INCOMPLETE SYLLOGISMS AND IRREGULAR ARGUMENTS.
=1. ENTHYMEME.=
_An enthymeme is a syllogism in which one of the three propositions is
omitted._
Suppressing the major premise gives an enthymeme of the _first order_;
whereas if the minor premise be suppressed, the enthymeme becomes one
of the _second order_; while omitting the conclusions gives an
enthymeme of the _third order_.
_Illustrations_:
Complete syllogism.
All true teachers are just,
You are a true teacher,
(Hence) You are just.
Enthymeme of _first order_; major premise omitted.
..........................
You are a true teacher,
(Hence) You are just.
Enthymeme of _second order_; minor premise omitted.
All true teachers are just,
..........................
(Hence) You are just.
Enthymeme of the _third order_; conclusion omitted.
All true teachers are just,
(And) You are a true teacher,
..........................
To argue in terms of the complete syllogism is the _unusual_, not the
_usual_ method. We have a way of abbreviating our remarks; expressing
only the necessary and leaving the obvious to be taken for granted.
Thus the enthymeme becomes the natural form of expression. But the mere
fact that a part of the argument is omitted, makes it more essential
for the student to think clearly and with careful continuity, that no
error may intrude itself.
Probably the most common enthymemes are those of the first order.
This may be explained by the fact that the major premise is usually the
most universal of the three propositions, and, in consequence, the one
which would be the most generally understood. The following represent
enthymemes of this order, gleaned from the ordinary conversation of
ordinary people:
(1) “Your beets won’t grow, because you are planting them in the
wrong time of the moon.”
(2) “You, being a member of the Sunday School, should be ashamed of
such language.”
(3) “Being the son of your father, you ought to have some pride in
this matter.”
(4) “We are going to have an open winter, because I have observed
that the hornets’ nests are near the ground.”
(5) “You had better put in lots of coal, for I have noticed that the
squirrels have gathered in more nuts than usual.”
Judging from these enthymemes, it would seem to be more natural to
assert the conclusion and follow this by a reason in the form of a
minor premise, leaving the _major_ to the intelligence of the auditor.
The enthymeme of the second order occurs only infrequently, since
it seems to be an unnatural mode of expression, though sometimes it
appears to lend emphasis to the conclusion; e. g., “All untrustworthy
boys come to a bad end, and I predict that you will come to a bad end.”
Enthymemes of the third order are commonly used for the sake of
emphasis, as the following make evident:
(1) “No business man wants an indolent boy, and you are indolent.”
(2) “All successful teachers are interested in their work, and you
plan to be a successful teacher.”
(3) “Humility is a sign of greatness, and Lincoln possessed this
quality.”
=2. EPICHEIREMA.=
_An epicheirema is a syllogism in which one or both of the premises is
an enthymeme._ To put it in another way: An epicheirema is a syllogism
in which one or both of the premises is supported by a reason.
When one premise is an enthymeme the syllogism is termed a _single_
epicheirema; whereas when both premises are enthymemes it becomes a
_double_ epicheirema.
_Single epicheirema._
All men are mortal, _because all men die_,
Socrates was a man,
∴ Socrates was mortal.
_Double epicheirema._
All men are mortal, _because all men die_,
Socrates was a man, _because he was a rational animal_,
∴ Socrates was mortal.
It is obvious that supporting each premise with a reason lends strength
to the argument. This justifies the use of the epicheirema.
=3. POLYSYLLOGISM.=
_A polysyllogism is a series of syllogisms in which the conclusion of
a preceding syllogism becomes a premise of a succeeding one._
The syllogism in the series whose conclusion becomes a premise of the
succeeding syllogism is termed a _prosyllogism_; while the syllogism
which uses as one of its premises the conclusion of the preceding
syllogism is called an _episyllogism_.
_Illustrations._
{ A quadruped is an animal, }
{ A dog is a quadruped, } _Prosyllogism_
_Polysyllogism_ {∴ A dog is an animal. } }
{ Fido is a dog, } _Episyllogism_
{∴ Fido is an animal. }
{ All who libel an associate }
{ are unprofessional, }
{ This teacher has libelled } _Prosyllogism_
{ her associate, }
_Polysyllogism_ {∴ This teacher is } }
{ unprofessional. } }
{ All who are unprofessional } _Episyllogism_
{ should be disciplined, }
{∴ This teacher should be }
{ disciplined. }
=4. SORITES.=
_A sorites is a series of syllogisms in which all of the conclusions
are omitted except the last one._
Just as the epicheirema is a combination of enthymemes of the _first_
and _second orders_, so the sorites is a combination of enthymemes of
the _third order_. If each conclusion were written, the sorites would
take the form of prosyllogisms and episyllogisms. Two forms of the
sorites are recognized by logicians. These are the _progressive_ or
Aristotelian, and the _regressive_ or Goclenian.
_Illustrations._
_Progressive_
_Symbolized._ _Put in Word Form._
All A is B Thomas Arnold was a teacher,
All B is C A teacher is a man,
All C is D A man is a biped,
All D is E A biped is an animal,
Hence all A is E Hence Thomas Arnold was an animal.
_Regressive_
All A is B A biped is an animal,
All C is A A man is a biped,
All D is C A teacher is a man,
All E is D Thomas Arnold was a teacher,
Hence all E is B Hence Thomas Arnold was an animal.
When regarded from the viewpoint of extension, the progressive sorites
proceeds from the smaller to the larger while the regressive is the
converse of this. The point may be illustrated by circles:
Illustration: FIG. 15.
Circle 1 stands for Thomas Arnold.
Circle 2 stands for teacher.
Circle 3 stands for man.
Circle 4 stands for biped.
Circle 5 stands for animal.
The progressive sorites proceeds from the smaller circle to the larger,
thus:
All of circle 1 belongs to 2
All of circle 2 belongs to 3
All of circle 3 belongs to 4
All of circle 4 belongs to 5
Hence, All of circle 1 belongs to 5
The regressive sorites proceeds from the larger to the smaller; i. e.:
All of circle 4 belongs to 5
All of circle 3 belongs to 4
All of circle 2 belongs to 3
All of circle 1 belongs to 2
Hence, All of circle 1 belongs to 5
Other differences become apparent when the omitted conclusions are
expressed.
_Progressive_
_Symbolized_ _Word Form_
All A is B T. Arnold was a teacher, (_A_)
All B is C A teacher is a man, (_A_)
∴ All A is C ∴ T. Arnold was a man. (_A_)
All C is D A man is a biped, (_A_)
∴ All A is D ∴ T. Arnold was a biped. (_A_)
All D is E A biped is an animal, (_A_)
∴ All A is E ∴ T. Arnold was an animal. (_A_)
In the three completed syllogisms it becomes evident that the
progressive sorites uses the minor as its first premise and in
consequence takes the form of the fourth figure, though the reasoning
is according to the first figure.
The progressive sorites must conform to the following rules:
(1) The first premise may be universal or particular, all the others
_must_ be universal.
(2) The last premise may be affirmative or negative; all the others
_must_ be affirmative.
A violation of the first rule would result in undistributed middle;
whereas a violation of the second rule would give illicit major. These
rules may be illustrated by giving attention to the symbols of the
foregoing completed syllogisms.
The first completed syllogism of the sorites is:
All A is B
All B is C
∴ All A is C
Securing a logical arrangement by interchanging the major and minor
premises gives:
(M) (G)
(A) All B is C (First premise universal)
―
(S) (M)
(A) All A is B
(S) (G)
(A) ∴ All A is C
―
Applying the rules we find this syllogism valid, or we may recall that
A
A is valid in the first figure.
A
Let us now make the first premise of the sorites _particular_ and test.
Some A is B
All B is C
∴ Some A is C
_Arranged logically_:
(M) (G)
(A) All B is C
―
(S) (M)
(I) Some A is B
(S) (G)
(I) ∴ Some A is C
_Proof_:
Since one premise is particular the conclusion must be particular.
(Rule 7) As there are no negatives in the argument, only one conclusion
is possible; namely, a particular affirmative (I). Thus, instead of
the conclusion, “All A is C,” which is an (A), it must be, “Some A
is C,” or an (I). Underscoring the distributed term, it is seen that
the middle term is distributed in the major premise and that no term
is distributed in the conclusion. Thus the mood is valid. This is
“checked” when we recall that AII is always valid in the first figure.
We have now shown that the first premise of a progressive sorites may
be _universal_ or _particular_. Let us further proceed to prove that
all the other premises must be universal.
_Data_: Given the first completed syllogism of the sorites:
All A is B
All B is C
∴ All A is C
_Proof_: Let any other premise, such as the second, be particular; this
gives the following:
All A is B
Some B is C
∴ Some A is C
_Arranged logically_: Mood, figure, and distribution indicated.
(M) (G)
(I) Some B is C
(S) (M)
(A) All A is B
―
(S) (G)
(I) ∴ Some A is C
We note at once that the middle term is undistributed, hence the
I
mood A is invalid in the first figure; reference to the valid moods in
I
figure _one_ “checks” this conclusion. Since no premise, other than the
first, can be particular, then all save the first must be universal.
The truth of the first rule has been demonstrated, and now we may
follow a similar plan to prove the truth of the second rule.
_Problem_: To prove that the last premise may be negative.[11]
_Data_: Given the _last_ completed syllogism:
{ All A is D
{ All D is E
{ ∴ All A is E
Let us make the last premise negative (E) and test the result. (As all
but the first must be universal we cannot use an O.)
All A is D
No D is E
∴ No A is E
_Arranged logically and symbolized_:
(M) (G)
(E) No D is E
― ―
(S) (M)
(A) All A is D
― ―
(S) (G)
(E) ∴ No A is E
― ―
_Proof_: Negative premise; negative conclusion. No particulars. Middle
term distributed in major premise. No term distributed in conclusion
which is not distributed in premise where it occurs. Syllogism valid.
We must now prove that all the other premises must be affirmative.
_Problem_: To prove that no other premise can be negative, or that all
others must be affirmative.
_Data_: Given last syllogism of sorites with the first premise negative.
(Any other may be taken.)
No A is D
All D is E
∴ No A is E
_Arranged logically and symbolized_:
(M) (G)
(A) All D is E
―
(S) (M)
(E) No A is D
(S) (G)
(E) ∴ No A is E
― ―
_Proof_: “G” is distributed in the conclusion but not in the major
premise. Fallacy of illicit major. Hence no other premise can be
negative.
We may now consider the completed syllogisms of the _regressive_
sorites.
All A is B
All C is A
∴ All C is B
All D is C
∴ All D is B
All E is D
∴ All E is B
By examining the foregoing it becomes apparent that the regressive
sorites, both in form and in the reasoning, adapts itself to the first
figure.
The rules of the regressive sorites are just the reverse of the
progressive. These are:
(1) The first premise may be negative; all the others must be
affirmative.
(2) The last premise may be particular; all the others must be
universal.
It would be a valuable exercise for the student to test these rules
according to the plan pursued in treating the progressive sorites.
=5. IRREGULAR ARGUMENTS.=
It has been intimated that a syllogistic argument, in order to be
logical, should be made to conform to the _rules of the syllogism_. It
must not be inferred from this, however, that all deductive reasoning
is included by the logical forms here treated. There seem to be
arguments which yield valid conclusions, and yet which are not logical
in the strict sense of the word. The following illustrate some of these
forms:
(1) _ Quantitative Arguments._
John is taller than James,
Albert is taller than John,
∴ Albert is taller than James.
Here, apparently, is a fallacy of four terms: these four terms are
(1) John, (2) taller than James, (3) Albert, (4) taller than John. Yet
we know that the argument is valid. There is not a particle of doubt in
the mind relative to the truth of the conclusion that “Albert is taller
than James.” We are consequently forced to the inference that such
quantitative arguments lie outside the field of syllogistic reasoning.
The argument involves this new principle, “Whatever is greater than a
_second thing_ which is greater than a _third thing_ is itself greater
than a third thing.”
There are many other arguments similar to this which are not
syllogistic in nature. To wit: A equals B, B equals C, C equals D;
A equals D. A is a brother of B, B is a brother of C, C is a brother
of D; A is a brother of D. A is west of B, B is west of C, C is west
of D; A is west of D.
(2) _Plurative Arguments._
These are arguments in which the propositions are introduced by _more_
or _most_; e. g.:
Most (more than half) of the team are seniors,
Most (at least half) of the team are under twenty,
∴ Some students under twenty are seniors.
I
Here we have an I which is evidently valid. No term distributed and yet
I
the conclusion is unquestionably true. This is due to the fact that the
propositions are so worded as to force an overlapping of the major and
minor terms. The student may illustrate this relation by circles.
=6. OUTLINE.=
INCOMPLETE SYLLOGISMS AND IRREGULAR ARGUMENTS.
(1) Enthymeme.
First, second and third orders.
Natural form.
(2) Epicheirema.
Single, double.
(3) Polysyllogism.
Prosyllogism, episyllogism.
(4) Sorites.
Progressive, regressive.
Two rules of each.
(5) Irregular Arguments.
Quantitative, plurative.
=7. SUMMARY.=
(1) An enthymeme is a syllogism in which one of the three propositions
is omitted. Suppressing the major premise gives an enthymeme of the
_first_ order; omitting the minor gives one of the _second_ order;
while omitting the conclusion gives one of the _third_ order.
The enthymeme is really the natural form of expression. Enthymemes of
the first order are the most _common_ while those of the third order
are the most _emphatic_.
(2) An epicheirema is a syllogism in which one or more of the premises
is an enthymeme. An epicheirema is said to be _single_ when but one
premise is an enthymeme, and _double_ when both premises are enthymemes.
(3) A polysyllogism is a series of syllogisms in which the conclusion
of the _preceding_ syllogism becomes a premise of the _succeeding_ one.
The one of the series whose conclusion becomes a premise is termed a
prosyllogism; while the one which uses the conclusion as a premise is
called an episyllogism.
(4) A sorites is a series of syllogisms in which all the conclusions
are omitted except the last one.
The two kinds of sorites are the progressive and regressive. The
progressive uses the “minor” as its first premise and adopts the form
of the _fourth_ figure, whereas the regressive uses the “major” as its
first premise and adopts the form of the _first_ figure.
The two rules of the progressive sorites are, (1) “The first premise
may be particular, all the others must be universal”; (2) “The last
premise may be negative, all the others must be affirmative.”
The two rules of the regressive are, (1) “The first premise may be
negative, all the others must be affirmative”; (2) “The last premise
may be particular, all the others must be universal”.
(5) Irregular arguments are such as yield valid conclusions and yet do
not conform to the syllogistic rules.
The quantitative argument expresses quantity and contains four terms.
This argument is based on the principle, “What ever is greater than a
second thing which is greater than a third thing is itself greater than
a third thing.”
Plurative arguments are introduced by “more” or “most” and give in
consequence a valid conclusion from two particulars. This is due to the
overlapping of the major and minor terms.
=8. REVIEW QUESTIONS.=
(1) Define and illustrate an enthymeme.
(2) Illustrate the enthymemes of the three orders and point out their
distinct uses.
(3) Why should the enthymeme demand closer thought than the ordinary
syllogism?
(4) Define and illustrate the epicheirema.
(5) Of what use is the epicheirema? Illustrate.
(6) Define and illustrate a prosyllogism and an episyllogism.
(7) Why are polysyllogisms so called?
(8) Define and illustrate the sorites.
(9) Relate the sorites and the epicheirema to the enthymeme.
(10) Illustrate the two forms of sorites.
(11) Explain the two forms of sorites by means of a diagram.
(12) Prove the truth of the two rules of the progressive sorites.
(13) Illustrate two kinds of irregular arguments and show that they
are valid.
(14) Complete the five enthymemes of page 248 and indicate their mood
and figure.
=9. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=
(1) Why should enthymemes of the second order be less common than
those of the first?
(2) You desire to make it evident to a child that a small beginning
often leads to a momentous ending; do so in terms of the
enthymeme of the first order.
(3) Show that prosyllogism and episyllogism are relative terms.
(4) When the common premise of the “pro” and “epi” syllogism is
omitted what abbreviated form results?
(5) From the viewpoint of your definition criticise this: “A sorites
is a series of prosyllogisms and episyllogisms in which all of
the conclusions are suppressed except the last.”
(6) Prove the truth of the two rules of the regressive sorites.
(7) Show that the prosyllogism and the episyllogism may be
progressive or regressive.
(8) “Reasoning from cause to effect”――is such progressive or
regressive? Explain.
(9) Which is inductive in nature, the progressive form of reasoning
or the regressive? Explain.
(10) Test the validity of the enthymemes on pages 248 and 249.
(11) “A sorites is at least as immediately convincing as the chain of
syllogisms into which it can be decomposed.” Discuss this.