A disjunction does not settle the truth values of its disjuncts, but it says enough that if we know also that one is false we know that the other is true. This principle is called modus tollendo ponens. Since each disjunct entails the disjunction, if we know that one disjunct is false, then the other disjunct adds the same information as the disjunction, an idea implemented in a further rule for exploiting disjunctions, also known as Modus Tollendo Ponens (MTP). The not-both form ¬ (φ ∧ ψ) is analogous to disjunction and analogous principles apply. A principle modus ponendo tollens tells us that ¬ (φ ∧ ψ) together with the assertion of one of φ and ψ entails the denial of the other. And, since the denial of either φ or ψ entails ¬ (φ ∧ ψ), we can have a rule Modus Ponendo Tollens (MPT) for exploiting not-both forms. The rules MTP and MPT are examples of detachment rules. The resource exploited in each is its main resource and the additional resource that must be available is the auxiliary resource.

We will refer to as weakening the principles that disjunctions and not-both forms are entailed by assertions of components (in the case of disjunctions) or their denials (in the case of the not-both form). These principles provide the basis for two further attachment rules, both called Weakening (Wk), that license the addition of inactive resources. Since the second resource needed for a detachment rule may be inactive, attachment rules can prepare for the use of detachment rules for gap-closing rules.