-logic with universal generalizations-

© Lyle Crawford [ 35 ]

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-LOGIC WITH UNIVERSAL GENERALIZATIONS-

i. DEDUCTION, VALIDITY, AND LOGIC

Validity

Deductive arguments are those that are supposed to be valid. In a valid deductive

argument, the premises support the conclusion in a special way: they absolutely guarantee

it. This means that if the premises are true, then it is absolutely impossible – literally

unthinkable, or unimaginable – that the conclusion could be false. Any deductive argument

that is not valid is called invalid.5

A valid argument can have false premises. However, if it does, the truth of the conclusion is

no longer guaranteed: the conclusion could be true or false. Valid deductive arguments are

“truth-preserving”: if we put truth “into” the argument (in its premises), we get truth “out”

(in its conclusion). This means that if we know that its conclusion is false, we can know that

it has at least one false premise.

Logic

Logic is the study of good argument patterns, patterns of inference that reliably lead to a

true conclusion when we start from true premises. “Logic” usually means “deductive logic”,

the study of valid argument patterns. Here are two examples:

Argument 3.1: All A’s are B’s. And X is an A. Therefore X is a B.

Argument 3.2: If P, then Q. And P is true. Therefore Q is true.

5 Vocabulary Alert: In ordinary language, the word “deduce” often means simply “infer”, not

necessarily with a logically deductive argument. And in ordinary language, “valid” often means “true”, “relevant”, “legitimate”, “justified”, or “good”.


In Argument 3.1, the letters stand for things; in Argument 3.2, the letters stand for

statements. It does not matter what statements we fill in for P and Q, or what things we fill

in for A, B, and X. Any argument that follows one of these patterns (or others that we will

learn) is valid. These kinds of short deductive arguments are called syllogisms. Valid

syllogisms re-organize or re-combine information from their premises for the conclusion.

ii. UNIVERSAL GENERALIZATIONS

The ancient Greek philosopher Aristotle (384–322 BCE) invented the first kind of logic. We

will learn a simplified version of this logic, which constructs valid syllogisms using universal

generalizations. A universal generalization relates one type of thing to another type of

thing. There may be lots, or just one, or none at all, of each type of thing. “Universal” means

that the statement gives a rule that allows not even one exception (case where it is false).

We’ll look at universal generalizations constructed with the quantifiers “all”, “only”, or “no”.

Every universal generalization can be written in this form:

[Quantifier] A’s are B’s.

A and B are plural nouns (things we can count).

Statement 3.3: All disasters are earthquakes.

Statement 3.4: Only professional athletes are celebrities.

Statement 3.5: No films are comedies.

Not every universal generalization is written in this form, but every one can be written in this

form. Writing generalizations in this form will be necessary for doing logic with them. To write

them this way correctly, we need to pay close attention to the structure of sentences.


Subject and Predicate

Every sentence has two parts: subject and predicate. The subject is what the sentence is

about. The predicate is what the sentence says about the subject; it begins with the verb.

Subject Predicate

Sentence 3.6: Bob runs.

Subject Predicate

Sentence 3.7: Bob and Abby run away from the zombies and don’t look back.

In a generalization, the subject of the sentence gives the quantifier and the first type of thing; the

predicate gives the second type of thing.

Subject Predicate

Sentence 3.8: Only birds fly.

“Fly” is a verb, not a plural noun. But we can easily turn it into a plural noun.

Statement 3.8: Only birds fly. = Only birds are flying things.

“Flying things” is a plural noun; we can count flying things. The method for writing a

generalization in this form is: 1) find the predicate, 2) turn the predicate into a plural noun.

Predicate

Statement 3.9: All smartphones have an operating system (OS).

All smartphones are things with an OS.

Predicate

Statement 3.10: Only planets with water support life.

Only planets with water are places that support life.


All A’s are B’s = Only B’s are A’s

Once we’ve identified the two types related in the statement, we can draw a simple diagram

of the generalization. Each type is shown with a circle.

Statement 3.8 Statement 3.9

FT Birds SP TWOS

These generalizations each show one type of thing entirely “contained” within another type of

thing. The diagrams say what the statements say: all flying things (FT) are birds, and all

smartphones (SP) are things with an OS (TWOS).

We can make these same statements using the word “only” instead of “all”. To convert a

generalization from one form to the other, we need to switch the order of the two things in

the sentence.

Statement 3.8: Only birds are FT.

All FT are birds.

Statement 3.9: All SP are TWOS.

Only TWOS are SP.

Every “all/only” universal generalization works this way.

All A’s are B’s. = Only B’s are A’s.

A’s B’s


All and only A’s are B’s

It is also possible to combine an “all” generalization with an “only” generalization: All and only

A’s are B’s. To diagram this, we combine the diagrams for “All A’s are B’s” and “Only A’s are

B’s”. The A’s circle and the B’s circle overlap.

All A’s are B’s. All B’s are A’s. All and only A’s are B’s.

Only B’s are A’s. Only A’s are B’s.

+ = A’s

A’s B’s B’s A’s B’s

No A’s are B’s

We can also re-write and diagram generalization that use the quantifier “no”.

Predicate

Statement 3.11: No cities have more than 15 million people.

No cities are places with more than 15m people (PWMT15MP).

No PWMT15MP are cities.

Cities PWMT15MP

Every “no” universal generalization works this way.

No A’s are B’s. = No B’s are A’s.

A’s B’s


iii. COUNTER-EXAMPLES

Universal generalizations state a rule that allows not even one exception. An exception to a

generalization is called a counter-example. Since a generalization relates two things, a

counter-example has a two-part description.

Generalization Counter-example (X)

All A’s are B’s. = Only B’s are A’s. X = A that is not a B.

No A’s are B’s. = No B’s are A’s. X = A that is also a B.

Statement 3.9 is true because there are no counter-examples. There is no smartphone that

not a thing with an OS. Statement 3.8 is false because there is a counter-example: flying fish.

These are flying things that are not birds.

For any counter-example, we can name it and describe it.

Counter-example to “Only birds are FT.” = “All FT are birds.”

Name Description

Flying fish FT that is not a bird

Flying fish show that some flying things are not birds. The FT circle is not contained within

the Birds circle – it must extend outside of it to include the counter-examples. We can draw

a corrected diagram with X’s to show the counter-examples.

Generalization Corrected Diagram

Only birds are FT. (Some FT are not birds.)

All FT are birds. FT Birds

FT Birds Flying fish X


What about a penguin? A penguin is 1) a bird, but 2) not a flying thing. This is not a counter-

example; it fits into the original, uncorrected diagram.

No correction necessary.

FT Birds

X

(A penguin would be a counter-example to a different generalization: All birds are FT.)

Statement 3.11 is also false. Again, there are counter-examples. Shanghai, Delhi, and Lagos

are 1) cities, and 2) also places that have more than 15 million people. Again we can correct

the diagram of the generalization.

Counter-examples to “No cities are PWMT15MP.” = “No PWMT15MP are cities.”

Name Description

Shanghai

Delhi

Lagos

City that is also a PWMT15MP


No cities are PWMT15MP. (Some cities are also PWMT15P.)

No PWMT15MP are cities.

Cities PWMT15MP

Cities PWMT15MP X

X X

Shanghai Delhi Lagos


“All and only…” combines two generalizations, so two sorts of counter-examples apply to it.

Predicate

Statement 3.12: All and only bacteria cause disease in normally healthy humans.

All and only bacteria are things that cause disease in normally

healthy humans (TTCDINHH).

Statement 3.12 is doubly false. The “all” generalization is false, and so is the “only”

generalization. Bacteria in our gut and on our skin, as well as bacteria in soil, do not cause

disease in normally healthy humans. And some viruses (e.g. influenza virus, which causes

flu), genetic abnormalities (e.g. a mutated form of the gene for CFTR protein, which causes

cystic fibrosis), and many of the causes of cancer (e.g. ultraviolet radiation, which causes skin

cancer) cause disease in normally healthy humans, but are not bacteria.

Counter-examples to “All and only bacteria are TTCDINHH.”

Name Description

Gut bacteria (GB)

Skin bacteria (SkB)

Soil bacteria (SB)

Bacteria that is not a TTCDINHH

Influenza virus (IV)

CFTR gene mutation (CGM)

Ultraviolet radiation (UVR)

TTCDINHH that is not bacteria


All and only bacteria are TTCDINHH. (Some bacteria are not TTCDINHH.)

(Some TTCDINHH are not bacteria.)

Bacteria GB X X IV

SkB X X CGM

TTCDINHH SB X X UVR

Bacteria TTCDINHH


Rejecting a Counter-example

Suppose that Bob believes Statement 3.8, “Only birds fly”, and Abby proposes (suggests) the

counter-example of an owl. Bob should not be convinced that some flying things are not birds.

An owl is a flying thing. However, an owl is also a bird. (How did Abby not know that?)


FT Birds

X

Suppose Abby claims that a cloud is a counter-example. Again, Bob should not be convinced.

Clouds are not birds. However, a cloud is really not a FT, either, even though it is in the sky.

It is just floating, not flying.


FT Birds

X

Suppose Abby suggests a dragon, a giant, flying, fire-breathing lizard. A dragon is a counter-

example because it is a FT that is not a bird. However, again Bob should not be convinced.

The problem with Abby’s suggestion, of course, is that there are no dragons. They are

mythical creatures that do not really exist.


FT Birds


Revising a Generalization

Bob should not be convinced that Statement 3.8 is false by owls, clouds, or dragons. However,

as we’ve seen, he should be convinced by the flying fish counter-example. The generalization

diagram must be corrected to include flying fish.

Corrected Diagram

(Some FT are not birds.)

FT Birds

Flying fish X

Bob may revise the generalization in to avoid the flying fish counter-example. He can do that

in two ways: restrict (make smaller) the FT type or expand (make larger) the Birds type.

Restrict FT to Animals that Fly by Flapping

their Wings (AFFW). Flying fish jump from

the water and glide a short distance. They

do not push themselves through the air by

flapping their wings; they are not AFFW.

Revised Generalization (FT Restricted)

Only birds are AFFW.

All AFFW are birds.

FT Birds

X

AFFW

Expand birds to animals. Flying fish are FT

and also animals.

Revised Generalization (Birds Expanded)

Only animals are FT.

All FT are animals.

FT Birds

X

Animals


These revised generalizations avoid the flying fish counter-example. However, this does not

mean that either new statement is true! There are other counter-examples that apply to each.

Bats and flying insects both fly by flapping

their wings, as did pterosaurs, which went

extinct many millions of years ago. But these

are not birds.

New Corrected Diagram

(Some AFFW are not birds.)

Birds

Bat X

Flying insect X

Pterosaur X

X

Flying fish AFFW

Airplanes and remote controlled drones

(RCD) both fly. But they are not animals.

New Corrected Diagram

(Some FT are not animals.)

FT

Airplane X

RCD X X

Flying fish Animals

We could do something similar with Statement 3.11 to avoid the counter-examples we found

to it, except in this case, it will do no good to expand either type. The only options are to

restrict one or the other. For example, we could restrict Cities to North American cities (NAC).

Or we could restrict PWMT15MP to “places with more than 30 million people” (PWMT30MP).


Corrected Diagram

(Some cities are also PWMT15P.)

Cities PWMT15MP

X

X X

Shanghai Delhi Lagos

Revised Generalization

(Cities Restricted)

No NAC are PWMT15MP.

NAC

Cities PWMT15MP

X

X X

Revised Generalization

(PWMT15MP Restricted)

No cities are PWMT30MP.

PWMT30MP

Cities PWMT15MP

X

X X

These revised generalizations avoid the counter-examples of Shanghai, Delhi, and Lagos.

Once again, this does not mean that either new statement is true. There could be other

counter-examples (although there are not any).

We’ve now seen what counter-examples are, as well as the different ways that we can respond

to a proposed counter-example.

How to respond to a proposed counter-example?

Reject the

Generalization

Reject the

Counter-example

Revise the

Generalization

“The statement is

false.”

“That thing

does not

exist.”

“That thing has

not been correctly

described.”

Modify (restrict

or expand) one

type.

Modify (restrict or

expand) the other

type.

-logic with universal generalizations-

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