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

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.

Boolean negation and non-conservativity I: Relevant modal logics

“…Neither paper provide any argument for defining the Boolean extension to include B3, the reason for including it, it seems, is to get a so-called reduced semantics. 15 It should, however, be noted that B3 is rather different from the other two Boolean axioms: whereas B1 and B2 simply express that any instance of a Boolean excluded middle is entailed by every formula and a Boolean contradiction entails everything, B3 expresses that any relevant conditional either fails to be true, or the Boolean material conditional is true, or to put it equivalently; either the premises of any instance of modus ponens holds, or its conclusion does. Since B3 is equivalent to A ∧ (A → B) = B, and A → B A = B is a derivable rule of even CBB, it follows that the axiom is derivable in any logic with the pseudo modus ponens axiom A ∧ (A → B) → B which, again, is interderivable in BB with the rule of contraction, i.e.…”

“…(1) the meta-rule of reasoning by cases, (2) the derivability of γ in the Boolean extension, 3 There are no exceptions for sublogics of R. There are, however, exceptions. One such is exhibited in [15].…”

“…This is the first of in all three essays on Boolean negation and nonconservativeness pertaining to relevant logics. The second essay, [15], deals with the question whether the variable sharing property is always preserved when extending a logic with Boolean negation, whereas the third essay, [16], 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.…”

Boolean negation and non-conservativity III: the Ackermann constant

“…This is the third and last in a series of essays on Boolean negation and non-conservativeness pertaining to relevant logics. The first essay, [15], dealt with modal relevant logics, whereas the second essay, [16], dealt with the question whether the variable sharing property is always preserved when extending a logic with Boolean negation. Together the three essays paint a picture of relevant logics being quite often non-conservatively extended by Boolean negation.…”