 One kind of approach without cut is developed and defended in Weir 2005, 2015 (and for naive set theory in Weir 1998, 1999), and is there dubbed ‘neoclassical’. On this approach, the third value in the models is taken to be neither true nor false, and a countermodel to an argument from \(\Gamma\) to \(B\) must either: make every sentence in \(\Gamma\) true and \(B\) untrue, or else make \(B\) false and every sentence but one in \(\Gamma\) true, while making that remaining sentence in \(\Gamma\) unfalse. The motivating idea is that valid arguments must preserve truth, and must also preserve falsity backwards in a certain sense: if a valid argument has all its premises but one true and its conclusion false, then the remaining premise must be false. This allows for counterexamples to cut, but not to simple transitivity, and allows for consistency to be maintained. The resulting logic is weaker than classical logic. In our version of the liar paradox, the trouble is at LEM: Weir’s approach allows for counterexamples to excluded middle.