-logic with universal generalizations-
TRANSCRIPT
© Lyle Crawford [ 35 ]
~ 3 ~
-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”.
© Lyle Crawford [ 36 ]
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.
© Lyle Crawford [ 37 ]
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.
© Lyle Crawford [ 38 ]
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
© Lyle Crawford [ 39 ]
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
© Lyle Crawford [ 40 ]
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
© Lyle Crawford [ 41 ]
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
Generalization Corrected Diagram
No cities are PWMT15MP. (Some cities are also PWMT15P.)
No PWMT15MP are cities.
Cities PWMT15MP
Cities PWMT15MP X
X X
Shanghai Delhi Lagos
© Lyle Crawford [ 42 ]
“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
Generalization Corrected Diagram
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
© Lyle Crawford [ 43 ]
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?)
No correction necessary.
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.
No correction necessary.
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.
No correction necessary.
FT Birds
© Lyle Crawford [ 44 ]
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
© Lyle Crawford [ 45 ]
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).
© Lyle Crawford [ 46 ]
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.