Some remarks on the validity of the principle of explosion in intuitionistic logic

Abstract

The formal system proposed by Heyting (1930, 1936) became the standard formulation of intuitionistic logic. The inference called ex falso quodlibet, or principle of explosion, according to which anything follows from a contradiction, holds in intuitionistic logic. However, it is not clear that explosion is in accordance with Brouwer's views on the nature of mathematics and its relationship with logic. Indeed, van Atten (2009) argues that a formal system in line with Brouwer's ideas should be a relevance logic. We agree that explosion should not hold in intuitionistic logic, but a relevance logic requires more than the invalidity of explosion. The principle known as ex quodlibet verum, according to which a valid formula follows from anything, should also be rejected by a relevantist. Given ex quodlibet verum, the inference we call weak explosion, according to which any negated proposition follows from a contradiction, is proved in a few steps. Although the same argument against explosion can be also applied against weak explosion, rejecting the latter requires the rejection of ex quodlibet verum. The result is the loss of at least one among reflexivity, monotonicity, and the deduction theorem in a Brouwerian intuitionistic logic, which seems to be an undesirable result. 1 The first author acknowledges support from CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), a doctorate scholarship. The second author acknowledges support from CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), research grant 311911/2018-8. The authors would like to thank Rodolfo Ertola-Biraben, André Porto and Wagner Sanz for some helpful comments and suggestions. They are not responsible, however, for any infelicities that may remain.