Wednesday, 10 February 2016

On the adoption of logical principles

Two weeks ago I had the pleasure of attending a one-day workshop on The Nature of Logic organized by the University of York. The focus of the day was Saul Krikpe's unpublished works on the 'adoption problem', an interpretation of Lewis Carroll's "What the Tortoise Said to Achilles". "What the Tortoise Said to Achilles" is probably my favorite piece of philosophy, ever; York is a day-trip away from Durham; and it was a chance to hear Kripke speak in the flesh, all three reasons to expect a very interesting and enjoyable day, and the workshop did not disappoint.

The talks were all thought-provoking, but it was the first, by Romina Padró, that set the stage for the day and also triggered the thoughts that I want to try to articulate here today. Padró recently completed her dissertation on What the Tortoise Said to Kripke: the Adoption Problem and the Epistemology of Logic. The "Adoption Problem" is detailed in S. 2.2, but the basic issue of this: Suppose you are confronted with someone, call him Harry, who has "no notion of the principles in question [modus ponens and universal instantiation] and has never inferred in accordance with them" (p. 31). Surely Harry has an impoverished reasoning ability and it would be useful to introduce him to these logical principles, such that he accepts them and can henceforth go on to reason according to them. This is the adoption of a logical principle:

By 'adopt' here we mean that the subject, Harry in this case, picks up a way of inferring according to, say, UI, something he wasn't able to do before, on the basis of the acceptance of the corresponding logical principle (p. 31, emphasis in the original).

The adoption problem is then whether such principles as MP and UI can be adopted. Padró's talk at the workshop was directed at arguing that they cannot: That in order to apply MP after it has been accepted, one must already be able to appeal to a notion of modus ponens. This is precisely what the Tortoise is pointing out to Achilles in Carroll's classic piece.

I remain unconvinced by Padró's argument, in part because it seems to me that Harry can accept a principle without applying it, and that once he has accepted it, he can then go on to apply it -- if he cannot apply it, then I would argue he hasn't in fact accepted it, contrary to assumption. But I will leave this point aside, and assume that there are some principles which cannot be adopted, and that MP and UI are, if anything are, prime candidates for such principles. The questions that I had -- and they are only questions, I don't have any idea how one would go about answering them, which is part of why I'm writing this, in case the collective power of the internet is smarter than me (it almost certainly is) -- stem from generalising the issue.

Padró's talk focussed on whether or not MP and UI are adoptable, and mentioned briefly other logical principles that may be similar, such as &I and &E, as well as some that likely can be adopted, such as disjunctive syllogism. This raises a general methodological point: How does one determine if a principle is adoptable? If every logical principle is adoptable, then we have no problem; if no logical principle is adoptable, then we have no problem. But if some are and some are not, then it would be useful to have a principled way of identifying them, preferably in advance. The argument for MP and UI is that in order to apply them, one must invoke the principles themselves:

If someone who never inferred in accordance with MPP were to be told that "For any A and B, if A then B, and A, then B," the subject wouldn’t be in a better position to perform a MPP inference. For the principle to be of use with any particular inference, she will need to infer in accordance with the MPP pattern that she does not use in the first place: in any particular case, she will only get to B from her premises by performing a MPP inference on the instantiation of 'For all A and B, if A, and if A then B, then B,' but that is exactly what she couldn't do to begin with (p. 36).

It seems then that one could argue that &I and &E cannot be adopted, since one must already have a concept of conjunction in order to introduce or eliminate conjunctions. But surely this is a matter of how the rule is formulated: With sufficient cleverness, I'm sure I could define &I and &E in a way that doesn't use 'and' at all, but only 'or' and 'not'. Would the principle then be adoptable, because it is formulated without appeal to the notion it is purporting to introduce?

If the answer is yes, then it immediately raises this question: If whether a principle can be adopted depends on how it is formulated, how do we know that MP and UI cannot be reformulated in a way that doesn't invoke them? For example, surely one could formulate MP in such a way that all Harry needs to know is disjunction and negation. If one wishes to maintain that MP-formulated-with-conditionals is not adoptable while MP-formulated-with-disjunction-and-negation is, then there is good reason to think that one must maintain that these are distinct logical principles. In that case, we're left with what I suspect is an extremely difficult question to answer: What are the identity conditions of logical principles?

At this point, I have no good intuitions about how to begin answering these questions.

© 2016 Sara L. Uckelman.


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