The logic of relative systems

Abstract

This study aims to develop the logic of relativism, then to apply that logic to the question of self-refutation in relativism.

After first defining generic relativism as radical indexed pluralism, the study identifies three substantive theses following as a consequence of this definition that any instance of relativism must meet: (1) the formal requirements for relativity, (2) objective equity, and (3) incommensurability. Each of the three theses is developed in detail to determine precisely what must be demonstrated by a claim of relativism.

Next, the study develops five separate logical systems to represent the logic of relativism, each adding progressively more complex relativistic features. The first system is modelled on basic modal logic. The second models multiple kinds of relativity. The third models relativised accessibility relations. The fourth combines features of the second and third systems. And the fifth is modelled on a non-normal modal logic. Finally, the charge of self-refutation is evaluated with regard to each system in turn.

I argue that while all five systems initially seem to support models that support fully relativistic perspectives, when an operator is added to the language to express whether a sentence is true for a perspective, four of the systems can no longer support fully relativistic perspectives, since some sentences including that operator must be evaluated to be absolute within all perspectives. The fifth system, however, formulated analogously to a non-normal modal logic, does support fully relativistic perspectives even with the additional operator. I argue further that the nature of relativism provides some motivation for considering this last system to be the logic of global relativism.

While these formal, structural arguments do not fully address all the varieties of self-refutation arguments levelled against relativism, in particular, those based upon pragmatics, incoherence, and the nature of language, I argue that there are reasons to think that these latter kinds of argument may not be completely successful, and I suggest various ways that these arguments might be strengthened against relativism.