Stewart Shapiro - Varieties of Logic

The aim of this book is to articulate various kinds of pluralism and relativism concerning logic, and to defend some of them. There is controversy over what logic is. Indeed, a certain flexibility concerning the nature of logic drives at least some aspects of the project. Although we will broach more general matters from time to time, the main focus is on rigorous deduction, within various mathematical theories. Our questions include these: Is there more than one logic that is employedor can be employed, or should be employedin legitimate mathematical theories? Is there more than one extension for the relation of logical consequence, or validity, depending on the background theory (or depending on something else)? If so, is the extension of logical consequence somehow relative to something?

In the philosophical literature, there is no consensus on terms like pluralism, relativism, and contextualism. So I must begin by briefly showing how I use the indicated terms. That is the business of the succeeding sections of this chapter, which will also help motivate the material to follow. For now, we will keep the terminological matters tacit, hoping that the resulting confusion is minimal.

develops a number of different senses in which one can be a relativist and/or a pluralist concerning logical consequence and validity. I argue, first, that the terms logical consequence and validity are polysemous or, perhaps better, they express cluster concepts. A number of different, closely related notions go by those names. They invoke matters of modality, meaning, effectiveness, justification, rationality, and form. The upshot is a kind of pluralism: there are different, mutually incompatible, but equally legitimate ways to sharpen or further articulate the intuitive notion(s) of logical consequence and validity. Absent a specific theoretical purpose, there is no need to choose between the various articulations and sharpenings.

I suggest that at least much of the clash between proof-theoretic and model-theoretic approaches to logic can be understood in these terms. Each camp is onto a different aspect of the intuitive notion of logical consequence, or, perhaps better, each camp proposes a different sharpening of the intuitive notion. The same goes for at least some of the clash between advocates of higher-order logic and those who insist on restricting logic to first-order, a debate that has occupied me for some time (Shapiro []). If this is correct, then in both cases, there is no real clash, unless it is over which of the various aspects of the intuitive notion(s) are primary, or, perhaps, which are relevant to this or that theoretical purpose.

The pluralism defended by J. C. Beall and Greg Restall [] can be located in the taxonomy. They articulate a pluralism entirely within the broadly model-theoretic camp, arguing that even the model-theoretic notions can be articulated, or sharpened, in different, equally legitimate ways. The field for this sort of pluralism is rich, but it is only one sort of pluralism.

A different sort of pluralism arises from a perspective toward logic sometimes called logic-as-model. The idea is that a logical systema formal language together with a deductive system and/or a model-theoretic semanticsis a mathematical model of norms that underlie inferential practice and consistency (or whatever else it is that underlies logical consequence). The pluralism emerges from the observation that, with models, there are tradeoffs to be negotiated. One model may be simpler and easier to work with, but more idealized; another more realistic but more cumbersome.