(3) To mistake co-existent phenomena for cause and effect:  as when a man, wearing an amulet and escaping shipwreck, regards the amulet as the cause of his escape.  To prove his point, he must either get again into exactly the same circumstances without his amulet, and be drowned—­according to the method of Difference; or, shirking the only satisfactory test, and putting up with mere Agreement, he must show, (a) that all who are shipwrecked and escape wear amulets, and (b) that their cases agree in nothing else; and (c), by the Joint Method, that all who are shipwrecked without amulets are drowned.  And even if his evidence, according to Agreement, seemed satisfactory at all these points, it would still be fallacious to trust to it as proof of direct causation; since we have seen that unaided observation is never sufficient for this:  it is only by experiment in prepared circumstances that we can confidently trace sequence and the transfer of energy.

There is the reverse error of mistaking causal connection for independent co-existence:  as if any one regards it as merely a curious coincidence that great rivers generally flow past great towns.  In this case, however, the evidence of connection does not depend merely upon direct Induction.

(4) Post hoc, ergo propter hoc:  to accept the mere sequence of phenomena, even though often repeated, as proving that the phenomena are cause and effect, or connected by causation.  This is a very natural error:  for although, the antecedents of a phenomenon being numerous, most of them cannot be its cause, yet it is among them that the cause must be sought.  Indeed, if there is neither time nor opportunity for analysis, it may seem better to accept any antecedent as a cause (or, at least, as a sign) of an important event than to go without any guide.  And, accordingly, the vast and complicated learning of omens, augury, horoscopy and prophetic dreams, relies upon this maxim; for whatever the origin of such superstitions, a single coincidence in their favour triumphantly confirms them.  It is the besetting delusion of everybody who has wishes or prejudices; that is, of all of us at some time or other; for then we are ready to believe without evidence.  The fallacy consists in judging off-hand, without any attempt, either by logic or by common sense, to eliminate the irrelevant antecedents; which may include all the most striking and specious.

(5) To regard the Co-Effects (whether simultaneous or successive) of a common cause as standing in the direct relation of cause and effect.  Probably no one supposes that the falling of the mercury in his thermometer causes the neighbouring lake to freeze.  True, it is the antecedent, and (within a narrow range of experience) may be the invariable antecedent of the formation of ice; but, besides that the two events are so unequal, every one is aware that there is another antecedent, the fall of temperature, which causes both.