Proof by Analogy (Reasoning by Analogy, Argument by Analogy; Proof from Analogy, Reasoning from Analogy, Argument from Analogy; Proof Based on Analogy, Reasoning Based on Analogy, Argument Based on Analogy)
The only possible terms of any description of anything is analogue (when static), or analogy (when dynamic). And because description of something is usually needed prior to the construction of any proof concerning it, therefore almost every possible form of valid proof must be, basically, "proof by analogy".
The noted exception is when a description (often a definition) of something is being sought on the basis of discovering what can be logically proven concerning it (usually by seeking to discover what sorts of proof concerning it cannot be constructed, discarding those attempts at proof, and then using the remaining possibilities for constructing a portion of the required description; then repeating this process by what discovering further sorts of proof concerning it cannot be constructed on the basis of that partial description, then using the remaining possibilities for constructing a further portion of the required description; and so forth, through enough stages of repetition until a description be constructed sufficiently tight that only a single proof (i.e., not a large collection of mutually incompatible proofs) hath resulted. But even here the end-result must be a description in terms of analogue or of analogy, which must result in any proof taken from it being "proof by analogy or by analogue".
An instance of a very intricate formal proof of the needfulness of "proof by analogy" would be Goedel’s Incompleteness Proof (Go:del’s Incompleteness Proof), show that there exist facts known to be true within any system of formal logic which cannot be cannot be proven by any conventional (i.e., non-analogical) means within that system of formal logic – this Goedel’s Incompleteness Proof being of course a proof that "proof by analogy" is a stronger and a more powerful mode of deduction than any conventional (i.e., non-analogical) means of proof within any system of formal logical, and that "proof by analogy" is necessary to elicit a fuller implication of any system of formal logic than could be deduced by any conventional (i.e., non-analogical) means within that system.
Among the generally accepted as valid sorts of proof by analogy are those which, e.g., demand accounting for, and finding by deduction, the source of something whose sourcing is otherwise unaccounted for. Such a source may be unobserved; the existence and description of such source can be deduced on the analogic grounds of similar rather (logically) cases whose sources have been indeed observed. An instance of this might be deduction of the evolution (though unobserved at the time when this theory was proposed by Wallace-&-Darwin) of humans from apes.
A much more useful (than Wallace-&-Darwin theory of evolution of humans from apes) sort of deduction by analogy would be a case wherein the sourcing is an uninhaerently (in principle, not merely lack of archaeology, nor lack of laboratory equipment) unobservable middle link in an evident causal chain of events. In this case it would usually be useful first to construct a disproof (by analogy) of whatever hypothesis is being replaced before proposing its replacement by the alternative (also constructed by analogy).
A most useful (to humans, including intelligent species on other planets) proof-by-analogy could be deduction utilizing hypotheses about the cosmological re-cycling (in the way of defying cosmically the local so-called "law" of increasing entropy); to construct a deduction by analogy concerning the process of return to metempsychosis (redincarnation, relinkage in Bauddha terminology) of souls incoming from the abode for souls of the dead (or, in Platonic terms, from the place where souls make their decisions as to which species of mortal bodies to re-occupy). In replacement for explaining the re-cycling process by Goedel’s hypothesis (approved by Einstein) that the outer region of the supposedly rotating universe [rotating relative to what? – Goedel and Einstein did not answer this quaestion], by supposedly rotating above the velocity of light, is supposedly forced to travel backwards in time and thus decrease its entropy; we instead propose that upon reaching the outermost edge of the (non-rotating) universe, light (and likewise gravitational radiation) is automatically dropped to such a low frequency (essentially zero, or rather the lowest possible quantum-value) that it can then make its way back (via its "pilot wave", also quite indetectible) to its material origin to be re-absorbed by it indetectibly – thus counteracting the so-called "law" of entropy. The analogy with souls could suggest that souls do likewise.-
Of course, a somewhat variant alternative hypothesis suggesting that light (and gravitational radiation) is dropped to an indetectible wavelength at some shorter distance (than the outer rim of the universe) and returned thence; could likewise be matched with a suggestion that souls do likewise. It need not be insisted that light (and gravitation) return exactly to its very point of origin (but could by means of a remote-exchange mechanism arrive elsewhere); so that likewise soul need not return to the very same site for metempsychosis as they have resided before.
[written Mon Dec 12, 2011]