From “What Is The Name Of This Book” by Raymond Smullyan
Chapter 8 pages 99 to 103
Many of the puzzles in this chapter deal with so-called
conditional statements: statements of the form
“If P is true then Q is true,”
where P, Q are statements under consideration. Before turning to
puzzles of this type, we must carefully clear up some
ambiguities which might arise. There are certain facts about
such statements which everyone agrees on, but there are others
about which there appears to be considerable disagreement.
Let us turn to a concrete example. Consider the following
statement:
(1)If John is guilty, then his wife is guilty.
Everyone will agree that if John is guilty and if statement (1)
is true, then is wife is also guilty.
Everyone will also agree that if John is guilty and his wife is
innocent, then statement (1) must be false.
Now, suppose it is known that his wife is guilty, but it is not
known whether John is guilty or innocent. Would you then say
that statement (1) is true or not? Would you not say that
whether John is guilty or whether he is innocent, his wife is
guilty in any case? Or would you not say: If John is guilty then
his wife is guilty, and if John is innocent then his wife is
guilty?
Illustrations of this use of language abound in the literature:
In Rudyard Kipling’s story Riki-Tiki-Tavi, the cobra says to the
terrified family, “If you move I will strike, and if you don’t
move I will strike.” This means nothing more less than: “I will
strike.” There is also the story of the Zen-master Tokusan, who
used to answer all questions as well as nonquestions, with blows
from his stick. His famous saying is: “Thirty blows when you
have s something to say; thirty blows just the same when you
have nothing to say.”
The upshot is that if a statement Q is true outright, then so is
the statement, “If P then Q” (as well as the statement, “If not
P, then Q”).
The most controversial case of all is this: Supposing P, Q are
both false. Then is the statement, “If P then Q” true or false?
Or does it depend on what P and Q are? Returning, to our
example, if John and his wife are both innocent, then should
statement (1) be called true or not? We shall return to this
vital question shortly.
A related question is this: We have already agreed that if John
is guilty and his wife innocent, then statement (1) must be
false. Is the converse true? That is, if statement (1) is false,
does it follow that John must be guilty and his wife innocent?
Put otherwise, is it the case that the only way that (1) can be
false is that John be guilty and his wife innocent? Well,
according to the way most logicians, mathematicians, and
scientists use the words “if...then,” the answer is “yes,” and
this is the convention we shall adopt. In other words, given any
two statements, P and Q, whenever I write “If P then Q” I shall
mean nothing more nor less than “It is not the case that P is
true and Q is false.” In particular, this means that if John and
his wife are both innocent, then statement (1) is to be regarded
as true. For the only way the statement can be false is that
John is guilty and his wife is innocent, and this state of
affairs can’t hold if John and his wife are both innocent.
Stated otherwise, if John and his wife are both innocent, then
it is certainly not the case that John is guilty and his wife is
innocent, therefore the statement cannot be false.
The following is an even more bizarre example:
(2) If Confucius was born in Texas, then I am Dracula.
All statement (2) is intended to mean is that it is not the case
that Confucius was born in Texas and that I am not Dracula. This
indeed is so, since Confucius was not born in Texas. Therefore
statement (2) is to be regarded as true.
Another way to look at the matter is that the only way (2) can
be false is if Confucius was born in Texas and I am not Dracula.
Well, since Confucius was not born in Texas, then it can’t be
that Confucius was born in Texas and that I am not Dracula. In
other words, (2) cannot be false, so it must be true.
Now let us consider two arbitrary statements P, Q, and the
`following statement formed from them:
(3) If P then Q.
This statement is symbolized: P->Q, and is alternatively read:
“P implies Q.” The use of the word “implies” may be somewhat
unfortunate, but it has found its way into the literature in
this sense. All the statement means, as we have seen, is that it
is not the case that P is true and Q s false. Thus we have the
following facts:
Fact
Fact
Fact
true
1: If P is false, then P->Q is automatically true.
2: If Q is true, then P->Q is automatically true.
3: The one and only way that P->Q can be false is that P is
and Q is false.
Fact 1 is sometimes paraphrased: “A false proposition implies
any proposition.” This statement came as quite a shock to many a
philosopher. Fact 2 is sometimes paraphrased: “A true
proposition is implied by any proposition.”
A Truth-Table Summary
Given any two statements, P, Q, there are always exactly four
possibilities: (1) P, Q are both true; (2) P is true and Q is
false; (3) P is false and Q is true; (4) P, Q are both false.
One and only one of these possibilities must hold. Now let us
consider the statement, “If P then Q” (P->Q). Can it be
determined in which of the four cases it holds and in which ones
it doesn’t? Yes it can, by the following analysis:
Case 1: P and Q are both true. In this case Q is true; hence P>Q is true by Fact 2
Case 2: P is true and Q is false. In this case, P->Q is false by
Fact 3.
Case 3: P is false and Q is true. Then P->Q is true by Fact 1
(also by Fact 2)
Case 4: P is false and Q is false. Then P-> Q is true by Fact 1.
These four cases are all summarized in the following table,
called the truth-table for implication.
P
T
T
F
F
Q
T
F
T
F
P -> Q
T
F
T
T
The first row, T,T,T (true, true, true), means that when P is
true and Q is true, P->Q is true. The second row, T, F, F means
that when P is true and Q is false then P->Q is false. The third
row, F,T,T (false, true, true) says that when P is false and Q
is true, P-> Q is true, and the fourth row, F,F,T (false, false,
true) says that when P is false and Q is false, then P->Q is
true.
We note that P->Q is true in three out of four of those cases;
only in the second is it false.
Another Property of Implication
Another important property of implication is this: To show that
a statement “If P then Q” holds, it suffices to assume P as
premise and then show that Q must follow. In other words, if the
assumption of P leads to Q as a conclusion, then the statement
“If P then Q” is established. We shall henceforth refer to this
fact as Fact 4
Boolean functions are like other functions
F(x,y)= x + y
F(1,2) = ?
F(1,2) = 1 + 2 = 3
Propositional Logic
What are the truth tables for
P
T
T
F
F
Q
T
F
T
F
Q unless P
F
T
T
T
~Q or ~P
F
T
T
T
P
T
T
F
F
Q
T
F
T
F
Q if P
T
F
T
T
P
T
T
F
F
Q
T
F
T
F
P is sufficient for Q
T
F
T
T
Q or ~P | or(Q, ~P)
T
F
T
T
P->Q
T
F
T
T
P
T
T
F
F
Q
T
F
T
F
P is necessary for q
T
T
F
T
~Q or P
T
T
F
T
~P->~Q
T
T
F
T
Q->P
T
T
F
T
P
T
T
F
F
Q
T
F
T
F
Q only if P
T
T
F
T
~Q or P
T
T
F
T
~P->~Q
T
T
F
T
Q->P
T
T
F
T
P
Q
p is necessary and
sufficient for q
p->q and q->p
p ≡ q
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
T
T
F
F
T
Q if
and
only if
P
T
F
F
T
q if and only if p
What is logical equivalence?
p is a sufficient and necessary conditions for q
Predicate Logic
A singular statement makes an assertion about a specifically
named person, place, thing, or time.
Socrates is mortal Ms
Fred is mortal Mf
Jane is mortal Mj
Tokyo is populous Pt
The Sun-Times is a newspaper Ns
King Lear is not a fairy tale ~F
Berlioz was not a German ~Gb
If Paris is beautiful then Andre told the truth Bp -> Ta
Irene is either a doctor or a lawyer Di or Li
Lx means x is a lawyer but you don’t know what x represents
Senator Wilkins will be elected only if he campaigns Ew ->Cw
General Motors will prosper if either Nissan is crippled by a
strike or Subaru declares bankruptcy (Cn or Ds)-> Pg
(C or D) -> P rewrites the predicate logic as propositional
logic
Indianapolis gets rain if and only if Chicago and Milwaukee gets
snow Ri ≡ (Sc * Sm)
Quantifiers
Existential Quantifier
Some s have property P
Universal Quantifier
All s have property P
What is the relation between negation and the universal and
existential quantifiers?
~∀𝑥∃𝑦𝑃(𝑥, 𝑦) ≡ ∃𝑥~∃𝑦𝑃(𝑥, 𝑦) ≡ ∃𝑥∀𝑦𝑃(𝑥, 𝑦)
~∃𝑥∀𝑦𝑃(𝑥, 𝑦) ≡ ∀𝑥~∀𝑦𝑃(𝑥, 𝑦) ≡ ∀𝑥∃𝑦𝑃(𝑥, 𝑦)
To translate statements involving quantifiers, first translate
the statement as a singular statement and then try to add the
quantifiers.
All triangles have 180 degrees
D means “triangle has 180 degrees”
Dt means the triangle t has 180 degrees
∀𝑥𝐷𝑥 where x is a triangle
There exist triangles with 180 degrees
∃𝑥𝐷𝑥 where x is a triangle