quote:If I had listed 1000 encounter with Crowes, would that make it any more persuasive?
Doesn't this all depend on how many Crowe's we think exist? Therefore, the degree of certainty to which our inductive generalization is supported depends on how general our hypothesis is. This is similar to Hempel's solution to the "Paradox of the Ravens."
(β) Whatever confirms a hypothesis also confirms a logically equivalent one
(α) A generalization of the form “All F are G” is confirmed by its positive instances-i.e., by cases of F that have been found to be G.
(p) All ravens are black
(p*) All non-black things are non-raven
According to (α), a non-black non-raven will support (p*); according to (β), any observation that supports (p*) will support (p). Therefore, a non-black non-raven will support (p). A red apple will support the proposition that “All ravens are black.”
Hempel’s solution is that both a black raven and a non-black non-raven support the proposition that “All ravens are black,” but they do so to different degrees. The non-black non-raven supports (p) to a much smaller degree than a black raven would. In order to understand this, imagine the similar proposition that “All coins in this container are pennies” and imagine that we know there are 100 coins in the container. Now imagine Joe checks one coin and discovers it is a penny. Jack then checks 99 coins in the container and they are also all pennies. According to (α), both these positive instances support the original proposition. Yet, would not Jack have much more confidence in the proposition than Joe? Is it reasonable to say that they supported the proposition to varying degrees? The answer seems to be yes.
To model it mathematically, if the claim is that “All F are G,” then the degree of confirmation (assuming there are no negative instances) is the ratio of the number of positive instances (F+) to the number of (F)s:
Degree of Confirmation (DoC) = [ (F+)/F]
A degree of confirmation of 1 would represent absolute confirmation. In the penny example, Jack would have a degree of confirmation of .99, whereas Joe would only have a degree of confirmation of .01. Bringing this back to the “Paradox of the Ravens,” let us use this formula to get a feel for the different degrees with which a black raven and a non-black non-raven would support (p). In both cases the numerator would be 1. The difference comes in the denominator. The set of all non-black things in the universe is gigantic. Therefore, the non-black non-raven would confirm (p*) only to an infinitesimally small degree. Since the degree of confirmation is the same for two logically equivalent statements (β), a non-black non-raven would support (p) as much as it supports (p*), which is practically zero. On the other hand, the set of black ravens in the universe is much smaller than the set of non-black things. The DoC associated with observing a black raven would therefore be much higher than the DoC associated with observing a non-black non-raven.
So, the degree to which we accept a hypothesis would depend on its DoC. I know this is kind of hard to think about in terms of major generalizations (e.g., all mass is correlated with curved spacetime), but as long as there are no counter-instances you have to accept the generatlization with the highest DoC, imo.
quote:In the often presented example of crows, the standard literature does not suggest that you need to know how many crows exist.
Are we maybe equivocating on the term "induction?" When you refer to induction, are you referring to the Baconian view that scientists can just gather a bunch of objective facts about the world and these facts would almost automatically permit a generalization?
I like Darwin's quote about this form of induction:
How odd it is that anyone should not see that all observation must be for or against some view if it is to be of any service.
I think everyone else here is referring to the hypothetical-deductive (H-D) method or the deductive-nomological (D-N) method. Hemple writes, "[H-D] is inductive in a wider sense, inasmuch as it involves the acceptance of hypothesis on the basis of data that afford no deductively conclusive evidence for it, but lend it only more or less strong 'inductive support', or confirmation."
I think this is the form of inductive logic Chiropter (and maybe Modulus) are referring to, and I think it is the form most people mean when they say science relies on inductive logic.
quote:Which exactly proves my point. You can't even begin to deduct until the axioms are supplied; thus, it's not a process you can complete in your head. You can't even begin to deduct absent an external source of information; the same with measurement.
Wouldn't this be quicker if you just showed the deduction?
1. A particular platinum-iridium rod is defined as one meter (not anymore, obviously)
2. My ruler is the same length as the rod
3. My desk is 10 rulers long
Ergo, my desk is 10 rulers * (1 rod/1 ruler)*(1 meter/rod)= 10 meters.