Patterns of Deductive Thinking• In deductive thinking we reason from a broad claim to some specific conclusion that can be drawn from it.

•In inductive thinking we reason from specific instances to some generalization based upon those instances.

•If the premises are general in nature (e.g., “All dogs bark”) and the conclusion involves a particular item (e.g., “Dusty barks”), the reasoning is deductive.

•If the premises are particular in nature (e.g., Dusty is a dog that barks”) and the conclusion is a generalization (e.g., “All male dogs bark.”), the reasoning is inductive.

Using Categorical Arguments

• They typically begin with “All” or “No,” make some categorical claim, then reach some specific conclusion. For example,

– All Planets are bodies that orbit the sun.

Venus is a planet.

Venus is a body that orbits the sun.

Upon examination of this syllogism, we can see that the statements consist of three terms as subject and predicate, that is, “planets,” “bodies that orbit the sun,” and “Venus.” Planet is called the middle term, and it is recognizable because it appears twice in the premises. The verbs “are” and “is” are called copulas, and they function to connect the terms.

Using Categorical Arguments II• When we are uncertain whether a conclusion follows from

the premises we have to use strict procedures to test the validity of the reasoning.

• The words “all,”, “no,” and “some” are called quantifiers because they specify how much of the subject class is included or excluded from the predicate class. The first form above asserts that the whole subject class is included in the predicate class, the second that the whole subject class is excluded in the predicate class, and so on.

Inductive and Deductive Reasoning

• Both forms of reasoning have subclassifications. Deductive thinking has three patterns: categorical, hypothetical, and disjunctive. Inductive reasoning has four types: analogy, causation, generalization, and hypothesis.

• When a deductive argument is not just broad based but begins with a universal claim, it is referred to as categorical in nature. The major premise is not surrounded by qualifications, exceptions, or alternatives but asserts that something is the case universally. The deductive arguments we have discussed so far are mainly of this pattern.

Universal, Particular, Affirmative or Negative

Sentence Standard Form Attribute

All snakes are slimy. A All S is P. Universal affirmative

No snakes are slimy. E No S is P. Universal negative

Some snakes are slimy. I Some S is P. Particular affirmative

Some snakes are not slimy.

O Some S is not P. Particular negative

Table 8.1 in the textbook, page 155

Distribution• Distribution is an attribute of the terms (subject and

predicate) of propositions. A term is said to be distributed if the proposition makes an assertion about every member of the class denoted by the term; otherwise, it is undistributed. In other words, a term is distributed if and only if the statement assigns (or distributes) an attribute to every member of the class denoted by the term. Thus, if a statement asserts something about every member of the S class, then S is distributed; otherwise S and P are undistributed.

All S are P

• Here is another way to look at All S are P.

The S circle is contained in the P circle, which represents the fact that every member of S is a member of P. Through reference to this diagram, it is clear that every member of S is in the P class. But the statement does not make a claim about every member of the P class, since there may be some members of the P class that are outside of S.

All S are P II

• Thus, by the definition of “distributed term”, S is distributed and P is not. In other words for any (A) proposition, the subject term, whatever it may be, is distributed and the predicate term is undistributed.

No S are P• “No S are P” states that the S and P class are separate, which

may be represented as follows:

This statement makes a claim about every member of S and every member of P. It asserts that every member of S is separate from every member of P, and also that every member of P is separate from every member of S. Both the subject and the predicate terms of universal negative (E) propositions are distributed.

Some S are P• The particular affirmative (I) proposition states that at least one

member of S is a member of P. If we represent this one member of S that we are certain about by an asterisk, the resulting diagram looks like this:

Since the asterisk is inside the P class, it represents something that is simultaneously an S and a P; in other words, it represents a member of the S class that is also a member of the P class. Thus, the statement “Some S are P” makes a claim about one member (at least) of S and also one member (at least) of P, but not about all members of either class. Thus, neither S or P is distributed.

Some S are not P• The particular negative (O) proposition asserts that at least one

member of S is not a member of P. If we once again represent this one member of S by an asterisk, the resulting diagram is as follows:

Since the other members of S may or may not be outside of P, it is clear that the statement “Some S are not P” does not make a claim about every member of S, so S is not distributed. But, as may be seen from the diagram, the statement does assert that the entire P class is separated from this one member of the S that is outside; that is, it does make a claim about every member of P. Thus, in the particular negative (O) proposition, P is distributed and S is undistributed.

Two mnemonic devices for distribution

• “Unprepared Students Never Pass”

Universals distribute Subjects.

Negatives distribute Predicates.• “Any Student Earning B’s Is Not On Probation”

– A distributes Subject.

– E distributes Both.

– I distributes Neither.

– O distributes Predicate.

Table 8.2 Distribution in Four Standard Types of Statements, page

155 of the textType of Sentence Subject Predicate

All parrots are birds. (A All S is P.)

Distributed Undistributed

No wars are profitable.

(E No S is P.)

Distributed Distributed

Some diseases are tropical. ( I Some S is P.)

Undistributed Undistributed

Some New Englanders are not friendly. (O Some S is not P.)

Undistributed Distributed

• Once we understand affirmative and negative and the concept of distribution, we can apply the rules governing the validity of deductive arguments of a categorical type. There are four rules:

1. At least one of the premises must be affirmative.2. If a premise is negative then the conclusion must be

negative, and if the conclusion is negative then a premise must be negative.

3. The middle term must be distributed at least once.4. Any term distributed in the conclusion must also be

distributed in a premise.

From page 156 in the textbook:

From these four rules we can judge that the following syllogisms are invalid:

Violates Rule 1:No Australians are poor swimmers.Some poor swimmers are not sailors.No sailors are Australians.

Violates Rule 2:No fish is fattening food.All fattening food is tasty.Some fish is tasty.

• Violates Rule 3:

All feminists are pro-choice.

Some Communists are pro-choice.

Some feminists are Communists.

• Violates rule 4:

All stars are bright.

No planets are stars.

No planets are bright.

• Having understood the summary parts of a syllogism and how to judge the soundness of a formal argument, we can now apply what we know to the arguments we meet and to the ones we make.

• The steps are:1. Separate the conclusion from the premises (the claims from the

warrant).2. Paraphrase the sentences into standard form A, E, I, O.3. Arrange the statements into a categorical syllogism, completing any

enthymemes.4. Judge the validity of the syllogism in terms of the four rules, using the

factors of affirmative or negative distribution.5. Determine whether the premises and conclusion are true and the

argument sound.

Please turn to page 157 in the textbook. We are going to read and analyze the argument about feminists in the book.•Please turn to page 161 of your textbook.

Hypotheticals: The If/Then Form

• Hypothetical arguments are usually more obvious than categorical ones. A hypothetical argument has an “if/then” pattern. It is conditional rather than making some absolute claim. We say that, provided one thing is true, then another thing would follow. For instance, if the ground is wet then it must have rained; if the bells are chiming, then I must be late for class; if he is the starting quarterback, then he must be off the injured list. An assumption is made at the start and the argument then carries out the implications of that assumption.

Hypotheticals: The If/Then Form II

• The first part of the major premise, from “if” to “then” is called the antecedent, and the second part, from “then” to the end of the sentence, is called the consequent. Antecedent and consequent mean nothing more than the part that goes before and the part that goes afterward.

• Take the following as a typical example of a valid hypothetical syllogism:

If Emily is a doctor, then she can cure bronchitis. Emily is a doctor.

She can cure bronchitis.

Hypotheticals: The If/Then Form III

• The argument is perfectly valid because, in the minor premise, we have affirmed the antecedent “Emily is a doctor,” then drawn the conclusion that follows from it, that “she can cure bronchitis.”

• Another valid form would be:If Emily is a doctor, then she can cure bronchitis.Emily can’t cure bronchitis.Emily is not a doctor.• Here we have denied the consequent, and although the

reasoning might be more difficult to see, it is also correct. The assumption is that every doctor can cure bronchitis, and if Emily is unable to do this then she cannot be a doctor.

Hypotheticals: The If/Then Form IV• These arguments are arranged in two different patterns but

in both cases the conclusion follows from the premises. From this we can generalize that the two valid forms of hypothetical thinking are affirming the antecedent and denying the consequent.

• In contrast to these valid forms, take the following two syllogisms:

• If Emily is a doctor, then she can cure bronchitis. Emily is not a doctor. Emily can’t cure bronchitis.

Hypotheticals: The If/Then Form V

• Here the conclusion does not follow logically, for although Emily is not a doctor, that does not mean she cannot cure bronchitis. Although all doctors can cure bronchitis, we do not know that only doctors (and no one else) can cure bronchitis.

• In this process of reasoning, we have denied the antecedent, which is an invalid form of a hypothetical argument.

Hypotheticals: The If/Then Form VI• Another invalid argument:

• If Emily is a doctor, then she can cure bronchitis.

Emily can cure bronchitis.

Emily is a doctor.

• This thinking is also incorrect, for just because Emily can cure bronchitis that does not make her a doctor. Although all doctors can cure bronchitis, that does not mean only doctors can cure bronchitis. This error is known as affirming the consequent.

Hypotheticals: The If/Then Form VII

• Please turn to pages 166 and 167 in your textbooks.

Disjunctives: Either/Or Alternatives• In a disjunctive sentence two possibilities are presented, at

least one of which is true (although both might be). If we say, for example, “Either we will stay at home or we will go to the movies tonight,” that is a disjunct. So are the sentences, “Either you are in class or you are absent,” and “The man is either fat or skinny.”

• One of the disjuncts has to be true, so if we know one of the alternatives to be false, we can declare the other to be true and produce a valid argument. It does not matter which disjunct we eliminate; the one remaining must be true.

Disjunctives: Either/Or Alternatives II• In diagram form, then a valid disjunctive argument would

appear this way:• Either P or Q not P.

Therefore Q• Now we said that at least one alternative is true, but in fact

both could be. That means we would not get a valid argument by affirming one part of the disjunct in a minor premise and denying the other in our conclusion. Since both parts might be true, one disjunct is not eliminated when we affirm the other.

Disjunctives: Either/Or Alternatives III

• For example:• Either I am paranoid or someone is out to get me.

My therapist says I am paranoid. Therefore No one is out to get me.

• The fallacy is that I could be paranoid and someone may be out to get me.

• Another example, “Either it is Monday or we are in Critical Thinking class.” Actually, both might be true. Affirming one does not rule out the other. In diagram form the mistake looks like this:– Either P or Q

P not Q

Disjunctives: Either/Or Alternatives III

• This leads us to the two rules about disjunctives: In a valid disjunctive argument we deny one of the disjuncts to affirm the other. An invalid disjunctive argument is one in which we affirm one of the disjuncts and deny the other.

Disjunctives: Either/Or Alternatives IV

• One qualification should be mentioned. In some types of disjuncts we do eliminate one part by affirming the other:

• Either I am in Critical Thinking class today or I am absent.

I am in Critical Thinking class today.

I am not absent.

• This is not a rule, though. Do not count on it to always hold.

• It is important to note that the word “or” has two possible senses. In its exclusive sense, the word “or” eliminates or excludes one of the possibilities. For example, if a waiter tells you, “You can have soup or salad,” he usually means that you can have either soup or salad but not both. In its non-exclusive sense, the word “or” does not exclude either possibility. For example, your advisor may inform you, “To fulfill your science requirements, you can take biology or chemistry.” What your advisor usually means is that you can take biology, chemistry, or both.

• In which sense are we supposed to understand the word “or” for the purpose of logic? In logic, the convention is to take the word “or” in its nonexclusive sense. A disjunction such as “The steak is good or the salad is fresh,” is true if either the steak is good, or the salad is fresh, or both.

• Therefore, saying the steak is good does not proven anything about the freshness of the salad. Since the steak is good makes the statement true, it will still be true whether the salad is fresh or not.

• However, if we say the steak is not good, then we can conclude that the salad is fresh. Because one of the disjuncts (but not necessarily only one) must be true.

• A disjunction is false if and only if both of its disjuncts are false.

The steps for judging disjunctive arguments are similar to those for hypotheticals, namely:

1. Arrange the statements into disjunctive form.2. Judge the argument’s validity in terms of the

rules.3. Determine whether the premises and conclusion

are true, and the argument sound.

• Please turn to page 173 of your textbook.