Inductive and Deductive Reasoning

Many people distinguish between two basic kinds of argument: inductive and deductive. Induction is usually described as moving from the specific to the general, while deduction begins with the general and ends with the specific; arguments based on experience or observation are best expressed inductively, while arguments based on laws, rules, or other widely accepted principles are best expressed deductively. Consider the following example:

Adham is using inductive reasoning, arguing from observation, while Rizik is using deductive reasoning, arguing from the law of gravity. Rizik's argument is clearly from the general (the law of gravity) to the specific (this kick); Adham's argument may be less obviously from the specific (each individual instance in which he has observed balls being kicked up and coming back down) to the general (the prediction that a similar event will result in a similar outcome in the future) because he has stated it in terms only of the next similar event--the next time he kicks the ball.

As you can see, the difference between inductive and deducative reasoning is mostly in the way the arguments are expressed. Any inductive argument can also be expressed deductively, and any deductive argument can also be expressed inductively.

Even so, it is important to recognize whether the form of an argument is inductive or deductive, because each requires different sorts of support. Adham's inductive argument, above, is supported by his previous observations, while Rizik's deductive argument is supported by his reference to the law of gravity. Thus, Adham could provide additional support by detailing those observations, without any recourse to books or theories of physics, while Rizik could provide additional support by discussing Newton's law, even if Rizik himself had never seen a ball kicked.

The appropriate selection of an inductive or deductive format for a specific first steps toward sound argumentation.

Exercises for Induction and Deduction

1. Which of the following claims would be best expressed by inductive reasoning?

Your first quiz grade usually indicates how you will do in the course.

The final exam accounts for 30% of the course grade.

Late papers will not be accepted.

Introduction to Deduction

As we explain in the Introduction to Induction and Deduction, an argument is inductive if its major premise is based on observation or experience, and deductive if its major premise is based on a rule, law, principle, or generalization. In general, there are two distinct ways of expressing a deductive argument: as a syllogism, or as a conditional. Any deductive argument can be expressed as either a syllogism or a conditional, though some arguments may seem to lend themselves more naturally to one form or the other. Similarly, tests for the validity of syllogisms and conditionals may appear quite different, but do essentially the same thing.

Syllogisms: The major premise of a syllogism states that something, Y, is or is not true for all or part of some group, X; the minor premise affirms or denies that some group or individual, Z, is part of X; and the argument then concludes whether that thing Y (from the major premise) is true or not true for that group or individual Z (from the minor premise). One form of a syllogism can be expressed by the following paradigm:

All X are Y

Z is X

Therefore, Z is Y

Consider the following example:

Everyone in class today received instructions for writing the essay. Mandia was in class today. Therefore, Mandia received instructions for writing the essay.

You might think that "everyone in class today received instructions for the essay" sounds like an observation, but it is a generalization: no observer is identified, and no process of observation is recounted. By using a generalization, we focus attention more directly on the truth of an assertion (and less on the manner of its verification); this is especially effective when the generalization is widely accepted, or when there is strong evidence to support it.

We can restate the argument as follows:

[Major:] "Receiving instructions" is true for all of the group "in class today."

[Minor:] "Mandia" is a member of the group "in class today."

[Conclusion:] "Receiving instructions" is true for "Mandia."

Notice that, twice, the phrase in the original example, "received instructions for writing the essay," became in the restatement, "receiving instructions." There are two reasons for this. First, a restatement of an argument should eliminate or shorten unnecessary terms, to make the argument more comprehensible. Here, we shortened "instructions for writing the essay" to "instructions"; if significant, the phrase's original form can be resubstituted in the conclusion.

Second, in order to avoid confusion, it is always best to use a state-of-being verb (for example, forms of the verb "to be") in the restatement of an argument, and convert the original verbs to other parts of speech. In this case, "received" has become a participial phrase, "receiving instructions," that functions as a noun.

Conditionals: The other common form of a deductive argument, a conditional, expresses that same reasoning in a different way. The major premise is, If something is true of P, then something is true of Q. The minor premise either affirms that it is true of P, or denies that it is true of Q. In the former case, the argument concludes that the something is true of Q; in the latter, that something is not true of P. One form of a conditional is expressed by the following paradigm:

If P then Q

P

Therefore, Q

The above example could be given in the form of a conditional as follows:

If Mandia was in class today, he received instructions for writing the essay. Mandia was in class. Therefore, he received instructions for writing the essay.

In the form of the paradigm above, this conditional can be restated as follows:

[Major:] If "in class" is true, then "received instructions" is true.

[Minor:] "In class" is true.

[Conclusion:] "Received instructions" must be true.

Notice that a conditional seems to use only two terms (P and Q), while a syllogism uses three (X, Y, and Z). But the third term is actually there. In our example, it is Mandia who is "in class," and Mandia who "received instruction."

Summary. Consider this example:

Jerzy claims that all his test scores have been good, and so his course grade should be good, too.

We can express that argument as a syllogism or a conditional:

Syllogism:

Therefore, Jerzy gets a good grade.

Conditional:

Therefore, good grade.

These two arguments reach the same conclusion, and their minor premises are similar, but their major premises appear to be rather different. In fact, "All good tests get good grades" and "If good test then good grade" are just two ways of expressing a relationship between good test scores and good course grades.

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Introduction to Conditional Arguments

The first premise of a conditional argument can be expressed in the form "If p, then q," where "p" is the antecedent and "q" is the consequent. The first premise establishes the condition--the relationship between the antecedent and the consequent. Consider the following examples:

• If Chinua arrives late, he will miss the bus.
• Chinua will miss the bus if he comes late.
• Chinua, if he arrives late, will miss the bus.

Notice that the word order can change, but the sentence retain that same meaning, as long as the same phrase is introduced by "if." Logically, all three can be expressed by the claim, "If Chinua arrives late, then he will miss the bus." For economy, we might shorten that to "If arrive late, then miss bus." In this case, p=arrive late, and q=miss bus.

The second premise of a valid conditional argument does one of two things: it affirms the antecedent (p), or denies the consequent (not q). Thus, the two valid second premises for the conditional above are: "he arrived late" (p), and "he did not miss the bus" (not q). And each of these valid conditional arguments has a valid conclusion:

If arrives late (p), then

miss bus (q).

If arrives late (p), 

then miss bus (q).

In other words, if the original conditional is true, we can draw the following valid conclusions: Chinua arrived late and therefore missed the bus; or Chinua did not miss the bus, and therefore he must not have arrived late. These two valid conditional arguments are expressed by the following paradigms:

(Modus Ponens/Affirming the Antecedent)

(Modus Tollens/Negating the Consequent)

These are the only two valid forms for a conditional argument. The only valid possibilites are a second premise of p, concluding q, and a second premise of not q, concluding not p.

Example 1. "If Chinua arrives late, he will miss the bus. And he does arrives late. Therefore, he misses the bus." This is a valid argument, because it fits one of the two forms for a valid conditional (in this case, modus ponens):

If p (arrives late), then q (misses bus).

Example 2. "If Chinua arrives late, he will miss the bus. And he does miss the bus. Therefore, he must have arrived late." This is an invalid argument, because it does not fit one of the two valid forms. In a valid conditional, the second premise must be either p or not q. In this case, the second premise ("miss bus") would be q, so no valid conclusion can be drawn. We say this second premise "affirms the consequent," which is invalid. (This may sound like a good argument, but it is easy to see why it is not, because the conditional says nothing about what might happen when Chinua does not miss the bus. Perhaps he arrived on time, or perhaps he got there late and the bus was delayed--we have insufficient information to conclude anything.)

Example 3. "If Chinua arrives late, he will miss the bus. But he does not arrive late. Therefore, he did not miss the bus." This is an invalid argument, because it does not fit one of the two valid forms. In a valid conditional, the second premise must be either p or not q. In this case, the second premise ("not arrive late") would be not p, so no valid conclusion can be drawn. We say this second premise "negates the antecedent," which is invalid. (Again, this may sound like a good argument, but it is easy to see why it is not, because the conditional says nothing about what will happen if Chinua arrives on time. Perhaps he did catch the bus, perhaps he fell asleep and missed it anyway--we have insufficient information to conclude anything.)

Example 4. "If Chinua arrives late, he will miss the bus. But he does not miss the bus. Therefore, he did not arrive late." This is a valid argument, because it fits one of the two valid forms (in this case, modus tollens):

If p (arrives late), then
                                    q (misses bus).

Introduction to Induction

As covered in the section on Inductive and Deductive Reasoning, inductive arguments are usually based on experience or observation. In effect, then, inductive arguments are all comparisons between two sets of events, ideas, or things; as a result, inductive arguments are sometimes called analogical arguments. The point of those comparisons, or analogies, is to establish whether the two sets under consideration, similar in a number of other ways, are also similar in the way of interest to the argument. Consider this example:

Mariko says, "Every time I've seen a red-tinted sunset, the next day's weather has been beautiful. Today had a red-tinted sunset, so tomorrow will be beautiful."

Essentially, Mariko is comparing one set of events (observed red-tinted sunsets and each following day's weather) with another (today's observed sunset and tomorrow's predicted weather). These sets are similar in an important way (red-tinted sunsets), and the inductive argument is that they will also be similar in another way (nice weather on the following day). In this case, Mariko is arguing from particular cases in the past to a particular case in the present and future, but she could also argue inductively from those particular cases to a general one, such as "It's always beautiful the day after a red-tinted sunset."

The strength of such an argument depends in large part on three of its elements:

1. how accurate and comprehensive the previous observations are;
2. how strong the causal link seems to be;
3. how similar the two cases are.

In Mariko's argument, to satisfy the first element, we would want to be sure that she's seen many such sunsets, and that "redness" and "beauty" have been judged consistently. To satisfy the second, we would want to feel confident that there is a strong correlation between weather patterns on successive days. To satisfy the third, we would want to know whether there are any significant differences between the observation of today's sunset and of the previous ones. A difference in season, a difference in geographical or topographical location, a difference in climate, or any other significant variation might affect the comparability of the two sets of observations.

In fact, we should always understand the second premise of an inductive argument to contain a claim like "there is otherwise no significant difference." The second premise of Mariko's argument, then, might read, "Today's sunset was red-tinted (and there were no significant differences between this and previous red-tinted sunsets)." Keeping such a disclaimer in mind is important, because this is where many inductive arguments are weakest.

Because we argue inductively from the particular to the general, such arguments are often called generalizations, or inductive generalizations. Other kinds of arguments with a similar format include causal arguments.

Exercises on Induction

1. Every time Jorge has seen a baseball game between the Giants and the Dodgers at Candlestick Park, the Giants have won. Tomorrow, the Giants play the Dodgers at Candlestick. Which of the following is least significant when arguing that the Giants will win tomorrow?

Jorge has only seen the Giants play the Dodgers twice.

Both teams have many new players.

Jorge won't be going to the game tomorrow.

The field at Candlestick will be unusually muddy tomorrow.