AppTheories_T3

1. Introduction to formal
logic
Tutorial 3

2. Summary Syllogistic Logic

3. D: Drug dealers in the U.S.
P: People who protested in Eastern Germany
S: Students in the course Application of Theories
Language
Capital letters are used for categories
Small letters are used for individuals
d: Tom the drug dealer
c: Charlie the protester
s: Silvia the student

4. 5 words
To formulate a correct wff, you need only five words:
all
no
some
is
not

5. 8 Wff’s (woofs)
There are only eight (8) forms of wffs:
all A is B
no A is B
some A is B
some A is not B
x is B
x is not B
x is y
x is not y

6. More notation
a ≠ A
A ≠ A’ ≠ A’’

7. Use of capital letters
All syllogistic wff’s have the verb is
Wff’s beginning with a word (not a letter) use two capital letters
Wff’s beginning with a letter (not a word) begin with a small
some C is D
g is C
If a wff begins with a letter, the second letter can be either capital
or small
(g is D) or (g is d)

8. The Star Test

9. The Star Test (1)
A syllogism is a vertical sequence of one or more wff’s 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)
no P is B
some C is B
----------------
some C is not P

10. Distributed: An instance of a letter is distributed in a wff if it occurs just
after all or anywhere after no or not.
To test validity, use the star test. Star premise letters that are
distributed and conclusion letters that are not distributed.
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.
all A is B
some C is A
----------------
some C is B
Valid or not?
The Star Test (2)

11. all A* is B
some C is A
----------------
some C* is B*
A, B and C have only one *
Only one star on the right hand side (B*)
The Star Test (3)
The argument is valid

12. Translation

13. Different ways to say: all A is B
Every (each, any) A is B
Whoever is A is B
A’s are B’s
Those who are A are B
If a person is A, then he or she is B
If your are A, then you are 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 is not A unless it is B
It is false that some A is not B

14. A’s are not B’s
Ever (each, any) A is non-B
Whoever is A is not B
If a person is A, then he or she is not B
If you are A, then you are not B
No one that is A is B
There is not a single A that is B
Not any A is B
It is false that there is an A that is B
It is false that some A is B
Different ways to say: no A is B

15. 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
Different ways to say: some A is B

16. One or more A’s are not B’s
There are A’s that are not B’s
Not all A’s are B’s
It’s false that all A is B
Different ways to say: some A is not B

17. Contradictions: If the one is false, the other one is true
no A is B is contradictory to some A is B.
all A is B is contradictory to some A is not B

18. Deriving Conclusions

19. Steps to derive conclusions
Step 1: Translate the premises and star
Step 2: Figure out the letters in the conclusions
(the two letters that occur just once in the premises)
Step 3: Figure out the form of the conclusion
Step 4: Add the conclusion and do the star test

20. Form of the conclusion
If all premises are universal (all, no), then use (all A is B) or (no A is B)
Otherwise, use (some A is B) or (some A is not B)
Use (x is A) or (x is not A)
Use (x is y) or (x is not y)
If both conclusion letters are capital
If just one conclusion letter is small
If both conclusion letters are small
Use no or not if there are any negative premises

21. Idiomatic arguments
Hence, thus, so, therefore...
It must be, it can’t be...
This proves (or shows) that...
These often indicate premises
These often indicate conclusions
Because, for, since, after all...
I assume that, as we know...
For these reasons...

22. Problem Set A

23. (1) Rewrite the arguments as wf's (verbally), (2) translate to syllogisms, (3) test
validity by means of (3.a) the star test, and (3.b) a venn diagram
1. No physical actions are chance occurrences, so no random events are physical
actions, since all chance occurrences are random events.
!
2. All mental decisions are things describable by science, because all events are
things describable by science, and all mental decisions are events.
!
3. No free decisions are things describable by science; therefore, no free decisions
are predictable events, inasmuch as all things describable by science are
predictable events.

24. 1
No
physical
ac,ons
(P)
are
chance
occurrences
(C)
All
chance
occurrences
(C)
are
random
events
(R)
No
random
events
(R)
are
physical
ac,ons
(P)

25. 1
No
physical
ac,ons
(P)
are
chance
occurrences
(C)
All
chance
occurrences
(C)
are
random
events
(R)
No
random
events
(R)
are
physical
ac,ons
(P)
No
P
is
C
All
C
is
R
No
R
is
P

26. 1
No
physical
ac,ons
(P)
are
chance
occurrences
(C)
All
chance
occurrences
(C)
are
random
events
(R)
No
random
events
(R)
are
physical
ac,ons
(P)
No
P
is
C
All
C
is
R
No
R
is
P

27. 1
No
physical
ac,ons
(P)
are
chance
occurrences
(C)
All
chance
occurrences
(C)
are
random
events
(R)
No
random
events
(R)
are
physical
ac,ons
(P)
No
P*
is
C*
All
C*
is
R
No
R
is
P

28. 1
No
physical
ac,ons
(P)
are
chance
occurrences
(C)
All
chance
occurrences
(C)
are
random
events
(R)
No
random
events
(R)
are
physical
ac,ons
(P)
No
P*
is
C*
All
C*
is
R
No
R
is
P
Invalid
argument

29. 1
No
physical
ac,ons
(P)
are
chance
occurrences
(C)
All
chance
occurrences
(C)
are
random
events
(R)
No
random
events
(R)
are
physical
ac,ons
(P)
No
P*
is
C*
All
C*
is
R
No
R
is
P
Invalid
argument
2
All
events
(E)
are
things
describable
by
science
(D)
All
mental
decisions
(M)
are
events
(E)
All
mental
decisions
(M)
are
things
describable
by
science
(D)
All
E
is
D
All
M
is
E
All
M
is
D
All
things
describable
by
science
(D)
are
predictable
events
(P)
No
free
decisions
(F)
are
things
describable
by
science
(D)
No
free
decisions
(F)
are
predictable
events
(P)
3
All
D
is
P
No
F
is
D
No
F
is
P

30. 1
No
physical
ac,ons
(P)
are
chance
occurrences
(C)
All
chance
occurrences
(C)
are
random
events
(R)
No
random
events
(R)
are
physical
ac,ons
(P)
No
P*
is
C*
All
C*
is
R
No
R
is
P
Invalid
argument
2
All
events
(E)
are
things
describable
by
science
(D)
All
mental
decisions
(M)
are
events
(E)
All
mental
decisions
(M)
are
things
describable
by
science
(D)
All
E
is
D
All
M
is
E
All
M
is
D
All
things
describable
by
science
(D)
are
predictable
events
(P)
No
free
decisions
(F)
are
things
describable
by
science
(D)
No
free
decisions
(F)
are
predictable
events
(P)
3
All
D
is
P
No
F
is
D
No
F
is
P

31. 1
No
physical
ac,ons
(P)
are
chance
occurrences
(C)
All
chance
occurrences
(C)
are
random
events
(R)
No
random
events
(R)
are
physical
ac,ons
(P)
No
P*
is
C*
All
C*
is
R
No
R
is
P
Invalid
argument
2
All
events
(E)
are
things
describable
by
science
(D)
All
mental
decisions
(M)
are
events
(E)
All
mental
decisions
(M)
are
things
describable
by
science
(D)
All
E*
is
D
All
M*
is
E
All
M
is
D*
All
things
describable
by
science
(D)
are
predictable
events
(P)
No
free
decisions
(F)
are
things
describable
by
science
(D)
No
free
decisions
(F)
are
predictable
events
(P)
3
All
D*
is
P
No
F*
is
D*
No
F
is
P

32. 1
No
physical
ac,ons
(P)
are
chance
occurrences
(C)
All
chance
occurrences
(C)
are
random
events
(R)
No
random
events
(R)
are
physical
ac,ons
(P)
No
P*
is
C*
All
C*
is
R
No
R
is
P
Invalid
argument
2
All
events
(E)
are
things
describable
by
science
(D)
All
mental
decisions
(M)
are
events
(E)
All
mental
decisions
(M)
are
things
describable
by
science
(D)
All
E*
is
D
All
M*
is
E
All
M
is
D*
All
things
describable
by
science
(D)
are
predictable
events
(P)
No
free
decisions
(F)
are
things
describable
by
science
(D)
No
free
decisions
(F)
are
predictable
events
(P)
3
All
D*
is
P
No
F*
is
D*
No
F
is
P
Valid
argument
Invalid
argument

33. (1) Rewrite the arguments as wf's (verbally), (2) translate to syllogisms, (3) test
validity by means of (3.a) the star test, and (3.b) a venn diagram
1. All mental decisions are things describable by science, but no things describable by science are
uncaused happenings; it follows that no mental decisions are uncaused happenings.
2. Some natural processes are not free choices, because some natural processes are not caused
occurrences, and no free choices are caused events.
3. All free decisions are uncaused happenings, and no mental decisions are uncaused happenings;
consequently, no mental decisions are free decisions.
4. All uses of free will are decisions, yet some decisions are careful reflections, hence some uses of
free will are careful reflections.
5. All persons who decide least are persons who are most free, so, since all persons who are most
free are persons who decide most, all persons who decide least are persons who decide most.
6. Some coerced decision are not free choices, hence some desires are coerced decisions, as some
desires are free choices.
7. All uncaused events are free decisions, but no mental decisions are free decisions, from which it
follows that no uncaused events are mental decisions.
8. All acts of free will are uncaused events, since all mental decisions are uncaused events, and all
acts of free will are mental decisions.
9. All acts of free will are uncaused events, so no acts of free will are existent things, because no
uncaused events are existent things.

34. Problem Set B

35. (1) Rewrite the arguments (verbally), (2) translate to wffs, (3) come up
with a conclusion that is valid (include implicit assumptions if necessary.
1. Ceausescu’s ban on abortion was designed to achieve one of his major aims: to
rapidly strengthen Romania by boosting its population.

36. 1
A
boost
of
the
popula,on
(B)
strengthens
a
country
(S)
The
ban
on
abor,on
(A)
gives
a
boost
to
the
popula,on
of
a
country
(B)
Therefore,
a
ban
on
abor,on
(A)
strengthens
a
country
(S)
All
B
is
S
All
A
is
B
All
A
is
S
1. Ceausescu’s ban on abortion was designed to achieve one of his major aims: to
rapidly strengthen Romania by boosting its population.

37. 1
A
boost
of
the
popula,on
(B)
strengthens
a
country
(S)
The
ban
on
abor,on
(A)
gives
a
boost
to
the
popula,on
of
a
country
(B)
Therefore,
a
ban
on
abor,on
(A)
strengthens
a
country
(S)
All
B
is
S
All
A
is
B
All
A
is
S
1. Ceausescu’s ban on abortion was designed to achieve one of his major aims: to
rapidly strengthen Romania by boosting its population.

38. 1
A
boost
of
the
popula,on
(B)
strengthens
a
country
(S)
The
ban
on
abor,on
(A)
gives
a
boost
to
the
popula,on
of
a
country
(B)
Therefore,
a
ban
on
abor,on
(A)
strengthens
a
country
(S)
All
B*
is
S
All
A*
is
B
All
A
is
S*
1. Ceausescu’s ban on abortion was designed to achieve one of his major aims: to
rapidly strengthen Romania by boosting its population.

39. 1
A
boost
of
the
popula,on
(B)
strengthens
a
country
(S)
The
ban
on
abor,on
(A)
gives
a
boost
to
the
popula,on
of
a
country
(B)
Therefore,
a
ban
on
abor,on
(A)
strengthens
a
country
(S)
All
B*
is
S
All
A*
is
B
All
A
is
S*
1. Ceausescu’s ban on abortion was designed to achieve one of his major aims: to
rapidly strengthen Romania by boosting its population.
Valid
argument

40. (1) Rewrite the arguments (verbally), (2) translate to wffs, (3) come up
with a conclusion that is valid (include implicit assumptions if necessary.
1. Government agents sardonically known as the Menstrual Police regularly rounded up women
in their work places to administer pregnancy tests. If a woman repeatedly failed to conceive,
she was forced to pay a steep “celibacy tax.”
2. On Christmas day of 1989 crime was at its peak in the United States... Experts were predicting
darker scenarios.
3. The evidence linking increased punishment with lower crime rates is very strong. Harsh prison
terms have been shown to act as both deterrent (for the would-be criminals on the street) and
prophylactic (for the would-be criminals who is already locked up)
4. Researchers found that in the instances where the woman was denied an abortion, she often
resented her baby and failed to provide it with good home... The researchers found that these
children were more likely to become criminals. (for the solution generalize this example from
MORE LIKELY to ALL and focus it in the case of unwanted children)