I am having trouble understanding Hintikka's Scandal of Deduction, as depicted in D'Agostino's article. According to this account, the problem stems from the fact that, while first order logic is undecidable, the completeness theorem shows that there is a semi-decision procedure. The problem is, even if given a set of premises a conclusion is indeed the logical consequence, there is no guarantee that we can obtain that proof within bounded resources (I take it as there being a practical resource limitation). And there is certainly no such procedure if the conclusion is not a logical consequence.
There have already been threads about this as listed below, and I have indeed read them all; but still I am not understanding it.
Is there a deduction analog to the problem of induction?
What is the difference between depth and surface information?
Is the problem of logical omniscience intractable?
Q1: First confusion I have concerns depth information; according to D'Agostino it is defined to be equivalent to semantic information (ie. in terms of possible worlds excluded), whereas the answer defines it as '...an ideal limit, all that we can, in principle, obtain from the armchair, without making any new observations.' It also seems to distinguish depth info from surface info by depth in terms of counts of quantifiers. I don't see how the latter's definition is in anyway related to being equivalent to semantic information.
Regarding the ideal limit bit, here is what I think it is saying but I am not at all sure: it concerns the cognitive limitation of human to see logical consequences that are of certain degree of complexity, i.e. The depth info of a sentence according to this formulation, are ALL logical consequences that are theoretically derivable, regardless of the limitation of human cognitive power.
For example, the Last Fermat theorem is a part of depth information derivable by sentences of number theory, but because of its complexity it is beyond human's cognitive power to immediately deduce it just by looking at sentences of number theory.
Q2: The quantifiers are the most confusing part: I don't at all understand the relevance of quantifier in this context; is this supposed to be an objective measure of how complicated the information a sentence conveys? e.g. ∀n∈N(n=n) is more complicated than ∃x∈N∀y∈N (x≤y) because the latter has two quantifiers? And surface information are those that have no quantifier (since 'Surface information is depth 0')?
I don't really see why quantifier is involved; for me the two sentences above, while having different counts of quantifiers, are pretty much equal in terms of being trivially true. Also if surface info are supposed to increase after we have known a new theorem through deduction, then I don't see how the quantifier is involved - I just cannot make the connection.
