§ 560. (4.) There is the Syllogism of Equivalence,—the reasoning from equal to equal. This is the Unfigured Syllogism of Hamilton the Expository Syllogism of others. The former is wider than the latter, which referred only to Singulars; but Hamilton, by making equivalents in quantity, widened its scope. There is not only reasoning from this to that, or individual A to individual B, but from the equivalence of all of one class to some of another. The formula of the Syllogism of Equivalence is, however, in all cases the same. What are equivalent, or non-equivalent, to a common third term, are equivalent or non-equivalent to each other.

If X be equivalent to Y,

and Y to X,

X is equivalent to Y.

If all X be equivalent to some Y,
and all Z be equivalent to all X,
all Z is equivalent to some Y.

§ 561. (5.) To these I am disposed to add a fifth formwhat I would call the Syllogism of Collection. Here we literally gather into one in the conclusion what we stated separately, yet as implicated, in the premisses. Thus :— The crops this season are good in quality; The crops this season are good in quantity;

Therefore the crops this season are good both in quality and in quantity.

So negatively:

The crops this season are not good in quality;

They are not good in quantity;

Therefore they are not good either in quality or quantity. This is a perfectly simple form of reasoning,-in common use, though not fitting into any of the received formula,nay, in the negative form, even apparently violating the rule against two negative premisses. The law may be generalised thus: Where the same middle term admits of predicates of opposite kinds or genera, these, when both positively related, may be affirmed, or, when both negatively related, may be denied, of the middle term as subject of the conclusion. This reasoning differs from the ordinary forms by admitting middle as subject of the conclusion, and in the negative

form the rule against double negatives does not apply, for the comparison has been instituted not through comparing major and minor through the middle, but collating major and minor in succession with the middle. The middle again appearing as subject of conclusion, with the gathered predicates, constitutes the conclusion naturally and simply a collectio-collection.

§ 562. According mainly to the manner of enouncement or expression, a reasoning may be Simple or Complex, Complete or Incomplete. A reasoning is simple in nature when it contains three and only three related propositions, constituting a single reasoning. It is simple in expression when these propositions are explicitly stated in the order either of Extension, Comprehension, or Equivalence. This is properly a Monosyllogism—that is, a single independent reasoning.

§ 563. But Syllogisms may be connected in a succession or series, and thus stand to each other in the relation of antecedent and consequent. This is regarded as a composite or complex reasoning, and is called a Polysyllogism, also a Chainsyllogism or Chain of Reasoning.

§ 564. In a Chain of Reasoning the order may be either that of thing proved and reason, or of reason and thing proved. In other words, "each successive syllogism is the reason of that which precedes it, or the preceding syllogism is the reason of that which follows it." The former order is called the Analytic or Regressive; the latter is the Synthetic or Progressive. The reason-containing Syllogism is called. the Prosyllogism; the consequent-containing Syllogism is called the Episyllogism.1 If the Chain of Reasoning be composed of more than two links, the same syllogism may be, in different relations, prosyllogism and episyllogism.

§ 565. A polysyllogism, not explicitly enounced, is made. 1 Cf. Krug, Logik, § iii. ; and Hamilton, Logic, iii. 364.

up either of partially complete and partially abbreviated syllogisms, or of syllogisms all equally abbreviated. In the former case we have what logicians call the Epicheirema (mixeípnua); in the latter the Sorites. Of the Epicheirema or Reason-rendering Syllogism, the following is an example :— X is Y;

But Z is X, for it is D;
Therefore Z is also Y.

It is permissible to take the life of a man who lays an ambush with the purpose of taking yours;

Milo, therefore, was justified in killing Clodius, for Clodius had laid an ambush against Milo's life.2

§ 566. The Chain-syllogism proper or Sorites (σwpeiтns, coacervatio, congeries, gradatio, climax, de primo ad ultimum) arises when we carry on the principle of Inference beyond the part of the highest part, and take in the part of that part, and so on through a series of successive parts. Thus a simple syllogism would run :

(All) B is a part of A;

(All) C is a part of B; .. (All) C is a part of A.

But we may proceed thus:

B is A-i.e., A contains B;
C is B-i.e., B contains C;
D is C-i.e., C contains D;
E is D-i.e., D contains E;

Therefore E is A-i.e., A contains E.

In this case we have the Chain-syllogism or Sorites, and this example in Extension. The predicate is the containing whole.

But the ordinary logical Sorites-sometimes called the Aristotelian-really proceeds in Comprehension, and this is the more natural form. Thus:

1 Esser, Logik, § 104; Hamilton, Logic, iii. 364.

2 Cicero, pro Milone. See Port Royal Logic, p. 231.

3 See especially Hamilton, Logic, iii. L. xix., who gives the best analysis of this form of reasoning, and who for the first time accurately stated its history.

E is D-i.e., has the mark D;
D is C-i.e., has the mark C;
C is B-i.e., has the mark B;
B is A―i.e., has the mark A;
Therefore E is A-i.e., has the mark A.

Here the subject is the containing whole, and the predicate the contained part. Both of these forms are Progressive, in the sense of proceeding from whole to part in the respective quantities.1

A concrete example in Comprehension is found in the following:

Every body is in space;

What is in space is in one part of space;

What is in one part of space may be in another;

What may be in another part of space may change its space; What may change its space is movable;

Therefore every body is movable.2

(a) Sorites, a heaper, is from owpòs, a heap, and originally designated the sophism named by Cicero acervalis. The Sorites, as the name for a form of reasoning, is not to be found in Aristotle. Nor was the form of reasoning afterwards designated Sorites developed by him, though it is improperly named the Aristotelian form.-(See the reference in An. Pr., i. 25.) The name was probably first applied to the reasoning by Valla in his Dialectica Disputationes, published after the midddle of the fifteenth century.-(See Hamilton, Logic, iii. p. 377.) Mark Duncan thinks this form is called the heaper, because as grain is superadded to grain in a heap, so proposition is superimposed on proposition in the reasoning. His definition of it is "an argumentation in which the attribute of every prior proposition is the subject of the posterior until, through several middles, we reach the term to be connected with the subject of the first proposition. It contains as many syllogisms as there are propositions between the first and the last."-(Inst. Log., L. iv. c. vii. § 6.)

§ 567. It is easy enough to state each of these in a Regressive form.

Hamilton lays down the rules: "In the Progressive Sorites of Comprehension and in the Regressive Sorites of Extension, the middle terms are the predicates of the prior premisses and the subjects of the posterior; the middle term is here in position intermediate between the extremes. On the contrary, in the Progressive Sorites of Extension and in the 1 Hamilton, Logic, iii. p. 366. 2 Hamilton, Logic, iii. p. 381.