Logic Page
Section
1:
PROPOSITIONS:
SIMPLE AND COMPLEX
PROPOSITIONS make a claim or statement that is either
true or false. Propositions are either SIMPLE or COMPLEX. SIMPLE propositions
contain only one subject and predicate. COMPLEX propositions have one or more
subjects and one or more predicates.
(a) SIMPLE PROPOSITIONS: One class of simple
statements is CATEGORICAL propositions. Categorical propositions are either
particular or general, and either affirmative or negative. Categorical
propositions can be classified in the following way:
EXAMPLES
Universal Affirmative: All
humans are rational.
Universal Negative: No humans are rational.
Particular Affirmative: Some human is rational.
Particular Negative: Some human is not rational.
We assign simple names to each type of proposition so
that they can be dealt with more easily. The names correspond to the first four
vowels: A, E, I, and O.
A: Universal Affirmative
E: Universal Negative
I: Particular Affirmative
O: Particular Negative
Here are a
few instances that require more thought:
"Most
humans are rational" -- this is an I since it is
not universal.
"Humans are rational"-- this
is an A since it is universal.
"Not all humans are rational."
--this is an O.
"Socrates is rational."-- propositions with proper names are universal; this is an
A.
"Socrates is not rational."-- this is an E.
Section
2:
Square of
Opposition
There are certain relationships that hold between the
four types of propositions mentioned above. A device that is helpful in
learning these relationships is the SQUARE OF OPPOSITION.

If A is true, then O is
false, and vice versa.
If I is true, then E is
false, and vice versa.
If A is true, then I is
true.
If E is true, then O is true.
Remember:
Contradictories: Always opposite truth values.
Contrary: Cannot both be true.
Keep in mind the fact that the above classification is
based upon the form of propositions; remember that logic is primarily concerned
with the form of propositions
and arguments.
Section
3:
Categorical
Syllogism
When trying to identify arguments within passages,
always look for the main conclusion first. Identify inference signs introducing
conclusions and circle them. Having done so, you can find the premises that
support the conclusions by circling inference signs introducing them.
Overview of the
Categorical Syllogism
SYLLOGISMS: One type of argument composed entirely of
simple statements is the CATEGORICAL SYLLOGISM. A SYLLOGISM is a set of three
propositions, one of which is the conclusion and two of which are the PREMISES.
It will suffice to think of premises as reasons. A CATEGORICAL SYLLOGISM
contains only categorical propositions. Categorical syllogisms have only three
terms: the major term, the minor term and the middle term. The MIDDLE TERM is
the term occurring in each of the two premises; it is of primary importance in
the method we will use for assessing arguments.
(Examples)
All humans are
rational.
All plumbers are human.
So, all plumbers
are rational.
No lover of fine art is
uncultured.
Every educated person is a lover of fine art.
Thus, no educated person is uncultured.
Are these arguments good? The categorical syllogisms
above strike one immediately as being correct. A technique discussed in class
and employed throughout the term should provide a quick and easy procedure for
testing categorical syllogisms. This method will help you become familiar with
the relationships between classes in some arguments. Traditional methods for
assessing the validity of syllogisms rely on identifying their FIGURE and MOOD.
FIGURE is determined by the position of the middle term. There are three:
Figure, Mood and Validity for the Syllogism
Fig.
1
Fig. 2 Fig. 3
M is
P
P is M M is P
S is
M
S is M M is S
S is
P
S is P S is
P
MOOD is determined by assigning the appropriate
letter-name to each proposition. A syllogism which consists
of Universal Affirmative propositions has the mood
AAA:
All mammals are
animals.
Fig. 1
All humans are mammals
AAA
All humans are animals.
FOUR IMPORTANT FIRST FIGURE MOODS
AAA: Barbara
EAE: Celarent
AII: Darii
EIO: Ferio
Four general rules for the validity of
categorical syllogisms:
1) If an argument is Figure 1, and AAA, EAE, EIO, or
AII, then it is valid.
- This does not imply that an argument is invalid if it lacks one of these
features. There are many other valid moods in figure 1, but these four
are the most common.
2) For a categorical syllogism to be valid it must
have a negative conclusion if it has a negative premise and vice versa.
- This does not imply that a syllogism with a negative
premise and conclusion is valid.
3) A valid argument in Figure 2 must have at least one
negative premise.
- This implies that the conclusion of a valid figure 2 syllogism
will be negative. (from rule #2)
4) Valid categorical syllogisms cannot have two
negative premises.
- This implies that a syllogism with two negative
premises is invalid.
Section
4:
Enthymemes
ENTHYMEMES are syllogisms with a missing proposition.
Since SYLLOGISMS are arguments composed of three propositions, it follows that
ENTHYMEMES only have two propositions.
(Example)
Socrates is
human. {It is your task to fill in
the missing premise}
Socrates is rational.
Steps for
constructing a valid syllogism from a conclusion:
1) Find the
main conclusion from the assigned reading.
Example:
Capital punishment is immoral.
2) Next
determine whether it is an A, E, I or O proposition.
Write it in that form.
Example: All capital punishment is immoral.
3) Set the
argument up in Figure 1. Use M to stand for the middle term.
Determine which of the 4 moods to use from the conclusion.
Example:
Anything that is M is immoral.
All capital punishment is M.
This
argument is Figure 1 AAA
All capital punishment is
immoral.
4) Now
fill in for M by finding the main point from the reading that supports the conclusion.
Example:
Anything that fails to treat humans with dignity
is immoral.
All capital punishment is fails to treat humans with
dignity.
All capital punishment is
immoral.
Section
5:
COMPLEX
PROPOSITIONS AND ARGUMENTS
COMPLEX PROPOSITIONS: We will treat complex
propositions of the following form:
CONDITIONAL: If humans are
rational, then they are free.
NEGATIVE: It is not the case that humans are
free.
We will symbolize propositions which
occur in complex arguments by assigning a letter to the proposition or some
part of it (any letters will suffice as long as we are consistent). Negative
propositions will be introduced by a minus sign '-'.
CONDITIONAL PROPOSITIONS contain the logical
connective If...then.... . For example, the
proposition "If humans are rational, then they are free" can be
symbolized "If R, then F". There are two argument patterns commonly
associated with conditional statements; the first is MODUS PONENS:
If P, then Q
P
Therefore, Q
Logicians refer to what comes before the arrow in a
conditional statement as the ANTECEDENT, and what comes after the arrow as the
CONSEQUENT. Modus ponens is an argument pattern which
affirms the antecedent.
The second argument pattern which
involves conditional propositions is called MODUS TOLLENS:
If P, then Q
-Q
Therefore, -P
Modus tollens is based upon
the denial of the consequent.
IMPORTANT REMINDERS:
'unless' always introduces a negated antecedent of a
conditional.
'only if' introduces the
consequent of a conditional.
`if' always introduces the
antecedent of a conditional.
Help with important reminders.
Complex argument practice exercises
Section
6:
VALIDITY/SOUNDNESS
One general goal of logic is to provide some means of
determining whether an argument is good or not. Since logic deals with the form
of propositions and arguments, a good argument will be one with a correct form.
But there are at least two important questions that need to be answered. 1)
What does it mean to say that an argument has good form? 2) Is having correct
form enough to ensure that an argument is good in the ordinary sense?
The first question can be answered by reflecting on
the concept of logical validity. An argument is valid if and only if its
conclusion follows from its premises. Another way of stating this is that an
argument is valid if and only if its form is such that its premises necessitate
its conclusion. Given this definition of validity, it will follow that if the
premises of a valid argument are true, then the conclusion would have to be
true is well. In other words, validity preserves truth. It is important to
remember, however, that the premises of a valid argument need not be true. Correct
form is the criterion for validity.
Is correct form enough to ensure that an argument is
good? This brings us to question (2). Answering this question depends on what
we mean by 'good' in this context. The following argument is logically valid,
but it does not satisfy our usual conception of a good argument:
All humans
are male.
All
teachers are human.
All
teachers are male.
The reason that this argument is not "good"
is that it has a false first premise and a conclusion that is false as well.
This points to something we should expect to find in "good" arguments
that we may not find in valid arguments-- true premises and a true conclusion.
Soundness is a notion in logic that captures are ordinary sense of what a good argument
is. An argument is sound if and only if it is valid and has true premises.
Since validity preserves truth, it follows that the conclusion of a sound
argument will be true also. We can see then that logical validity is only a
minimal condition for good arguments.
