2

Syllogistic Logic

Aristotle, the first logician (§16.1), invented syllogistic logic, which features arguments using “all,” “no,” and “some.” This logic, which we’ll take in a non-traditional way, provides a fine preliminary to modern logic (Chapters 6–14).

2.1 Easier translations

We’ll now create a “syllogistic language,” with rules for constructing arguments and testing validity. Here’s how an English argument goes into our language:

· All logicians are charming.

· Gensler is a logician.

· ∴ Gensler is charming.

· all L is C

· g is L

· ∴ g is C

Our language uses capital letters for general categories (like “logician”) and small letters for specific individuals (like “Gensler”). It uses five words: “all,” “no,” “some,” “is,” and “not.” Its grammatical sentences are called wffs, or well-formed formulas. Wffs are sequences having any of these eight forms, where other capital letters and other small letters may be used instead:¹

¹ Pronounce “wff” as “woof” (as in “wood”). We’ll take upper and lower case forms (like A and a) to be different letters, and letters with primes (like A′ and A″) to be additional letters.

all A is B	x is A
no A is B	x is not A
some A is B	x is y
some A is not B	x is not y

You must use one of these exact forms (but perhaps using other capitals for “A” and “B,” and other small letters for “x” and “y”). Here are examples of wffs (correct formulas) and non-wffs (misformed formulas):

· Wffs: “all L is C,” “no R is S,” “some C is D,” “g is C”

· Non-wffs: “only L is C,” “all R is not S,” “some c is d,” “G is C” 0007

Our wff rule has implications about whether to use small or capital letters:

Wffs beginning with a word (not a letter) use two capital letters:

Correct: “some C is D”

Incorrect: “some c is d”

Wffs beginning with a letter (not a word) begin with a small letter:

Correct: “g is C”

Incorrect: “G is C”

A wff beginning with a small letter could use a capital-or-small second letter (as in “a is B” or “a is b”). Which to use depends on the second term’s meaning:

Use capital letters for general terms, which describe or put in a category:

B = a cute baby

C = charming

F = drives a Ford

Use capitals for “a so and so,” adjectives, and verbs.

Use small letters for singular terms, which pick out a specific person or thing:

b = the world’s cutest baby

t = this child

d = David

Use small letters for “the so and so,” “this so and so,” and proper names.

· Will Gensler is a cute baby = w is B

· Will Gensler is the world’s cutest baby = w is b

An argument’s validity can depend on whether upper or lower case is used.

Be consistent when you translate English terms into logic; use the same letter for the same idea and different letters for different ideas. It matters little which letters you use; “a cute baby” could be “B” or “C” or any other capital. I suggest that you use letters that remind you of the English terms.

Syllogistic wffs all use “is.” English sentences with a different verb should be rephrased to make “is” the main verb, and then translated. So “All dogs bark” is “all D is B” (“All dogs is [are] barkers”); and “Al drove the car” is “a is D” (“Al is a person who drove the car”).

2.1a Exercise: LogiCola A (EM & ET)¹

¹ Exercise sections have a boxed sample problem that’s worked out. They also refer to LogiCola computer exercises (see Preface), which give a fun and effective way to master the material. Problems 1, 3, 5, 10, 15, and so on are worked out in the answer section at the back of the book.

Translate these English sentences into wffs.

John left the room.

j is L

1. This is a sentence.

2. This isn’t the first sentence.

3. No logical positivist believes in God.

4. The book on your desk is green. 0008

5. All dogs hate cats.

6. Kant is the greatest philosopher.

7. Ralph was born in Detroit.

8. Detroit is the birthplace of Ralph.

9. Alaska is a state.

10.Alaska is the biggest state.

11.Carol is my only sister.

12.Carol lives in Big Pine Key.

13.The idea of goodness is itself good.

14.All Michigan players are intelligent.

15.Michigan’s team is awesome.

16.Donna is Ralph’s wife.

2.2 The star test

Syllogisms, roughly, are arguments using syllogistic wffs. Here’s an English argument and its translation into a syllogism (the Cuyahoga is a Cleveland river that used to be so polluted that it caught on fire):

· No pure water is burnable.

· Some Cuyahoga River water is burnable.

· ∴ Some Cuyahoga River water isn’t pure water.

· no P is B

· some C is B

· ∴ some C is not P

More precisely, syllogisms are vertical sequences 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):

(If you imagine the two instances of each letter being joined, it’s like a chain.)

· no P is B

· some C is B

· ∴ some C is not P

The last wff is the conclusion; other wffs are premises. Here are three more syllogisms:

· a is C

· b is not C

· ∴ a is not b

· some G is F

· ∴ some F is G

· ∴ all A is A

The last example is a premise-less syllogism; it’s valid if and only if it’s impossible for the conclusion to be false.

Before doing the star test, we need to learn the technical term “distributed”:¹

¹ §16.2 mentions the meaning of “distributed” in medieval logic. Here I suggest that you take a distributed term to be one that occurs just after “all” or anywhere after “no” or “not.”

An instance of a letter is distributed in a wff if it occurs just after “all” or anywhere after “no” or “not.”

0009 The distributed letters below are underlined and bolded:

all A is B	x is A
no A is B	x is not A
some A is B	x is y
some A is not B	x is not y

By our definition:

· The first letter after “all” is distributed, but not the second.

· Both letters after “no” are distributed.

· Any letter after “not” is distributed.

Once you know which letters are distributed, you’re ready to learn the star test for validity. The star test is a gimmick, but a quick and effective one; for now, it’s best just to learn the test and not worry about why it works.

The star test for syllogisms goes as follows:

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.

As you learn the star test, use three steps: (1) underline distributed letters, (2) star, and (3) count the stars. Here are two examples:

· (1) Underline distributed letters (here only the first “A” is distributed):

o all A is B

o some C is A

o ∴ some C is B

· (2) Star premise letters that are underlined and conclusion letters that aren’t underlined:

o all A* is B Valid

o some C is A

o ∴ some C* is B*

· (3) Count the stars. Here every capital letter is starred exactly once and there is exactly one star on the right-hand side. So the first argument is VALID.

· (1) For our next argument, again underline distributed letters (here all the letters are distributed – since all occur after “no”):

o no A is B

o no C is A

o ∴ no C is B

· (2) Star premise letters that are underlined and conclusion letters that aren’t underlined:

o no A* is B* Invalid

o no C* is A*

o ∴ no C is B

· (3) Count the stars. Here capital “A” is starred twice and there are two stars on the right-hand side. So the second argument is INVALID.

A valid syllogism must satisfy two conditions: (a) each capital letter is starred in one and only one of its instances (small letters can be starred any number of times); and (b) one and only one right-hand letter (letter after “is” or “is not”) 0010 is starred. Here’s an example using only small letters:

· (1) Underline distributed letters (here just ones after “not” are distributed):

o a is not b

o ∴ b is not a

· (2) Star premise letters that are underlined and conclusion letters that aren’t underlined:

o a is not b* Valid

o ∴ b* is not a

· (3) Count the stars. Since there are no capitals, that part is automatically satisfied; small letters can be starred any number of times. There’s exactly one right-hand star. So the argument is VALID.

Here’s an example without premises:

· (1) Underline distributed letters:

o ∴ all A is A

· (2) Star conclusion letters that aren’t underlined:

o ∴ all A is A* Valid

· (3) Count the stars. Each capital is starred exactly once and there’s exactly one right-hand star. So the argument is VALID.

When you master this, you can skip the underlining and just star premise letters that are distributed and conclusion letters that aren’t. After practice, the star test takes about five seconds to do.¹

¹ The star test is my invention. For why it works, see http://www.harryhiker.com/star.htm or my “A simplified decision procedure for categorical syllogisms,” Notre Dame Journal of Formal Logic 14 (1973): pp. 457–66.

Logic takes “some” to mean “one or more” – and so takes this to be valid:²

² In English, “some” can also mean “two or more,” “several,” “one or more but not all,” “two or more but not all,” or “several but not all.” Only the one-or-more sense makes our argument valid.

· Gensler is a logician.

· Gensler is mean.

· ∴ Some logicians are mean.

· g is L Valid

· g is M

· ∴ some L* is M*

Similarly, logic takes this next argument to be invalid:

· Some logicians are mean.

· ∴ Some logicians are not mean.

· some L is M Invalid

· ∴ some L* is not M

If one or more logicians are mean, it needn’t be that one or more aren’t mean; maybe all logicians are mean.

2.2a Exercise – No LogiCola exercise

· Which of these are syllogisms?

· no P is B

· some C is B

· ∴ some C is not P

This is a syllogism. (Each formula is a wff, each letter occurs twice, and the letters form a chain.)

0011

1. all C is D

∴ some C is not E

2. g is not l

∴ l is not g

3. no Y is E

all G is Y

∴ no Y is E

4. ∴ all S is S

5. k is not L

all M is L

some N is M

Z is N

∴ k is not Z

2.2b Exercise: LogiCola BH

Underline the distributed letters in the following wffs.

some R is not S

some R is not S

1. w is not s

2. some C is B

3. no R is S

4. a is C

5. all P is B

6. r is not D

7. s is w

8. some C is not P

2.2c Exercise: LogiCola B (H and S)

Valid or invalid? Use the star test.

· no P is B

· some C is B

· ∴ some C is not P

· no P* is B* Valid

· some C is B

· ∴ some C* is not P

1. no P is B

some C is not B

∴ some C is P

2. x is W

x is not Y

∴ some W is not Y

3. no H is B

no H is D

∴ some B is not D

4. some J is not P

all J is F

∴ some F is not P

5. ∴ g is g

6. g is not s

∴ s is not g

7. all L is M

g is not L

∴ g is not M

8. some N is T

some C is not T

∴ some N is not C

9. all C is K

s is K

∴ s is C

10.all D is A

∴ all A is D

11.s is C

s is H

∴ some C is H

12.some C is H

∴ some C is not H

13.a is b

b is c

c is d

∴ a is d

14.no A is B

some B is C

some D is not C

all D is E

∴ some E is A

2.3 English arguments

Most arguments in this book are in English. Work them out in a dual manner. First use intuition. Read the argument and ask whether it seems valid; sometimes this will be clear, sometimes not. Then symbolize the argument and do a validity 0012 test. If your intuition and the validity test agree, then you have a stronger basis for your answer. If they disagree, then something went wrong; reconsider your intuition, your translation, or how you did the validity test. This dual attack trains your logical intuitions and double-checks your results.

When you translate into logic, use the same letter for the same idea and different letters for different ideas. The same idea may be phrased in different ways;¹ often it’s redundant or stilted to phrase an idea in the exact same way throughout an argument. If you have trouble remembering which letter translates which phrase, underline the phrase in the argument and write the letter above it; or write out separately which letter goes with which phrase.

¹ “Express the same idea” can be tricky to apply. Consider “All Fuji apples are nutritious” and “All nutritious apples have vitamins.” Use the same letter for both underlined phrases, since the first statement is equivalent to “All Fuji apples are nutritious apples.”

Translate singular terms into small letters, and general terms into capital letters (§2.1). Capitalization can make a difference to validity. This first example uses a capital “M” (for “a man” – which could describe several people) and is invalid:

· Al is a man.

· My father is a man.

· ∴ Al is my father.

· a is M Invalid

· f is M

· ∴ a* is f*

This second example uses a small “m” (for “the NY mayor” – which refers to a specific person) and is valid:

· Al is the NY mayor.

· My father is the NY mayor.

· ∴ Al is my father.

· a is m Valid

f is m

· ∴ a* is f*

We’ll more likely catch capitalization errors if we do the problems intuitively as well as mechanically.

2.3a Exercise: LogiCola BE

Valid or invalid? First appraise intuitively. Then translate into logic and use the star test to determine validity.

· No pure water is burnable.

· Some Cuyahoga River water is burnable.

· ∴ Some Cuyahoga River water isn’t pure water.

· no P* is B* Valid

· some C is B

· ∴ some C* is not P

1. All segregation laws degrade human personality.

All laws that degrade human personality are unjust.

∴ All segregation laws are unjust. [From Dr Martin Luther King.]

2. All Communists favor the poor.

All Democrats favor the poor.

∴ All Democrats are Communists. [This reasoning could persuade if expressed emotionally in a political speech. It’s less likely to persuade if put into a clear premise–conclusion form.] 0013

3. All too-much-time penalties are called before play starts.

No penalty called before play starts can be refused.

∴ No too-much-time penalty can be refused.

4. No one under 18 is permitted to vote.

No faculty member is under 18.

The philosophy chairperson is a faculty member.

∴ The philosophy chairperson is permitted to vote. [Applying laws, like ones about voting, requires logical reasoning. Lawyers and judges need to be logical.]

5. All acts that maximize good consequences are right.

Some punishing of the innocent maximizes good consequences.

∴ Some punishing of the innocent is right. [This argument and the next give a mini-debate on utilitarianism. Moral philosophy would try to evaluate the premises; logic just focuses on whether the conclusion follows.]

6. No punishing of the innocent is right.

Some punishing of the innocent maximizes good consequences.

∴ Some acts that maximize good consequences aren’t right.

7. All huevos revueltos are buenos para el desayuno.

All café con leche is bueno para el desayuno.

∴ All café con leche is huevos revueltos. [To test whether this argument is valid, you don’t have to understand its meaning; you only have to grasp the form. In doing formal logic, you don’t have to know what you’re talking about; you only have to know the logical form of what you’re talking about.]

8. The belief that there’s a God is unnecessary to explain our experience.

All beliefs unnecessary to explain our experience ought to be rejected.

∴ The belief that there’s a God ought to be rejected. [St Thomas Aquinas mentioned this argument in order to dispute the first premise.]

9. The belief in God gives practical life benefits (courage, peace, zeal, love, …).

All beliefs that give practical life benefits are pragmatically justifiable.

∴ The belief in God is pragmatically justifiable. [From William James.]

10.All sodium salt gives a yellow flame when put into the flame of a Bunsen burner.

This material gives a yellow flame when put into the flame of a Bunsen burner.

∴ This material is sodium salt.

11.All abortions kill innocent human life.

No killing of innocent human life is right.

∴ No abortions are right.

12.All acts that maximize good consequences are right.

All socially useful abortions maximize good consequences.

∴ All socially useful abortions are right.

13.That drink is transparent.

That drink is tasteless.

All vodka is tasteless.

∴ Some vodka is transparent. 0014

14.Judy isn’t the world’s best cook.

The world’s best cook lives in Detroit.

∴ Judy doesn’t live in Detroit.

15.All men are mortal.

My mother is a man.

∴ My mother is mortal.

16.All gender-neutral terms can be applied naturally to individual women.

The term “man” can’t be applied naturally to individual women. [We can’t naturally say “My mother is a man”; see the previous argument.]

∴ The term “man” isn’t a gender-neutral term. [From Janice Molton.]

17.Some moral questions are controversial.

No controversial question has a correct answer.

∴ Some moral questions don’t have a correct answer.

18.The idea of a perfect circle is a human concept.

The idea of a perfect circle doesn’t derive from sense experience.

All ideas gained in our earthly existence derive from sense experience.

∴ Some human concepts aren’t ideas gained in our earthly existence. [This reasoning led Plato to think that the soul gained ideas in a previous existence.]

19.All beings with a right to life are capable of desiring continued existence.

All beings capable of desiring continued existence have a concept of themselves as a continuing subject of experiences.

No human fetus has a concept of itself as a continuing subject of experiences.

∴ No human fetus has a right to life. [From Michael Tooley.]

20.The bankrobber wears size-twelve hiking boots.

You wear size-twelve hiking boots.

∴ You’re the bankrobber. [This is circumstantial evidence.]

21.All moral beliefs are products of culture.

No products of culture express objective truths.

∴ No moral beliefs express objective truths.

22.Some books are products of culture.

Some books express objective truths.

∴ Some products of culture express objective truths. [How can we make this valid?]

23.Dr Martin Luther King believed in objective moral truths (like “Racism is wrong”).

Dr Martin Luther King disagreed with the moral beliefs of his culture.

No people who disagree with the moral beliefs of their culture are absolutizing the moral beliefs of their own culture.

∴ Some who believed in objective moral truths aren’t absolutizing the moral beliefs of their own culture.

24.All claims that would still be true if no one believed them are objective truths. “Racism is wrong” would still be true if no one believed it.

“Racism is wrong” is a moral claim.

∴ Some moral claims are objective truths. 0015

25.Some shivering people with uncovered heads have warm heads.

All shivering people with uncovered heads lose much heat through their heads.

All who lose much heat through their heads ought to put on a hat to stay warm.

∴ Some people who have warm heads ought to put on a hat to stay warm.

2.3b Mystery story exercise – No LogiCola exercise

Herman had a party at his house. Alice, Bob, Carol, David, George, and others were there; one or more of these stole money from Herman’s bedroom. You have the data in the box, which may or may not give conclusive evidence about a given suspect:

1. Alice doesn’t love money.

2. Bob loves money.

3. Carol knew where the money was.

4. David works for Herman.

5. David isn’t the nastiest person at the party.

6. All who stole money love money.

7. All who stole money knew where the money was.

8. All who work for Herman hate Herman.

9. All who hate Herman stole money.

10.The nastiest person at the party stole money.

Did Alice steal money? If you can, prove your answer using a valid syllogism with premises from the box.

Alice didn’t steal money:

· a is not L* – #1

· all S* is L – #6

· ∴ a* is not S

1. Did Bob steal money? If you can, prove your answer using a valid syllogism with premises from the box.

2. Did Carol steal money? If you can, prove your answer using a valid syllogism with premises from the box.

3. Did David steal money? If you can, prove your answer using a valid syllogism with premises from the box.

4. Based on our data, did more than one person steal money? Can you prove this using syllogistic logic?

5. Suppose that, from our data, we could deduce both that a person stole money and that this same person didn’t steal money. What would that show?

2.4 Harder translations

Suppose we want to test this argument: 0016

· Every human is mortal.

· Only humans are philosophers.

· ∴ Every philosopher is mortal.

· all H is M

· all P is H

· ∴ all P is M

Here we need to translate “every” and “only” into our standard “all,” “no,” and “some.” “Every” just means “all.” “Only” is trickier; “Only humans are philosophers” really means “All philosophers are humans,” and so it symbolizes as “all P is H” (switching the letters).

This box lists some common ways to say “all”:

“all A is B” =

Every (each, any) A is B.

Whoever is A is B.

A’s are B’s.¹

¹ Logicians standardly take “A’s are B’s” to mean “all A is B” – even though in ordinary English it also could mean “most A is B” or “some A is B.”

Those who are A are B.

If a person is A, then he or she is B.

If you’re A, then you’re B.

Only B’s are A’s.

None but B’s are A’s.

No one is A unless he or she is B.

No one is A without being B.

A thing isn’t A unless it’s B.

It’s false that some A is not B.

“Only” and “none but” require switching the order of the letters:

· Only dogs are collies = All collies are dogs

· only D is C = all C is D

So “only” translates as “all,” but with the terms reversed; “none but” works the same way. “No … unless” is tricky too, because it really means “all”:

· Nothing is a collie unless it’s a dog = All collies are dogs

· nothing is C unless it’s D = all C is D

Don’t reverse the letters here; only reverse with “only” and “none but.”

This box lists some common ways to say “no A is B”:

“no A is B” =

A’s aren’t B’s.

Every (each, any) A is non-B.

Whoever is A isn’t B.

If a person is A, then he or she isn’t B.

If you’re A, then you aren’t B.

No one that’s A is B.

There isn’t a single A that’s B.

Not any A is B.

It’s false that there’s an A that’s B.

It’s false that some A is B.

Never use “all A is not B.” Besides not being a wff, this form is ambiguous. “All 0017 cookies are not fattening” could mean “No cookies are fattening” or “Some cookies are not fattening.”

These last two boxes give ways to say “some”:

some A is B =

A’s are sometimes B’s.

One or more A’s are B’s.

There are A’s that are B’s.

It’s false that no A is B.

some A is not B =

One or more A’s aren’t B’s.

There are A’s that aren’t B’s.

Not all A’s are B’s.

It’s false that all A is B.

Formulas “all A is B” and “some A is not B” are contradictories: saying that one is false is equivalent to saying that the other is true. Here’s an example:

Not all of the pills are white = Some of the pills aren’t white

Similarly, “some A is B” and “no A is B” are contradictories:

It’s false that some pills are black = No pills are black

Such idiomatic sentences can be difficult to untangle. Our rules cover most cases. If you find an example that our rules don’t cover, puzzle out the meaning yourself; try substituting concrete terms, like “pills” and “white,” as above.

2.4a Exercise: LogiCola A (HM & HT)

Translate these English sentences into wffs.

Nothing is worthwhile unless it’s difficult.

all W is D

1. Only free actions can justly be punished.

2. Not all actions are determined.

3. Socially useful actions are right.

4. None but Democrats favor the poor.

5. At least some of the shirts are on sale.

6. Not all of the shirts are on sale.

7. No one is happy unless they are rich.¹

8. Only rich people are happy.

9. Every rich person is happy.

10.Not any selfish people are happy. 0018

11.Whoever is happy is not selfish.

12.Altruistic people are happy.

13.All of the shirts (individually) cost $20.

14.All of the shirts (together) cost $20.

15.Blessed are the merciful.

16.I mean whatever I say.

17.I say whatever I mean.

18.Whoever hikes the Appalachian Trail (AT) loves nature.

19.No person hikes the AT unless he or she likes to walk.

20.Not everyone who hikes the AT is in great shape.

¹ How would you argue against 7 to 9? Would you go to the rich part of town and find a rich person who is miserable? Or would you go to the poor area and find a poor person who is happy?

2.5 Deriving conclusions

This next exercise gives you premises and has you derive a conclusion that follows validly. Do the problems in a dual manner: first try intuition, then use rules. Using intuition, read the premises slowly, say “therefore” to yourself, hold your breath, and hope that the conclusion comes. If you get a conclusion, write it down; then symbolize the argument and test for validity using the star test.

The rule approach uses four steps based on the star test:

1. Translate the premises, star, see if rules are broken.

2. Figure out the conclusion letters.

3. Figure out the conclusion form.

4. Add the conclusion, do the star test.

(1) Translate the premises into logic, star the distributed letters, and see if rules are broken. If you have two right-hand stars, or a capital letter that occurs twice without being starred exactly once, then no conclusion validly follows – so you can write “no conclusion” and stop.

(2) The conclusion letters are the two letters that occur just once in the premises. So if your premises are “x is A” and “x is B,” then “A” and “B” will occur in the conclusion.

(3) Figure out the form of the conclusion:

· If both conclusion letters are capitals: use an “all” or “no” conclusion if every premise starts with “all” or “no”; otherwise use a “some” conclusion.

· If at least one conclusion letter is small: the conclusion will have a small letter, “is” or “is not,” and then the other letter.

· Always derive a negative conclusion if any premise has “no” or “not.”

Here are examples using “all” and “no”:

· From premises “all” and “all,” derive an “all” conclusion.

· From “all” and “no,” derive “no.” (The order of the premises doesn’t matter; 0019 so from “no” and “all,” also derive “no.”)

Any “some” premise gives you a “some” conclusion:

· From “all” and “some” (positive), derive “some.”

· From “all” and “some is not,” derive “some is not.”

· From “no” and “some” (positive), derive “some is not.”

If the premises have a small letter but the conclusion has to have two capitals, derive “some”:

· From “x is A” and “x is B,” derive “some A is B.”

· From “x is A” and “x is not B,” derive “some A is not B.”

And if at least one conclusion letter has to be small, then the conclusion will have a small letter, “is” or “is not,” and then the other letter:

· From “a is b” and “b is c,” derive “a is c.”

· From “a is b” and “b is C,” derive “a is C.”

· From “a is C” and “b is not C,” derive “a is not b.”

Always derive a negative conclusion if any premise has “no” or “not.”

(4) Add the conclusion and do the star test; if it’s invalid, see if you can make it valid by reversing the letters in the conclusion (e.g., changing “all A is B” to “all B is A” – or “some A is not B” to “some B is not A” – the order matters with these two forms). Finally, put the conclusion back into English.

Suppose we want to derive a valid conclusion using all the English premises on the left. We first translate the premises into logic and star:

· Some cave dwellers use fire.

· All who use fire have intelligence.

· some C is F

· all F* is I

No rules are broken. “C” and “I” will occur in the conclusion. The conclusion form will be “some … is ….” We find that “some C is I” follows validly, and so we can conclude “Some cave dwellers have intelligence.” Equivalently, we could conclude “Some who have intelligence are cave dwellers.”

Or suppose we want to derive a valid conclusion using all of these next premises. Again, we first translate the premises into logic and star:

· No one held for murder is given bail.

· Smith isn’t held for murder.

· no M* is B*

· s is not M*

Here “M” is starred twice and there are two right-hand stars, and so rules are broken. So no conclusion follows. Do you intuitively want to conclude “Smith is given bail”? Maybe Smith is held for kidnapping and so is denied bail. 0020

Let’s take yet another example:

· Gensler is a logician.

· Gensler is mean.

· g is L

· g is M

No rules are broken. “L” and “M” will occur in the conclusion. The conclusion form will be “some … is ….” Since “some L is M” follows validly, and we can conclude “Some logicians are mean.” Equivalently, we could conclude “Some who are mean are logicians.”

2.5a Exercise: LogiCola BD

Derive a conclusion in English (not in wffs) that follows validly from and uses all the premises. Write “no conclusion” if no such conclusion validly follows.

· No pure water is burnable.

· Some Cuyahoga River water is not burnable.

· no P* is B*

· some C is not B*

· no conclusion

Do you want to conclude “Some Cuyahoga River water is pure water”? Maybe all of the river is polluted by something that doesn’t burn.

1. All human acts are determined (caused by prior events beyond our control).

2. Some human acts are free.

3. All acts where you do what you want are free.

4. All men are rational animals.

5. All philosophers love wisdom.

6. Luke was a gospel writer.

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7. All cheap waterproof raincoats block the escape of sweat.

8. All that is or could be experienced is thinkable.

All that is thinkable is expressible in judgments.

All that is expressible in judgments is expressible with subjects and predicates.

9. All moral judgments influence our actions and feelings.

10.No feelings that diminish when we understand their origins are rational.

11.I weigh 180 pounds.

12.No acts caused by hypnotic suggestion are free.

13.All unproved beliefs ought to be rejected.

14.All unproved beliefs ought to be rejected.

15.Jones likes raw steaks.

16.Some human beings seek self-destructive revenge.

No one seeking self-destructive revenge is motivated only by self-interest.

17.All virtues are praised.

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18.God is a perfect being.

All perfect beings are self-sufficient.

19.God is a perfect being.

All perfect beings know everything.

20.All basic moral norms hold for all possible rational beings as such.

21.All programs that discriminate simply because of race are wrong.

22.Some racial affirmative action programs are attempts to make amends for past injustices toward a given group.

23.Some actions approved by reformers are right.

24.Some wrong actions are errors made in good faith.

25.All moral judgments are beliefs whose correctness cannot be decided by reason.

Here 1–3 defend three classic views on free will: hard determinism, indeterminism, and soft determinism; 8 and 20 are from Immanuel Kant; 9 is from David Hume; 10 is from Richard Brandt; 17 and 18 are from Aristotle; and 19 is from Charles Hartshorne.

2.6 Venn diagrams

Having learned the star test, we’ll now learn a second test that’s more difficult but also more intuitive. Venn diagrams have you diagram the premises using three overlapping circles. We’ll apply Venn diagrams only to traditional syllogisms (two-premise syllogisms with no small letters). 0023

Here’s how to do the Venn-diagram test:

Draw three overlapping circles, labeling each with one of the syllogism’s letters. Then draw the premises as directed below. The syllogism is valid if and only if drawing the premises necessitates drawing the conclusion.

First, draw three overlapping circles:

Circle A contains all A things, circle B contains all B things, and circle C contains all C things.

The central area, where all three circles overlap, contains whatever has all three features (A, B, and C). Three middle areas contain whatever has only two features (for example, A and B but not C). Three outer areas contain whatever has only one feature (for example, A but not B or C). Each of the seven areas can be empty or non-empty. We shade areas known to be empty. We put an “×” in areas known to contain at least one entity. An area without either shading or an “×” is unspecified; it could be either empty or non-empty.

Draw the premises as follows:

“no A is B”

Shade wherever A and B overlap.

“No animals are beautiful” = “nothing in the animal circle is in the beautiful circle.”

“some A is B”

“×” an unshaded area where A and B overlap.

“Some animals are beautiful” = “something in the animal circle is in the beautiful circle.”

“all A is B”

Shade areas of A that aren’t in B.

“All animals are beautiful” = “everything in the animal circle is in the beautiful circle.”

“some A is not B”

“×” an unshaded area in A that isn’t in B.

“Some animals are not beautiful” = “something in the animal circle is outside the beautiful circle.”

0024 Shading means the area is empty while “×” means it contains something.

Follow these four steps (for now you can ignore the italicized complication):

1. Draw three overlapping circles, each labeled by one of the letters.

2. First draw “all” and “no” premises by shading.

3. Then draw “some” premises by putting an “×” in some unshaded area. (When “×” could go in either of two unshaded areas, the argument is invalid; to show this, put “×” in an area that doesn’t draw the conclusion. I suggest you first put “×” in both areas and then erase the “×” that draws the conclusion.)

4. If you must draw the conclusion, the argument is valid; otherwise, it’s invalid.

Here’s a valid example:

· all H is D Valid

· no F is D

· ∴ no H is F

We draw “all H is D” by shading areas of H that aren’t in D. And we draw “no F is D” by shading where F and D overlap. Here we’ve automatically drawn the conclusion “no H is F” (we’ve shaded where H and F overlap).

So the argument is valid.

Here’s an invalid example:

· no H is D Invalid

· no F is D

· ∴ no H is F

We draw “no H is D” by shading where H and D overlap. We draw “no F is D” by shading where F and D overlap. Here we haven’t automatically drawn the conclusion “no H is F” (we haven’t shaded all the areas where H and F overlap).

So the argument is invalid.

Here’s a valid argument using “some”:

· no D is F Valid

· some H is F

· ∴ some H is not D

We draw “no D is F” by shading where D and F overlap. We draw “some H is F” by putting “×” in some unshaded area where H and F overlap. But then we’ve automatically drawn the conclusion “some H is not D” – since we’ve put an “×” in some area of H that’s outside D.

So the argument is valid. (Recall that we draw “all” and “no” first, and then “some.”) 0025

I earlier warned about a complication that sometimes occurs: “When ‘×’ could go in either of two unshaded areas, the argument is invalid; to show this, put ‘×’ in an area that doesn’t draw the conclusion. I suggest you first put ‘×’ in both areas and then erase the ‘×’ that draws the conclusion.” Here’s an example:

· no D is F Invalid

· some H is not D

· ∴ some H is F

We draw “no D is F” by shading where D and F overlap. We draw “some H is not D” by putting “×” in both unshaded areas in H that are outside D (since either “×” would draw the premise).

We then erase the “×” that draws the conclusion “some H is F.” So then we’ve drawn the premises without drawing the conclusion. So it’s invalid.

Since it’s possible to draw the premises without drawing the conclusion, the argument is invalid. Since this case is tricky, you might reread the explanation a couple of times until it’s clear in your mind.

2.6a Exercise: LogiCola BC

· Test for validity using Venn diagrams.

· no P is B

· some C is B

· ∴ some C is not P

1. no B is C

all D is C

∴ no D is B

2. no Q is R

some Q is not S

∴ some S is R

3. all E is F

some G is not F

∴ some G is not E

4. all A is B

some C is B

∴ some C is A

5. all A is B

all B is C

∴ all A is C

6. all P is R

some Q is P

∴ some Q is R

7. all D is E

some D is not F

∴ some E is not F

8. all K is L

all M is L

∴ all K is M

9. no P is Q

all R is P

∴ no R is Q 0026

10.some V is W

some W is Z

∴ some V is Z

11.no G is H

some H is I

∴ some I is not G

12.all E is F

some G is not E

∴ some G is not F

2.7 Idiomatic arguments

Our arguments so far have been phrased in a clear premise–conclusion format. Real-life arguments are seldom so neat and clean. Instead we may find convoluted wording or extraneous material. Key premises may be omitted or only hinted at. And it may be hard to pick out the premises and conclusion. It often takes hard work to reconstruct a clearly stated argument from a passage.

Logicians like to put the conclusion (here italicized) last:

“Socrates is human. All humans are mortal. So Socrates is mortal.”

· s is H

· all H is M

· ∴ s is M

But people sometimes put the conclusion first, or in the middle:

“Socrates must be mortal. After all, he’s human and all humans are mortal.”

“Socrates is human. So he must be mortal – since all humans are mortal.”

Here “must” and “so” indicate the conclusion (which goes last when we translate into logic). Here are some words that help us pick out premises and conclusion:

These often indicate premises:

· Because, for, since, after all …

· I assume that, as we know …

· For these reasons …

These often indicate conclusions:

· Hence, thus, so, therefore …

· It must be, it can’t be …

· This proves (or shows) that …

When you don’t have this help, ask yourself what is argued from (these are the premises) and what is argued to (this is the conclusion).

In reconstructing an argument, first pick out the conclusion. Then symbolize the premises and conclusion; this may involve untangling idioms like “Only A’s are B’s” (which translates as “all B is A”). If some letters occur only once, you may have to add unstated but implicit premises; using the “principle of charity,” interpret unclear reasoning to give the best argument. Then test for validity.

Here’s a twisted argument – and how it goes into premises and a conclusion: 0027

“You aren’t allowed in here! After all, only members are allowed.”

· Only members are allowed in here.

· ∴ You aren’t allowed in here.

· all A is M

· ∴ u is not A

Since “M” and “u” occur only once, we need to add an implicit premise linking these to produce a syllogism. We add a plausible premise and test for validity:

· You aren’t a member. (implicit)

· Only members are allowed in here.

· ∴ You aren’t allowed in here.

· u is not M* Valid

· all A* is M

· ∴ u* is not A

2.7a Exercise: LogiCola B (F & I)

First appraise intuitively. Then pick out the conclusion, translate into logic (using correct wffs and syllogisms), and determine validity using the star test. Supply implicit premises where needed; when two letters occur only once but stand for different ideas, we often need an implicit premise that connects the two.

Whatever is good in itself ought to be desired. But whatever ought to be desired is capable of being desired. So only pleasure is good in itself, since only pleasure is capable of being desired.

· all G* is O Valid

· all O* is C

· all C* is P

· ∴ all G is P*

The conclusion is “Only pleasure is good in itself”: “all G is P.”

1. Racial segregation in schools generates severe feelings of inferiority among black students. Whatever generates such feelings treats students unfairly on the basis of race. Anything that treats students unfairly on the basis of race violates the 14th Amendment. Whatever violates the 14th Amendment is unconstitutional. Thus racial segregation in schools is unconstitutional. [This was the reasoning behind the 1954 Brown vs. Topeka Board of Education Supreme Court decision.]

2. You couldn’t have studied! The evidence for this is that you got an F– on the test.

3. God can’t condemn agnostics for non-belief. For God is all-good, anyone who is all-good respects intellectual honesty, and no one who does this condemns agnostics for non-belief.

4. Only what is under a person’s control is subject to praise or blame. Thus the consequences of an action aren’t subject to praise or blame, since not all the consequences of an action are under a person’s control.

5. No synthetic garment absorbs moisture. So no synthetic garment should be worn next to the skin while skiing.

6. Not all human concepts can be derived from sense experience. My reason for saying this is that the idea of “self-contradictory” is a human concept but isn’t derived from sense experience. 0028

7. Analyses of humans in purely physical-chemical terms are neutral about whether we have inner consciousness. So, contrary to Hobbes, we must conclude that no analysis of humans in purely physical-chemical terms fully explains our mental activities. Clearly, explanations that are neutral about whether we have inner consciousness don’t fully explain our mental activities.

8. Only what is based on sense experience is knowledge about the world. It follows that no mathematical knowledge is knowledge about the world.

9. Not all the transistors in your radio can be silicon. After all, every transistor that works well at high temperatures is silicon and yet not all the transistors in your radio work well at high temperatures.

10.Moral principles aren’t part of philosophy. This follows from these considerations: Only objective truths are part of philosophy. Nothing is an objective truth unless it’s experimentally testable. Finally, of course, moral principles aren’t experimentally testable. [From the logical positivist A. J. Ayer.]

11.At least some women are fathers. This follows from these facts: (1) Jones is a father, (2) Jones had a sex change to female, and (3) whoever had a sex change to female is (now) a woman.

12.Only language users employ generalizations. Not a single animal uses language. At least some animals reason. So not all reasoners employ generalizations. [From John Stuart Mill.]

13.Only pure studies in form have true artistic worth. This proves that a thing doesn’t have true artistic worth unless it’s abstract, for it’s false that there’s something that’s abstract but that isn’t a pure study in form.

14.Anything that relieves pressure on my blisters while I hike would allow me to finish my PCT (Pacific Crest Trail) hike from Mexico to Canada. Any insole with holes cut out for blisters would relieve pressure on my blisters while I hike. I conclude that any insole with holes cut out for blisters would allow me to finish my PCT hike from Mexico to Canada. [So I reasoned – and it worked.]

15.We know (from observing the earth’s shadow on the moon during a lunar eclipse) that the earth casts a curved shadow. But spheres cast curved shadows. These two facts prove that the earth is a sphere.

16.Whatever is known is true, and whatever is true corresponds to the facts. We may conclude that no belief about the future is known.

17.No adequate ethical theory is based on sense experience, because any adequate ethical theory provides necessary and universal principles, and nothing based on sense experience provides such principles. [From Immanuel Kant.]

18.At least some active people are hypothermia victims. Active people don’t shiver. It follows that not all hypothermia victims shiver. [From a ski magazine.]

19.Iron objects conduct electricity. We know this from what we learned last week – namely, that iron objects are metallic and that nothing conducts electricity unless it’s metallic.

20.Only things true by linguistic convention are necessary truths. This shows that “God exists” can’t be a necessary truth. After all, existence claims aren’t true by linguistic convention.

21.No bundle of perceptions eats food. Hume eats food, and Hume is a human person. From this it follows (contrary to David Hume’s theory) that no human person is a bundle of perceptions. 0029

22.Any events we could experience as empirically real (as opposed to dreams or hallucinations) could fit coherently into our experience. So an uncaused event couldn’t be experienced as empirically real. I assume that it’s false that some uncaused event could fit coherently into our experience. [From Immanuel Kant.]

23.I think I’m seeing a chair. But some people who think they’re seeing a chair are deceived by their senses. And surely people deceived by their senses don’t really know that they’re seeing an actual chair. So I don’t really know that I’m seeing an actual chair.

24.No material objects can exist unperceived. I say this for three reasons: (1) Material objects can be perceived. (2) Only sensations can be perceived. Finally, (3) no sensation can exist unperceived. [Bertrand Russell criticized this argument for an idealist metaphysics.]

25.Only those who can feel pleasure or pain deserve moral consideration. Not all plants can feel pleasure or pain. So not all plants deserve moral consideration.

26.True principles don’t have false consequences. There are plausible principles with false consequences. Hence not all true principles are plausible.

27.Only what divides into parts can die. Everything that’s material divides into parts. No human soul is material. This shows that no human soul can die.

2.8 The Aristotelian view

Historically, “Aristotelian” and “modern” logicians disagree about the validity of some syllogism forms. They disagree because of differing policies about allowing empty terms (general terms that don’t refer to any existing beings).

Compare these two arguments (unicorns don’t really exist, even though some myths speak of such one-horned horse-like animals):

· All cats are animals.

· ∴ Some animals are cats.

· All unicorns are animals.

· ∴ Some animals are unicorns.

The first seems valid while the second seems invalid. Yet both have the same form – one that tests out as “invalid” using our star test:

· all C* is A Invalid

· ∴ some A* is C*

When we read the first argument, we tend to presuppose that there’s at least one cat. Given this as an assumed additional premise, it follows validly that some animals are cats. When we read the second argument, we don’t assume that there’s at least one unicorn. Without this additional assumption, it doesn’t follow that some animals are unicorns.

So “all C is A ∴ some A is C” is valid if we assume as a further premise that there are C’s; it’s invalid if we don’t assume this. The Aristotelian view, which assumes that each general term in a syllogism refers to at least one existing 0030 being, calls the argument “valid.” The modern view, which allows empty terms like “unicorn” that don’t refer to existing beings, calls the argument “invalid.”

I prefer the modern view, since we often don’t presuppose that our general terms refer to existing entities. If your essay argues that angels don’t exist, your use of “angel” doesn’t presuppose that there are angels. If you tell your class “All with straight-100s may skip the final exam,” you don’t assume that anyone will get straight-100s. On the other hand, we sometimes can presuppose that our general terms all refer; then the Aristotelian test makes sense.

Suppose we have an argument with true premises that’s valid on the Aristotelian view but invalid on the modern view. We should draw the conclusion if we know that each general term in the premises refers to at least one existing being; otherwise, we shouldn’t. Consider this pair of arguments with the same form (a form that’s valid on the Aristotelian view but invalid on the modern view):

· All cats are mammals.

· All cats are furry.

· ∴ Some mammals are furry.

· All square circles are squares.

· All square circles are circles.

· ∴ Some squares are circles.

The first inference is sensible, because there are cats. The second inference isn’t sensible, because there are no square circles.

Some logic books use the Aristotelian view, but most use the modern view. It makes a difference to very few cases; all the syllogisms in this chapter prior to this section test out the same on either view.

To adapt the star test to the Aristotelian view, word it so that each capital letter must be starred at least once (instead of “exactly once”). To adapt Venn diagrams to the Aristotelian view, add this rule: “If you have a circle with only one unshaded area, put an ‘×’ in this area”; this assumes that the circle isn’t empty.