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Say you have a conclusion, and you have drawn the parameters of your argument such that it bounds a certain set of hypotheticals. From the simple Bayesian perspective, it's basically just $A|B$. You've proven $A|B$, you've established $B$, and therefore you've proven $A$.

But then Bob comes along and makes an objection comes up that doesn't contest $B$, but instead some larger context $K$. He's talking about $(A|B)|K$, and that K is false (or not always true).

Is there a set of terms for their two perspectives? I'm not talking about just uncovering unwarranted assumptions or finding problems in proofs. I'm talking about the relative concept of being inside or outside of an argument. It doesn't matter whether Bob has a good point - he might be right or wrong, or he might be completely obliterating a proof for resting on a bad implicit premise, or he might be an annoying distraction for legitimately being out of bounds. But I am wondering if there is a set of terms for these two inside-versus-outside perspectives, kind of like "intensional/extensional" for semantics.

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  • $\begingroup$ WLOG is my first thought ( without loss of generality). $\endgroup$ – user451844 Aug 16 '17 at 23:50
  • $\begingroup$ Your post is very imprecise. First of all, what are you interpreting $A|B$ to mean? Is that supposed to be a conditional probability? If so, where's the probability? Suppose the probability that $A$ happens given that $B$ happens is vanishgly small, and you have good reason for thinking $B$: then you might have good reason for thinking $A$ is false, not true. Alternatively, your use of the word "proven" in "therefore you've proven $A$" makes it seem that you are reading the notation as the conditional $B\rightarrow A$. You should clarify whether you are talking deduction or abduction or... $\endgroup$ – symplectomorphic Aug 17 '17 at 2:24
  • $\begingroup$ Perhaps you're just trying to capture the fact that our credence in a proposition $p$ (modeled Bayesian-ly) depends on what we condition on. This is a well-known problem that comes in many forms. One is the "problem of priors." Another is the "reference class problem." $\endgroup$ – symplectomorphic Aug 17 '17 at 2:29
  • $\begingroup$ Yes, apologies, I was using the Bayesian example lazily, literally as "A given B". I meant it the same as "B implies A" - no probability intended. I am wondering if there is a term for being inside the the argument and another term for being outside the argument. $\endgroup$ – tunesmith Aug 17 '17 at 5:53
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It is called generalization... the undisputed favorite of mathematicians everywhere. ..

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