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Unformatted text preview: Notes for Week 10 of Confirmation 11/07/07 Branden Fitelson 1 Some Preliminary Cautionary Remarks on Formalization Whenever one applies formal theories, one runs the risk of over-formalization (or mis-formalization). For instance, consider sentential logic. If all we have to work with is sentential logic, then many “intuitively logically valid” (hereafter, simply valid ) arguments will “come out invalid”. This is because sentential validity is a property that many valid arguments do not have. Roughly, an argument is sententially valid if it instantiates a valid sentential form. But, if an argument does not instantiate a valid sentential form, it does not follow that the argument is invalid (for instance, the argument expressed by: “Socrates is wise. Therefore, someone is wise.”). So, whenever we use a formal logical theory, we must be careful not to infer too much from what that particular theory says about something informal that we’re applying it to. Probabilistic methods are also subject to this kind of cautionary remark. For instance, I might try to apply sentential probability theory to cases involving relationships that are intuitively non-sentential in nature. This isn’t necessarily verboten . It just has to be done carefully, and in the right way. Here’s a salient example. Let “ A ” be interpreted as ( ∀ x)(Rx ⊃ Bx) , “ B ” be interpreted as Ra , “ C ” be interpreted as Ba , and “ K ” be interpreted as some “background corpus”, which captures what we take ourselves to know about the predicates R and B (as featured in our A , B , and C ). To be more precise, we can also specify that “ R ” stands for ravenhood and “ B ” stands for blackness. Thus, we are assuming that “ A ” stands for the proposition that all ravens are black . In this sense, “ A ” (and even “ B ” and “ C ”, for that matter) stands for something with monadic-predicate-logical ( i.e. , non-sentential) structure. This is OK, as long as we keep in mind that: + The systematic logical structure of any sentential probability model ( i.e. , its sentential structure) will not be sensitive to any non-sentential logical relations among its “interpreted atomic sentences”. As a result, any such relations must be explicitly represented by extra-systematic constraints on the model. In our present example, there are extra-systematic ( viz ., non-sentential) logical relations among our “atomic sentences” — under their intended interpretations . For instance, ( ∀ x)(Rx ⊃ Bx) and Ra jointly entail Ba — in monadic predicate logic . But, this “entailment” is extra-systematic , and it will not be captured by any systematic sentential-logical relations between A , B , and C . After all, A,B ø C — in sentential logic . This is not necessarily a barrier to sentential formalization. In order to model things properly in this example, we’ll need to add (at least) the following extra-systematic constraint to our sentential probability model:...
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- Fall '06
- Logic, First-order logic, Oa Ra, D. Goodman, extra-systematic constraints