7.1ex I

Think about the following interpretation:

 


UD = all inner planets of our solar system, Earth, Mars, Venus, Mercury.

a: Mercury, b: Venus, c: Earth, d: Mars,

Sx: "x has a natural satellite (a moon)."
I.e., 'Sc' and 'Sd' are true as 'S' is associated only with the planets having moons: Earth (c) and Mars (d).
Tx: "x is a terrestrial (rocky) planet".
I.e. 'Sa','Sb','Sc', and 'Sd' are all true because the inner planets (unlike the outer ones) are terrestrial as opposed to gassy or icy.
Ixy: "x orbits inside y".
So, for examples, 'Iab' is true: Mercury orbits inside Venus. So are: 'Iac',Iad','Ibc','Ibd',and 'Icd'.
Now consider the sentence,

(*) (^x)[(Sx&Tx)>Ixd].

What does it mean for (*) to be true?

  1. Each substitution instance of (*) is true.
  2. Exactly one substitution instance of (*) is true.
  3. At least one substitution instance of (*) is true

 

 

7.1ex I page2

So,

(*) (^x)[(Sx&Tx)>Ixd].

is true if all it's substitution instances are true. Keeping in mind that the names for all members of the UD are 'a', 'b', 'c', and 'd', then identify all these substitution instances below.

  1. (^x)[(Sa&Ta)>Iad]
  2. (Sa&Ta)>Iad
  3. (Sb&Tb)>Ibd
  4. (Sa&Tb)>Icd
  5. (Sc&Tc)>Icd
  6. (Sd&Td)>Idd

7.1ex I page3

So,

(*) (^x)[(Sx&Tx)>Ixd].

is true if all it's substitution instances are true.

It's substitution instances follow. Click on each which is true — and don't just guess!
  1. (Sd&Td)>Idd
  2. (Sc&Tc)>Icd
  3. (Sb&Tb)>Ibd
  4. (Sa&Ta)>Iad

(Remember the interpretation:

a: Mercury, b: Venus, c: Earth, d: Mars,

Sx: "x has a natural satellite (or moon)".
I.e., 'Sc' and 'Sd' are true as 'S' is associated only with the satellite bearing planets Earth (c) and Mars (d).
Tx: "x is a terrestrial (rocky) planet".
I.e. 'Sa','Sb','Sc', and 'Sd' are all true because the inner planets (unlike the outer ones) are terrestrial as opposed to gassy or icy.
Ixy: "x orbits inside y".
So, for examples, 'Iab' is true: Mercury orbits inside Venus. So are: 'Iac',Iad','Ibc','Ibd',and 'Icd'. But, of course, 'Idd' is false: Mars does not have an orbit inside it's own!)

7.1ex I page4

Finally, then,

(*) (^x)[(Sx&Tx)>Ixd].

is true false (click one) because some of (*)'s substitution instances are true but some are not.

Not all planets make '(Sx&Tx)>Ixd' true. Only the three innermost do. Mars does not. So, (*) is false.

Now, what else does this tell us? Click on all true sentences below...

  1. ~(^x)[(Sx&Tx)>Ixd]
  2. (%x)[(Sx&Tx)>Ixd]
  3. ~(%x)[(Sx&Tx)>Ixd]

7.1ex II
PL Semantics
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem. Use the interpretation from the tutorial.

UD: George Washington and John Adams
g: George Washington
j: John Adams
Mx: x is Male
Fx: x is the first US president.
Sx: x is the second US president.
Bxy: x was president before y.

1. Which of the following is TRUE on the given interpretation?
a. (%x)Mx b. (^x)Sx c. (^x)Sx=(%x)Mx
2. Which of the following is TRUE on the given interpretation?
a. (%x)~Mx b. (^x)~Sx c. (^x)Sx=~(%x)Mx

3. Which of the following is TRUE on the given interpretation?
a. (^x)(Mx>Fx) b. (^x)(Fx>Mx) c. (^x)(Fx=Mx)

4. '(^x)(Mx>Fx)' is FALSE because
a. the substitution instance 'Mg>Fg'is false. b. the substitution instance 'Mj>Fj'is false.

5. '(^x)(Mx=Fx)' is FALSE because
a. the substitution instance 'Mg=Fg' is false. b. the substitution instance 'Mj=Fj' is false.

6. '(^x)(Fx>Mx)' is TRUE because
a. the substitution instance 'Fg>Mg' is true. b. the substitution instance 'Fj>Mj' is false. c. both substitution instances are true.

7. Which of the following is true?
a. (^x)(%y)Byx b. (%x)(%y)(Byx&~Mx) c. (%x)(%y)(Byx&Fx) d. (%x)(%y)(Bxy&Fx)

8. '(^x)(%y)Byx' is FALSE because
a. 'Bgj' is a true substitution instance of '(^x)(%y)Byx'. b. 'Bjg' is a false substitution instance of '(^x)(%y)Byx'. c. '(%y)Byg' is a false substitution instance of '(^x)(%y)Byx'.

9. '(%x)(%y)(Byx&~Mx)' is FALSE because
a. There is no way to substitute in for x and make 'Byx' true. b. There is no way to substitute in for x and make '~Mx' true.

10. '(%x)(%y)(Byx&Fx)' is FALSE because
a. Substituting in for x to make 'Fx' true requires that x be set to g, the first president. But then there is no way to to truthfully substitute in for y. b. There is no way to substitute in for x and y to make 'Bxy' true.

7.1ex III
Concepts
Multiple Choice: Click on the correct answer and the page will jump forward to the next problem.

1. A PL sentence P is logically true in PL if and only if ...
a. P is not false on any truth value assignment. b. P is not true on any truth value assignment. c. P is not false on any interpretation. d. P is not true on any interpretation.

2. A PL sentence P is logically false if and only if ...
a. ~P is logically true. b. P is not logically false. c. P is not logically indeterminate.

3. If two PL sentences P and Q are logically equivalent, then ...
a. there is no interpretation on which one of P and Q is true while the other is false. b. an argument with P amongst its premises and Q as its conclusion is valid in PL. c. if P is logically true, Q is also logically true. d. All of the above.

4. If an argument has either a logically inconsistent set of premises or a logically true conclusion, then ...
a. that argument is valid. b. that argument is invalid. c. Neither of the above.

5. If P is logically false, then ...
a. an argument with P as conclusion must be invalid. b. a set including P must be inconsistent. c. Both of the above. d. Neither of the above.

7.1ex IV
Semantics

Matching. Drag sentences from the right to the correct location in the proof box.

UD = the counting numbers: 1,2,3, etc.
Gxy: x is greater than y

Now, show that the sentence

(1) '(%w)(^z)Gzw' is FALSE.


Assume          __________

Then           _________

But          ________
So,           _________

But          ________

(for contradiction) that
'(%w)(^z)Gzw' is true
.
some substitution instance of (1)
    (^z)Gzn
must also be true (where 'n' is picked to name the appropriate number).
because '(^z)Gzn' is true, all it's substitution instances must be true.
'(^z)Gzn's substitution instance
     Gnn
is true.

this last result that 'Gnn' is is true is an absurdity for no number is greater than itself!

Hence, the initial assumption is wrong and (*) is false.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7.1ex IV
Semantics

Write out proofs for each of the following propositions.

Each proof below is finished by giving examples: either to show that and existential proposition is true or that a general proposition fails to hold in all cases. Your job will be to come up with particular examples of numbers to prove your point. In each case, you will use names for the numbers in substitution instances.

UD = the counting numbers: 1,2,3, etc.
Gxy: x is greater than y

(2) (%z)(%w)Gzw

This sentence is true false
Proof:

 

 

(3) (^z)(%w)Gzw

This sentence is true is false.
Proof:

 

 

7.1ex V

Show that

(^x)Px&(^y)Rya
(^x)Px

is valid.

First do this by means of a derivation (just as you would in SD).

Does this mean that there is no interpretation on which the premise is true and the conclusion false.

Of course, derivations only show that arguments are valid. They never show that an argument is invalid? Think about it.


To show an argument invalid, one needs to give an interpretation on which all premises are true and the conclusion false.

For example:

Argument 1: (^x)Sx
(%y)Py  
(^x)(Sx&Px)

Give an interpreation that shows this.

Now do the same for:

Argument 4:< (^x)(Qx>Px)
(^x)[Px>(%y)(Gyx&(Py&Qy))]
(^x)(~Qx>~Px)

The exercises will help you construct these.

7.2ex I

Symbolize as before.

UD: all people
Hx: x is healthy
Px: x is a philosopher
Rx: x is rich (or wealthy)
Wx: x is wise

  1. Some philosopher is wise.
  2. Somebody is a philosopher.
  3. All philosophers are wealthy (rich).
  4. Every philosopher is healthy, wealthy, and wise.
  5. Some healthy philosopher is wise.
  6. At least one wise philosopher is unhealthy.
  7. All philosophers are healthy but poor. All wise people are philosophers.

Notice that 7 does not say "Not all philosophers are wealthy" but implies more, that all philosophers are non-wealthy.

7.2ex II

Answer the following using Ax: x is an accountant, Bx: x is brave, Cx: x is capricious.

1. Which of the following could be used to symbolize "All accountants are brave"?
a) (^x)(Ax&Bx)
b) (%x)(Ax&(^y)(By>Ay))
c) (^x)(Ax>Bx)

2. Which of the following could symbolize "No capricious accountants are brave".
a) (%x)((Cx&Ax)>~Bx)
b) ~(%x)((Cx&Ax)&Bx)
c) ~(^x)(Cx>(Ax&~Bx))

3. Which of the following could NOT symbolize "All brave accountants are capricious"?
a) ~(%x)((Bx&Ax)&~Cx)
b) ~(%x)((~Bx&Ax)&Cx)
c) (^x)((Bx&Ax)>Cx)

4. Which of the following could NOT symbolize "Some brave accountants are capricious"?
a) (%x)((Bx&Ax)&Cx)
b) (^x)~((Bx&Ax)&~Cx)
c) (%y)(By&(Ay&Cy))

7.2ex III

Symbolize the following as categorical statements.

UD: all people
Cx: x is a U.S. citizen
Rx: x is a U.S. resident
Ox: x is a voter
(remember, no 'V's in PL)

  1. Some U.S. citizens are voters. (hint)
  2. All U.S. citizens are voters. (hint)
  3. No U.S. citizens are voters. (hint)
  4. Some U.S. citizens are not voters. (hint)
  5. Some U.S. residents are voters.
  6. No U.S. residents who are not citizens are voters. (hint)
  7. All nonvoting U.S. residents are non-citizens. (hint)
  8. No voters are U.S. residents.
  9. Only citizens are voters. (hint)

More Symbolization — 7.2ex IV

Symbolize the following. If an example looks unfamiliar, try to figure it out before going to hints or answers. And if you still feel confused, read on to tutorial 7.3 for more comments on complex symbolization.

UD: all living things
s: Shamu (the Seaworld orca)
g: the General Sherman tree (the
largest tree in the world)
Ax: x is an animal
Bx: x is a beluga
Cx: x is a cat
Fx: x is a fish
Mx: x is a mammal
Nx: x lives in North America
Sx: x is a sea creature
Wx: x is a whale
Cxy: x climbs y
Lxy: x is larger than y

  1. No fish is a whale.
  2. Cats are mammals.(Hint)
  3. Only mammals are whales. (Hint)
  4. Shamu is larger than any beluga. (Hint)
  5. Cats are mammals and so are whales. (Hint)
  6. All belugas and cats are mammals. (Hint)
  7. No cat from North America climbs the General Sherman tree.
  8. The only belugas living in North America are smaller than Shamu. (Hint)
  9. Any sea creature is a mammal or a fish.(Hint)
  10. All belugas but no fish are mammals.
  11. Shamu is smaller than no fish from North America.
  12. Some living thing is larger than the General Sherman tree.(Hint)
  13. Some animal is bigger than the General Sherman tree.
  14. Shamu climbs no living thing.
  15. If no whale is a fish, then any fish is no beluga.

Which Quantifier? — 7.3ex I

Symbolize the following. The main concern for these problems is to determine whether a universal or existential quantifier is to be used.

UD: all living things
s: Shamu (the Seaworld orca)
g: the General Sherman tree (the
largest tree in the world)
Ax: x is an animal
Bx: x is a beluga
Cx: x is a cat
Fx: x is a fish
Mx: x is a mammal
Nx: x lives in North America
Sx: x is a sea creature
Wx: x is a whale
Cxy: x is climbing y
Lxy: x is larger than y

  1. Every beluga is a whale.
  2. A beluga is a whale.(Hint)
  3. A cat is climbing the General Sherman tree. (Hint)
  4. If any cat is climbing the General Sherman tree, then that cat lives in North America.
  5. Whales are mammals. (Hint)
  6. Cats live in North America. (Hint)
  7. If any cat is climbing the General Sherman tree, then there are cats living in North America.

7.3ex II

Symbolize the following. All these problems require a second quantifier within the scope of the first.

UD: all living things
s: Shamu (the Seaworld orca)
g: the General Sherman tree (the
largest tree in the world)
Ax: x is an animal
Bx: x is a beluga
Cx: x is a cat
Fx: x is a fish
Mx: x is a mammal
Nx: x lives in North America
Sx: x is a sea creature
Wx: x is a whale
Cxy: x is climbing y
Lxy: x is at least as large as y

  1. Some living thing is as as large as any other.
  2. There is no living thing of smallest size. (Hint)
  3. Some cat is as large as some whale.
  4. Some whale is at least as large as every cat. (Hint)
  5. There is a mammal at least as large as any fish. (Hint)
  6. Some cat climbing the General Sherman tree is at least as large as some sea creature.
  7. All whales are as large or larger than all fish.
  8. Some cat is a climber. (Hint)

7.3ex III
Complex Subjects and Predicates

Symbolize the following.

UD: everything
g: George W. Bush
u: the United States
Cx: x is a country
Fx: x is female
Gx: x has a government Hx: x is heroic
Px: x is a person
Yx: x is a capital city
Ixy: x is located in y
Lxy: x is the leader of y
Kxy: x knows y

  1. Every country has some person as leader. (Hint)
  2. No country has a heroic leader. (Hint)
  3. Some countries have a female leader.
  4. No country has a leader who is not a person. (Hint)
  5. Every country has a capital city. (Hint)
  6. Every country has a capital city and a government.
  7. Every country with a capital city has a leader. (Hint)
  8. No country has a government yet no leader. (Hint)
  9. Some one person is the leader of absolutely every country.

More Symbolizations — 7.3ex IV

Symbolize the following.

UD: everything
g: George W. Bush
u: the United States
Cx: x is a country
Fx: x is female
Gx: x has a government Hx: x is heroic
Px: x is a person
Yx: x is a capital city
Ixy: x is located in y
Lxy: x is the leader of y
Kxy: x knows y

  1. All countries have a government.
  2. No country has a government. (Hint)
  3. Some country has a govenment.
  4. Some country has no govenment.
  5. No country is led by George W. Bush.
  6. Every capital city is located in some country.(Hint)
  7. Some country has a capital city but no leader. (Hint)
  8. Some country has a capital city and is led by George W. Bush.
  9. If someone is the leader of the United States, then everyone knows that person. (Hint)
  10. Someone is the leader of the US, but no one knows that person.