In his extensive writings, the prestigious philosopher Richard Swinburne makes a useful distinction between two types of inductive arguments. Let B be our background information or evidence; E be the evidence to be explained; and H be an explanatory hypothesis.
“C-inductive argument”: an argument in which the premisses confirm or add to the probability of the conclusion, i.e., P(H | E & B) > P(H | B).
“P-inductive argument”: an argument in which the premisses make the conclusion probable, i.e., P(H | E & B) > 1/2.
It seems to me that there is a third type of inductive argument which should go between C-inductive and P-inductive arguments. I’m going to dub it the “F-inductive argument.”
“F-inductive argument”: an argument in which the evidence to be explained favors one explanatory hypothesis over one or more of its rivals, i.e., P(E | H1 & B) > P(E | H2 & B). Explanatory arguments are F-inductive arguments and have the following structure.1. E is known to be true, i.e., Pr(E) is close to 1.
2. H1 is not intrinsically much more probable than H2, i.e., Pr(|H1|) is not much greater than Pr(|H2|).
3. Pr(E | H2 & B) > Pr(E | H1 & B).
4. Other evidence held equal, H1 is probably false, i.e., Pr(H1 | B & E) < 0.5.
Good F-inductive arguments show that E is prima facie evidence — that is why (4) begins with the phrase, “Other evidence held equal.” They leave open the possibility that there may be other evidence which favors H1 over H2; indeed, they are compatible with the situation where the total evidence favors H1 over H2.
F-inductive arguments are “stronger” than C-inductive arguments insofar as they show E not only adds to the probability of H2, but that E is more probable on the assumption that H2 is true than on the assumption that H1 is true. They are weaker than P-inductive arguments, however, because they don’t show that E is ultima facie evidence — they don’t show that E makes H2 probable.
One final point. Although I believe I am the first to give F-inductive argument a name and place within Swinburne’s taxonomy of inductive arguments, the structure for such arguments is not mine. Paul Draper deserves the credit for that.