Sunday, December 16, 2007

Standard Conclusions from 'Vague' Sets of Premises

A thought about tentative arguments. Suppose we have a pair of premises like the following:

-S+M
-P+M

That is, "All S is M" and "All P is M". These can't directly render the conclusion +S+P, "Some S is P". In fact, the barriers are pretty serious here: trying to draw +S+P directly from the premises commits the fallacies of undistributed middle, illicit major, and illicit minor at one time. Nonetheless, there are cases where, given the information in the above premises, we would want to draw the conclusion +S+P as at least a tentative conclusion. (One would prefer to draw the conclusion +S+P, as being weaker, rather than the even more dubious -S+P, "All S is P".) What conditions govern these cases?

One would often say that -S+M, combined with -P+M, allows for no conclusion. But this is not strictly true; each premise contributes genuine information, and from this information conclusions can be drawn. For instance, given these premises there can be no SP that is not M. So the premises given restrict the possible conclusions that can be drawn. The following pairs of conclusions are therefore possible given the premises (if we are 'presupposing existence', as they say):

-S+P and +P+S
-S+P and -P+S
+S+P and +P+S
+S+P and -P+S
-S-P and -P-S

Other combinations are ruled out. So we can draw a (tentative) conclusion if we have a principle that allows us to prefer one of these pairs over all the others. The most plausible such principle is a principle of simplicity, namely, if we should prefer the conclusion that allows for the simplest characterization of the world. Of the above pairs there are two that are candidates for this: +S+P with +P+S and -S-P with -P-S. Knowing that we can reach either member of the pair from the other by immediate inference, we can completely characterize each by just one member, e.g., +S+P or -S-P. That reduces the choices to two; but it leaves us no way of choosing between "Some S is P" and "No S is P", which is unhelpful. But we can choose between them if we accept the possibility of subalternation as an immediate inference. This allows us to recognize that the original premises, -S+M and -P+M, also tell us that +S+M and +P+M. In that context we can read +S+P as saying:

It's a world where some S is P.

But -S-P has to be read as saying:

It's a world where no S is P, and some S is non-P and some P is non-S.

This is very clear if we diagram it using Carroll's literal diagrams. Thus +S+P is the simplest characterization of the world among the choices. Thus, if -S+M and -P+M, and we have a rule for tentatively accepting the simplest conclusion containing all the information of the premises but not ruled out by them, we can conclude +S+P. The only question left would be when it is legitimate to have such a rule.

The reason we can do this, I suppose, is much the same reason why Carroll is able in Symbolic Logic to dismiss, mockingly, the common claim that negative premises do not yield a conclusion, and also why Tom is right that we can have legitimate parasyllogisms (and here also). It's actually rather absurd to say that the premises allow no conclusion; for the premises to allow us to draw no conclusion they would have to be contentless, and carry no information about the domain. If they do carry information about the domain, they at least rule out some conclusions, and the fact that they rule out some conclusions means that we can draw conclusions about the domain from them. What we actually mean is that the conclusions that can be drawn directly lack certain desirable characteristics (they cannot be expressed in a form that is taken to be standard). Thus our premises above, for example, leave us with a conclusion about the domain that is not expressible in a single standard form, because they leave open the possibility of several different states of the domain that are each expressible in mutually exclusive standard forms. They are, in that limited sense, vague. This is the result of assuming that none of the possible states is to be preferred to the others; no 'standard' conclusion can be drawn. But if we reject this assumption and provide some rule for preferring one over the others, we can go even further and draw standard conclusions from such premises -- tentatively, of course, since we are restricted by how much confidence we can place in the reliability of our rule of preference.

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