Philosophy assignment 10 questions
Nancy 00001
1583433839_578__15_Everything.pdf
Conjecture: Every card that has an even number on one side is red on the other side.
Which cards does one have to turn over to find out whether the conjecture is true?
PHIL 110; Spring 2020; Lecture 15 1
Every card has a colour on one side and a number on the other.
Is this a valid inference?
Premise: Every person at the party was a twentysomething.
Conclusion: Every person at the party who was wearing a jacket was
a twentysomething.
Valid! Not valid!
PHIL 110; Spring 2020; Lecture 15 2
13: Everything PHIL 110; Spring 2020; Tom Donaldson
Things to be getting on with
• Take it easy – relax after the midterm.
• There will be an assignment next week.
PHIL 110; Spring 2019; Lecture 13 4
1: Beyond Statement Logic
Beyond Statement Logic
• There are certain inferences which cannot be adequately evaluated using the tools we’ve discussed so far.
• Let’s look at some examples.
PHIL 110; Spring 2020; Lecture 15 6
Tense Logic
Premise: Ashni will swim and Ben will swim, but Ashni won’t
swim while Ben swims.
Conclusion: Either Ashni will swim and then Ben will, or Ben will
swim and then Ashni will.
PHIL 110; Spring 2020; Lecture 15 7
Deontic Logic
Premise: You may have coffee.
Premise: You may have tea.
Conclusion: You may have coffee and tea.
Premise: C
Premise: T
Conclusion: (C & T)
PHIL 110; Spring 2020; Lecture 15 8
The Logic of Quantification
Premise: Every dog is a mammal.
Premise: Fido is a dog.
Conclusion: Fido is a mammal.
PHIL 110; Spring 2020; Lecture 15 9
We’ll focus on the logic of quantification …
• Tense isn’t relevant in (pure) mathematics.
• Deontic notions (such as obligation and permission) are also not relevant.
• But “every” is everywhere in mathematics! • Every natural number has a unique prime factorization. • Every polynomial of degree three has a real root. • Every polynomial is differentiable.
• The negation of an “every” statement is equivalent to a “some” statement.
• So we’ll focus on “every” and “some”.
PHIL 110; Spring 2020; Lecture 15 10
2: Introducing “Every”
Universal Generalizations
Universal generalizations in English often contain the word “every”, or “everything” or “everyone”, or “any”, or “all”:
• Every whale is a mammal.
• Everything is broken.
• All dogs are hairy.
But there are exceptions:
• Dogs have four legs.
• A bear is a mammal.
• Man is born free, but everywhere he is in chains.
PHIL 110; Spring 2020; Lecture 15 12
The Need for Symbols
Compare:
• A bear is a mammal.
• A bear goes through my trash can every night.
As we said earlier in the term, English is extremely complicated, so in logic we need to use artificial symbols instead.
We won’t introduce any new symbols today, however.
PHIL 110; Spring 2020; Lecture 15 13
Strict vs. Loose
• There are two sorts of universal generalization – strict and loose.
• Strict: “Every single dog without exception is a mammal.”
• Loose: “Dogs have four legs.”
• A strict universal generalization can be refuted by just one example, a “counterexample”. • For example, if someone claims that all birds can fly, you can prove him
wrong by showing him a single penguin.
PHIL 110; Spring 2020; Lecture 15 14
Strict vs. Loose
• There are two sorts of “every” statement – strict and loose.
• Strict: “Every single dog without exception is a mammal.”
• Loose: “Dogs have four legs.”
• A strict universal generalization can be refuted by just one example, a “counterexample”.
• Loose universal generalizations are not so easily refuted.
• It is sometimes unclear whether a universal generalization is strict or loose. Consider: “Abortions are immoral.”
• When doing philosophy, it is a good idea often to ask, “Is that strict or loose?”
PHIL 110; Spring 2020; Lecture 15 15
Domains of Quantification
• When one says “everything”, it is rare that one means to consider every single thing in the whole universe without restriction. • Example: “Every beer bottle is empty!”
• Example: “Every number is either odd or even.”
• Typically, one means to consider only the things within a certain “domain of quantification”.
PHIL 110; Spring 2020; Lecture 15 16
Vacuous Generalizations
• The universal generalization “Every A is a B” is said to be “vacuous” if there are no A’s. Consider: • Every unicorn has a horn.
• Every witch wears a black hat.
• Logicians assume that all vacuous universal generalizations are true.
• This might seem a bit odd at first. (Think about “All the kryptonite in Vancouver is stored in my basement.”)
PHIL 110; Spring 2020; Lecture 15 17
Premise: Every person at the party was a twentysomething.
Conclusion: Every person at the party who was wearing a jacket was
a twentysomething.
Premise: Every A is C.
Conclusion: Every A that is B is C.
PHIL 110; Spring 2020; Lecture 15 18
Existential Generalizations
Existential generalizations often contain “some” or “there is” or “a”:
• A dog is barking in the garden.
• Some dog is barking in the garden.
• There is a dog barking the garden.
The negation of a universal generalization is equivalent to an existential generalization:
• It is not true that everyone enjoyed the party.
• Someone didn’t enjoy the party.
The negation of an existential generalization is equivalent to a universal generalization:
• It is not true that one of the men at the party was unmarried.
• All of the men at the party were married.
PHIL 110; Spring 2020; Lecture 15 19
3: Venn Diagrams
PHIL 110; Spring 2020; Lecture 15 21
Famous people
PHIL 110; Spring 2020; Lecture 15 22
Famous people
Denzel Washington
PHIL 110; Spring 2020; Lecture 15 23
Famous people
Denzel Washington
Kim Kardashian
PHIL 110; Spring 2020; Lecture 15 24
Famous people
Denzel Washington
Kim Kardashian
Tom Donaldson
PHIL 110; Spring 2020; Lecture 15 25
Famous people
Denzel Washington
Kim Kardashian
People who should be famous
Tom Donaldson
PHIL 110; Spring 2020; Lecture 15 26
Dogs
Black things
x
There is a dog that isn’t black.
PHIL 110; Spring 2020; Lecture 15 27
Dogs
Black things
x
Some dog is black.
PHIL 110; Spring 2020; Lecture 15 28
Dogs
Black things
x
Something is black.
x
PHIL 110; Spring 2020; Lecture 15 29
No dog is black.
PHIL 110; Spring 2020; Lecture 15 30
Every dog is black.
PHIL 110; Spring 2020; Lecture 15 31
Canadians Singers
Talented People
x
PHIL 110; Spring 2020; Lecture 15 32
Canadians Singers
Talented People
x
PHIL 110; Spring 2020; Lecture 15 33
Canadians Singers
Talented People
x
x
PHIL 110; Spring 2020; Lecture 15 34
PHIL 110; Spring 2020; Lecture 15 35
3: Four Kinds of Statements
Code Form Examples
A All A are B. Every zebra is a mammal. All men are mortal. Every single member of the club was at the party.
E No A are B. No person can hold their breath for thirty minutes. Not one person in this room is honest. Expensive moisturizing creams are never worth buying.
I Some A is B. Some foxes live in Iceland. There are people who can run a mile in four minutes. At least one singer was off key.
O Some A is not B. Some logicians are not well groomed. Some famous people do not deserve to be famous. There are basketball players who aren’t tall.
PHIL 110; Spring 2020; Lecture 15 37
PHIL 110; Spring 2020; Lecture 15 38
All A are B.
PHIL 110; Spring 2020; Lecture 15 39
No A are B.
PHIL 110; Spring 2020; Lecture 15 40
A
B
Some A is B.
x
PHIL 110; Spring 2020; Lecture 15 41
A
B
Some A is not B.
x
4: Carrol l Diagrams
Venn Diagrams With Four Categories …
… are rather hard to draw.
PHIL 110; Spring 2020; Lecture 15 43
Venn Diagrams with Five Categories …
… are even harder!
PHIL 110; Spring 2020; Lecture 15 44
This is where Carroll Diagrams come in handy! Carroll diagrams work just like Venn diagrams, except they use rectangular grids rather than overlapping ellipses.
PHIL 110; Spring 2020; Lecture 15 45
B
A
PHIL 110; Spring 2020; Lecture 15 46
B
A
C
PHIL 110; Spring 2020; Lecture 15 47
B
A
C
D
5 : Eva lua t i ng In f e r ences Us ing Ve nn D ia g ra ms
Evaluating Inferences Using Venn Diagrams
I recommend the following procedure for evaluating inferences using Venn diagrams: 1. Write down a list all of the premises and the negation of the
conclusion. 2. Try to draw a Venn diagram depicting a situation in which all the
statements on your list are true. 3. If you succeed, you have shown that the inference is invalid. 4. If you find that it is impossible to depict such a situation, this is an
indication that the inference is valid. A tip for step three: When drawing your diagram, deal with the universal generalizations first. Then think about the existential generalizations. To put it another way: Do your shading first!!
PHIL 110; Spring 2019; Lecture 13 49
PHIL 110; Spring 2020; Lecture 15 50
A B
C
All A are B.
No B are C.
Therefore:
No A are C.
PHIL 110; Spring 2020; Lecture 15 51
A B
C
All A are B.
No B are C.
Therefore:
No A are C.
All A are B.
No B are C.
Some A is C.
PHIL 110; Spring 2020; Lecture 15 52
A B
C
Some A is B.
All B are C.
Therefore:
Some A is C.
PHIL 110; Spring 2020; Lecture 15 53
A B
C
Some A is B.
All B are C.
Therefore:
Some A is C.
Some A is B.
All B are C.
No A is C.
PHIL 110; Spring 2020; Lecture 15 54
A B
C
Some A is B.
Some B is C.
Therefore:
Some A is C.
PHIL 110; Spring 2020; Lecture 15 55
A B
C
Some A is B.
Some B is C.
Therefore:
Some A is C.
Some A is B.
Some B is C.
No A is C
Some for you to try
(1) No A is B. (2) All A are B.
No B is C. All B are C.
Therefore: Therefore:
No A is C. All B are C
Valid! Invalid!
PHIL 110; Spring 2020; Lecture 15 56
Two More
(1) No A are B. (2) All A are B.
Some A is C. No C are A.
Therefore: Therefore:
Some C is not B. No C are B.
Valid! Not valid!
PHIL 110; Spring 2020; Lecture 15 57
One More
Every A is either B or C.
Everything that is not C is not B.
Something is A.
Therefore:
Something is C.
Valid! Not valid!
PHIL 110; Spring 2020; Lecture 15 58
