A Note on Deduction Rules with Negative Premises

The so called frame axioms problem or frame problem is well-known to anybody who is interested in automatic problem solving.Some information concerning this problem can be found in (1) or (2).In (2) the author proposes to eliminate the frame axioms in such a way that they are replaced by a new deduction rule denoted as UNLESS-operator.In fact,this operator is a rule enabling to derive that something concerning the environment is valid at the present situation supposing it was valid in a past situation and it is not provable that a change concerning the validity of this statement has occur-ed.For our reasona only the formally logical aspect of this solution is interesting es it leais immediately to a new and very interesting modification of the notion of formalized theory. The aspect just mentioned consists in in the fact that in (2) the author formalizes his UNL_.SS-operator in the following form of a deduction rule: If a formula A is deducible and a formula B is not deducible ; then a formula C is de-ducible.Written in symbols (1) where,of course."not" and " " are symbols of an appropriate metatheory,not the investigated theory itself. The use of a deduction rule of this type may involve some doubts whether we are justified to do so.The first objection arises from the fact that any deduction rule,including those of the type (1), is an integral part of the definition of the notion "deducible formula",ao this notion itself.neither in its negative form "not l-B",is not allowed to occur in a deduction rule.It is a matter of fact that in "usual" formalized theories the deduction rules are also written in the form using the symbol ,e.g. if the modus ponens rule is considered. However,it is a well-known fact that in this case the symbol can be eliminated so that the definition of the notion "provable formula" should be correct.lt is the g08l of this paper to investigate the deduction rules of the type (1) in order to see,wheher a set of deduction rules,containing at least one rule of the type (1) and considered together with a recursive set of axioms defines unambiguously and correctly e set of de-ducible formulas.The second objection concerning the deduction rules of the type (1) arises from the fact that when a proof is constructed we have at our disposal in every step only a finite number of formulas proved already to be theorems,so …