Logical Pluralism

Truth versus Truth in a Structure#

Graham Priest, A note on mathematical pluralism and logical pluralism.pdf

There is a crucial distinction to be drawn between the preservation of truth (simpliciter) and the preservation of truth-in-a-structure

Truth versus Truth in a Structure

But the canonical application of logic is not about truth-in-a-structure-preservation. It is about truth-preservation. When we reason, we are interested in whether, given that our premises are true [or assuming them to be true] our conclusion is [would be] so as well.

Now, why?

But the canonical application of logic is not about truth-in-a-structure-preservation. It is about truth-preservation. When we reason, we are interested in whether, given that our premises are true [or assuming them to be true] our conclusion is [would be] so as well. That the canonical application of logic is about truth preservation is not a profound claim; in some sense, it is a simple truism. Of course, it is a contentious matter as to how to spell out exactly what it means. Nor is it even clear what machinery is best employed to articulate the thought: proof procedures, set-theoretic interpretations, modal notions, probability theory? These matters are not pertinent here, though. The point is the simple distinction between truth-preservation, however one understands this notion, and the preservation of truth-in-a-structure. And once this distinction between the two is noted, it is clear that the fact that there are different ways to preserve truth in a structure, depending on the structure, does not imply that there are many ways to preserve truth, simpliciter.

"Truth in a structure is like truth in a fiction". From outside a fictional universe, a statement about truth in the fiction is truth simpliciter, but inside it the "same" (in what sense) truth is just truth simpliciter. Mathematical pluralism is saying that different mathematical universes can coexist, but this statement itself, that different mathematical universes can coexist, is a statement that is not mathematical, except that all mathematical universes agree on this - but that's already an absolute statement that is irrelevant to the precise details of any mathematical universe.

Namely, the preservation of proof-theoretic notion of truth; but what is proof-theoretic truth? It depends on whether there's an absolute ground for truth, and whether it is possible that there's only relative truths. 

Note that, "there's only relative truths" is itself an absolute statement. A relative statement is an absolute statement from a larger universe where the relative statement is included in the universe.

Challenges#

A way to challenge this: 

Truth preservation simpliciter is validity. Validity is truth preservation in all interpretations. Interpretations are the same thing as structures. So validity is just truth preservation in all structures.

But this 

1. presupposes a model-theoretic account of validity. One would have to facce the fact that in many logics validity is not defined in terms of truth preservation but in other terms (e.g. in many-valued logics it is defined in terms of the preservation of designated values.) And one would have to defend the anything but obvious claim that structure and interpretations are the same thing.
2. Even assuming these points can be adequately addressed, model-theoretic validity is not truth-preservation in all interpretations. Different logics have different kinds of interpretations. Their model theories therefore provide an understanding of truth preservation in the appropriate kind of structure - the internal logic of the structure - not validity simpliciter.

A better challenge: truth is determined by reality and reality is just another structure. 

The answer to this challenge, I think, is weaker: 

But it’s not any old structure. It is highly privileged. […] Reality, then, is not simply a structure, on a par with other structures.

Further Notes#

Cf. Michael Dummett, The Logical Basis of Metaphysics, Chapter 1. Semantics Values, Model Theory et seq. ; also Chapter 2. Inference and truth, Is Truth Really the Salient Notion for Logic?