Hypotheticals: The If/Then Form

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Hypotheticals: The If/Then Form.

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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.
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:
• Arrange the statements into disjunctive form.
• Judge the argument’s validity in terms of the rules.
• Determine whether the premises and conclusion are true, and the argument sound.