Boolean negation and non-conservativity III: the Ackermann constant

Logic Journal of the IGPL

2020

Self Cite

Many relevant logics can be conservatively extended by Boolean negation. Mares showed, however, that E is a notable exception. Mares’ proof is by and large a rather involved model-theoretic one. This paper presents a much easier proof-theoretic proof which not only covers E but also generalizes so as to also cover relevant logics with a primitive modal operator added. It is shown that from even very weak relevant logics augmented by a weak K-ish modal operator, and up to the strong relevant logic R with a S5 modal operator, all fail to be conservatively extended by Boolean negation. The proof, therefore, also covers Meyer and Mares’ proof that NR—R with a primitive S4-modality added—also fails to be conservatively extended by Boolean negation.

Section: Main Theoremmentioning

confidence: 94%

Section: Main Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Boolean negation and non-conservativity I: Relevant modal logics

Logic Journal of the IGPL

2020

Self Cite

“…This is the second of in all three essays on Boolean negation and nonconservativeness pertaining to relevant logics. The first essay, [11], dealt with modal relevant logics, whereas the third essay, [12], deals with the question whether relevant logics with the truth-constant known as the Ackermann constant can be conservatively extended by Boolean negation. Together the three essays paint a picture of relevant logics being quite often non-conservatively extended by Boolean negation.…”

Section: Introductionmentioning

confidence: 99%

Boolean negation and non-conservativity II: The variable-sharing property

Logic Journal of the IGPL

2020

Self Cite

Many relevant logics are conservatively extended by Boolean negation. Not all, however. This paper shows an acute form of non-conservativeness, namely that the Boolean-free fragment of the Boolean extension of a relevant logic need not always satisfy the variable-sharing property. In fact, it is shown that such an extension can in fact yield classical logic. For a vast range of relevant logic, however, it is shown that the variable-sharing property, restricted to the Boolean-free fragment, still holds for the Boolean extended logic.

Non-Boolean classical relevant logics II: Classicality through truth-constants

2021

Synthese

Self Cite

This paper gives an account of Anderson and Belnap’s selection criteria for an adequate theory of entailment. The criteria are grouped into three categories: criteria pertaining to modality, those pertaining to relevance, and those related to expressive strength. The leitmotif of both this paper and its prequel is the relevant legitimacy of disjunctive syllogism. Relevant logics are commonly held to be paraconsistent logics. It is shown in this paper, however, that both E and R can be extended to explosive logics which satisfy all of Anderson and Belnap’s selection criteria, provided the truth-constant known as the Ackermann constant is available. One of the selection criteria related to expressive strength is having an “enthymematic” conditional for which a deduction theorem holds. I argue that this allows for a new interpretation of Anderson and Belnap’s take on logical consequence, namely as committing them to pluralism about logical consequence.