[T]he judge is not bound always to make a finding one way or the other with regard to the facts averred by the parties. He has open to him the third alternative of saying that the party on whom the burden of proof lies in relation to any averment made by him has failed to discharge that burden.
[Rhesa Shipping Co SA v Edmunds (The Popi M) [1985] 1 WLR 948 at 955; [1985] 2 All ER 712 (Rhesa Shipping v Edmunds) at 718.]
A failure to find a matter alleged does not establish the truth of the contrary of that which is alleged. …
(Citations omitted; see also Hobbs v Tinling (CT) & Company, Limited [1929] 2 KB 1 at [21] per Scrutton LJ.)
118 With respect, Lord Brandon of Oakbrook (with whom the other members of the House of Lords agreed) in Rhesa Shipping v Edmunds stated the point with great clarity when he said the following (at 718):
My Lords, the late Sir Arthur Conan Doyle in his book The Sign of Four, describes his hero, Mr Sherlock Holmes, as saying to the latter's friend, Dr. Watson: "How often have I said to you that, when you have eliminated the impossible, whatever remains, however improbable, must be the truth?" It is, no doubt, on the basis of this well-known but unjudicial dictum that Bingham J decided to accept the shipowners' submarine theory, even though he regarded it, for seven cogent reasons, as extremely improbable.
In my view there are three reasons why it is inappropriate to apply the dictum of Mr Sherlock Holmes, to which I have just referred, to the process of fact-finding which a judge of first instance has to perform at the conclusion of a case of the kind here concerned.
The first reason is one which I have already sought to emphasise as being of great importance, namely, that the judge is not bound always to make a finding one way or the other with regard to the facts averred by the parties. He has open to him the third alternative of saying that the party on whom the burden of proof lies in relation to any averment made by him has failed to discharge that burden. No judge likes to decide cases on burden of proof if he can legitimately avoid having to do so. There are cases, however, in which, owing to the unsatisfactory state of the evidence or otherwise, deciding on the burden of proof is the only just course for him to take.
…
In my opinion Bingham J adopted an erroneous approach to this case by regarding himself as compelled to choose between two theories, both of which he regarded as extremely improbable, or one of which he regarded as extremely improbable and the other of which he regarded as virtually impossible. He should have borne in mind, and considered carefully in his judgment, the third alternative which was open to him, namely, that the evidence left him in doubt as to the cause of the aperture in the ship's hull, and that, in these circumstances, the shipowners had failed to discharge the burden of proof which was on them.
(See also Jackson v Lithgow City Council [2008] NSWCA 312 at [9]–[10] per Allsop P with whom Basten JA and Grove J agreed.)