Answer:
The Vertical-Horizontal Illusion is interesting because it is relevant to debates about modularity, cognitive penetration, and the nature of experience. To explain: on the hypothesis that the mind is modular, a mental module is a kind of semi-independent department of the mind which deals with particular types of inputs, and gives particular types of outputs, and whose inner workings are not accessible to the conscious awareness of the person – all one can get access to are the relevant outputs. So, in the case of the Vertical-Horizontal Illusion, a standard way of explaining why experience of the illusion persists even though one knows that one is experiencing an illusion is that the module, or modules, which constitute the visual system are ‘cognitively impenetrable’ to some degree—i.e. their inner workings and outputs cannot be influenced by conscious awareness. For a general discussion of cognitive penetration, see Macpherson (2012).
Philosophers have also been interested in what illusions like the Vertical-Horizontal Illusion can tell us about the nature of experience. For example, in the case of experiencing the Vertical-Horizontal Illusion, it would seem to be that one can know that the two lines are the same length whilst at the same time experience them as different lengths. If so, then this might count against the claim the perceptual states are belief-like, because if perceptual states were belief-like then, when experiencing the Vertical-Horizontal Illusion one would simultaneously believe that the lines were, and were not, the same length. This would seem to entail that one was being irrational, because one would simultaneously be holding contradictory beliefs. But it seems highly implausible that one is being irrational when under going this illusion. For discussion of this general point about whether perceptions are like beliefs, see Crane & French (2016).