Syllogisms and Venn Diagrams

Contents
 Syllogisms and Venn Diagrams
  • syllogism is two-premise argument whose premises and conclusion have the forms:
    • Every A is a B
    • No A is a B
    • At least one A is a B.
    • At least one A is not a B.
  • Venn Diagrams are sets of overlapping circles used to determine the validity of syllogisms.
Example
  • The Venn Diagram to the right shows the following argument valid
    • Some elements have an atomic number greater than 92.
    • No element with an atomic number greater than 92 occurs naturally.
    • Therefore, some elements don’t occur naturally.
  • In the diagram E stands for elements, G for substances with atomic numbers greater than 92, and N for substances occurring naturally.
  • The first premise is represented by the tethered Xs.
  • The second premise is represented by the blacked-out common area of circles N and G
  • The conclusion is therefore true because of the X within circle E but outside circle N.
Procedure
  • Steps in determining validity by Venn Diagram:
    • Draw a blank Venn diagram
    • Represent the premises
    • See if the conclusion is true.
  • Consider the syllogism:
    • All fish have two-chambered hearts.
    • All bass are fish.
    • Therefore, all bass have two-chambered hearts.

Fish, bass, and creatures with two-chambered hearts are classes to which individuals can belong in eight possible combinations.  The combinations are represented by a 3-circle Venn diagram:

The F circle represents fish; the H circle creatures with 2-chambered hearts; and the B circle bass.  The overlapping circles form eight atomic regions collectively occupying the entire box and individually having no subdivisions.

Atomic regions represent groups of individuals that, for each class, all belong to it or all don’t.  For example, region 2 represents F’s that are neither H nor B, i.e. fish that are neither bass nor have two-chambered hearts.  Region 8 represents F’s that are H and B, i.e. fish that are bass and have two-chambered hearts.  Region 1 represents individuals that are neither F, nor H, nor B.  The regions can also be represented by a truth table:

The second step is to represent the premises.  All fish having two chambered-hearts means regions 2 and 5 are empty.  The first premise is represented by shading those regions, indicating nothing’s there:

The second premise, that all bass are fish, means nothing’s in regions 4 and 7.  It’s represented by those regions being shaded.  The premises are thus represented by:

Now the key question: does the diagram make the conclusion true?  The conclusion, that all bass have two-chambered hearts, is true if regions 4 and 5 are empty, which they are.  The conclusion is thus true and the argument valid.

So the procedure for determining the validity of a syllogism is:

  1. Draw and label the overlapping circles (or ellipses or any closed curves) representing the classes involved.
  2. Represent the premises.
  3. See if the resulting diagram makes the conclusion true: if so, the syllogism is valid; otherwise, invalid.

Another example:

  • Some elements have an atomic number greater than 92.
  • No element with an atomic number greater than 92 occurs naturally.
  • Therefore, some elements don’t occur naturally.

Using E for elements, G for substances with atomic numbers greater than 92, and N for substances occurring naturally, we start with:

The existence of elements with atomic number greater than 92 means there’s something in the area common to circles E and G.  Using X to signify the existence of at least one individual, the first premise is represented by:

X’s tethered across atomic regions means that something’s in the combined area though, for each region, it’s unknown whether something’s in it.  The premise asserts the existence of an element with atomic number larger than 92, but doesn’t indicate whether the element is naturally occurring.  The tethered X’s of the diagram thus mean something’s in the area common to E and G though the exact whereabouts are unknown.

The second premise, that no element with atomic number greater than 92 occurs naturally, means nothing’s in the area common to G and N and is represented by shading it. Superimposing the second premise’s representation on the first yields:

The question is whether the diagram makes the conclusion true.  The tethered X’s mean something’s in the area common to E and G.  Since the lower region is shaded, something must exist in the upper, meaning some elements don’t occur naturally.  The Venn diagram thus renders the conclusion true, validating the argument.

A final example:

  • Only natural born citizens are eligible to be President.
  • John Oliver isn’t a natural born citizen.
  • Therefore, he’s not eligible to be President.

The argument introduces a complication: John Oliver.  Unlike natural born citizens and people eligible to be President, John Oliver is an individual, not a class.  He’s thus not represented by a circle.  Rather, individuals are denoted by lower-case letters.  Using N for natural born citizens, P for people eligible to be President, and (shortly) s for John Oliver, the blank Venn Diagram is:

Now the first premise: only natural born citizens are eligible to be President.  We’ve dealt all B’s are F’s, no N’s are G’s, and some E’s are G’s.  But what does only N’s are P’s mean?  Consider a more concrete example: only women belong to the LPGA, the Ladies Professional Golf Association.  Only women belong to the LPGA means that anyone belonging to the LPGA is a woman.  Thus only women belong to the LPGA is the reverse of all women belong to the LPGA. The premise that only natural born citizens are eligible to be President is thus equivalent to the proposition that everyone eligible to be President is a natural born citizen:

For the second premise, that John Oliver isn’t a natural born citizen, the lower-case s denoting Oliver is placed outside the N circle, indicating he’s not natural born.  Since the rightmost portion of the P circle is shaded, Oliver’s little s can only go outside both N and P circles, its precise location irrelevant:

The resulting Venn diagram indicates that John Oliver isn’t eligible to be President, s lying outside the P circle.  The conclusion is thus true and the syllogism valid.

A Closer Look

Having seen them in action, let’s look at Venn diagrams in more detail.  As we’ve seen, there are three steps:

  1. Draw a Venn diagram for the classes used by the syllogism, using capital letters to label the closed curves (circles, ellipses, etc.) representing the classes.
  2. Represent the premises, superimposing representations.
  3. Determine whether the Venn diagram makes the conclusion true: if so, the syllogism is valid; otherwise, not.
A Closer Look at Step 1
Draw a Venn diagram for the classes used by the syllogism, using capital letters to label the closed curves (circles, ellipses, etc.) representing the classes

Syllogisms implicitly refer to classes.  Consider these (independent) statements:

  1. All men are mortal. Socrates is a man. Therefore Socrates is mortal.
  2. All and only conscious beings have souls.  Dogs don’t have souls. Therefore dogs aren’t conscious beings.
  3. Leptons and quarks have half-integer spin.  Photons don’t.  Therefore photons are neither leptons nor quarks.

The first refers to two classes (people and mortal beings), the second to three (conscious beings, dogs, and beings with souls), and the third to four (leptons, quarks, photons, and particles with half-integer spin).  2-class, 3-class, and 4-class diagrams look like this:

Atomic regions, numbered for reference, are areas with no subdivisions.  Their number depends on the number of classes, the number being 2 times itself for each class.  Thus, for a 2-class diagram there are 4 atomic regions (2 X 2); for a 3-class diagram 8 regions (2 X 2 X 2); for a 4-class diagram 16 regions (2 X 2 X 2 X 2).  Venn diagrams exist for more than 4 classes, but the number of atomic regions (32, 64, 128, …) takes the fun out of doing them.

It is important to distinguish individuals like Socrates from classes.  Classes are groups of individuals; individuals are members of classes

A Closer Look at Step 2
Represent the premises within the Venn diagram, superimposing representations. 

Propositions involving classes are represented by Venn diagrams, meaning a proposition and its representation have the same information content.  Thus, no A’s are B’s is represented by only the leftmost of the following diagrams.

The middle diagram comes up short, failing to depict that nothing’s in region 6.  The rightmost diagram goes too far, going beyond no A’s are B’s in portraying that nothing resides in regions 5 and 7.  A key point: though the rightmost diagram fails to represent no A’s are B’s, it still makes it true, regions 6 and 8 being shaded.  A Venn diagram can therefore make a proposition true without representing it.  But a diagram can’t represent a proposition without making it true.  The leftmost diagram represents no A’s are B’s as well as making it true.

Venn diagrams are built by superimposing representations of the premises.  Consider the argument-form:

  • No A’s are B’s.
  • No B’s are C’s.
  • So, no A’s are C’s.

The first premise is represented by:

The second by:

Placing one on top of the other yields:

The form-of-argument is invalid because the final diagram fails to make the conclusion true, that no A’s are C’s, since region 5 isn’t shaded.

Representations of Typical Propositions involving Classes
All A’s are B’s

Represented by shading the portion of A outside B.

No A’s are B’s

Represented by shading the area common to A and B

Only A’s are B’s

Represented by shading the portion of B outside A.

Only A’s are B’s means that B’s are restricted to A’s, that no B’s are non-A’s.  Thus only A’s are B’s is logically equivalent to all B’s are A’s.

All and only A’s are B’s

Represented by shading the portion of A outside B and the portion of B outside A.

All and only A’s are B’s means that A’s and B’s are the very same things.  The proposition implies that all A’s are B’s and all B’s are A’s, since the diagrams make both propositions true.

Some A’s are B’s

Represented by an X or tethered X’s in every atomic region of the area common to A and B.

The tethered X’s in regions 6 and 8 of the 3-class diagram indicate that something exists in at least one of those regions, though it’s not known which.  Compare the diagram to:

This diagram makes some A’s are B’s true, but it doesn’t represent it, since the X’s aren’t tethered.  Rather, it represents the stronger proposition that some A’s that are B’s are not C’s and some A’s that are B’s are C’s.

Not all A’s are B’s

Not all A’s are B’s is logically equivalent to some A’s are not B’s.  Thus, not everyone with a law degree practices law is equivalent to some with law degrees do not practice.

Individual s is an A

Represented by an s or tethered s’s appearing in every atomic region of A.

Thus s appears in regions 2, 5, 6 and 8 of the 3-class Venn diagram, indicating that s exists in one of those regions, though it’s not known which.

Individual s is an A or a B but not both

Represented by tethered s’s in the atomic regions in the portion of A outside B and the portion of B outside A.

Tethers apply only to atomic regions where their letters appear.  The tether of the 2-class diagram means s is in region 2 or 3.  The fact that it crosses region 4 means nothing, since s isn’t in the region.  Similarly, the tether of the 3-class diagram means s is in region 5, 2, 3 or 7, s occurring in each.  The tether spanning region 6 signifies nothing.

A Closer Look at Step 3
Determine whether the Venn diagram makes the conclusion true: if so, the syllogism is valid; otherwise, not.

A Venn diagram makes a proposition true if the proposition is certain given the informational content of the diagram.  Consider:

The following must be true, given the diagram’s content:

  • Individual m is an A but not a B or a C.
  • No A’s are B’s.
  • No B’s are C’s.
  • A’s exist.
  • m is not a B.
  • m is not a C.
  • No B is an A or C.

The following might be false:

  • No A’s are C’s.
  • Some A’s are C’s.
  • B’s exist.
  • B’s don’t exist.
  • C’s exist.
  • C’s don’t exist.

Each might be false because the Venn diagram can be extended, without erasing any part, so the resulting diagram makes it false.  For example, no A’s are C’s is made false by putting an X in region 5; some A’s are C’s by shading region 5; B’s exist by shading region 3.  A Venn diagram thus makes a proposition true if it can’t be consistently extended to make the proposition false.

Venn Diagrams can also be used to determine whether propositions are logically equivalent.  (Two propositions are logically equivalent if it’s self-contradictory for one to be true and the other false.) The procedure is to represent the propositions with separate Venn diagrams and compare; if the diagrams look the same, the propositions are equivalent, otherwise not.  Consider the propositions (a) all A’s are both B and C and (b) all A’s are B’s and all A’s are C’s.  The diagram for (a) is:

The diagrams for the two parts of (b) are:

Superimposing the diagrams yields the diagram for (a).  Propositions (a) and (b) are thus logically equivalent.  Other useful equivalences, verifiable by Venn diagrams, include:

Aristotle vs Venn Diagrams

Aristotle, who invented Logic, thought the following argument valid:

  • Everyone caught shoplifting at Macy’s last year was prosecuted.
  • Therefore, some people were caught shoplifting at Macy’s last year.

But this Venn diagram indicates the argument is invalid:

  • S: people caught shoplifting at Macy’s last year
  • P: people who were prosecuted

With no X in the S circle, the Venn diagram fails to make the conclusion true.  So who’s right: Aristotle or the Venn diagram?  The question is whether all A’s are B’s implies A’s exist.  If it does, A’s don’t exist should imply not all A’s are B’s.  But it doesn’t.  Indeed, we can prove that A’s don’t exist implies all A’s are B’s:

The following propositions are obvious:

  1. If not all A’s are B’s, some A’s are not B’s.
    • E.g. if not all people convicted of a felony committed the crime, some people convicted of a felony did not commit the crime.
  2. If some A’s are not B’s, A’s exist.
    • E.g. if some people convicted of a felony committed the crime, there are people convicted of a felony.

Proof that A’s don’t exist implies all A’s are B’s:

  • Assume that no A’s exist
  • It’s therefore false that some A’s are not B’s
    • By principle #2 and modus tollens
    • Modus tollens is the valid deductive argument form:
      • P → Q,
      • ~Q
      • Therefore ~P
  • It’s therefore false that not all A’s are B’s
    • By principle #1 and modus tollens
  • Therefore all A’s are Bs
    • Because all A’s are Bs = it’s false that not all A’s are B’s
  • Thus, no A’s exist implies all A’s are B’s.

The Venn diagram is therefore correct and Aristotle wrong.

In Sum
  • To determine whether a syllogism is valid:
    1. Draw a Venn diagram for the classes used by the syllogism, labeling with capital letters the closed curves (circles, ellipses, etc.) representing the classes.
    2. Represent the premises, superimposing representations.
    3. Determine whether the Venn diagram makes the conclusion true: if so, the syllogism is valid; otherwise, not.
  • Atomic regions are areas with no subdivisions.
  • For an n-class Venn diagram there are 2n atomic regions.
  • A shaded area means nothing exists in the represented subclass.
  • An X in an atomic region means something exists in the represented subclass.
  • Tethered X’s mean something exists in at least one of the subclasses represented by the atomic regions in which the X’s appear.
  • A lowercase letter in an atomic region means the denoted entity exists in the represented subclass.
  • Tethered lowercase letters mean the denoted entity exists in at least one of the subclasses represented by the atomic regions in which the letters appear.
  • A blank area means it’s unknown whether anything exists in the represented subclass.
  • A Venn diagram representing a proposition makes it true; but not every proposition made true by a Venn diagram is represented by it.
  • To determine whether propositions are logically equivalent:
    1. Draw separate Venn diagrams representing the propositions.
    2. See whether the diagrams look the same