1)
What is the formal system of Chapter 2 called?
R: The formal system presented in this
chapter is called pq-system.
2)
What are the distinct symbols of this formal
system?
R: p
q -.
3)
How many axioms in the pq-system?
R:
Infinite number of axioms.
4)
Write down the axiom schema for the pq-system.
R: xP
– Qx – is an axiom, whenever x is composed of hyphens only.
5)
What is a “schema”? Define the term.
R:
A mold to define something. How to do something.
6)
Write down the three shortest axioms in the
pq-system.
R:-
p - q- - / - -p - q - - - / - - - p - q - - - -
7)
Write down the sole rule of production for the
pq-system
R: Suppose x, y and z all stand for
particular strings containing only hyphens. And suppose that x p y q z is known
to be a theorem. Then x p y – q z – is a theorem
8)
Show that - -p - -q - - - - is a theorem of the
pq-system. That is, derive it from an axiom and repeated application of the
rule.
R: (1) - - p - q - - - axiom
(2) - - p - - q - - - - from (1)
9)
Show that - - - - -p- - - -q- - - - - - - - - is
a theorem of the pq-system. That is, derive it from an axiom and repeated
application of the rule.
R:
(1) - - - - - p - q - - - - - - axiom
(2) - - - - -p - - q - - - - - - - from(1)
(3) - - - - -p - - - q- - - - - - - from(2)
(4) - - - - -p - - - -q - - - - - - - - - from(3)
10)
Write down a string of symbols in the pq-system
which is not well formed
R: -
- p - - p - -p - - q - - - - - - - -
11)
State a decision procedure for the pq-system
R:
Take your theorem, iterate backwards, find the beginning and check whether it`s
an axiom or not, if it is an axiom, then it is by definition a theorem, and the
test is over.
12)
In the longest paragraph on page 48, Hofstadter
engages in some “top-down” reasoning. In one sentence, articulate exactly what
it is that he demonstrates with his top-down reasoning in this paragraph?
R: How
to find the beginning of all theorems, the axiom schemata or maybe you will
find something that is not an axiom.
13)
In one sentence, characterize “top-down”
reasoning
R: Start
in the theorems and find the axioms.
14)
In one sentence, characterize “bottom-up”
reasoning
R: Start
in the axioms and find the theorems
15)
Consider the procedure for generating theorems
of the pq-system given at the top of page 49. What will be in the bucket after
executing statements (1a) and (1b) and (2a) and (2b) and(3a) and (3b) – after
all six of these statements have been executed!
R: (1a) - p - q - -
(1b) - p - - q - - -
Bucket: - p - q
- -; - p - - q - - -
(2a) - - p - q - - -
(2b) - p - - q - - - ; - p - - - q - - -
-; - - p - - q - - - -
Bucket: - p - q - -; - p - - q - - -;- p - - - q - - - -; - - p - q - - -; -- p - - q - - -- ;
(3a) - - - p - q - - - -
(3b) - p - q - -; - p - - q - - -;
- p - - - q - - - -; - p - - - - q - - - - - ; - - p - q - - -; - - p -
- q - - - -;
; - - p - - - q - - - - -; - - -p - q - -
- -; - - -p - -q - - - - -;
16)
16 What role does the procedure introduced on
the top of page 49 ply in Hofstander`s presentation of the pq-system and
related matters? Answer in just one sentence!
17)
What is an isomorphism?
R: Two
complex structures can be mapped onto each other, in such a way that to each
part of one structure there is a corresponding part in the other structure.
18)
What is an interpretation in the context of a
formal system?
R: The
symbol-word correspondence that you give to your formal system.
19)
When was Linear B deciphered?
20)
How many meaningful interpretations of the
pq-system did Hofstadter present in this chapter?
R:
2
21)
How many meaningless presentations of the
pq-system are there?
R: Infinite.
22)
In 50 plus or minus 20 words, summarize what
Hofstadter says in the section titled “Formal systems and reality.”
R: Hofstadter
says that is possible combine reality and formal systems, and maybe the reality
is just a big formal system. Affirm that the reality is deterministic is
something hard to accept.
PDF
