With reference to Aristotle's idea of rejection (demolishing) of certain propositions on the basis of others, the notion of rejected proposition of a system, e.g. Aristotle's syllogistic, is inductively defined by Lukasiewicz in the following way:

1. Some propositional expressions formulated in the language of a certain system have been classified as rejected axioms.

2. The other expressions are rejected by means of expressions which are already rejected as well as by means of the following special rules of rejection:
a) The rule of rejection by substitution: any expression is rejected if one of its substitutions is a rejected expression;
b) The rule of rejection by detachment: if there exists such a rejected expression 'b' that the conditional sentence built of 'a' as the antecedent and 'b' as the consequent is the thesis of the system under consideration (is asserted), then 'a' is the rejected expression.

It follows that if the rejected axioms are false then all the rejected expressions are false.

The notion of decidability, as well as that of consistency of Aristotle's syllogistic and other deductive systems were formulated by Lukasiewicz by means of exemplification, without providing any clear definition, however, as indicated by contexts, the meaning intended by him - and accepted by his disciple Jerzy Slupecki - was as follows:
3. The system is decidable if every of its expressions which is not a thesis is rejected on the basis of a finite number of axiomatically rejected expressions;
4. The system is consistent if none of its thesis is rejected.

The biaspectual formalization of Aristotle's syllogistic done by Lukasiewicz, comprising - beside "ordinary" axioms and inference rules - rejected axioms and the rules of rejection (a) and (b), made it impossible for him to obtain a satisfactory solution of the question of decidability of this system. The solution was found by his disciple Jerzy Slupecki. Having added to the syllogistic system a new rule of rejection, specific to the syllogistic, Slupecki proved that the syllogistic system, enriched in this way, is both decidable and consistent. The result achieved by Slupecki - in the words of Lukasiewicz [1939] - "organically united with the researches of the author ... the author regards as the most significant discovery made in the field of syllogistic since Aristotle."

It is intended in this paper:
1) to remind and discuss the above achievements of Lukasiewicz and Slupecki,
2) to present a general overview of the results of research concerning the question of decidability as understood by Lukasiewicz-Slupecki,
3) to outline certain studies of the very notion of decidability in the above mentioned sense, as well as of generalization of the notion of the rejected expression in the form of a function defined on any set of propositions of the language of the given system and with the values in the sets of such propositions. The latter notion, originated by Slupecki [1959] as a so called function or consequence of rejection, was set and examined on the basis of the general theory of deductive systems formulated by A.Tarski [1930].

The definition of this function is as follows: y is rejected on the ground of propositions of the set X if and only if at least one of the propositions of this set is derivable (deductible) from y. The above definition is associated with the following intuition: If a set of propositions X, on the basis of which propositions are rejected, is a set of false propositions then the value of the function of rejection for X is a set of false propositions, too. Under the supervision of J.Slupecki, research on the function and the relative notions was continued first by the author of this paper and then by G.Bryll [1969], which contributed greatly to formulation of the theory of rejected propositions.