The simple consistency of a set theory based on the logic ${\rm CSQ}$.

Curry’s Paradox and ω -Inconsistency

“…17.4, and [3]. Most of this discussion also extends to relevant approaches to the paradoxes ( [2], [8], [1]) since the ω-consistency of BCKN and BCKD would imply the ω-consistency of their strong relevant cousins BCIN and BCID, where I is the principle: φ → φ.…”

“…The first condition ensures that one can formulate an adequate account of the syntax. 3 The second condition ensures that the truth predicate behaves disquotationally in all contexts. If one made the further logical assumption that every instance of φ → φ was in the theory then one would be able to derive from the second condition the more traditional disquotational principle, the T-schema.…”

“…There are well known logics which do not contain the rule, such as intuitionistic logic and relevant logic. However these are not suitable for a naïve truth theory since they both contain the contraction axiom, W. Other logics not containing the rule are known to support a naïve truth predicate, such as Field's logic in [5] and Brady's CTQ [3] 8 , as well as the proposals based on relevant logic in [2], [8] and [1]. However these are all based on conditionals which are are substantially weaker than the conditional in L ∞ .…”

A New Conditional for Naive Truth Theory

“…4 I am indebted to Cian Dorr here for useful discussion of this fact. 5 Note that φ → (φ * φ) by RQ2 and φ → φ. Now consider the Curry sentence, γ ↔ (γ → ⊥).…”

“…The subsequent proposals have employed instead the revision theoretic techniques first outlined by Brady in [6] -and indeed there is a rich literature tied to this approach (see, for example, Brady [6], [5], [4], Priest [12], Yablo [15], Field [7], and Beall [2].) 1 However even the logics generated by the revision theoretic techniques lack a number of natural logical principles.…”