Varieties of Logic by Stewart Shapiro

Author:Stewart Shapiro [Shapiro, Stewart]
Language: eng
Format: epub, pdf
ISBN: 9780199696529
Publisher: Oxford University Press
Published: 2014-09-04T00:00:00+00:00

5 Option 1: Constant Meaning of Logical Terms

Recall that for the time being, we are leaving it open whether the logical particles have the same meaning when used in the different theories. So we will operate under various assumptions. Suppose, first, that—pace Dummett, pace Quine, and pace Carnap—logical particles do have the same meaning in different theories, even if the logics are different. So, in particular, we assume that this meaning is not given by truth-tables, inference rules, or the like. We will maintain that supposition throughout this section, and then turn to its negation in the next.

Earlier in this chapter (§2), I suggested that words like “valid” and “logical consequence” are terms-of-art. Their use, even by philosophical logicians, is not sufficiently uniform. In writing a logic textbook, an author will typically just stipulate (perhaps implicitly) what these words mean. So we are after something more like a proposed sharpening, rather than an analysis of a word or concept in use.

In that same section, I coined a term, “super-valid,” which is intended to be non-context-sensitive. An argument form is super-valid just in case it is valid in all legitimate structures and theories. Notice that a similar stipulation can be made for most context-sensitive terms, no matter how the context-sensitivity is articulated—contextualist, assessment-sensitive, whatever. A food is “super-tasty” if it is tasty to everyone; an object is “super-left” if is on the left according to every vantage point; a person is “super-ready” if she is ready for anything; a person is a super-enemy just in case he is an enemy of everyone; a pub is super-local if it is local to everywhere. In many cases, of course, the coined term will be completely useless. A super-local pub would be quite handy, but, alas, there aren’t any.

Present focus is on phrases like “valid” and “logical consequence,” when they are not used in this context-independent way, but rather within the context of a particular mathematical structure or theory. 14 That is, we are not concerned with super-validity.

Under our present supposition that logical terms have the same meaning in all of the contexts, we have at least a folk-relativism concerning “valid” and “logical consequence” and a corresponding pluralism. For example, let Φ be a statement of the intermediate value theorem and consider the instance of excluded middle Φ∨¬Φ. On the assumption in question, that sentence has the same meaning in classical analysis and in intuitionistic analysis. It expresses the same proposition in both contexts, if you will. Yet Φ∨¬Φ is a logical consequence of the empty set in the former and not a logical consequence of the empty set in the latter. In other words, the very same argument is valid in one context and invalid in the other. So, in the slogan of folk-relativism, there is no such thing as “simply being valid.” Rather there is validity-in-classical-theories, validity-in-intuitionistic-theories, etc. Swoyer’s General Relativistic Schema is directly à propos: the dependent variable is for validity and the independent variable is for a theory or structure.

So

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