Strong negation is a well-known alternative to the standard negation in intuitionistic logic. It is defined virtually by giving falsity conditions to each of the connectives. Among these, the falsity condition for implication appears to unnecessarily deviate from the standard negation. In this paper, we introduce a slight modification to strong negation, and observe its comparative advantages over the original notion. In addition, we consider the paraconsistent variants of our modification, and study their relationship with non-constructive principles and connexivity.

This paper uses eye-tracking while reading to examine Standard English speakers’ processing of sentences with two syntactic negations: a negative auxiliary and either a negative subject (e.g., Nothing didn’t fall from the shelf) or a negative object (e.g., She didn’t answer nothing in that interview). Sentences were read in Double Negation (DN; the ‘she answered something’ reading of she didn’t answer nothing) and Negative Concord (NC; the ‘she answered nothing’ reading of she didn’t answer nothing) biasing contexts. Despite the social stigma associated with NC, and linguistic assumptions that Standard English has a DN grammar, in which each syntactic negation necessarily contributes a semantic negation, our results show that Standard English speakers generate both NC and DN interpretations, and that their interpretation is affected by the syntactic structure of the negative sentence. Participants spent more time reading the critical sentence and rereading the context sentence when negative object sentences were paired with DN-biasing contexts and when negative subject sentences were paired with NC-biasing contexts. This suggests that, despite not producing NC, they find NC interpretations of negative object sentences easier to generate than DN interpretations. The results illustrate the utility of online measures when investigating socially stigmatized construction types.

In two recent articles, Norihiro Kamide introduces unusual variants of Nelson’s paraconsistent logic and its classical extension. Kamide’s systems, IP and CP, are unusual insofar as double negations in these logics behave as intuitionistic and classical negations, respectively. In this article we present Hilbert-style axiomatizations of both IP and CP. The axiom system for IP is shown to be sound and complete with respect to a four-valued Kripke semantics, and the axiom system for CP is characterized by four-valued truth tables. Moreover, we note some properties of IP and CP, and emphasize that these logics are unusual also because they are contra-classical and inconsistent but nontrivial. We point out that Kamide’s approach exemplifies a general method for obtaining contra-classical logics, and we briefly speculate about a linguistic application of Kamide’s logics.