The Star Test

Why does the star test work?

Preliminaries
Syllogistic logic studies arguments whose validity depends on "all," "no," "some," and similar notions. In symbolizing such arguments, we use capital letters for general categories (like "logician") and small letters for specific individuals (like "Gensler"). We also use these five words: "all," "no," "some," "is," and "not." These vocabulary items can combine to form "wffs," or grammatical sequences. A wff (well-formed formula) is a sequence having any of these eight forms (where other capital letters and other small letters may be used instead):

all A is B
some A is not B

some A is B
no A is B

x is A
x is not A

x is y
x is not y

For later use, note that the two forms in each little box above are contradictories, which means that they must have opposite truth values.

A syllogism is a series of one or more wffs in which each letter occurs twice and the letters "form a chain" (each wff has at least one letter in common with the wff just below it, if there is one, and the first wff has at least one letter in common with the last wff). The last wff in a syllogism is the conclusion. The other wffs (if any) are premises. Here are two examples of syllogisms:

      no P is B
      some C is B
  ∴ some C is not P

      a is F
      a is G
  ∴ some F is G

An instance of a letter is distributed in a wff if it occurs just after "all" or anywhere after "no" or "not." The distributed letters here are underlined:

all A is B
some A is not B

some A is B
no A is B

x is A
x is not A

x is y
x is not y

The star test for syllogisms has two steps:

Star premise letters that are distributed and conclusion letters that aren't distributed.
Then the syllogism is VALID if and only if every capital letter is starred exactly once and there is exactly one star on the right-hand side.

One strategy is first to underline all the distributed letters – and then to star premise letters that are underlined and conclusion letters that aren't underlined. Here are three examples:

no P* is B* some C is B ∴ some C* is not P	Valid – every capital starred once and one right-hand star
no P* is B* some C is not B* ∴ some C* is P*	Invalid – P and B starred twice and three right-hand stars
a is F a is G ∴ some F* is G*	Valid – since small letters can be starred any number of times

The star test gives a quick and easy gimmick to test the validity of a syllogism. But why does the star test work? Why does it give the correct answer about the validity and invalidity of a syllogism? To answer this, we first need to learn about antilogisms.

Antilogisms
An antilogism is a series of one or more wffs in which each letter occurs twice and the letters "form a chain" (as explained before). Every syllogism has a corresponding antilogism; their statements are the same except that the last statements are contradictories of each other. Here's an example:

    SYLLOGISM
       all M is P
       all S is M
   ∴ all S is P

ANTILOGISM

all M is P
all S is M
some S is not P

Now to call an argument valid means that it would be impossible (inconsistent) to have its premises all true and its conclusion false. Thus a syllogism is valid if and only if its antilogism is inconsistent.

Mirroring the star test for syllogisms is the antilogism test:

Underline distributed letters in the antilogism.
Then the antilogism is INCONSISTENT if and only if every capital letter is underlined exactly once and there is exactly one underlined letter on the right-hand side.

This is how we'd work out an example, using the star test for syllogisms and the antilogism test:

    SYLLOGISM
(valid)
       all M* is P
       all S* is M
   ∴ all S is P*

ANTILOGISM
(inconsistent)

all M is P
all S is M
some S is not P

Note how the two tests mirror each other:

The syllogism is VALID on the star test, because every capital letter is starred exactly once and there is exactly one star on the right-hand side.
The antilogism is INCONSISTENT on the antilogism test, because every capital letter is underlined exactly once and there is exactly one underlined letter on the right-hand side.

The antilogism test is simpler. The antilogism test treats all statements the same – while the star test has to star the conclusion differently, in order to give the same result. The star test works because the antilogism test works. And we can show that the star test works by showing that the antilogism test works.

Showing that the antilogism test works
Let's suppose that we have an antilogism in which we've underlined just the distributed letters. Then, according to the test, the antilogism will be INCONSISTENT if and only if two conditions hold:

every capital letter is underlined exactly once, and
there is exactly one underlined letter on the right-hand side.

Note that if the second condition holds then the antilogism will have exactly one negative statement (since just the negative statements have an underlined letter on the right-hand side).

We'll consider three groups of antilogisms: (a) those with only capital letters, (b) those with only small letters, and (c) those with both small and capital letters. We'll try to show, for each group, that antilogisms of that group are inconsistent if and only if they satisfy the antilogism test's two conditions. If we show this, then, since every antilogism belongs to one of the groups, we'll have shown that any antilogism is inconsistent if and only if it satisfies these two conditions.

Antilogisms with only capital letters
Now we'll try to show that the antilogism test works for antilogisms with only capital letters. To help us show this, we'll develop a proof procedure that is adequate to show the inconsistency of any such antilogism that in fact is inconsistent.

Our proof procedure first takes each statement in the antilogism that starts with "some" and replaces it with a pair of statements. For each statement of the form "some A is B," we substitute a pair of statements of the form "x is A" and "x is B" – where we use a small letter for "x" that has not yet occurred in the antilogism. And for each statement of the form "some A is not B," we substitute a pair of statements of the form "x is A" and "x is not B" – where we use a small letter for "x" that has not yet occurred in the antilogism. Thus we would go from the antilogism in the middle box to the one on the right:

    SYLLOGISM
(valid)
       all M* is P
       all S* is M
   ∴ all S is P*

ANTILOGISM
(original)

all M is P
all S is M
some S is not P

ANTILOGISM
(modified)

all M is P
all S is M
a is S
a is not P

(Note that this modification doesn't change which capital letters are underlined.) Now the original antilogism will be consistent if and only if the thus-modified antilogism is consistent.

Our proof procedure then derives further statements from the thus-modified antilogism using these four inference rules (which hold for any small letter "x" and any capitals "F" and "G"):

all F is G, x is F ⇒ x is G

all F is G, x is not G ⇒ x is not F

no F is G, x is F ⇒ x is not G

no F is G, x is G ⇒ x is not F

These four rules give the only valid inference forms that argue from a combination of a universal statement and a statement that begins with a small letter. Using these rules:

At least one premise has to be positive.
If both premises are positive, then so is the conclusion.
If at least one premise is negative, then so is the conclusion.
The capital letter common to both premises has to be underlined exactly once.
The capital letter in the conclusion inherets its underlining-state from the premises.

If we take the modified antilogism above, we can use these four rules to derive a contradiction:

    SYLLOGISM
(valid)
       all M* is P
       all S* is M
   ∴ all S is P*

ANTILOGISM
(original)

all M is P
all S is M
some S is not P

ANTILOGISM
(modified)

all M is P
all S is M
a is S
a is not P

Further inferences:

∴ a is M {from lines 2 and 3}
∴ a is P {from "a is M" and line 1}
– and this contradicts line 4

Here's an example using two statements:

    SYLLOGISM
(valid)
      no A* is B*
  ∴ no B is A

ANTILOGISM
(inconsistent)

no A is B
some B is A

    ANTILOGISM
(modified)

    no A is B
    x is B
    x is A

Further inferences:

∴ x is not A {from lines 1 and 2}
– and this contradicts line 3

Here's an example using only one statement:

SYLLOGISM
(valid)
∴ all A is A*

ANTILOGISM
(inconsistent)

some A is not A

ANTILOGISM
(modified)

x is A
x is not A

Further inferences:

not needed, since lines
1 and 2 contradict

And here's an example using four statements:

    SYLLOGISM
(valid)
      some A is B
      all B* is C
      no C* is D*
  ∴ some A* is not D

ANTILOGISM
(inconsistent)

    some A is B
    all B is C
    no C is D
    all A is D

    ANTILOGISM
(modified)

    x is A
    x is B
    all B is C
    no C is D
    all A is D

Further inferences:

    ∴ x is C {from 2 and 3}
    ∴ x is not D {from "x is C" and 4}
    ∴ x is not A {from "x is not D" and 5}
– and this contradicts line 1

An antilogism with only capital letters is inconsistent, and can be shown to be inconsistent by our proof strategy, if and only if these three conditions hold of the antilogism:

Exactly one statement must start with "some."
There is exactly one negative statement.
Each capital letter is distributed exactly once.

These conditions mirror the antilogism test. We've seen that (2) is equivalent to "there is exactly one underlined letter on the right-hand side" (since just the negative statements have an underlined letter on the right-hand side). And (3) is equivalent to "every capital letter is underlined exactly once." But what about (1) – the condition that exactly one statement must start with "some"?

This condition is guaranteed by the other conditions. Suppose that every capital is underlined exactly once and there is exactly one underlined letter on the right-hand side. Then there has to be exactly one statement with a left-hand capital letter that isn't underlined (since otherwise some letter would be underlined twice). So there has to be exactly one statement that starts with "some" (since just the statements that start with "some" have a left-hand letter that isn't underlined).

Let ANT be an antilogism with only capital letters. ANT is inconsistent if and only if it satisfies the three conditions just noted. But ANT will be called "inconsistent" on the antilogism test if and only if it satisfies these three conditions. Therefore, ANT is inconsistent if and only if it will be called "inconsistent" on the antilogism test. So the antilogism test works for antilogisms with only capital letters.

Antilogisms with only small letters
Now we'll try to show that the antilogism test works for antilogisms with only small letters. To help us show this, we need to broaden our proof strategy. We need to add an inference rule about statements of the form "a is b":

Given "a is b," we can interchange
"a" and "b" in any statement.

So if we have "a is F" and also have "a is b," then we can derive "b is F." And if we have "c is not b" and also have "a is b," then we can derive "c is not a." As with our earlier four rules:

At least one premise has to be positive (the "a is b" premise).
If both premises are positive, then so is the conclusion.
If at least one premise is negative, then so is the conclusion.
Any capital letter in the conclusion inherets its underlining-state from the premises.

However, the small letter common to both premises needn't be underlined exactly once. (This is why our tests require of capital letters, but not of small letters, that they be starred or underlined exactly once.)

So far, deriving a contradiction has involved deriving a pair of statements of the form "x is A" and "x is not A." But now, deriving the self-contradiction "a is not a" will be a second way to derive a contradiction.

We can use these rules to derive a contradiction from this two-line antilogism with only small letters:

    SYLLOGISM
(valid)
      a is b
  ∴ b* is a*

    ANTILOGISM
(inconsistent)

    a is b
    b is not a

Further inferences:

∴ a is not a {from line 2 using line 1}
– and this is a self-contradiction

And here's how we work out a one-line antilogism with only small letters:

SYLLOGISM
(valid)
∴ a* is a*

ANTILOGISM
(inconsistent)

a is not a

Further inferences:

not needed, since the line
is a self-contradiction

An antilogism with only small letters is inconsistent, and can be shown to be inconsistent by our proof strategy, if and only if it has exactly one negative premise. And this mirrors the antilogism test's condition that "there is exactly one underlined letter on the right-hand side" (since just the negative statements have an underlined letter on the right-hand side). In antilogisms with only small letters, the antilogism test's other condition (about every capital letter being underlined exactly once) is automatically satisfied, since there are no capital letters.

Let ANT be an antilogism with only small letters. ANT is inconsistent if and only if it has exactly one negative statement. But ANT will be called "inconsistent" on the antilogism test if and only if it has exactly one negative statement. Therefore, ANT is inconsistent if and only if it will be called "inconsistent" on the antilogism test. So the antilogism test works for antilogisms with only small letters.

Antilogisms with both small and capital letters
Now we'll try to show that the antilogism test works for antilogisms with both small and capital letters. To help us show this, we use the proof procedure that we've developed in the two previous sections.

Here's how we can use this procedure on a longer antilogism with both small and capital letters:

    SYLLOGISM
(valid)
      x is A
      y is x
      y is B
      all B is C
  ∴ some A is C

ANTILOGISM
(inconsistent)

x is A
y is x
y is B
all B is C
no A is C

Further inferences:

    ∴ x is B {from line 3 using line 2}
    ∴ x is C {from "x is B" and 4}
    ∴ x is not A {from "x is C" and 5}
– and this contradicts line 1

An antilogism with both small and capital letters is inconsistent, and can be shown to be inconsistent by our proof strategy, if and only if these two conditions hold of the antilogism:

There is exactly one negative statement.
Each capital letter is distributed exactly once.

These conditions mirror the antilogism test. We've seen that (1) is equivalent to "there is exactly one underlined letter on the right-hand side" (since just the negative statements have an underlined letter on the right-hand side). And (2) is equivalent to "every capital letter is underlined exactly once."

An implication of these two conditions is that antilogisms with both small and capital letters, in order to be inconsistent and be capable of being shown to be inconsistent by our proof strategy, must not contain a "some" or a statement of the form "x is not y." I spare you the complex details.

Let ANT be an antilogism with both small and capital letters. ANT is inconsistent if and only if it satisfies the two conditions just noted. But ANT will be called "inconsistent" on the antilogism test if and only if it satisfies these two conditions. Therefore, ANT is inconsistent if and only if it will be called "inconsistent" on the antilogism test. So the antilogism test works for antilogisms with both small and capital letters.

Why the star test works
The last three sections have shown that the antilogism test works for all three groups of antilogisms: (a) those with only capital letters, (b) those with only small letters, and (c) those with both small and capital letters. So the antilogism test works for all antilogisms.

Given that the antilogism test works, we can argue that the star test works.

Let SYL be any syllogism and ANT the corresponding antilogism. SYL is valid if and only if ANT is inconsistent. ANT is inconsistent if and only if it will be called "inconsistent" on the antilogism test. But ANT will be called "inconsistent" on the antilogism test if and only if SYL will be called "valid" on the star test. Therefore, SYL is valid if and only if it will be called "valid" on the star test.

So the star test works.

And the star test works, again, because the antilogism test works and because the star test is just an indirect way to do the antilogism test.

Put more simply, the star test checks whether, if you replace the conclusion with its contradictory, you'd have every capital letter distributed exactly once and exactly one negative statement – which are precisely the conditions needed to derive a contradiction.

Further comments
The star test is my invention. I proposed it in a 1973 article in the Notre Dame Journal of Formal Logic (click here or here to download this article in Acrobat format). I argued that the star test, when applied to traditional syllogisms, gives the same results about validity as do the traditional medieval rules.

Christine Ladd-Franklin (1847-1930), an American logician and psychologist, proposed this brilliant and useful idea of antilogisms in 1883, as part of her doctoral dissertation. She was unfortunately delayed for 44 years in getting her Ph.D. from Johns Hopkins, even though she fulfilled all the requirements, because the school did not grant this degree to women. Bummer!

Teachers: If you want to teach proof techniques with syllogistic logic, you might have students prove syllogisms that are valid (as determined by the star test) using the antilogism technique explained here. And if you want to teach some metalogic about syllogisms, you might have students read this Web page and then go over the basic ideas with them.

E-mail me about errors or suggestions (to gensler@jcu.edu). Thanks!

by Harry J. Gensler, last modified on 21 December 2009