Classifying material implications over minimal logic

Abstract

The so-called paradoxes of material implication have motivated the development of many nonclassical logics over the years [2][3][4][5]11]. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency.

The Drinker Paradox and its Dual

“…However, such an embedding is not possible if the domain contains a single element. It was shown above that DP holds in models with branches if the domain contains only one term, while in [6] it is shown that DGP holds only in v-free models. Therefore there can be no way of deriving instances of DGP from DP without an embedding using two or more elements in the domain.…”

“…A n hold for every formula. A full model, as described in [6], is sufficient. A full model is one where we can freely create predicates, as long as they satisfy the usual monotonicity requirements.…”

“…5 Also known as "explosion". 6 Also known as the "principle of excluded middle" and as "tertium non datur". 7 The name DGP was introduced in [6], and is a literary reference to the novel [1], whose main character believes in "the fundamental interconnectedness of all things".…”

“…6 Also known as the "principle of excluded middle" and as "tertium non datur". 7 The name DGP was introduced in [6], and is a literary reference to the novel [1], whose main character believes in "the fundamental interconnectedness of all things". DGP is otherwise also known as (weak) linearity, and is the basis for Gödel-Dummett logic [13].…”

“…It is a natural question to ask whether we can find similar results in the absence of EFQ. Previous work by a subset of the authors [6] has investigated the case of propositional schemata, but has left the predicate case untouched. Similar work can also be found in [7,10].…”