Roy T Cook: There is No Paradox of Logical Validity

A number of arguments have recently appeared (independently in work by Beall & Murzi, Shapiro, and Whittle) for the claim that the logical validity predicate, when added to Peano Arithmetic (PA), is inconsistent in much the same manner as the addition of an unrestricted truth predicate to PA leads to a contradiction. In this paper I show that there is no genuine paradox of logical validity. Along the way a number of rather important, rather more general, lessons arise, including:

1. Whether or not an operator is logical depends not only on what content that operator expresses, but the way that it expresses that content (e.g. as a predicate versus as a logical connective).
2. Different paradoxes (or purported paradoxes) require different assumptions regarding the status of the principles involved (e.g. mere truth preservation versus logical validity).
3. A previous analysis of the purported paradox, due to Jeffrey Ketland, fails to properly locate the root of fallacious reasoning involved (apportioning the blame equally to the assumption that PA is logically valid and to the assumption that the validity rules are themselves logically valid).

As a result, there is no paradox of logical validity. More importantly, however, these observations lead to a number of novel, and important, insights into the nature of validity itself.