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Tore Fjetland Øgaard | Boolean negation and non-conservativity I-III | Logic Journal of the IGPL

Boolean negation and non-conservativity I: Relevant modal logics

Tore Fjetland Øgaard

ABSTRACT

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 NRR with a primitive S4-modality added—also fails to be conservatively extended by Boolean negation.

KEYWORDS: Boolean negation, non-conservative extension, entailment, modality, relevant logics.

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

Tore Fjetland Øgaard

ABSTRACT

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.

KEYWORDS: Boolean negation, non-conservative extension, relevant logics, variable sharing.

Boolean negation and non-conservativity III: the Ackermann constant

Tore Fjetland Øgaard

ABSTRACT

It is known that many relevant logics can be conservatively extended by the truth constant known as the Ackermann constant. It is also known that many relevant logics can be conservatively extended by Boolean negation. This essay, however, shows that a range of relevant logics with the Ackermann constant cannot be conservatively extended by a Boolean negation.

KEYWORDS: Ackermann constant, Boolean negation, non-conservative extension, relevant logics.