Spandrels of Truth

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. A full comparison, however, would require taking disjunction, conjunction and negation into account.…”

“…1 See, for example, [2], [5], [1]. 2 Similar principles can be added if one wished to include further connectives.…”

“…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 ∞ .…”

“…When people are talking about the Lukasiewicz predicate logic, they do not usually mean some particular axiomatic system but the logic validated by the Lukasiewicz connectives over [0,1]. There is no recursive axiom system that characterises this consequence relation, so there is no question of simply "dropping" the rule from Lukasiewicz logic understood this way.…”

“…What the absence of this rule from BCKN and BCKD demonstrates is that what we know we can have from a conditionalas seen in the theories of [2], [5], [1] for example -is substantially weaker than 10 It is worth noting that BCKD is closely related to the implicational fragment of basic logic, BL. Given the rule φ → χ, ψ → χ (φ ∨ ψ) → χ, which the model above also validates, you can prove in BCKD, F: ((φ → ψ) → χ) → (((ψ → φ) → χ) → χ), which constitutes the basis of the implicational fragment of BL -see [4] for further details.…”