Bulletin of the Section of Logic Volume 20:3/4 (1991), pp. 129–135 reedition 2005 [original edition, pp. 129–137] Timothy Williamson A RELATION BETWEEN NAMESAKES IN MODAL LOGIC Abstract The closure under a certain rule of the modal system usually called “K4” is shown to be the modal system called “K4” by Soboci´ nski. A modal system S provides double cancellation iff for all sentences A and B, if ⊢ S LA ≡ LB and ⊢ S MA ≡ MB then ⊢ S A ≡ B. It is of course not required that ⊢ S ((LA ≡ LB)&(MA ≡ MB)) ⊃ (A ≡ B). If S is any modal system, there is a smallest normal extension S+ of S providing double cancellation (terminology as in [1]). For reasons given in [3], the double cancellation rule may be of some philosophical interest. This note extends the results of [3] and [4] on double cancellation in normal systems. The name “K4” is ambiguous. It is now customarily used for the smallest normal system containing the schema LA ⊃ LLA; this practice is followed here. Soboci´ nski used the same name for the smallest system containing KT 4 (= S4) and the schemata A ⊃ (MLA ⊃ LA) and MLA ≡ LMA ([2]); this normal system is here referred to as K4 Sob . The relation to be established between these namesakes is that K4+ = K4 Sob ; it will also be shown that K4 Sob is the only normal extension of K4 providing double cancellation in which there is not modal collapse. This extends the results of [3], in which it was shown that KD4!+ = K4 Sob , where KD4! is the smallest normal system containing the formula MT (T being a constant tautology) and the schema LLA ≡ LA, and [4], in which it is shown that KW + is inconsistent, where KW is “provability logic”, the smallest normal system containing L(LA ⊃ A) ⊃ LA, an extension of K4. Lemma A. ⊢ K4+ (A&M (A&LA)) ⊃ LA.