Syllogism Part 2: Practical

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[If you have already read Part 1 of this article, there will be some repetition here, but presented from a different point of view and it will provide you with the opportunity to practice what you have learned.  If you are reading this part first, I recommend that you go on to read Part 1 later, for it will show you the conceptual basis that underlies this new method.  Neither part assumes any prior knowledge of the syllogism.]

 

Sometimes you see arguments like this:

“Sherman is pathetic!  Manipulative people are really pathetic, and Sherman is as manipulative as they come.”

This argument type is called the syllogism, and it has been around for over 2000 years going back at least to Aristotle.   It is an example of the simplest form of the syllogism, which even has a name, Barbara, and it is so obviously valid that it hardly needs a test, but let’s see what we would have to know if we were to test it:

Premise 1           All s is m                             s > m

Premise 2           All m is p                             m > p

Conclusion         So, all s is p                        s > p

The symbolic expressions on the right use the relationship symbols I created to simplify the process of determining whether syllogisms and polysyllogisms are validAn argument form is valid if true premises will always produce a true conclusion.

 

Let’s see what’s happening here.  We know that the conclusion is “Sherman is pathetic” because it is a statement and is followed by the exclamation point.  (Usually, we need to look for a conclusion marker such as “since” which I’ve supplied, below.  And I’ve also provided the letters to use when the argument is written out in symbolic form.)

(All) Sherman (s) is pathetic (p) since (All) Sherman (s) is a manipulative person (m), and (All) manipulative people (m) are pathetic (p).”

The terms of a syllogism are usually classes, like “manipulative people” but “Sherman” can be treated as if it is a class with a single member since this Sherman is uniquely the person we are talking about.  In the symbols, this is:

s  >  p ! s  >  m  >  p

We are using (s) and (p) because we are talking about the Subject and the Predicate of the conclusion, which are “Sherman” and “Pathetic.”  (I chose “Sherman” and “Pathetic,” just to simplify this example.)  The exclamation point (!) stands for the conclusion marker, “since.”  Now for the test.  Drop out the middle term (m), which happens to be “Manipulative,” and see if the symbols can be pushed together so that they nest hand-in-glove.

s  >  p ! s      >>     p

With just another little nudge these will nest perfectly to give us the same symbol we have in the conclusion; therefore, the argument is valid.  If you think about it for only a second, you will agree that the conclusion is inherent in the premises of a valid syllogism.  This test merely makes that obvious.

 

Some syllogisms can be very complicated, in part because they can be made up of four types of statements  (the symbols below are type-compatible forms for symbols that can be hand-written with a single line, with the exception of [><]) :

All  a is b                              a > b

No  a is b                             a >< b

Some a is b                         a <> b

Some a is not b                 a <>< b

 

Here is another simple example.

“Because all intelligent people are booklovers, we can say that some students are not intelligent, for some students are not book-lovers.”

The conclusion marker is “we can say that” for it precedes something that follows from the other statements, and the premise markers are “because” and “for,” words that introduce premises.  We can rewrite this:

“Some students (s) are not intelligent people (i), since some students (s) are not booklovers (b), and all intelligent people (i) are booklovers (b).”

I’ve provided the letters to use in the argument, so we can write this out with symbols.  Notice that the symbol for the last premise as shown below is read  “all (i) is (b)” from right to left.  More about this later.

s  <><  i  !  s  <><  b  <  i

Drop out the middle term and push together the relation symbols.

s  <><  i ! s      <><<      i

Again, it is clear that the with just a little nudge the relationship symbols in the premises can be pushed together to nest hand-in-glove to match the relationship symbol in the conclusion, so this argument is clearly valid and it may amuse you to know that it, too, has a name, Baroco.  All of the valid syllogistic forms have names.

 

Now, we need to look at the relationship symbols more closely.  The “all a is b” symbol [>] may be used in statements reading either right or left, but it must point in the direction you are reading.  Thus, we can have “all a is b”:   [a > b] or   [b < a].  The sign for “some a is not b” [<><]can also be written to read in either direction, but in this case the “open” side must be on the side in the direction you are reading, so “some a is not b” can be written [a <>< b] and “some b is not a” can be written [a ><> b].  The symbols for “no a is b” (><) and “some a is b” (<>) are symmetrical, and that means that they can be read from either side; in other words, they also mean “no b is a” and “some b is a.”

The symbols for “no a is b” (><) and  “some a is not b” (<><) are interesting in another way—both contain a negative, and these negatives can be detached to produce “all a are not-b” [a  >  <b] and “some a are not-b” [a  <> <b].  (“Not-b” could be “un-happy” or “in-appropriate,” etc.)  They can also be re-attached, so if we had “all a are un-b” [a  >  <b] we could rewrite it as “no a are b” [a ><  b] and vice versa.  And “some a are un-b” could be rewritten as “some a are not b.”

 

“Since no sarcastic people are mild mannered, and all pain-ridden people are ill-tempered, some sarcastic people are pain-ridden.”

At a glance, this may look as if it might be a good argument but that’s only because we might think that the premises and conclusions might both be true (sort of).  And we should always remember that an argument can be valid even if the premises and the conclusion are false.  Look at Barbara.  It is obviously valid, but we could use it to argue that all sports fans are baboons since all sports fans drink Gatorade and all Gatorade drinkers are baboons.  A valid argument form gives us a true conclusion only if the premises are true, and only then can it be called a good argumentvalid does not by itself, mean good!  Some of the examples given below will be valid, but will have false premises.  Let’s spell out the procedure we’ll follow in four steps, using the example above to illustrate them.

For Step One, we need to identify the conclusion and write it in a blank schema (framework).  “Since,” at the beginning of a statement, is a premise marker so the first statement is a premise—and the “and” before the second statement tells us that it, too, is a premise.  Thus, the conclusion is “some sarcastic people are pain-ridden.”  We can use “s” for “sarcastic (people)” and “p” for “pain-ridden (people).”  Since there is only one middle term [m] “mild mannered” we can put it directly into the schema.

Step One:            s <> p ! s           m            p

Then, we need to put in the other relation symbols:

Step Two:           s <> p ! s   ><   m>   <    p

We will assume that “ill-tempered” means “not mild mannered,“ so “all pain-ridden people are ill-tempered” is [p   >  <m], or [m>  <   p], as we have it here.  In step three we must attach any detached negatives, and this gives us [m  ><  p].  If we had two negation signs negating the same term and also had a relationship symbol we would have dropped both negation signs because a double negation is positive in English, not negative.  (For example, “he was not unkind” is a weakened way of saying “he was kind.”)  We can’t go on to step four until we have nothing but relationship symbols—no detached negation signs, no double-negatives–and [ m ><p ] has one relationship symbol.

Step Three:        s <> p ! s   ><    m   ><   p

This reads, “some s is p since no s is m and no m is p.”

Clearly these premises will not nest hand-in-glove to give us the conclusion, so we don’t need to write out step four, but we will do it to make another point.

Step Four:          s <> p ! s    >< ><    p

The syllogism is not valid, and in fact, no conclusion can be drawn from these premises at all:  the fact that “s” and “p” are both unrelated to “m” doesn’t tell us that they are related to each other.

 

Now it gets to be fun

 

Let’s look at some more examples and you will find that this will all get easier and easier.  These are taken from Lewis Carroll’s book Symbolic Logic.  As well as being the writer of Alice Wonderland, he was a mathematician and logician.  As you would expect, these are nonsensical, but then, we are not interested in the truth of the statements, but in the validity of the argument forms.

We will follow the same four steps: 1) identify the conclusion and write it into a blank schema (together with any necessary negation signs); 2) place the other terms and the necessary relation symbols into the schema (together with any necessary negation signs); 3) simplify by removing double-negatives and attaching negatives to relation signs as needed; and 4) test the syllogism by dropping the middle term and seeing if the symbols will nest, hand-in-glove.

 

“Every eagle can fly, and some pigs cannot fly, so some pigs are not eagles.”

The “conclusion marker,” “so,” identifies the conclusion so we can immediately write the conclusion, “some pigs are not eagles,” and create a blank schema into which the other relation symbols can be put.  (Since there is only the one middle term we can put it in now, and since “can fly” is not a class, we rewrote it as “fliers” [f].)

Step One:            p <>< e ! p            f             e

Now we put in the symbols for the premises.

Step Two:           p <>< e ! p  <><  f      <       e

Note that the symbol in the second premise must be read backwards, “all eagles are fliers.”  Now, if there were any double negatives we would remove them, but we have nothing but relation signs so we skip to Step Four.  We drop out the middle terms) and see if we can push the symbols together to nest together, hand-in-glove, to make the same symbol that appears in the conclusion

Step Four:          p <>< e ! p         <><<         e

With an extra nudge they would nest perfectly, so the syllogism is valid (which, again, means that this argument form will produce a true conclusion if the premises are true).  Did you recognize Baroco.  The Medieval scholars memorized the valid syllogisms ‘names so they could recognize whether a syllogism was valid.  It’s much easier just to use relationship symbols and nest them!

 

Let’s try another.

“No emperors are dentists, and all dentists are dreaded by children, so no emperors are dreaded by children.”

The conclusion marker, “so,” tells us which statement the conclusion is so we go to the first step.  (I’ve substituted the synonym, “feared (by children),” for “dreaded (by children)” to avoid confusing “dentists” and “dreaded.”)

Step One:            e  ><  f ! e            d                   f

Now we need to put in the symbols for the premises.

Step Two:           e  ><  f ! e    ><    d         >        f

Again, we can skip Step Three, and proceed to the last step, dropping the middle term(s).

Step Four:          e  ><   f! e          ><   >               f

The symbols will not nest hand-in-glove to produce the same symbol that is in the conclusion so this syllogism is invalid, which is to say that the form will not produce a true conclusion, even if the premises are true.

 

Now let’s move to something slightly different.  Polysyllogisms (also called sorites) are arguments that have the character of syllogisms but have more than two premises.

“Babies (b) are illogical (i); nobody is despised (d) who can manage a crocodile(c); and illogical (i) people are despised (d).”

I’ve used the initial (d) for “despised people” and (c) for “crocodile managers.”  Notice that no conclusion is given.  Carroll has chosen to make it more difficult (and informative) by forcing us find the conclusion ourselves, but that is easier than you might think.  After all, the subject and predicate of the conclusion only appear once in the premises, so any premise with a term that doesn’t appear in any other premise is either the first premise or the last.  (You know that in a chain of premises each term except the first and last must be a part of both the preceding and following statement.  Think of them being overlapped.)   For example, “illogical” appears in both “Babies are illogical” and “illogical people are despised.”  “Babies,” on the other hand, only appears once.  So, we go from this:

b > i          d >< c          i > d

to this (by overlapping repeated terms–after all, they have the same meaning):

b > i > d >< c

We can go on to a version of step-two now, since we know that the subject of the conclusion is “b” and the predicate is “c.”

Step Two:           b      c ! b    >    i   >   d  ><  c

There are no double negatives to discard so we can go on to find out what the conclusion is in step four.

Step Four:          b >< c ! b     > > ><     c

Give them a slight nudge and the symbols from the premises nest nicely, hand-in-glove.  Thus, we know the only possible conclusion from those premises must be “No babies are crocodile managers,” or as Carroll puts it, “Babies cannot manage crocodiles.”

 

This polysyllogism illustrates clearly what I meant when I said the conclusion is—in a sense—inherent in the premises.  This symbolic method of working with syllogisms can be seen as a method of revealing the conclusions that are already present within them.

 

Let’s try two more.  But I will give you the conclusions—which is the way you would see arguments in real life—after all, people who make them are generally trying to support a position they have already taken.   But they do often leave out a premise, and the resulting argument is called an enthymeme.   Suppose that our first syllogism was presented to you as an enthymeme:  “Sherman is pathetic—he’s as manipulative as they come.”  You would know that he was assuming that manipulative people are pathetic, the missing premise.  Here we go.

“No one takes the Times unless he is well-educated:  consequently we know that hedgehogs don’t take the Times.  After all, we know that no hedgehogs can read, and those beings that cannot read are not well-educated.”

The conclusion marker is “consequently” and “after all” is a premise marker.  Let’s proceed directly to step two.

Step Two:            h  ><  t ! h   ><   r>   ><  e >  ><   t

We can read this as:  no (hedgehogs) are (Times takers) since no (hedgehogs) are (readers), and no (not readers) are (well-educated beings), and no (Times takers) are (not well-educated beings).  Now we need to detach and eliminate the double negatives.

Step Three:        h >< t ! h    ><      r     <     e     <      t

Step Four:          h >< t ! h                 ><<<                  t

Give these a slight nudge and they nest hand-in-glove perfectly to match the conclusion, so the polysyllogism is valid.  You wouldn’t actually have needed to rewrite the symbols for step three or four—you can just draw a circle around the double negatives in step two.  Then you should be able to see that the symbols will nest by simple inspection of the chain of premises.

 

“Guinea pigs never really appreciate Beethoven.  After all, no one who is hopelessly ignorant of music ever keeps silent when the “Moonlight Sonata” (M.S.) is being played and we know that no one who really appreciates Beethoven fails to keep silent when the M.S. is playing, and guinea pigs are  hopelessly ignorant of music.”

Here, we have a premise marker in “after all,” and it generally follows a conclusion, so let’s go directly to step two.  (I said “generally” because “after all” might follow a premise that has a connection with another premise.  Read carefully.)

Step Two            g >< b ! g   >   i    ><    s>  ><   b

We could read this as: “(Guinea Pigs) are not (Beethoven appreciators) since (Guinea Pigs) are all (ignorant of music) and no one (ignorant of music) is (silent when the M. S. is playing), and no (Beethoven appreciators) are not (silent when the M. S. is playing).”  And we simplify this by discarding (or encircling) the double negative in the last statement, which is equivalent to, “All Beethoven appreciators keep silent.”  This brings us to step three.

Step Three         g >< g ! g   >   i   ><   s    <    b

You can tell that it is valid by inspection of the nest of symbols in step four, but it is also obvious in step three, and you would even have been able to tell this from step two if we had encircled the double negatives.

Step Four            g >< b ! g             > >< <             b

It is clearly valid.

 

Finally, here is one to remind you of how watchful you must be if you are to recognize the conclusions.  (This one is not Carroll’s.)

“What do you mean, ‘(Good laws) aren’t (easy to obey)’?  (Fair laws) are (rooted in social patterns,) and (good laws) are (fair), since (laws rooted in social patterns) are (easy to follow).”

Everything after the question mark is an argument intended to show that the other person is wrong.  What’s more, the other person was apparently contradicting the speaker’s conclusion, so what the speaker said must have been, “Good laws are easy to obey.”  When you recognize this can you go on to analyze the polysyllogism, which is actually extremely easy.

gl > eo  !  gl > fl > sp > eo

This is nothing more than the argument written out in symbols.  They provide a kind of short-hand that allows you to quickly and clearly write down the bones of an argument.  This is very useful in itself.  In fact, when it is written out this way you can tell that this argument is valid just by looking at it, and no simplification or test procedure is necessary at all.

 

This one is a sort of expanded Barbara, one made up of two syllogisms, and you can think of the first two premises as being the premises of the first of these syllogisms.  Let’s look at it that way.  I’ve used curly brackets to make this a little clearer.  Here is the whole polysyllogism.

gl > eo !  {gl > fl > sp}  > eo

The curly brackets enclose the premises of first syllogism that we need to solve. And its conclusion (which we inferred) is given square brackets, below.

[gl  > sp]  !  {gl > fl > sp}

Now we write the conclusion of this syllogism as the first premise of the second syllogism.

gl > eo  !  [gl > sp] > eo

 

That’s why these are called polysyllogisms.  They can be treated as a string of syllogisms, but it sometimes requires a lot of thought to decide just where to begin.  This one was elementary.  And of course, it isn’t necessary to do this at all.

 

Notes

Lewis Carroll’s books, Symbolic Logic and Game of Logic have been published together by Dover Publications, Inc.  They don’t cover what we now think of as symbolic logic at all, but he did use diagrams for exploring statements of various kinds and for working with syllogisms.  The book is rather difficult and/or tedious, at least from my point of view.  There is also a little genteel anti-Semitism, by which I mean the sort of bigoted remarks that some people make, people who would probably insist (wrongly) that they aren’t bigoted in the least.  Some might say in Carroll’s defense that this was all long ago, but though this may make it easier to understand how people can hold such sentiments, it is no excuse for them.