Hopeful Monsters: A Note on Multiple Conclusions

Abstract

Arguments, the story goes, have one or more premises and only one conclusion. A contentious generalisation allows arguments with several disjunctively connected conclusions. Contentious as this generalisation may be, I will argue nevertheless that it is justified. My main claim is that multiple conclusions are epiphenomena of the logical connectives: some connectives determine, in a certain sense, multiple-conclusion derivations. Therefore, such derivations are completely natural and can safely be used in proof-theoretic semantics. Keywords multiple conclusions • defining rules • structure of derivations • logical inferentialism Delta: [.. . ] He seems engrossed in the production of monstrosities. But monstrosities never foster growth, either in world of nature or in the world of thought. Gamma: Geneticists can easily refute that. Have you not heard that mutations producing monstrosities play a considerable role in macro-evolution? They call such monstrous mutants 'hopeful monsters'. Lakatos, Proofs and refutations, pp. 21-22 1 Preamble An argument has one or more premises and one conclusion. Thus goes the orthodox stance. In this paper, I will defend the heretical claim that an argument may have

Inferential Quantification and the ω-Rule

“…16.3 below). 3 For a recent criticism of using multiple conclusions in a proof-theoretic framework see (Steinberger, 2011) and for a recent defence see (Dicher, 2020). 4 When sentential classes are considered, (Carnap, 1942(Carnap, : 38-39, 1943 introduced the following special cases: the universal class that comprises all sentences could be constructed either conjunctively (xV & y is the universal conjunctive, which is true if all sentences are true) or disjunctively (xV v y is the universal disjunctive, which is true if at least one sentence is true), and the null class that comprises no sentence could also be taken either conjunctively (xΛ & y is the null conjunctive, which by definition is true, since it contains no false statement) or disjunctively (xΛ v y is the null disjunctive, which by definition is false, since it contains no true statement).…”

Meta-inferences and Supervaluationism

“…To ensure that this system is also sound for global consequence, Cobreros continues, one must then restrict the locally but not globally valid rules of inference so that they can only be applied if Δ-Strengthening has not been previously applied in the derivation. 7 We have independently developed an approach based on restricting inferential moves to axiomatise global consequence in the context of the study of epistemic might [13], where global consequence is known as informational consequence [30]. In our earlier work, we provide a natural deduction system for epistemic modality in which only one primitive rule must be restricted and the restrictions are confined to hypothetical proof contexts.…”

“… Williams [26,§7] makes a similar observation from the model-theoretic perspective 7. Cobreros and Tranchini[6] extend Cobreros's strategy to cut-free, multiple conclusion calculi.…”

“…The existence of multiple-conclusion arguments in the wild is controversial, as are the credentials of multiple-conclusion systems as generalisations of actual inferential practice. For discussion, see[7,23] 3. The focus on single-conclusion arguments is crucial here.…”