10.
TESTING FOR VALIDITY
WITH COMPLETE TRUTH TABLES
Let's go back to the
beginning of the class, when you were presented with this idea:
We
are interested, then, in what cases are pattern-perfect. We do
this primarily by using letters to stand for statements or parts of
statements. For instance, we'll just say "F" for the complete
statement "logic is fun" or "Sa" for "Alice is a student" and "(x)Ax"
for "everyone is ambitious." We will want to see how we can test
patterns for deductive validity (we have a couple of ways to do this)
and we will want to be able to show how, given certain statements as
premises, we could work our way step by step to an intended
conclusion.
Now that you have seen how we could individually symbolize any of the
parts of an argument--the statements that make up the premises and the
conclusion--we need to see how we can test whether in fact we have a
perfect pattern--a case in which it would not be possible for the
premises to be true while the conclusion is false.
Let's start off by introductng some punctuation that will indicate an
argument form is intended. We will separate the premises with
commas and introduce the conclusion with this symbol: |- (we call it an
assertibility sign).
Let's work with a couple of examples.
We have these two arguments: (1) Logic is easy if it is fun, but
it is not fun, so it must not be easy.
(2) Logic is easy only if it is fun, but it is not fun, so it must not
be easy.
You already know how to symbolize each of the three statements making
up each argument, so we will now represent the complete arguments
symbolically.
(1) Logic is easy if it is fun, but
it is not fun, so it must not be easy. F->E, ~F |- ~E
(remember not to just follow the English word order here)
(2) Logic is easy only
if it is fun, but it is not fun, so it must not be
easy. E->F, ~F |- ~E
Now we are going to set up truth tables with the premises and the
conclusion side by side.
E
F
F->E
~F
~E
1
1
1
0
0
1
0
1
1
0
0
1
0
0
1
0
0
1
1
1
In this first truth table we
have what we will call a bad line: a line in which all our premises true
but the conclusion is false.
This tells us that we do not have a perfect pattern and that the
argument form in fact is not valid. (To understand the point here
imagine how being fun makes logic easy, but even though logic is not
fun--boring, even-- it turns out still to be very easy.)
E
F
E->F
~F
~E
1
1
1
0
0
1
0
0
1
0
0
1
1
0
1
0
0
1
1
1
In the second truth table we do not
have a bad line since whenever we have a false conclusion we also have
a false premise. This means we do have a perfect pattern or what
we call a deductively valid argument form. (Again, the idea is
that we cannot even imagine how the conclusion could be wrong without
contradicting the information in the premises.)
Let's take two more examples.
Unless Alice
studies she will not pass. Alice does not pass, so she must not
have been studying. ~Sa v Pa, ~Pa |- ~Sa
Pa
Sa
~Sa
v Pa
~Pa
~Sa
1
1
1
0
0
1
0
1
0
1
0
1
0
1
0
0
0
1
1
1
Here again there is no bad
line. The argument is deductively valid.
Everyone who studies will pass. Someone did not study.
Therefore, someone did not pass. (x)(Sx->-Px), (Ex)~Sx |- (Ex)~Px
Here we set up a truth table so that we are talking about at least two
individuals in a group (we would increase this for each new existential
quantifier in our premises). This will automatically increase the
number of lines we need in order to cover all possibilities. Please note that
this is used as an example of how we could work with an argument
involving quantifiers, but later we will see a much easier
technique. You will not be expected to work with truth tables for
this type of argument.
Pa
Pb
Sa
Sb
(x)(Sx->Px)
(Ex)~Sx
(Ex)~Px
1
1
1
1
1
0
0
1
1
1
0
1
1
0
1
1
0
1
1
1
0
1
1
0
0
1
1
0
1
0
1
1
0
0
1
1
0
1
0
1
1
1
1
0
0
1
0
1
1
1
0
0
0
1
1
1
0
1
1
1
0
0
1
0
1
1
0
0
1
1
0
1
0
1
1
1
1
0
1
0
0
1
1
1
0
0
1
1
0
0
1
0
0
1
0
0
1
1
0
0
0
1
0
1
1
0
0
0
0
1
1
1
Here we have two bad lines:
first when person "a" studies but passes anyway and then again when
person "b" studies but also manages to pass (imagine that neither of
them had to study since either they were very smart or the test was
very easy). You should be able to see the parallel with the first
example, which is an instance of the formal fallacy of denying the
antecedent.
In addition to what you have seen already, we are able to talk about
what are called zero-premise
arguments. These are expressions that exist as tautologies, statements that
are always true no matter what the truth of the atomic elements (a P or an Sa) that go into making them
up. Let's look at two of these.
|- P v ~P
P
P
v ~P
1
1
0
1
|- [(P->Q) & P]
-> Q
P
Q
[(P->Q)
& P] -> Q
1
1
11
1
1
0
00
1
0
1
10
1
0
0
10
1
If you think about it, you
will see that we are expressing the valid argument form just as a
conditional that would always have to be true. Looked at this
way, all valid argument forms can be expressed as tautologies.
EXERCISES (ON YOUR OWN)
Use
truth tables to test for the validity of these two argument
forms. (Remember that you need eight lines since there are three
letters in use.)
