Introduction to Logic

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Syllogism: valid moods in each figure

Syllogism:

Valid moods in each figure:

Table:

FIGURE 1	BARBARA, BARBARI, DARII, CELARENT, CELARONT, FERIO
FIGURE 2	CESARE, CESARO, CAMESTRES, CAMESTROS, FESTINO
FIGURE 3	DARAPTI, DATISI, DISAMIS, FELAPTON, FERISON, BOCARDO
FIGURE 4	BRAMANTIP, CALEMES, CALEMOS, DIMATIS, FESAPO, FRESISON

 

Reason & Impediments to Good Reasining

Reason, Rhetoric, and Argument Analysis

GOAL: to determine whether the premises of an argument provide good reason to believe that its conclusion is true. In real life, we’re not always interested in achieving this goal.

Sometimes, we might want to be comforted, or amused, or morally challenged by written (or spoken) words.

There are various non-rational ways of dealing with or responding to arguments, e.g.:

·       Credulity [accepting every argument]

·       Contradiction [rejecting every argument]

·       Dogmatism [maintaining beliefs in light of any argument]

·       Skepticism [not taking arguments seriously]

The rational way of responding to an argument involves trying (in good faith, and in a careful, reflective way) to determine whether the premises support the conclusion.

In order to do this, we must maintain an open mind, and think very carefully about (a) what is the strongest version the argument that has been given?, and (b) how strong is it?

Here are some impediments to good (rational) reasoning.

1.    Lacking an adequate vocabulary

In order to effectively analyze arguments, we need the right conceptual tools/vocabulary.

We’ve been mentioning “rational strength” and we have distinguished it from rhetorical power and literary merit.

But, we have not yet said precisely what the “rational strength” of an argument is (or consist in).

Knowing a precise set of concepts of things discussed in any argument, helps us to get a precise definition of “rational strength”.

Having that vocabulary will be crucial for our goal.

2.   The Desire to be “Tolerant” / “Open-Minded”

Being open-minded — in a sense — is important (and a good thing) for successful argument analysis.

But, perhaps ironically, one can be too open-minded.

When we analyze arguments, we have to be willing (in some cases) to say that various sorts of errors or mistakes have been made in the course of a passage/argument.

We may be hesitant to make such judgments — out of a desire to be “tolerant” or “open-minded” (in some sense).

But, there is no real conflict between making such judgments and being “tolerant” and “respectful” of others.

There is nothing intolerant or disrespectful about carefully explaining to others errors you see in their arguments.

The point of argument analysis is not to “put down” the arguments of others. Rather, it is to (ultimately) come to your own conclusion, based on the available evidence.

It might be helpful to think of the arguments we encounter in this class as being “given” to us from an unknown source.

Argument reconstruction is not a personal, but a rational activity — whose aim is to come up with the best arguments on both sides of issues we think about.

Ultimately, it is about rational inquiry into the truth.

3.   Misunderstanding the Point of Argument Analysis

So much of what we hear and read concerns the rhetorical power of arguments, and not their rational strength.

This may make it difficult to overcome the tendency to think of “arguments” in a non-rational way.

One must always remember the point of argument analysis.

Every argument aims to determine the degree to which the premises of an argument rationally support the truth of its conclusion.

Ultimately, we want to find the best arguments in favor of (and against!) any particular statement — with an eye toward rationally determining whether it is true or false.

4.     The Use of “Argument Stoppers”

There are various quick responses to arguments which have the effect of cutting-off discussion and preventing careful rational analysis of the argument in question.

These are called argument stoppers. Examples:

• “Well, that’s a matter of opinion.”

• “Who’s to say what the truth is about that?”

• “That’s a subjective judgment.”

Often, these quips are just shorthand for something like:

• “I would prefer not to think about what you said. I would prefer to continue believing what I have believed up to until now, so I’m going to ignore your argument.”

So, such statements are often (seemingly) polite ways of avoiding thinking about someone’s argument.

In some contexts, there may be some substance to the claim that a judgment is “subjective” or “mere opinion”.

But, these terms tend to be confusing and are oft abused.

Thus, one thing we must do is learn to use terms like “subjective” and “mere opinion” very carefully.

It’s something very subtle, and we must watch out for it.

significance of alphabets in the names of various moods of a syllogism:

To facilitate Reduction, the recognised Moods have all had names given them; which names, again, have been strung together into mnemonic verses of great force and pregnancy:

Barbara, Celarent, Darii, Ferioque prioris:
Cesare, Camestres, Festino, Baroco, secundæ:
Tertia, Darapti, Disamis, Datisi, Felapton,
Bocardo, Ferison, habet: Quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.

In the above verses the names of the Moods of Fig. I. begin with the first four consonants B, C, D, F, in alphabetical order; and the names of all other Moods likewise begin with these letters, thus signifying (except in Baroco and Bocardo) the mood of Fig. I., to which each is equivalent, and to which it is to be reduced: as Bramantip to Barbara, Camestres to Celarent, and so forth.

The vowels A, E, I, O, occurring in the several names, give the quantity and quality of major premise, minor premise, and conclusion in the usual order.

The consonants s and p, occurring after a vowel, show that the proposition which the vowel stands for is to be converted either (s) simply or (p) per accidens; except where s or p occurs after the third vowel of a name, the conclusion: then it refers not to the conclusion of the given Mood (say Disamis), but to the conclusion of that Mood of the first Figure to which the given Mood is reduced (Darii).

M (mutare, metathesis) means ‘transpose the premises’ (as of Camestres).

C means ‘substitute the contradictory of the conclusion for the foregoing premise,’ a process of the Indirect Reduction to be presently explained (see Baroco, Section 8).

The other consonants, r, n, t (with b and d, when not initial), occurring here and there, have no mnemonic significance.

What now is the problem of Reduction? The difference of Figures depends upon the position of the Middle Term. To reduce a Mood of any other Figure to the form of the First, then, we must so manipulate its premises that the Middle Term shall be subject of the major premise and predicate of the minor premise.

Now in Fig. II. the Middle Term is predicate of both premises; so that the minor premise may need no alteration, and to convert the major premise may suffice. This is the case with Cesare, which reduces to Celarent by simply converting the major premise; and with Festino, which by the same process becomes Ferio. In Camestres, however, the minor premise is negative; and, as this is impossible in Fig. I., the premises must be transposed, and the new major premise must be simply converted: then, since the transposition of the premises will have transposed the terms of the conclusion (according to the usual reading of syllogisms), the new conclusion must be simply converted in order to prove the validity of the original conclusion. The process may be thus represented (s.c. meaning ‘simply convert’)

The Ostensive Reduction of Baroco also needs special explanation; for as it used to be reduced indirectly, its name gives no indication of the ostensive process. To reduce it ostensively let us call it Faksnoko, where k means ‘obvert the foregoing premise.’ By thus obverting (k) and simply converting (s) (in sum, contrapositing) the major premise, and obverting the minor premise, we get a syllogism in Ferio, thus:

In Fig. III. the middle term is subject of both premises; so that, to reduce its Moods to the First Figure, it may be enough to convert the minor premise. This is the case with Darapti, Datisi, Felapton, and Ferison. But, with Disamis, since the major premise must in the First Figure be universal, we must transpose the premises, and then simply convert the new minor premise; and, lastly, since the major and minor terms have now changed places, we must simply convert the new conclusion in order to verify the old one. Thus:

Bocardo, like Baroco, indicates by its name the indirect process. To reduce it ostensively let its name be Doksamrosk, and proceed thus:

In Fig. IV. the position of the middle term is, in both premises, the reverse of what it is in the First Figure; we may therefore reduce its Moods either by transposing the premises, as with Bramantip, Camenes, and Dimaris; or by converting both premises, the course pursued with Fesapo and Fresison. It may suffice to illustrate by the case of Bramantip:

This case shows that a final significant consonant (s, p, or sk) in the name of any Mood refers to the conclusion of the new syllogism in the First Figure; since p in Bramantip cannot refer to that Mood’s own conclusion in I.; which, being already particular, cannot be converted per accidens.

Finally, in Fig. I., Darii and Ferio differ respectively from Barbara and Celarent only in this, that their minor premises, and consequently their conclusions, are subaltern to the corresponding propositions of the universal Moods; a difference which seems insufficient to give them rank as distinct forms of demonstration. And as for Barbara and Celarent, they are easily reducible to one another by obverting their major premises and the new conclusions, thus:

There is, then, only one fundamental syllogism.

Syllogism

While immediate inference contained two propositions, a premise and a conclusion, and thus, two and only two terms, a standard syllogism contains more. The familiar syllogism of men, mortals, and Socrates will again prove its value, providing the basis for introducing new terms and new definitions. The "\ " is read as "therefore."

All men are mortal.

Socrates is a man.

\Socrates is mortal.

 The Basic Elements

The standard syllogism must contain three and only three propositions, two of which are premises; the other is the conclusion. The two premises and the conclusion share three and only three terms. In the syllogism above, the three terms are men, mortals, and Socrates. Socrates in the conclusion and the second premise; mortal in the conclusion and the first premise; and men (or man) in the two premises. Each appears twice, but never twice in the same proposition. Each term must mean the same thing, that is to say must be univocal. For example, "mortal" in the conclusion and the premise must mean the same thing. Thus, a syllogism is an argument having two premises and a conclusion with the subject term of the conclusion in one of the premises, the conclusion's predicate term in the other premise, and a third term in both premises. The third term of the premises must never appear in the conclusion.

The Terms of the Syllogism

The syllogism above or any other standard syllogism, can be expressed as an implication.
A(mp) A(sm) < A(sp),
where, s stands for Socrates; p stands for mortal; m stands for man; and "< " stands for implies. The subject term of the conclusion is the minor term. The predicate of the conclusion is the major term. The term that appears in both premises, not the conclusion, is the middle term. The premise that contains the major term is the major premise, and is usually placed first. The premise that contains the minor term is the minor premise; it is placed after the major premise.
Thus, the conclusion of our syllogism is an inference from the major premise through the mediation of the minor premise.

Moods

The mood of an argument is an individual case of an inference, a mediated inference. For example, each of the propositions of the syllogism above are of the form All a is b -- the A Form. The mood, we say, is AAA; the first letter denotes the major premise, the second letter denotes the minor premise, and the third letter denotes the conclusion. Thus, the mood of a syllogism refers to the forms of the syllogism and the order of the forms beginning with the major and ending with the conclusion. Every standard syllogism has a mood of three and only three forms, but there is more.

The Figure of a Syllogism

The figure of a syllogism refers to the position of the middle term in the premises. Omitting any reference to the conclusions, there are four possible positions as shown below. It may be helpful to think of Figures 1 and 4 as mirror images of each other, as are Figures 2 and 3. (m stands for the middle term; p stands for the major term; s stands for the minor term.)

1st Premise	M-p	p-M	M-p	p-M
2nd Premise	s-M	s-M	M-s	M-s
Figure	1	2	3	4

By figure, then, we indicate the relative positions of the one term shared by both premises -- the middle term.

The Frame of a Syllogism

The frame of a syllogism is a name assigned to the combination of the mood and the figure of a standard syllogism. Thus, when we speak of the form of a syllogism, we mean the frame -- its mood and figure together. Our syllogism above has this frame: AAA-1.
Valid syllogistic frames were given names by logicians. In part, their purpose was the development of a system of frame-names in verses as a memory device to aid in identifying the different valid moods and figures of the syllogism. Some other characteristics of these names will be discussed in due course.
With four forms and four figures, there are 256 frames. (Four forms can be combined in pairs results in 16 different sets of premises; each pair has one of the four forms as a conclusion for a total of 64 (16 x 4); factor in 4 figures, for a total of 256 frames.) Of course, not all of these were named, only the valid ones. There are 24 valid frames, hardly as intimidating as 256, if one had to rely on memory. Fortunately, there are other means more reliable than memory.

Validity

As has been stated, valid is a quality of arguments in which the conclusion necessarily results from the premises. An argument is valid if the form of the conclusion is true every time the forms of the premises are true. This means essentially that if an argument is valid, then it is impossible for the premises to be true and the conclusion false. Obviously, the conclusion of a valid syllogism must not contain a term not in the premises.
The validity or invalidity of a syllogism can be determined by either the application of rules or the Method of Deduction. We start with rules since they are quite easy to apply once an argument's frame has been made explicit and leave the deductive method for the next section. The rules themselves are derived from the valid frames. There are five rules for testing the validity of syllogisms.

The Five Rules

Is a syllogism valid? It is if it does not violate these five rules.

Rule 1	Two premises in both of which the middle term is undistributed do not imply a conclusion.
Rule 2	Two premises with undistributed terms having a conclusion which distributes those same terms do not imply a conclusion
Rule 3	Two affirmative premises do not imply a negative conclusion.
Rule 4	Two negative premises do not imply a conclusion.
Rule 5	An affirmative and negative pair of premises do not imply an affirmative conclusion.

First, symbolize the syllogism. The choice of subject and predicate letters is arbitrary only take care to use them in consistent fashion. There must be only three such letters, since a standard syllogism contains three and only three terms each used univocally. The letters s, p, and m, used previously will serve. As before, s stands for "Socrates" and is the minor term; p stands for "mortal" and is the major term; and m stands for "man" and is the middle term. The subscripts "d" and "u" stand for distributed and undistributed, respectively.

Major Premise	All m d is p u	A(mp)
Minor Premise	All s d is m u	A(sm)
\Conclusion	\ All s d is p u	\ A(sp)

Second, apply each of the rules. The syllogism must satisfy each and all of the rules if it is to count as valid. The first rule states that the middle term must be distributed in at least one of the premises. Observe, the middle term is distributed in the major premise. The second rule compares terms in the premises with terms in the conclusion. If a term is undistributed in the premise, it must not be distributed in the conclusion. Check the major term P in the major premise. It is undistributed and it remains undistributed in the conclusion. The third rule states that two affirmative premises do not imply a negative conclusion. This syllogism has two A Form (affirmative) propositions as premises. The conclusion is also affirmative. The fourth rule makes reference to negative premises. The premises of this syllogism are affirmative in quality. The fifth rule says that an affirmative premise and a negative premise do not imply an affirmative conclusion. This syllogism ends with an affirmative conclusion, but it does not contain a negative premise. Therefore, this syllogism is valid. Indeed, all syllogisms having this form are valid. The frame AAA-1 is a valid frame, since it satisfies all of the rules.
 The rules themselves are both sufficient and necessary. They are sufficient since they leave untouched the 24 syllogisms proved valid by the deductive method, and prove the remaining ones invalid. The rules are also necessary since each applies to at least one invalid syllogism for which none of the others apply.
A study of the rules alone will eliminate a number of invalid frames. For example, by Rule #4 the syllogisms with premises EE, EO, OO, and OE (all negative) are invalid. Rule #1 declares invalid the syllogisms with premises II; OI, figures 1 and 3; and IO, figures 3 and 4, since these arrangements leave the middle term undistributed. A systematic study of the rules should eliminate as invalid all but 24 of the 256 frames. Thus the significance of necessary and sufficient rules is no mere detail.

The Method of Deduction

It is an unavoidable fact, though many try to skirt it, that every system of thought, philosophy, theology, or body of knowledge has starting points without which the system could not get off the ground. To put it another way: every system of thought or knowledge has an axiom or a set of axioms which are indemonstrable within that system. An axiom is a first principle or premise which cannot be demonstrated precisely because axioms themselves are used to demonstrate or prove other statements which we call theorems. A theorem is a proposition deduced from an axiom. Thus, first principles or axioms are the basis of all argument and demonstration.
The deductive method proves valid frames as theorems. To that end, seven of the 24 valid frames will be proved as theorems. These proofs should suffice to introduce a beginner to the significance of the deductive method in syllogistic reasoning. We state first the two axioms, then two (operational) rules. These are applied to the axioms to deduce theorems. They may also be applied to theorems to deduce additional theorems.
Axiom 1: A(ba) A(cb) < A(ca) Axiom 1 reads: All b is a & All c is b implies All c is a.
Axiom 2: E(ba) A(cb) < E(ca)  Axiom 2 reads: No b is a & All c is b implies No c is a.

Rule I DM	If in any valid implication the premise and the conclusion are interchanged and contradicted, the result is a valid implication. ("DM" stands for Deductive Method.)
Rule II DM	If in any valid implication its premise be strengthened or its conclusion be weakened, then a valid implication will result.

 Application of Rule I above is easily accomplished. It can be applied as often as necessary, first to one premise and conclusion, then to the other premise and conclusion, in any order. Application of Rule II may raise some concern about the meanings of strengthened form of the premise and weakened form of the conclusion. So let us define these. The premise of a valid mood and figure can be said to be a strengthened form of the conclusion, and the conclusion a weakened form of the premise. More explanation follows.
Some examples applying the rules might be helpful. From the square of opposition, we know that A(ab) implies I(ab) is a valid inference. If we interchange the premise and conclusion and contradict each (Rule I), we prove that E(ab) implies O(ab). Of course, the latter implication is valid based on subalternation, but here we illustrate the application of the Rule I. Now apply Rule II to the valid implication, A(ab) implies I(ab). A(ab) is the strengthened form of I(ab); and I(ab) is the weakened form of A(ab). Also, I(ba) is the weakened form of I(ab). By weakening the conclusion of A(ab) implies I(ab), we prove A(ab) implies I(ba). Of course, the latter is valid per accidens, but here we illustrate the application of Rule II. Confused? Well, just below starting with the Law of Identity as an axiom, we deduce two theorems to illustrate further the use of Rules I and II in proving theorems.

1	A(ab) < A(ab)	Using the Law of Identity as an Axiom
2	A(ab) < I(ab)	Theorem One, by Rule I, replacing the conclusion of #1, above, by its weakened form, I(ab).
3	E(ab) < O(ab)	Theorem Two by Rule II, interchanging and contradicting the premise and conclusion of Theorem One.

Deduction of Theorems

It should be obvious now that the process of deducing or proving theorems from axioms is not a difficult operation; although, it requires careful attention at each step. The order of the deduced theorems may vary from one person to another. For example, one person may apply Rule I to the minor premise and conclusion of an axiom first, then secondly to the major premise and conclusion of the axiom. Another may reverse the process. Whether you start with the minor and conclusion, or the major and conclusion is arbitrary. Likewise, whether you apply Rule I first, then Rule II, or vice versa is optional.

Axiom 1	A(ba) A(cb) < A(ca)
Axiom 2	E(ba) A(cb) < E(ca)
Theorem 1	A(ba) O(ca) < O(cb)	by Rule I on Axiom 1: Interchange and contradict Axiom 1's minor premise and conclusion.
Theorem 2	O(ca) A(cb) < O(ba)	by Rule I on Axiom 1: Interchange and contradict Axiom 1's major premise and conclusion.
Theorem 3	I(ca) A(cb) < I(ba)	by Rule I on Axiom 2: Interchange the contradictories of the conclusion and major premise.
Theorem 4	E(ba) I(ca) < O(cb)	by Rule I on Axiom 2: As above, only this time with the conclusion and the minor premise.

Theorems 1-4 are not in conventional format, and they must be if one is to ascertain the correct mood and figure. (By conventional format we mean a certain order based on a "ca" conclusion.) So, let us stipulate that "c" is the minor term; "a" is the major term and "b" is the middle term. Applying these conventions to Theorems 1-4, we obtain conventional format for each and thereby the correct mood and figure.

Theorem 1	A(ab) O(cb) < O(ca)	AOO-2
Theorem 2	O(ba) A(bc) < O(ca)	OAO-3
Theorem 3	I(ba) A(bc) < I(ca)	IAI-3
Theorem 4	E(ab) I(cb) < O(ca)	EIO-2

The conclusions of the two axioms are universal. A universal conclusion validly implies the corresponding particular. By weakening the form of the conclusion of Axiom 2, we deduce two additional theorems. Note that weakening the form of a conclusion and strengthening the form of a premise, function as replacements of one form by another logically valid form. Thus, the conclusion E(ca) can be weakened by replacement with E(ac) or O(ca). Axiom 2's premise, E(ba), can be strengthened by replacing it with E(ab).

Axiom 2	E(ba) A(cb) < E(ca)
Theorem 5	E(ba) A(cb) < E(ac)	by Rule II on Axiom 2: Weakened form of the conclusion. [E(ac) counts as weakened form of E(ca). One is the converse of the other.]
Theorem 6	E(ba) A(cb) < O(ca)	by Rule II on Axiom 2: Weakened form of the conclusion, E(ca).
Now deduce Theorem 7 from Theorem 5 still using Rule II
Theorem 7	E(ba) A(cb) < O(ac)	by Rule II on Theorem 5: Weakened form of the conclusion, E(ac).

Theorem 6 is in conventional format, 5 and 7 are not. This operation will require the re-ordering of the premises in Theorems 5 and 7. Recall that the premise with the major term (the same as the predicate term of the conclusion) is the major premise and is placed first; the minor premise, i.e., the premise with the minor term (the same as the subject term of the conclusion) is placed second.

Theorem 5	A(ab) E(bc) < E(ca)	AEE-4 (mood & figure)
Theorem 6	E(ba) A(cb) < O(ca)	EAO-1 (mood & figure)
Theorem 7	A(ab) E(bc) < O(ca)	AEO-4 (mood & figure)

The Deductive Method has proven seven theorems from two axioms all of which can be used to deduce additional theorems using the rules and the definitions provided. Theorems may be used at any stage together with the original axioms and rules to prove additional theorems. The problem now is to deduce the remaining theorems for a total of twenty four. Your deductions may prove theorems previously deduced, but keep trying until you have twenty four unique theorems each in conventional format. If you deduce one which is doubtful, appeal to the set of Five Rules to check your proof.

Frame Names

The valid frames of syllogistic logic were named and may be of more historical interest than practical. The vowels indicate the mood. Other lower case letters stand for certain operations we shall briefly describe in due course, but first, the names:

1st Figure	Barbara, Celarent, Darii, Ferio.
2nd Figure	Cesare, Camestres, Festino, Baroko.
3rd Figure	Darapti, Disamis, Datisi, Felapton, Bokardo, Ferison.
4th Figure	Bramantip, Camenes, Dimaris, Fesapo, Fresison.

These names designate nineteen valid frames. Five others are available on the basis that the universal conclusion of a valid frame implies the corresponding particular. Thus, from Barbara-1 or AAA-1, application of Rule II yields AAI-1, the Weakened Form of Barbara. Similarly, from Celarent-1 or EAE-1, Rule II application yields EAO-1, the Weakened Form of Celarent.
Chart 3.1 lists the names in the order of Theorems 1-7 already deduced.

Chart 3.1: Theorems and Frame Names

	Theorems	Mood & Figure	Names
1	A(ab) O(cb) < O(ca)	AOO-2	Baroko
2	O(ba) A(bc) < O(ca)	OAO-3	Bokardo
3	I(ba) A(bc) < I(ca)	IAI-3	Disamis
4	E(ab) I(cb) < O(ca)	EIO-2	Festino
5	A(ab) E(bc) < E(ca)	AEE-4	Camenes
6	E(ba) A(cb) < O(ca)	EAO-1	Celarent-1, weakened form
7	A(ab) E(bc) < O(ca)	AEO-4	Camenes-4, weakened form

                                      
The vowels of the names, as mentioned above, stand for the mood of the syllogism. The other lower case letters in the names of the first figure do not have any special meaning, but the "s," "p," and "k" of the names in figures two, three, and four do.

 "s"       stands for simple conversion of the preceding proposition. For example, if in Camenes you convert the conclusion E(ca) to E(ac) and change to conventional format, which in this case requires a reordering of the premises, you get Celarent, EAE-1. Similar conversions hold for Cesare, Camestres, Festino, Disamis, Datisi, Ferison, Dimaris, Fesapo, and Fresison, all having at least one "s" preceded by a letter standing for a standard form.

"p"        means to convert the preceding proposition by limitation or per accidens. If you apply this operation to Fesapo (EAO-4), you get Festino (EIO-2). Other frames that qualify are Darapti, Felapton, and Fesapo.

"k"       stands for reductio ad absurdum (RAA) or assuming the conclusion to be false as part of the premise set in order to deduce by valid inferences, step by step, a contradiction. In this manner, one demonstrates that the assumption of a false conclusion as premise was unwarranted, and the original implication, valid.

To illustrate, let us show that Bokardo (OAO-3) is valid by RAA proof.
            O(ba) A(bc) < O(ca)                Bokardo-3

1.	O(ba)		true premise		\ O(ca)
2.	A(bc)		true premise
Assume	3. O(ca) is false		RAA method
Then	4. A(ca) is true		contradictory of 3
Then	5. A(ca) A(bc) < A(ba)		4 & 2; Barbara-1
But	6. A(ba) cannot be true		contradictory of 1, O(ba)
So	7. A(ba) must be false		1 & 6 contradictory
But if	8. A(ba) is false		Step 7
Then	9. A(ca) or A(bc) is false.		5 & 8; Barbara-1
Option 1 Assume		10. A(ca) is false		From Step 9
Then		11. O(ca) in 3 can't be false		3 & 10 contradictories
Then		13. O(ca) is both true & false		3 & 11; Impossible!
Option 2 Assume		14. A(bc) is false		From Step 9
Then		15. A(bc) is both true & false		2 & 14; Impossible!

 16. So, in assuming that the true premises imply a false conclusion, we have deduced by valid inferences contradictions in Steps 13 and 15. Therefore, O(ca) must be true. The original implication (Bokardo-3) is valid.

Nonstandard Syllogisms

A syllogism may fail to be in standard form in a number of ways. The first pair of examples below are syllogisms containing more than three but not unrelated terms. Also, their propositions are not in the proper order: major premise, minor premise, then conclusion. The second set of examples discusses syllogisms with a suppressed premise or conclusion (enthymemes). Last, a third type of nonstandard syllogism, sorites, is described.

Syllogisms Containing More Than 3 Terms

1st Argument:

All inexpensive things are poorly constructed.

All German cars are expensive.

\ No poorly constructed things are German cars.

The terms are inexpensive things, poorly constructed (things), German cars, and expensive (things).

2nd Argument:

Some of the stolen books are not replaceable.

No irreplaceable things are deductible.

\ Some of the stolen books are non-deductible.

The terms are stolen books, replaceable (books), irreplaceable things, deductible (items), and non-deductible (items).
Both arguments, above, have more than three terms each. So, the first task is to reduce the number of terms to three, if possible, making certain that each term is used in the same sense. This can be accomplished quite easily by obverting the second premise of the first argument and the first premise and the conclusion of the second argument.

Phase 1, First Example:

All inexpensive things are poorly constructed.

No German cars are inexpensive. (by obversion)

\ No poorly constructed things are German cars.

The terms have been reduced to three, each used in the same sense.

Phase 1, Second Example:

Some of the stolen books are irreplaceable. (by obversion)

No irreplaceable things are deductible.

\ Some of the stolen books are not deductible. (by obversion)

Again, the terms have been reduced to three univocal terms.
Now change the order of the premises in each argument.

Phase 2, First Example:

Major	No German cars are inexpensive.
Minor	All inexpensive things are poorly constructed.
Conclusion	\ No poorly constructed things are German cars.

 INVALID, EAE-4, Rule #2 (The minor term, poorly-constructed-things, is undistributed in the premise but distributed in the conclusion.)

Phase 2, Second Example:

Major	No irreplaceable things are deductible.
Minor	Some of the stolen books are irreplaceable.
Conclusion	\ Some of the stolen books are not deductible.

 VALID, EIO-1, Ferio-1. The tests of Five Rules are met in this example.

Enthymeme

An otherwise perfectly valid categorical syllogism may appear not to be so when one of its propositions is suppressed or understood but not explicitly stated. Such an argument is known as an enthymeme. The first enthymeme has a suppressed major premise, the second, a suppressed minor premise, and the third, a suppressed conclusion.

Suppressed Major Premise

Some NFL quarterbacks are good passers because some NFL quarterbacks have strong throwing arms.
Identify the conclusion first, then classify the premise as either the major or minor. In this case, the premise is the minor premise, since it contains the minor term.

Missing Major	All persons with strong throwing arms are good passers. A(ba)
Minor	Some NFL quarterbacks have strong throwing arms. I(cb)
Conclusion	\ Some NFL quarterbacks are good passers. I(ca)

Complete Syllogism:     A(ba) I(cb) < I(ca). Valid: AII-1, Darii.

Suppressed Minor Premise

No one in his right mind claims infallibility, for only perfect persons can claim infallibility.

Major	All persons claiming infallibility are perfect persons. A(ab)
Missing Minor	No person in his right mind claims to be a perfect person. E(cb)
Conclusion	\ No person in his right mind claims infallibility. E(ca)

Complete Syllogism:     A(ab) E(cb) < E(ca). Valid: AEE-2, Camestres.

Suppressed Conclusion

No fair-minded person is capricious and some capricious people are irresponsible.

Major	No fair-minded person is capricious. E(ab)
Minor	Some capricious people are irresponsible. I(bc)
Missing Conclusion	\ Some irresponsible people are not fair-minded. O(ca)

 Complete Syllogism:    E(ab) I(bc) < O(ca). Valid: EIO-4, Fresison.

Sorites

Nonstandard categorical syllogisms may contain more than the required three forms. A sorites consists of a series of propositions in which the predicate of each is the subject of the next. The conclusion consists of the first subject and the last predicate. The chain of propositions is arranged in pairs of premises to make explicit the suppressed conclusion, thereby revealing the syllogism. The validity of the entire chain will depend on the validity of each syllogism in the chain. In this example, a = atheists; t = theologians; n = nihilists; s = scholars; and u = unreasonable (people). What can be concluded, given the following four propositions?

i	All atheists are nihilists.	A(an)
ii	All nihilists are misologists.	A(nm)
iii	All misologists are unreasonable.	A(mu)
iv	All unreasonable ones are fools.	A(uf)

                  
One interpretation takes "nihilists" in the first two propositions as the middle term and rearranging the premises yields the first syllogism.

Major	(ii)	All nihilists are misologists.	A(nm)
Minor	(i)	All atheists are nihilists.	A(an)
1st Conclusion		\ All atheists are misologists.	A(am) (made explicit)

 Using the 1st Conclusion as a premise in conjunction with the third proposition and rearranging the premises yields the second syllogism.

Major	(iii)	All misologists are unreasonable.	A(mu)
1st Conclusion (Minor)		All atheists are misologists.	A(am)
2nd Conclusion		\ All atheists are unreasonable.	A(au) (made explicit)

 Using the 2nd Conclusion as a premise in conjunction with the fourth proposition and rearranging the premises yields the third syllogism.

Major	(iv)	All unreasonable ones are fools.	A(uf)
2nd Conclusion (Minor)		All atheists are unreasonable.	A(au)
3rd Conclusion		\ All atheists are fools.	A(af) (made explicit)

As stated earlier, for a sorites to be valid each syllogism forming a part of the sorites must be valid; otherwise the sorites is invalid. Each syllogism above is an instance of AAA-1, Barbara. Therefore, the sorites as a whole is valid.
In evaluating a sorites, keep in mind these requirements:

1	If a conclusion is negative, then one and only one of the premises must be negative.
2	If the conclusion is affirmative, all of the propositions must be affirmative.
3	If the conclusion is universal, all of the premises must be universal.
4	A particular conclusion calls for not more than one particular premise.

Premise and Conclusion Indicator Words

You may have noticed that some of the arguments in this Study included such phrases as "because," "for," "so," etc. These words are known as indicator words or phrases. They introduce or otherwise indicate the presence of a premise or premises and a conclusion. Thus, the two lists of indicator words that follow.

Premise Indicators	Conclusion Indicators
... and ...	so
... but ...	thus
since ...	hence
because ...	therefore
however ..	consequently
assuming that ...	accordingly
inasmuch as ...	it follows that
nevertheless ...	which implies that
this is why ...	which means that
implied by ...	one can conclude that

                                             

Summation

Mediated inferences, that is, syllogisms, their elements, the arrangement of their forms and terms to determine their moods and figures have all been the subject matter of this Study. Next, the Five Rules for evaluating syllogistic frames as either valid or invalid were described. The Method of Deduction proved seven of the twenty four valid frames, using two axioms and two deductive method rules. Of more historical than practical interest are the names of the valid frames. The significance of lower case letters in some of the frame names was described. The RAA proof was illustrated in detail. Indicator words provide means for identifying premises and conclusions in arguments. Last, nonstandard syllogisms were described and methods for evaluating them were introduced. Of these, perhaps the most important is the enthymeme, since much of contemporary argumentation consists of enthymematic reasoning.
Of course, there is more. The use of diagrams for showing the validity of syllogisms is left for advanced study. Other aspects of syllogistic reasoning have been reserved for the last two studies.

Review

All of the syllogisms below are invalid. Each invalid argument illustrates the violation of one of the five rules for determining the validity of a syllogism. What rule is violated in each case? Does each example violate one and only one rule? Understanding the particular rule violated should suggest corrective strategies to convert an invalid syllogism into a valid one.

1 All hedonists are irrational. All irrationalists are misologists. \ Some misologists are not hedonists. ANSWER:______________	2 All men are intelligent. All men are bipeds. \ All bipeds are intelligent ANSWER:______________
3 Some fruit is not sweet. All pears are sweet. \ Some pears are fruit.ANSWER:______________	4 No dictators are benevolent. Some kings are not dictators. \ Some kings are not benevolent. ANSWER:______________
5 All men have two legs. All apes have two legs. \ All apes are men.ANSWER:______________