[This article has two parts. This part is historical and theoretical. It presents the metatheory underlying a new method of working with syllogisms. Part two is a short primer that shows you how to use this method without offering any theoretical background at all. If your interest is purely practical, you might want to look at Part two first. And if you begin with this part you, may still benefit from the different viewpoint and the exercises you will find in Part two. I have assumed that the reader of both parts has no background in syllogistic.]
Let’s take one last look at the syllogism before it descends into oblivion. We will see that even after two thousand years there are still some new wrinkles to be found. A categorical syllogism is an argument that contains three statements (two premises and a conclusion), and each statement deals with membership in categories. (We would be more likely to say “classes” now instead of “categories.”) Each statement has one of the following forms.
A All a are b
E No a are b
I Some a are b
O Some a are not b
A-type statements are called “universal affirmatives,” E-type statements are “universal negatives,” I-type statements are “particular affirmatives,” and O-type statements are “particular negatives.” The reasons for these names are rather obvious. Universals are sweeping “all” or “no” statements. Particulars are limited by the use of the word, “some” and negatives are limited by “no” or “not.” The A-I and E-O designations come from the first two vowels in the Latin words “affirmo“ (I agree) and “nego” (I deny.) In each case, the first vowel is used to designate the universal and the second to designate the particular.
The most common of the syllogistic arguments is “Barbara,” and this is the probably most famous example:
A All men are mortal
A Socrates is a man
A Therefore, Socrates is mortal
“Socrates is a man” is equivalent to saying that all of the Socrates’s we are concerned with are men and there is only one of them. (Purists call this a quasi-syllogism because “Socrates” is a single item, not a class. Nevertheless, it can function perfectly well as a syllogism.) The name, “Barbara,” is a mnemonic aid since it has three “a” vowels and the syllogism is made up of three A-type statements. The consonants in such names have meaning as well, but the system is too complicated to explain here and is actually passé. It was used by medieval philosophers who memorized the names of all the valid syllogistic patterns. The letter codes gave them hints on the means by which any given syllogism could be reduced to a simpler form, and the names identified the valid forms. Barbara is valid, which means that the conclusion is true if the premises are also true.
Barbara is traditionally symbolized as follows:
A All m are p
A All s are m
A Therefore all s are p
[s], [m], and [p] are called “terms”: here they represent “Socrates” [s]; “man” [m]; and “mortal (beings)” [p]. [s] and [p] are the subject and predicate of the conclusion—hence the symbols—and we need the middle term [m] to complete the premises. Since it is called the “middle” term, you might rightly guess that [s] and [p] are called “end terms.”
To see the other statement types in use, let’s look at Ferio which uses them all, as you can tell by its name:
E No m are p
I Some s are m
O Therefore some s are not p
Each of the patterns, such as the AAA that we see in Barbara and the EIO that we see in Ferio is called a “mood” of the syllogism. But there are also four “figures” of the syllogism to contend with. Barbara, Celarent, Darii, and Ferio are Figure 1 syllogisms, and have the middle term located in the positions in the premises that we have seen. It could just as well be in the subject position of the second premise and the predicate position of the first, or it could be in the subject position of both premises, or it could be in the predicate position of both. Each of these patterns constitutes a different figure. Here, for example, is the valid second figure argument, Festino. (Compare it with Ferio, above).
E No p are m
I Some s are m
O Therefore some s are not p
Proving the validity of a syllogism requires a set of rather arcane rules. One concept, “distribution,” is troublesome even for professional logicians. The other common approach is to draw Venn diagrams, but this is not always easy (especially when obversion—which you will learn about later—is involved) and it isn’t applicable to polysyllogisms (sorites), syllogisms of more than two premises, unless they are broken apart into separate sylllogisms. Euler diagrams are simple and almost self-evident when used for very simple syllogisms, but are of little help for more troublesome ones.
I’ve skimmed through this just to show you how complicated traditional syllogistic is. There is no unifying idea or metatheory to guide one in the traditional approach, and the terminology makes it difficult to make any headway . . . so, we will back away with our hands up and approach the syllogism through the notion of transitivity. This will seem complicated at times, but it is just background, and in the end we will throw it all away.
Transitive arguments, as I define them here, can be regarded as chains of elements, each element expressing a relationship between two terms and constituting a link of the chain. Familiar interpretations for such relationships are “is larger than,” “is the same as,” and “shares the attributes of.” Using the exclamation point [!] to represent the conclusion marker “since” and [R] to represent any relationship, we can represent the fundamental pattern of a transitive argument as:
(1) s R p ! s R m R p
(conclusion) ! (chain of premises)
This can be read in ordinary English as, “s bears relationship R to p, since s bears relationship R to m, and m bears relationship R to p.” In the “chain” of premises, both [s R m] and [m R p] are “links.” This is a familiar argument pattern in which [R] represents such relationships as “is taller than” or “is the same color as.”
The conclusion [s R p] is not a part of the chain but a statement of the end-relationship—the relationship between the end terms of the chain, [s] and [p] here. (This emphasizes the tautological character of such arguments.) If the conclusion follows from the premises, the chain is said to be “intact,” and a transitive relationship links each pair of terms. If transitivity does not exist at some point in the chain, the chain is said to be “broken,” and it isn’t possible to state an end-relationship (conclusion).
Of course, syllogisms aren’t quite as simple as this for we have to deal with negative relationships [R’] and the quantifiers, “all” [A] and “some” [S]. We can use this expanded notion of transitivity to represent Barbara as follows:
(2) s ARS p ! s ARS m ARS p
This can be read as, “all s bears relationship R to some p; since all s bears relationship R to some m; and all m bears relationship R to some p.” (The meaning of these statements is most easily demonstrated by an Euler diagram in which there are three concentric circles [s], [m], and [p] representing the argument. The one representing [s] is the innermost and the one representing [p] is the outermost. Thus, All of circle s bears relationship R to Some part of circle p since All of circle s bears relationship R to Some part of circle m and All of circle m bears relationship R to Some part of circle p. Notice how these circles nest together: the concept is important.) Here, we will take the meaning of [R] to be “shares the attributes of.” Thus, Barbara can be read as, “all s shares the attributes of some p since all s shares the attributes of some m, and all m shares the attributes of some p.” This is usually abbreviated as: “all s is p, since all s is m and all m is p.”
The Logic of Transitivity
Let’s turn our attention to the chain of premises and develop the rules governing this expanded notion of transitivity. There are three rules of quality:
S1: If a chain is intact and is constructed of positive links, the end-
relationship is positive.
S2: If a chain is intact and has one negative link, the end-relationship
S3: If the chain has two or more negative links, it is broken and no end-
relationship can be stated.
These rules seem to be obvious and can be taken to be axiomatic. Only the last needs any elaboration, but it simply states that we can’t know how two terms are related merely by knowing that they aren’t related to a third term.
We can also state three rules of quantity, but before we do we must define some terms. The complex of quantifiers and relation symbol such as [ARS] at the center of a link, is called a “node,” so each link consists of two terms and a node. Further, the juncture between two nodes (ignoring the intervening term) may be an [S–S] juncture like the juncture linking the nodes in [s ARS m SRA p] (I have underlined the relevant quantifiers to make this easier to see); an [A–A] juncture such as the one linking the nodes in [s SR’A m AR’S p]; or an [A–S] juncture such as the one linking the nodes in [s AR’A m SR’A p] or [s ARS m AR’A p]. The rules of quantity can be stated as follows:
Q1: If the chain is intact, the quantities associated with the end terms in the chain are those associated with them in the end-relation.
Q2: If there is an [S-S] juncture in the chain, the chain is broken.
Q3: If there is an [A-A] juncture in the chain, the chain is broken.
The first two of these can be taken as axiomatic. Q1 offers the only option we have for assigning quantities to the end terms (what other quantities could they take?). Q2 simply tells us that if there is an [S–S] juncture, we don’t know whether any of the members of the quantity represented by one Some are also members of the quantity represented by the other Some; thus, the chain is broken. Q3 brings up the existence assumption: if the term within the juncture does not exist, the chain is broken. (This assumption is based upon a bizarre parsing of “all” and “some.” It isn’t one of my favorite ideas, and we will eventually discard it.) I am including Q3 at this point to make the exposition of the system very conservative in terms of traditional syllogistic.) The application of these rules may be simplified by summarizing them into the following rule: General Rule of Transitivity: The chain is intact if rules S3, Q2, and Q3 do not apply; that is, if the chain is made up entirely of [A-S] junctures and has no more than one negative node. The “nesting” character of [A-S] junctures as represented by the Euler diagram discussed above is very important here.
We have four syllogistic relations to deal with, each characterized by a different node:
A All s is p s ARS p
E No s is p s AR’A p
I Some s is p s SRS p
O Some s is not p s SR’A p
Using this notation, we could write Ferio as follows:
(3) s SR’A p ! s SRS m AR’A p
This can be read as, “some s does not bear relationship R to any p, since some s bears relationship R to some m, and no m bears relationship R to any p.” (“Any” simply inventories “all” of the items in the class.) To say “no m is related to any p” is to say that m and p are exclusive. By the General Rule, the chain is intact since there is only one [A-S] juncture and one negative node. By Q1, the quantity associated with s is [S] and the quantity associated with p is [A]. And by S2, the relationship between the end terms must be negative. Thus, the conclusion is [s SR’A p] as shown.
These rules can apply to a chain of any length, allowing us to tell whether an end relationship can be stated and what it is. Consider the following polysyllogism (also called sorites):
(4) a SR’A f ! a SRA b SRS c ARS d AR’A e SRA f
Using the General Rule, we can tell by inspection that the chain is intact since the junctures are all [A-S] and there is only one negative node; moreover, because there is one negative node, we know that the relationship between the end terms must be negative, and since we know the quantities associated with the end terms, we can state the end relationship as [a SR’A f], as shown.
This is all much cleaner, theoretically, than the rules of traditional syllogistic, and it presents none of the difficulties presented by “distribution,” but the notation is extremely cumbersome. We can simplify it by using symbols to represent the nodes. There are eight possible forms to deal with:
— ARA : equality E AR’A >< cross
A ARS > right dart — AR’S ><> left quack
— SRA < left dart O SR’A <>< right quack
I SRS <> diamond — SR’S <><> double diamond
Table I. The eight possible nodes with their symbolic substitutes and names. (Note that these are typewriter-compatible forms. Each symbol should be drawn with a continuous line, with the exception of the equality and the cross.)
Each of these (with the exception of the equality) is an exact symbolic representation of the corresponding node. The indented sides of the symbols represent “all,” the pointed sides represent “some,” and an “ex” built into a symbol shows that it is negative. The pointed and indented sides preserve the “nesting” characteristic we observed earlier. The reversed symbols, the left dart and the left quack, cannot appear in conclusions, but are necessary in the premises where they are read from right to left; that is, [s < m] is read “all m are s,” and [s ><> m] is read, “some m are not s.” (To see this in use, look back at the first node of the chain in Example 4.) Furthermore, the design of the universal affirmative [>] and universal negative [><] signs has a certain mnemonic value, suggesting inclusion and negation.
The equality node [ARA] is trivial from the point of view of argument, but the double-diamond [SR’S] is a non-trivial relation that seems to have slipped through the cracks of traditional syllogistic. This may be because the development of syllogistic has been limited—always—by the ease with which certain kinds of statements are formed in ordinary language. It is easy to say “some s are not p,” and we have an immediate sense of its meaning, but “some s are not some p” is unfamiliar linguistically and as a result it is ignored. The doctrine of distribution is concocted to remedy this by giving an interpretation to certain statements, but it leads to its own problems. This issue, the quantification of the predicate, was the subject of a heated debate between the Hamiltonians and the anti-Hamiltonians in the 19th century.
Since we have preserved the “nesting” characteristic of [A-S] junctures in the shape of the symbols, we can use a new decision procedure. This is easily demonstrated by rewriting the problem diagrammed in Example 4 in the new symbols. Let’s imagine that it was posed in the following words:
all f are e, all c are d, and all b are a, so we can say that some a are not f, since some c are b and no e are d.
This looks horribly complicated. First, we must identify the conclusion in order to identify the terms of the end relationship. These terms are then written out to create a blank “schema” (shown below) consisting of the end relationship, the conclusion separator [!], and the terms at the beginning and end of the chain. (Here, the conclusion marker “so” in the argument helps us to identify the conclusion which is “some a are not f.”) :
(5) a <>< f ! a f
(If there are only three terms we could immediately place the middle term in its appropriate position.)
Then we fill out the schema by adding the links as needed to make the connections in the chain. The only other statement that uses “a” is “all b is a” so it must be the first link in the chain. This allows us to pick “some c are b” as the next link, and so it goes down the chain. (Note that there is only one way in which you can connect the links, so you could write out the only possible conclusion after creating the chain, even if it weren’t provided to you. This emphasizes the tautological character of syllogisms.)
(6) a <>< f ! a < b <> c > d >< e < f
We can read this as, “some a are not f, since all b are a, and some b are c, and all c are d, and no d are e, and all f are e.” The hand-in-glove “nesting test” is performed by dropping all of the terms except the end terms and pushing the symbols of the chain together to form the end relation (if the argument is valid).
(7) a <>< f ! a <<>> ><< f
With a slight nudge, these would come to nest exactly. This test works because nesting preserves the quantities adjacent to the end terms; it also preserves a negation as an “ex”; and all of the [A-S] junctures nest together. If there is more than one negation or any other violation of the rules, the result of the nesting test will not be one of the allowable symbols listed earlier. (You could have seen that this is valid by inspection of step 6 without dropping the terms!)
Barbara and the other Figure 1 forms are shown below, and you can see that their validity—as demonstrated by the nesting test—can be determined at sight. (Note that names of the arguments (such as Darii) are formed by reading the names of the symbols from right to left across the argument, from p to s.)
Barbara s > p ! s > m > p
Celarent s >< p ! s > m >< p
Darii s <> p ! s <> m > p
Ferio s <>< p ! s <> m >< p
Being symmetrical, the symbols used for E and I type statements can be read from either direction: thus, conversion is performed automatically in writing symbols into the schema. (Conversion is the transposition of the terms in a statement, subject for predicate, predicate for subject—something that is only possible for E and I type statements. In Ferio, for example, the first premise can be read “some s is m” or “some m is s,” and the second premise can be read as “no m is p” or “no p is m.”) As a consequence of this, seven of the fifteen syllogisms regarded as valid by traditional syllogistic take the forms shown above. This means that Festino and Ferio have the same form in these symbols and that is true of the other figures as well. This will be made very clear in Table II below.
In fact, the symbols are so revealing that it is possible to discover all of the valid argument forms just by dissecting the four basic symbols in a reversal of “nesting” while remembering that there are two possible statements for each occurrence of a symmetrical (convertible) symbol. This could not be said for any other decision procedure. Such a listing is shown in Table II below which begins with the fifteen forms generally agreed to be valid on the left, followed by Darapti, Fesapo, and Felapton, followed by the five weakened syllogisms, followed by Bramantip on the extreme right. The last nine are not regarded as valid by many logicians, and as you can see, their symbols are not created by “unnesting.” Darapti, Fesapo, and Felapton violate Q3 and the existence assumption. Weakened syllogisms draw a particular conclusion when a universal conclusion is warranted. I will leave Bramantip for you to figure out for yourself. We will come back to these nine syllogisms later.
Conclusion Possible Combinations of Symbols in the Premises
A [>] >> 1 — 0 — 0 — 0
I [<>] <<> <>> 4 < > 1 >> 1 << 1
E [><] ><< >>< 4 — 0 — 0 — 0
O [<><] <<>< <><< <> >< 6 < >< 2 ><< >>< 4 — 0
Totals 15 3 5 1
Table II. The twenty-four potentially valid syllogisms inferred from the symbols for the four possible conclusions. (In this table, the two convertible forms are underlined as a reminder that they can be read in each direction, which means that they can appear in different premises and must be counted twice, for example m <> p represents both “some m are p” and “some p are m” so it must be counted twice.)
Table II can be taken as a kind of proof of this symbol-system, a mapping of the arguments determined to be valid by this method onto the arguments known to be valid through other tests (in traditional syllogistic this was established by inspection of all 256 possible syllogisms). However, it is possible to go further and show that the rules of transitivity can be translated into the “rules” of syllogistic and vice versa. I will leave this to you but will lend a hand by pointing out that the indented side of any symbol is always adjacent to a distributed term and the pointed side is adjacent to a particular term: this is an immediate consequence of the meanings “all” and “some” which are represented by those aspects of the symbols. (A very concise set of syllogistic rules is: 1) The middle term must be distributed once and only once; 2) If an end term is distributed, it must be distributed in both its premise and the conclusion; and 3) If there are negative statements, both the conclusion and one premise (only) must be negative.)
An interesting and unusual proof is based upon the nesting decision procedure, itself. The right quack can be dissected into the left dart, diamond, right dart, and cross, as can be seen from Example 7. Further, all chains which nest to produce any of those symbols (the chains for all valid arguments) can be thought of as nesting to form part of the quack. Thus, it is possible to imagine dissecting portions of the quack in different ways to produce all possible chains for all valid arguments (requirement 1). If the dissection takes into account the requirement that all junctures be [A-S] junctures, no chains corresponding to invalid arguments can be produced (requirement 2). In a sense, Requirement 1 shows the completeness of the system, Requirement 2 shows its coherence, and the quack to which these apply is the point of reference which unifies the requirements.
The traditional names can be given to any of the valid forms by reading the letter designations (AEIO) for the symbols from right to left across the schema—although different figures of the same mood cannot be separated from each other, as is the case with Fesapo and Felapton in Table II, which are both “EAO.” (Look back at Ferio and Festino in example 3 to see how they read as “EIO.”) If it seems important to assign the traditional names to the syllogistic fallacies, this can be done as well, since the indented sides of the dart, cross, and quack are adjacent to distributed terms. For example, the “undistributed middle” is easily recognized as any argument in which the pointed sides of both relation symbols are adjacent to the middle term.
Simplification of Arguments
Up to this point, the use of this system is superior to either the traditional rules or Venn diagrams, since it is only necessary to fill in the schema with whichever of the six symbols (including the left dart and left quack) are appropriate, nest them, and compare the results with the conclusion. Obversion is more complicated since it introduces the step of simplifying statements, but this is where the method really shows its worth.
In traditional syllogistic, obversion is regarded as the process of negating the predicate term of a statement and changing the statement from positive to negative or vice versa. For positive statements this is accomplished by adding double negatives. Remember that negation is represented in the cross and the quack by an “ex.” The “ex” is created by an additional “dart” with the indented side adjacent to the negated term, and this dart is, itself, the negation sign. A cross is created from a right dart by adding a left-dart negation sign (and one could equally well say that the cross is created from a left dart by adding a right dart negation sign). Here lies the crux of the matter. To say that “no s is p” is to say that “all s is not-p,” and that is to say that “all p is not-s,” and that is to say that “no p is s.” This is one of the most remarkable things about this symbology, and its full consequences are shown by the table of immediate inferences, Table III. A great deal can be learned just by studying the permutations of form in this table.
A I E O
Statement a > b a <> b a >< b a <>< b
Converted — b <> a b >< a —
Obverted a >< <b a <>< <b a > <b a <> <b
Obverted Converse — b <>< <a b > <a —
Contrapositive b> >< a — — b> <> a
Obverted Contra. b> > <a — — b> <>< <a
Table III. Table of Immediate Inferences. These are inferences that can be made from a single statement. In other words, each column of the table is a list of statements that describe the same state of affairs. For example, the contrapositive, as you can see from the table, is a converted obverse.
There must always be one relation symbol in any statement. Any superfluous darts that have the indented side adjacent to a term are negation signs. If there is one dart adjacent to a diamond or a dart (pointed side to pointed side), the two symbols are to be joined to produce a quack or a cross. (This is not nesting which is only done in the final proof process, and which bridges across separate statements. This is simplification of a single statement.) When two such darts are adjacent, they are double negatives and are discarded, and this means that a negation sign may be detached from a quack or cross in order to do so, which is just obversion. If a group of symbols cannot be rationalized in these ways, it does not represent one of the basic statement types, and it cannot function in a syllogism.
Putting It to Use
Deciding whether a syllogism is valid, has four steps: 1) identify the conclusion and write its relation symbol (together with any additional negation signs) into a blank schema; 2) fill out the schema with the term(s), and the relation symbols and negation signs necessary to capture the sense of the English statements; 3) simplify the statements by removing double negatives (it is easiest to just circle them) and by attaching existing negatives to affirmative relation symbols where necessary; and 4) test the syllogism by dropping the middle term(s) and nesting the symbols in the chain. (As a practical matter, you can do all of this after ten minutes work spent learning the symbols and the procedures. No knowledge of nodes, quantifiers, links, junctures, obversion, or conversion is needed.) Let’s try it.
Since no masterful (thinkers) are un-philosophical, some scholarly (people) are philosophical, for we know that some masterful (thinkers) are not un-scholarly.
By a remarkable coincidence, the terms begin with the letters “s,” “m,” and “p” so we could rewrite this as: “Since no m are un-p, some s are p, for we know that some m are not un-s.” The conclusion is “some s are p.” The “un-“ prefixes are negation signs just as much as “no” and “not” are. (Note that we can always use the initials of the terms. It isn’t necessary that any of them be “s,” “m,” or “p.” We were just “lucky” here.)
1) s <> p ! s m p
2) s <> p ! s> ><> m >< <p
3) s <> p ! s <> m > p
4) s <> p ! s <>> p
In this example, we have the obverted converse of the first premise and the obverse of the second (see Table III), but this added complexity presents no problems and you don’t even have to use (or know) these terms. The symbols can be written directly into the schema as necessary to express the English meaning (step 2). In simplifying, the double negatives are discarded in both statements (step 3), and the remaining symbols can be nested to show that the syllogism is valid (step 4). (As mentioned before, the nesting test must not be done until obversion is completed; that is, it must not be done until all double-negatives are discarded.)
In practical reasoning, I would accept all 24 arguments shown in Table II. This only requires that we restrict our talk to cases in which the classes are not empty. The existence assumption is really of concern only to theorists preoccupied with the coherence of the system, and it is clear that weakened syllogism, are—in real life—perfectly good arguments.
In the medieval period, the task of remembering the various moods and figures of the syllogism were made easier by the following hexameter mnemonic:
Barbara Celarent Darii Feriosque prioris;
Cesare Camestres Festino Baroco secundi;
Tertia Darapti Disamis Datisi Felapton
Bocardo Ferison habet: quarta inusper addit
Bramantip Camenes Dimaris Fesapo Fresison.
We should, I suppose, be glad that the syllogism is no longer central to logic, and especially glad that we don’t have to use the complicated medieval system of dealing with such arguments. Nonetheless, there is a certain bizarre charm to the subject; after all, assigning personal names to argument types such a “Barbara” is a whimsical idea worthy of Lewis Carroll, himself. The importance of the categorical syllogism as the fundamental type of logical thought, however, has always been a gross exaggeration (though strongly adhered to), and while arguments can be (and have been!) structured rigidly around syllogistic patterns, they do not occur very often in everyday reasoning and Barbara occurs far more often than any other, with the possible exception of the Fallacy of the Excluded Middle.
In the late 70’s or early 80’s I was trying to contrive a symbolism that would make the conclusion implicit in the premises of syllogistic arguments obvious. I worked out this symbolism one summer. It didn’t take long since I had begun by thinking that the arguments would have to be seen as chains, and that the relationship markers would have to be written in forms that would somehow collapse or merge to produce the relationship marker seen in the conclusion. I began teaching it in my critical thinking classes the following year. The first time I did so, I made its use on the test optional and was delighted to find that students who used the symbols did much better than those who used Venn diagrams, despite the fact that the textbook gave detailed coverage to Venn diagrams while the symbols were introduced only on the blackboard. In fact, the symbols provide such a convenient shorthand for writing out arguments that students in subsequent classes almost invariably wrote out the arguments in symbols before drawing Venn diagrams, even when the Venn diagrams were required. I worked out the transitivity metatheory the following summer, and began to use it to provide a theoretical background for syllogistic. (Ironically, it was much harder to work out than the symbols had been.)
The only problem I have ever found is that students become so facile in using this approach that they perform the task of translating English statements into symbols and determining validity entirely by inspection of surface features; that is, they tend not to consider the meaning of the words at all. One early reviewer objected very intemperately to the fact that the dart and the negation symbol have the same form, but anyone who looks at the table of immediate inferences for E-type statements (Table 3) can see that this must be so, and it is rather elegant that it works out this way.
I’ve never published this (except in materials I handed out to my classes early on, and in a critical thinking booklet I wrote for my classes, later), but I have shared it with some of my colleagues at work and at various conferences.