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Substitution in relevant logics. (English) Zbl 1476.03027

The author concludes that substitution (Subs) (i.e., Leibniz’s law in the form \(s=t\wedge A(s)\rightarrow A(t)\)) ought to fail in relevant logics since “i) it trivializes naïve set theory; ii) it is invalid in Belnap’s test-model for relevance; iii) it is in conflict with one of the motivations behind relevant logics as such, namely the need to recognize impossible worlds – worlds at which logic itself is different.” (p. 679). Previously, the paper presents a thorough study of Subs in G. Priest’s [An introduction to non-classical logic. From if to is. Cambridge: Cambridge University Press (2008; Zbl 1148.03002)] (INCL) and [In contradiction. A study of the transconsistent. Dordrecht etc.: Martinus Nijhoff Publishers (1987; Zbl 0682.03002)] (IC). Consider the following thesis Subset Constraint (SC).
SC: any identity statement true at some world is true at the base world (\(=\) is, at every non-normal world in the model, interpreted to be a subset of the set of all identity pairs).
The criticism of Priest’s INCL and IC is based upon the presence of SC in the logics defined in these works. Although identity contracts in relevant logics in which Ackermann’s rule \(\delta \) (equivalently, rule Asser, [G. Robles, Log. Log. Philos. 22, No. 4, 411–427 (2013; Zbl 1321.03036)] is valid, the author shows that identity does not contract in quantified contractionless ticket entailment TW with identity (i.e., reflexivity and Subs). (\(\delta \) is \(A\rightarrow (B\rightarrow C),B\vdash A\rightarrow C\)); Asser is \(A\vdash (A\rightarrow B)\rightarrow B\)) Nevertheless, it is shown that in INCL identity contracts, due to SC, even in quantified basic relevant logic B with identity (i.e., reflexivity and Subs). Thus Subs is not SC. Moreover, SC does not entail an unwarranted version of cross-world substitution (§2, §3). In §4, the author studies the problem of validating Subs without validating SC or the rule weakening (i.e., \( A\vdash B\rightarrow A\)), while in §5, he investigates the origin of SC in IC. Finally, regarding Subs, he comes to the conclusions stated at the beginning of this review.

MSC:

03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03A05 Philosophical and critical aspects of logic and foundations
03B53 Paraconsistent logics
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
03E70 Nonclassical and second-order set theories

Software:

MaGIC
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References:

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