Non-Boolean classical relevant logics I

Abstract

Relevant logics have traditionally been viewed as paraconsistent. This paper shows that this view of relevant logics is wrong. It does so by showing forth a logic which extends classical logic, yet satisfies the Entailment Theorem as well as the variable sharing property. In addition it has the same S4-type modal feature as the original relevant logic E as well as the same enthymematical deduction theorem. The variable sharing property was only ever regarded as a necessary property for a logic to have in order for it to not validate the so-called paradoxes of implication. The Entailment Theorem on the other hand was regarded as both necessary and sufficient. This paper shows that the latter theorem also holds for classical logic, and so cannot be regarded as a sufficient property for blocking the paradoxes. The concept of suppression is taken up, but shown to be properly weaker than that of variable sharing.

Boolean negation and non-conservativity III: the Ackermann constant

“…13 Thus reasoning by cases holds for Giambrone and Meyer's strengthened CTW t 1 . 14 However, the counter-model to f ∨ (f ∧ t → A) in Fig. 5 also validates these two stronger Boolean principles.…”

“…This follows from Lem. 9 together with the fact that t is definable as A → A for an arbitrary A in Ł 3 , that reasoning by cases is derivable for Ł 3 7 Both Ackermann's Π , as well as the logic Π E presented in [14] are worth mentioning as exceptions. See the latter paper for a discussion of why relevant logics ended up being wrongly viewed as inherently paraconsistent.…”

Farewell to Suppression-Freedom

“…It was shown in [18] that even classical logic satisfies Anderson and Belnap's Entailment Theorem, and so the Routley tradition's claim in this regard was correct. Thus, since the variable sharing property fails for classical logic, satisfying the Entailment Theorem can't be upheld as a sufficient criterion for relevancy if, as all within the relevant school seem in agreement it ought be, the variable sharing property is upheld as necessary.…”

“…], which aimed at giving an account of what may be called progressive reasoning, was rejection of the decidedly non-progressive principle of Identity, A → A. Thus it is true that (1) If A → A then (as a matter of logic) the Peripatetic theory of implication 18 is wrong: i.e. (A → A) → B for the wff B given.…”

Boolean negation and non-conservativity I: Relevant modal logics

“…A B A ∨ C B ∨ C holds in any logic CL where L extends BB[A12], and in any logic C L where L extends BB, provided only adjunction and modus ponens are primitive rules L. 12 Ackermann's Π , as well as the logic Π E presented in [14] are worth mentioning as exceptions. See the latter paper for a discussion of why relevant logics ended up being wrongly viewed as inherently paraconsistent.…”