Logic Page

Section 1: 
PROPOSITIONS: SIMPLE AND COMPLEX

Introduction

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. 

Overview of the Square

     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. 

Square of Opposition Practice

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. 

Overview

(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 

Overview

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

Conditional 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.