Abstract
The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. 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.
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Notes
To align the two hierarchies, one may consider \(\mathbf {Lk}\) as corresponding to DNE, \(\mathbf {Li}\) to EFQ, \(\mathbf {Le}\) to PP. One can easily see that \(\mathbf {Le}^\prime \) is stronger than \(\mathbf {Lj}\)+WT, but the latter has \(\mathcal {W}_3^\prime \) as a model, whereas the former does not. Similarly DGP, LEM, WLEM, \(\text {DGP}^\rightarrow \) do not seem to have any correspondence in Odintsov’s hierarchy.
The version \(( \varphi \rightarrow \lnot \varphi )\rightarrow \lnot \varphi \) is used in [19]. However, since ex falso quodlibet falsum (\(\bot \rightarrow \lnot \phi \)) holds in minimal logic, that version is provable over minimal logic. This suggests that while the present work illustrates many distinctions, there are still more distinctions that are not apparent here.
Our name for this is based on the guiding principle of the protagonist of Douglas Adam’s novel Dirk Gently’s Holistic Detective Agency [1] who believes in the “fundamental interconnectedness of all things.” It also appears as an axiom in Gödel-Dummett logic, and is more commonly known as (weak) linearity.
The Wikipedia page at https://en.wikipedia.org/wiki/Consequentia_mirabilis actually lists this as equivalent to 6, however that is not quite correct as we can see below. If we add the assumption that \(\varphi \rightarrow \psi \equiv \lnot \varphi \vee \psi \), then they do turn out to be equivalent. However that statement—interpreting \(\rightarrow \) as material implication—is minimally at least as strong as LEM.
Connexive logics are closely related. There, instead, the antecedent is taken as axiom schema; thus connexive logics are entirely non-classical, since \(\lnot (\varphi \rightarrow \lnot \varphi )\) is not classically valid.
Since all of these axioms are not provable in minimal logic, deducing them classically is not mechanical and they are therefore good exercises for students.
Of course this is intuitionistically provable/definitional.
An axiom in Jankov’s logic, and De Morgan logic.
With judicious substitutions—in particular, using \(\varphi \equiv \top \) in the right-to-left implications.
A historical perspective can be found in e.g. [6], which establishes links with Diego’s Theorem. The present paper focuses on classifying several well-known principles into clearly worked out equivalences and distinctions.
Simply use \(\vee \)-introduction and the proposed translation; EFQ follows. To translate \(\varphi \vee \psi \) as \(\lnot \varphi \rightarrow \psi \) would therefore be disingenuous.
Note that negated conjunction is a special case.
It is not clear, however, that this makes no difference in proofs; more on this later.
It may be interesting to note that at least one instance of contraction is used in this proof.
Substitute \(\varphi \rightarrow \vartheta \) for \(\alpha \), and \(\psi \rightarrow \vartheta \) for \(\beta \).
This is a point of departure for many other non-classical logics, such as relevant logics, logics of formal inconsistency, and the like. As mentioned, this is a preliminary investigation into the realm of non-classical reverse mathematics more generally; so we stick fairly close to the usual, classical, interpretation of negation.
This holds if, for example, \(\mathcal {W}\) is finite, or one assumes Zorn’s Lemma.
The converse does not hold. See the model \(\mathcal {W}_4\) in Sect. 6.
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This research was carried out while McKubre-Jordens was partially supported on Marsden Grant UOC1205, funded by the Royal Society of New Zealand; and Diener and McKubre-Jordens were partially funded by the Royal Society of New Zealand’s Counterpart Funding Initiative PIRE for the FP7 Marie Curie project CORCON.
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Diener, H., McKubre-Jordens, M. Classifying material implications over minimal logic. Arch. Math. Logic 59, 905–924 (2020). https://doi.org/10.1007/s00153-020-00722-x
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DOI: https://doi.org/10.1007/s00153-020-00722-x