Immune Logics

Abstract

This article is concerned with an exploration of a family of systems—called immune logics—that arise from certain dualizations of the well-known family of infectious logics. The distinctive feature of the semantic of infectious logics is the presence of a certain “infectious” semantic value, by which two different though equivalent things are meant. On the one hand, it is meant that these values are zero elements for all the operations in the underlying algebraic structure. On the other hand, it is meant that these values behave in a value-in-value-out fashion for all the operations in the underlying algebraic structure. Thus, in a rather informal manner, we will refer to immune logics as those systems whose underlying semantics count with a certain “immune” semantic value behaving in a way that is somewhat dual to that of the infectious values. In a more formal manner, carrying out this dualization will prove to be not as straightforward as one could imagine, since the two characterizations of infectiousness discussed above lead to two different outcomes when one tries to conduct them. We explore these alternatives and provide technical results regarding them, and the various logical systems defined using such semantics.

Humberstone on Ayer’s Emotivism

“…For Humberstone, the OL semantics, especially the truth-table for conjunction, captures the main intent of Ayer's view about instantiated moral sentences. Since the moral sentence 'You acted wrongly in stealing the money' just means 'You stole the money and that action is wrong', it follows logic, see (Belikov, 2021) and (Fisch and Turquette, 1966); for Sobociński's logic, see (Da Ré and Szmuc, 2021) and (Joaquin, 2021). Cooper uses 'T', 'G', and 'F' to represent 'true', 'gap', and 'false', respectively (Cooper, 1968, 305).…”

“…On the other hand, Humberstone uses '1', '2', and'3' (Humberstone, 2011, 1044). For our purposes, we follow the semantic representation by (Joaquin, 2021) and (Da Ré and Szmuc, 2021).…”

Many-Valued Logics and Bivalent Modalities

“…5. A truth-value r is called immune [Da Ré and Szmuc, 2021] if, whenever it is an input of a truth-function along with a truth-value r ′ , r ′ is the output. M L be a matrix for an immune logic L such that 1 and d are the designated values of the matrix such that v(ϕ∧ψ…”

“…Immune logics are new in the literature. We refer the reader to[Da Ré and Szmuc, 2021] where they were first introduced.…”