Arnold, Jack & Shapiro, Stewart - Where in the (World Wide) Web of Belief is the Law of Non-contradiction?

07/13/2007

Nous, Vol 4 Issue 2, 2007

This is a paper that tries establish that there are two interpretations of Quine, or perhaps more accurately, that Quine was of two minds when it came to the status of logical truths. Authors believe there was a 'logic-friendly' Quinean empiricism, one that placed the rules of truth-preserving inference (logic) outside of the possibility of revision by recalcitrant experience. There was also 'radical' Quinean empiricism, dating back to 'Two Dogmas', that did not exclude any of the rules of logic from possible revision. This is a concern because it could mean that the law of non-contradiction and 'ex falso quodlibet' (explosion) are subject to revision. Because authors are classical logicians, this is troubling to them. Authors first lay out the possible conflict between the logic-friendly Quine and the radical Quine, but then mostly focus on the implications for the radical Quine. The thesis for the paper is that if we are to take these rules of logic on using the radical Quine's own principles of empirical confirmation, these rules will not have the robust status of universal rationality that many (most) logicians want them to have.

The first part of the paper entails the authors go to the original sources often and trying to work out the radical Quine's theory (pg 279-281). Ultimately the authors conclude that it is part of Quine's holism that logic is included in the 'web' of beliefs. It just so happens that the 'theories' of logic go very far to the inside of the web, making recalcitrant experience more likely to change beliefs on the edges rather than closer to the insides. This is complicated by other claims made by Quine that seem to say that logic cannot be changed since, any change in logic is a 'change of subject'. This is Quine's reply to dialetheists-- 'you're just changing the subject'. So, is a change in the theory of logic change the meaning of the theory? Quine says yes. (pg 280-281) Now we might be left to wonder how theories of logic could be changed at all without changing everything altogether.

The next matter for consideration is how the rules of logic should be used within the web of belief. Should they be used everywhere? One problem here is that Quine does not like to engage with normativity when discussing belief. Talk of 'should' would mean that the rules of logic have some sort of force other than merely helping to make predictions and avoid recalcitrant experience. Authors interpret Quine as talking about using causation (or constant conjunction) for belief formation and ordering (pg 283-4), not using normative logical rules.

Lastly, we have a problem of what 'recalcitrant experience' really is. Does it mean there are contradictions between one belief and another? If so, then it appears the law of non-contradiction has some priority and is immune from alteration. The authors use the same talk of data causing assent to Belief A, and they previously would have assented Belief ~A, which is impossible to do. No real talk of contradictions, just of beliefs that cannot be taken together because they are impossible. (pg 285-286)

The second portion of the paper involves using only a descriptive picture of the realm of science and the realm of ordinary every-day beliefs in discovering the status of the law of non-contradiction and explosion. Authors look at two 'chunks' of belief: everyday beliefs, and scientific theories. In both cases, authors argue that the law of non-contradiction applies in some areas and not in others, and that (in the case of everyday beliefs) humans have a 'knack' for intuiting where to apply it and where not to, and that (in the case of science) often we are willing to accept contradictions as long as we get the predictions right (pg 286-292). The case for explosion (ex falso quodlibet) is even worse. There is no widespread acceptance of this in everyday usage or even in science.

The last part of the paper discusses that we are left with. The only hope of establishing a robust notion of the law of non-contradiction is to assert it's epistemic usefulness. Certainly it fits into a logical system (classical logic) that is disciplined, consistent and orderly. But so is a paraconsistent system! This does not save the law of non-contradiction. (pg 292-293) Lastly, we might hope that the Minimum Mutilation Thesis preserves the truths of logic. Unfortunately, this seems consistent with paraconsistent/dialetheists as well.