Four Basic Logical Issues

Abstract

The paper addresses what we see as the four major issues in logic. The overriding issue is that of the choice of logic. We start with some discussion of the preliminary issue of whether there is such a 'one true logic,' but we reserve the main discussion for the first issue of 'classical logic versus non-classical logic.' Here, we discuss the role of meaning and truth, the relation between classical logic and classical negation, and whether and, if so, how classical logic should reside at the base world. Given the argument in favor of an overall use of non-classical logic, the second issue is that of the choice of non-classical logic. Brady's logic MC of meaning containment is argued for, with some comparison made with other relevant logics. For the remaining two issues, we make a case for relevant deduction, in comparison with classical deduction, and we explore possibilities for the appropriate meta-logic, comparing classical and non-classical approaches.

Semantics without Toil? Brady and Rush Meet Hallden

“…This, however, would not be what Brady and Rush have in mind, though one does have to look around a bit to see this. The discussion in (Brady and Rush 2009) immediately following that quoted in Section 1 is this (p. 505f.…”

“…(#/#) Under (2) in the quotation from (Brady and Rush 2009) given above, "not", "&", "∨" and "if . .…”

“…Since the logical devices required for homophonic model theory of the kind described as trivial in (Brady and Rush 2009) include not only a suitable "iff" but also the universal quantification (over interpretations I) used in the definition of validity, 11 if we want to retain the restriction to purely sentential languages, aspects of the first option will be needed to deal with that, whether or not the same route is taken in the case of "iff". (A formula A is defined to be valid in the quotation above if "I(A) = T , for all assignments to its variables," which more explicitly must be take to mean "I(A) = T -or "I |= A", on the =free notation preferred treatment here -for all interpretations I".)…”

“…The material under (2) of the following quotation from Section 5 (entitled "Classical meta-logic versus nonclassical meta-logic") of (Brady and Rush 2009) provides the stimulus for the present discussion; the rest of the quotation is included to provide context. Note that letters from the range p, q, r, .…”

“…Note also that it is not only in the logic of metaphysical necessity but also in the logic of vagueness that identifying the truth on an interpretation of a vague sentence with a something's being identical with something may raise difficulties for anyone thinking that identity statements can be vague - (Evans 1978) and all that.) Though we continue to use the I(A) = T notation when dealing with the explicit discussion in (Brady and Rush 2009), we return to the pitfalls invited by doing so in Remark 4.4 below. ◀…”