Infinite types and the principle of union

Øystein Linnebo and Agustïn Rayo argue that ‘plausible’ assumptions lead to the conclusion that one should countenance very high order languages whose ideology encompasses proper class-many types. This chapter assesses their argument for the thesis of Infinite Types. After ironing out a technical glitch with their characterization of order, I turn to a premiss in Linnebo and Rayo’s argument which has so far received little attention. The Principle of Union states that one should countenance any language which ‘pools together’ the expressive resources drawn from any set of languages already deemed legitimate. The premiss may be understood in two ways depending on how we disambiguate ‘legitimate’ but neither reading is both plausible and dialectically effective.