I argue that the strong transcendental strategy can offer us a serious and robust response the sceptic who doubts that we can have knowledge of the extra-mental. This sceptic is motivated by sceptical hypotheses to worry that I could have all the thoughts and experiences I do in fact have if the world were radically different to how I take it to be. Transcendental arguments start from a premise about our thoughts or experience and move on to show that something must be the case because it is a necessary condition of our having such thoughts or experience. As such, transcendental arguments are particularly well-positioned to answer this sceptic, as the premises of a transcendental argument are drawn from the mental propositions the sceptic accepts. Some philosophers have argued that the same concerns that drive the sceptic to doubt the extra-mental should also give her cause to doubt the mental. To prevent the sceptic retreating to this thought, I show that these arguments are only effective against propositions that were unlikely to form the basis of a transcendental argument. Strong transcendental arguments (STAs) are differentiated from weak transcendental arguments as being those that move from mental premises to conclusions about the extra-mental world. I defend STAs against Stroud’s objection that this is not possible, on the basis that the objection rests on an illicit assumption of dualism about mind and world. I argue for the plausibility of supervenience physicalism as a metaphysical picture upon which such inferences would be possible. I show how a dispositionally essentialist understanding of the laws of nature would plausibly support a metaphysically necessary psychophysical law, from which we could draw the bridging premise of an STA. This changes the dialectic, forcing the sceptic to defend specific metaphysical positions, such as resemblance nominalism, and to engage substantially with philosophy. The plausibility of strong transcendental arguments tells us something of what must be true of the world for the sceptic’s arguments to even get started.