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The authorities first divide Proof into two general classes; viz., (1) Direct Proof, or the process of sustaining a given proposition; and (2) Indirect Proof, or the process of refuting objections to a given proposition. Of course, the refuting of an objection to a given proposition does not directly prove the proposition itself, but it removes from the argument an obstacle to direct proof and is therefore of value, indirectly.

Another division of Proof is that of: (1) Proof by Deductive Argumentation; and (2) Proof by Inductive Argumentation. Proof by Deductive Argumentation is that in which Deductive Reasoning is employed—where the process of reasoning seeks to derive a particular truth from a general truth. Thus, from the general truth that “All men are mortal,” and the fact that

“Socrates is a man ,” we may prove by Deductive Argumentation that “Socrates is mortal.” Proof by Inductive Argumentation is that in which Inductive Reasoning is employed—where the process of reasoning seeks to derive a general truth from particular truths. Thus, from the particular truths that “Heat will expand iron, zinc, copper, gold, silver, etc.” we may prove by Inductive Argumentation that “Heat will expand all metals.”

Of course, all general truths must have been originally reasoned by induction from particular truths, and being so established have become bases for deductive reasoning from which particular truths may be established. There are certain general truths or principles which are generally accepted as axiomic or “self-evident truth,” without the necessity of argument. In Deductive Argumentation, the argument proceeds from an accepted general truth and attempts to establish the fact that a particular proposition comes under, or is included in, that general truth, and is thus a particular truth. In Inductive Argumentation, the argument proceeds to show the apparent existence of a general truth in a number of particular things, from which it may be supposed or reasonably believed that the general law exists as a truth. The rule in Inductive Reasoning is that: “The probability of a hypothesis is in proportion to the number of facts and phenomena it will explain.”

Arguments may be classified, according to form, as follows: (1) Argument based upon Cause and Effect; (2) Argument based upon Analogy; (3) Argument based upon Association.

Argument based upon Cause and Effect is sometimes known as “Argument from Antecedent Probability.” It is based upon the general principle or axiom that: “From the same causes, under the same circumstances, the same results will flow.” This axiom is based upon man’s belief in the uniformity of natural laws. From this, we arrive at the belief that given a certain cause, under certain circumstances, we may safely infer the presence of a certain effect; or given a certain effect, under certain circumstances, we may safely infer the presence of a certain cause. In order to reason or argue along these lines, however, we must understand that there are certain “Tests of Casual Agency” accepted by logicians, by which we may ascertain the casual agency in any particular case, and thus make an inductive inference. These “tests” are as follows:

I. The Method of Agreement : “If, whenever a given object or agency is present, without counteracting forces, a given effect is produced, there is strong evidence that the object or agency is the cause of the effect.”

II. The Method of Difference : “If, when the supposed cause is present the effect is present, and when the supposed cause is absent the effect is wanting, there being in neither case any other agents present to effect the result, we may reasonably infer that the supposed cause is the real one.”

III. The Method of Residues : “When in any phenomena we find a result remaining after the effects of all known causes are estimated, we may attribute it to a residual agent not yet reckoned.”

IV. The Method of Concomitant Variations : “When a variation in a given antecedent is accompanied by a variation of a given consequent, they are in some manner related as cause and effect.”

Atwater says of the above: “When either of these criteria is found, free from conflicting evidence, and especially when several of them concur, the evidence is clear that the cases observed are fair representatives of the whole class, and warrant a valid universal inductive conclusion.”

Argument based upon Cause and Effect may be employed in arguing the probability of the occurrence of a certain effect arising from an admitted cause; or in arguing the probability of the occurrence of a certain effect arising from an admitted cause; or in arguing the probability of the existence of a certain precedent cause by reason of a certain cause to effect; the other from effect to cause. Thus we may argue that from the admitted existence of certain casual agencies, belonging to a class of causes which have resulted in certain effect in the past, there will arise certain effects in the future. As for example, we may argue that as John Smith is drinking to excess, and that this cause has resulted in drunkenness and poverty in other cases in the past, therefore, John Smith is likely to become a drunkard and a pauper in the future. Or, that undue familiarity existing in a certain case, and familiarity having been known to “breed contempt” in the past, therefore, contempt is apt to arise from the present manifestation of familiarity. Likewise, we may argue that from the existence of certain admitted effects, there must have existed a certain precedent cause. Thus, we may argue that there being an egg before us, and from the fact that eggs have always been known to have been laid by birds or fowls, therefore, the present egg must have been laid by a bird or fowl. Or, that a coin having disappeared from a drawer, and persons having been known to steal coins in the past, therefore, some person must have stolen the coin. The doctrine of Causation, or the theory that nothing exists with a cause, and that everything proceeds from a precedent cause, and must, in itself, be the cause of a subsequent effect, is well grounded in human reason and experience, and argument based thereon is of the strongest character. The question of “motive” belongs to this form of argument.

Argument based upon Analogy is sometimes known as “Argument from Resemblance,” or “Argument from Example.” It is based upon the general principle or axiom that: “ If two or more things resemble each other in many points, they will probably resemble each other in many other points. ” The principle involved in this form of Argument may be gathered from the following quotations from two well-known authorities:

Jevons says: “If I see a machine with boiler, cylinder, air-pump, piston-rod, crank, and other parts exactly resembling those of a steam-engine, I do not hesitate to call it a steam-engine, to assert that it has a piston, valves, and other hidden parts, like all steam engines. It is in the same way that we reason about the substance of which anything is made. If a person offers me a shilling as change, how can I be sure that it is a good shilling, and made of silver? All that I can do is to examine the coin, and observe whether it has a fine pure white lustre where the surface is rubbed; whether there is in other parts of the surface the black tarnish peculiar to silver; whether the coin seems to be hard, and gives a sharp ringing sound when thrown down. If it has all these characters and, moreover, has a good impression exactly like that seen on other shillings issued from the mint, then it is doubtless made of silver, and is a true shilling, that is to say, it will show all the other properties of standard silver, when examined in a manner suited for showing them.”