Transcript Chapter 8

Chapter 8 Lecture Notes
Deductive Arguments:
Propositional Logic
Chapter 8
Categorical logic is the oldest logic that we have, but it is
no longer regarded as the most basic part of logic (216)
This distinction goes to propositional logic. This is the
logic of propositions (think sentences) and their
compound. There are four basic logical terms in
propositional logic:
not
and
or
and
if then
The are sometimes called logical connectives.
Chapter 8
If we want to test the validity of an argument, then we can
use letters to represent complete sentences and logical
symbols to represent not, and, or, and if then.
It is important to see that once a letter is used to represent
a statement in an argument, that letter cannot be used
to represent another statement in the argument. (216)
For example, if we had the following argument:
1. I like ice-cream and I like pie.
Therefore
2. I like ice-cream.
Chapter 8
This argument would look like this using letters to represent
sentences and using symbols to represent the logical
connectives.
1. I  P
Therefore
2. I
‘I’ represents “I like ice-cream” and ‘P’ represents “I like pie”
Now we see the logical form of the argument. What we
need to do now is define the symbols we will be using to
represent: not, and, or, and if then.
Chapter 8
The logical connectives not, and, or, and if then will be
represented by the following: –, , , and .
We are going to define these symbols using truth-tables
and thus they will have truth-table definitions. The point
of a truth-table definition is to provide all the possible
cases of truth or falsity for the values of the the sentence
letters.
Each connective will have its own definition.
Chapter 8
Not
Not functions to change the truth of a statement to its
opposite. The truth-table definition for not is quite
simple.
If P is true, the not P is false. If P is false, then not P is
true. This is our truth-table definition of not.
Chapter 8
And
And is a conjunction and connects two conjucts. The
conjunction is true when both conjuncts are true and
false ever where else. Here is our truth-table definition
of and.
The only time P  Q is true is when both P and Q are true.
Chapter 8
Or
Or is disjunction and it is true when either one or both of
the disjuncts are true and false when both disjuncts are
false. Here is the truth-table definition of or.
Because we have defined or a being true when both
disjuncts are true, this is called inclusive or
Chapter 8
If then
If then is a logical connective that has parts with names
and is represented by the horseshoe: ‘’. The parts of a
conditional are the antecedent and the consequent.
So in a conditional like: If it is raining, then my car is
wet.
It is raining is the antecedent and my car is wet is the
consequent. Since the truth of the whole conditional is
based on the truth of the parts, it is important to know the
parts.
Chapter 8
We are going to define the horseshoe as true whenever the
consequent is true or whenever the antecedent is false.
The only time a conditional statement is false is when
the antecedent is true, and the consequent is false. So,
if it is raining and my car isn’t wet, because it is in the
garage, then the prior conditional is false. Here is the
truth-table definition of the conditional.
Chapter 8
Formalization of arguments makes testing the argument for
validity much easier. When we turn English sentences
into formalized statements with sentence letters and
logical connectives we have to be careful.
There are stylistic variants of the conditional as well as
uses of and that cannot be expressed with the truth-table
definition.
One should take special care to make sure that
formalization is appropriate for a given proposition.
Chapter 8
Creating truth-tables to test argument for validity requires
knowing how many sentence letters are in a given
argument. We can use this formula:
2n
Where ‘n’ represents the number of distinct statement
letters. So, if there is one statement, there will be two
rows on the truth-table; two statement letters will get four
rows; three letters will need eight rows, and four letters
will require 16 rows.
Chapter 8
In order to account for every possible combination of
values for the statement letters, we need a standard way
of creating truth-tables. Here is how you should do it.
After determining how many rows will occur in the truthtable, begin at the left most statement letter and fill the
table’s top half with Ts and the bottom half with Fs.
Then move to the right one letter and alternate T and F
making sure that half the rows of T and F from the
column to the left are filled by half with Ts and Fs. The
final column to the far right should ultimately end up
alternating Ts and Fs. This will account for all the
possible cases.
Chapter 8
Imagine the following argument:
1. D  S
2. –D
Therefore,
3. –S
Figure 8.5 shows the truth-table for this argument. We can
see that it is possible for the premises to all be true and
the conclusion false, and thus, this inference is invalid.
This happens to be the fallacy of denying the
antecedent.
Chapter 8
Methods of shortening truth-tables can be useful. Since we
are looking for cases where all the premises are truth
and the conclusion is false, you only have to look at rows
on a truth-table that have a false conclusion.
Of course if there are no instances where all the premises
are true and the conclusion is false, the argument in
question is a deductively valid, propositional argument.
Let’s briefly look at some issues with formalization and
translation.
Chapter 8
Translating English into propositional logic can be a bit
tricky. We need to watch for special cases and
instances of stylistic variants.
Not requires that we have a instance where the sentence in
question has the opposite truth value, not merely as a
contrary. See page 229 for more examples.
Chapter 8
There are instances of and that are not propositional, and
there are other words that function as and.
Jim and Mary are married to each other is not an case
where we can use  to join two proposition, because
there is only one proposition.
Words like ‘but’ and ‘although’ also have a core meaning of
like and. The real issue is that one has to be careful
when translating from English to propositional logic. See
page 229-230 for more examples.
Chapter 8
Or can have two meanings: inclusive and exclusive.
The inclusive meaning of or is just that statements
constructed with or are true when one or both disjucts
are true.
Exclusive or on the other had is false when both disjuncts
are true. This can be expressed in English as: You can
have soup or salad, but not both.
We shall default to the inclusive usage of or unless there is
special reason to think otherwise. See page 230-1
Chapter 8
Conditional statements can get complicated. One such
conditional is the counterfacutal conditional. These
conditionals have false antecedents as a matter of
course. For example,
If Hitler had never been born, WWII would still have occurred.
Is true under are current understanding of the conditional.
There are problems with this interpretation for
counterfactual (or subjunctive mood) conditional, and
thus we won’t let the horseshoe stand for them. For
more information about conditional see pages 231-4.
Chapter 8
Translation tips:
Both…and… is P  Q.
Neither…nor… is –(P  Q)
Implies that… is P  Q
Provided that… is Q  P
Only if… P  Q
Unless is best understood as or -- .
See pages 236-41 for detailed examples.
Chapter 8
Necessary, Sufficient, and Necessary and Sufficient
conditions can be represented by conditionals.
(1) Being mercury is sufficient for being a metal.
(2) Passing the final exam is a necessary condition for passing the class.
Both of these statements can be expressed as
conditionals.
(1) M  M1 (M = being mercury and M1 = being a metal)
(2) C  E (C = Passing the class and E = passing the final exam)
For full definitions of necessary and sufficient conditions
see pages 240-1 and the glossary of chapter 8.
Chapter 8
Sometimes argument can get too big to draw a truth-table
for them. When this happens, simple proofs and logical
equivalencies are useful. There are about 20 different
valid and invalid logical moves that everyone should try
to memorize. Some are:
Modus ponens, modus tollens, transpostion, De Morgan’s
Rules, Double negation and many others. (A more
complete list can be found on page 224)
When you cannot create a direct proof, you can use the
indirect proof method of a conditional proof (227).
Chapter 8
Terms to review:
Affirming the consequent
Conditional proof
Conjunction
Contradictory statements
Counterfactual
Denying the antecedent
Disjunction
Horseshoe
Biconditional
conditional statement
consequent
contrary statements
denial
dilemma argument
exclusive disjunction
inclusive disjunction
Chapter 8
Indirect proof
Modus tollens
Propositional logic
Sufficient condition
modus ponens
necessary condition
reductio ad absurdum
truth table