I introduce a novel hybrid calculus H for the validities and invalidities of classical propositional logic. The calculus H is different in nature from other hybrid calculi that can be found in the literature in that it does not include specific anti-sequent rules. Instead, I add to the sequent rules of classical propositional logic only two structural rules that allow us to introduce and eliminate anti-sequents in our derivations. The resultant system is much simpler than the existing systems in the (...) literature. For one thing, it does not suffer from a proliferation of rules. For another, it is structurally complete: its admissible rules are all derivable. (shrink)

Strict-Tolerant Logic ($$\textrm{ST}$$ ST ) underpins naïve theories of truth and vagueness (respectively including a fully disquotational truth predicate and an unrestricted tolerance principle) without jettisoning any classically valid laws. The classical sequent calculus without Cut is sometimes advocated as an appropriate proof-theoretic presentation of $$\textrm{ST}$$ ST. Unfortunately, there is only a partial correspondence between its derivability relation and the relation of local metainferential $$\textrm{ST}$$ ST -validity – these relations coincide only upon the addition of elimination rules and only within (...) the propositional fragment of the calculus, due to the non-invertibility of the quantifier rules. In this paper, we present two calculi for first-order $$\textrm{ST}$$ ST with an eye to recapturing this correspondence in full. The first calculus is close in spirit to the Epsilon calculus. The other calculus includes rules for the discharge of sequent-assumptions; moreover, it is normalisable and admits interpolation. (shrink)

Against the backdrop of the frequent comparison of theories of truth in the literature on semantic paradoxes with regard to which inferences and metainferences are deemed valid, this paper develops a novel approach to defining a binary predicate for representing the valid inferences and metainferences of a theory within the theory itself under the assumption that the theory is defined with a classical meta-theory. The aim with the approach is to obtain a tool which facilitates the comparison between a theory (...) and its competitors within the theory itself, thereby expressing the disagreement between the theories within the theories. After discussing what we can and should require of an object-linguistic representation of a theory for that purpose, this paper proposes to restrict the representation of valid metainferences to locally valid metainferences, a requirement which turns out to be \-consistent and conservative over classical first-order arithmetic. This approach is then applied to four theories definable on strong Kleene models using a labelled nested sequent calculus. (shrink)