 The double use pointed to above will loom largest if these connectives are read multiplicatively: if 2d really is to do the work of \(\neg \Tr(\left\ulcorner L\right\urcorner)\) and \(\Tr(\left\ulcorner L\right\urcorner)\) together, then it really does use two copies of 2a, one for each conjunct. On an additive reading of 2d and 3d, this seeming double use need not be troubling, since 2d itself only needs to do the work of one of its conjuncts. Although this can be either one, whichever conjunct it is, a single use of 2a will suffice. On this additive reading, it is the principles LEM and EFQ that come into question; for example, with \(\wedge\) read additively it takes two occurrences of the same contradiction to entail an arbitrary sentence (since both conjuncts must be used), while the derivation above only yields one. (The situation for 3d and 3a is similar, in either case.) We do not here consider further details; for more on these choices and noncontractive approaches in general, see Beall and Murzi (2013), Grishin (1982), Petersen (2000), Restall (1994), Ripley (2015), L. Shapiro (2011a, 2015), and Zardini (2011, 2013). (Some of these focus on set-theoretic paradoxes rather than truth-theoretic paradoxes, but many of the issues are parallel. See also the entry on  Russell’s paradox.)