Reflections: What distinguishes a deductive argument is that the premises are claimed to be sufficient for the conclusion, that no further evidence is needed in order to draw the conclusion. In fact, of course, such claims are made (e.g., in criminal trials), without the evidence actually being sufficient. When it is asserted or implied that this set of premises sufficiently supports the conclusion, we've got a deductive argument. If, however, a conclusion is drawn in spite of missing pieces or gaps in the set of premises, the argument is inductive. The evidence offers only partial support for the conclusion and, consequently, you cannot be certain that the conclusion is true. Being able to tell these two basic argument forms apart from one another is crucial for our ability to assess their strength and persuasiveness. The chapter ends with a discussion of assessing inductive arguments. With these skills, your reading comprehension and analytical skills will get a big boost.

Goals for this chapter: My goal for this chapter is to help you acquire the tools of logic that are very helpful for critical thinking and especially for tackling arguments. After discussing the role of propositions in helping us set out the structure of an argument, I set out two major sections to the chapter:

First we survey the key deductive argument forms. By learning these patterns, you'll be able to spot common deductive arguments you come across. The next task is to evaluate deductive arguments to see if they are valid and sound. How this is done is set out here.

Secondly, we survey the key inductive argument forms. These are all around us and, so, it is most important that you be able to recognize them. And knowing that they cannot offer us certainty is key. The best we can hope for with inductive reasoning is an inference that rests on probability or likelihood. And yet that doesn't mean you can't have a strong inductive argument--as we see in this chapter.

The Logic Machine: Deductive and Inductive Reasoning

1. Propositions

The standard form of a proposition is "A is/were/will be B." In other words, we are saying of something (the subject, A) that it is/was/or will have some characteristic being predicated (B)--such as "Ice Cube (subject) is a person who is both an actor and a singer (predicate)." Note that the proposition isn't always strictly expressed in the above form--e.g., rhetorical questions can be rewritten as propositions.

A proposition is called a categorical proposition if it can be expressed starting with the words All, No, Some, or x% (where x is any number other than 0 or 100). For example, "All lions are ferocious beasts when hungry," "Some hungry beasts are carnivores," "87% of children are people who like to draw rainbows."

2. Deductive Reasoning

With deductive arguments, the conclusion comes right out of the premises. Think of all those movies you saw where the prosecutor tried to prove the case beyond a reasonable doubt. One important thing about deductive arguments: the focus is not on the truth or falsity of the claims, but whether or not the evidence suffices to make the case. A deductive argument is an argument in which the premises are claimed to be sufficient for the drawing of the conclusion. In that sense, a deductive argument is a closed set. In other words, just like a crossword puzzle, the clues (evidence) should be sufficient to complete the job (draw the conclusion without resorting to any other references or resources).

Applications of deductive reasoning: Even examples from arithmetic demonstrate deductive reasoning. Many arguments concerning moral and legal reasoning make use of deductive reasoning. For example, suppose someone said: "All who abuse children are cruel and inhumane people. All cruel and inhumane people should be punished. Therefore, all who abuse children should be punished." This argument is also self-contained. The conclusion, "All who abuse children should be punished" comes out of the two premises.

The key deductive argument forms are:

1. Categorical Syllogisms and Chains of Syllogisms: These are three-line arguments (or chains of them), consisting of two premises and a conclusion, with all of the propositions in the form of categorical propositions. These propositions can be expressed in one of the four possible forms: "All A are B;" "No A is B;" "Some A is B; " and "Some A is not B."
   
   
2. Modus Ponens. These are arguments of the form: "If A then B. A is the case. Therefore, B is true also." For example: If we see The Manchurian Candidate, then we can discuss politics over dinner. We saw The Manchurian Candidate. Therefore, we can discuss politics over dinner.
   
   
3. Modus Tollens: These are arguments of the form: "If A then B. B is not the case. Therefore, A is not true either." For example: If we see The Bourne Supremacy, then we can talk about spy films over coffee. We did not talk about spy films over coffee. Therefore, we did not see The Bourne Supremacy.
   
   
4. Disjunctive Syllogism. These are arguments of the form: "Either A or B. Not A. Therefore, not B." For example: "Either we go to the movies or we go on a drive to the beach. We did not go to the movies. Therefore, we went on a drive to the beach."
   
   
5. Hypothetical Syllogism. These are arguments of the form: "If A then B. If B then C. Therefore, if A then C. For example: If we go to the movies, we'll eat a lot of popcorn. If we eat a lot of popcorn, we'll need to get a large drink. Therefore, if we go to the movies, we'll need to get a large drink."
   
   
6. Constructive Dilemma. These take the form of: "If A then B, andif C then D. Either A or C is the case. Therefore, either B or D is the case." In other words, there's a choice between two options, where each option leads to some effect and you have to pick between either of the two options. This means, if you assume you'll pick one or the other option, then either of the two effects will happen. For example: If Ali studies Spanish, he'll go to Madrid over Christmas, but if he takes French, he'll fly to Montreal in January. Either Ali will take Spanish or he'll take French. So either he'll go to Madrid over Christmas or go to Montreal in January.
   
   
7. Variations of Modus Ponens and Modus Tollens:
   
   A. One, variation is an argument of the form: "A unless B. B is not the case. Therefore, A." Another is "A unless B. Not A. Therefore B." For example: "Ali will take a class in Chinese unless they offer Swahili. They did not offer Swahili, so Ali took Chinese."
   
   B. Application of a Rule: Another variation of modus ponens is in the form of the application of a rule to things that satisfy a set of criteria: "rule X applies to any cases with characteristics A, B, C, and D. Individual case P has characteristics A, B, C, and D. Therefore, rule X applies to case P.
   For example: The rule at the park is no child under 3 feet tall can go on the "Wild Toad" ride. Adam is under 3 feet tall, so he cannot go on the "Wild Toad" ride.
   
   → Compounding the Terms of the Argument. Be aware that all of the argument forms above, the A and B could each stand for a compound statement. A compound statement is one containing any of these words: "not," "and," "or," "ifÉthen," and "if and only if."

   So, for example, you could have an argument in the form of Modus Ponens that had compound terms, such as "If (P or Q) then (R and S). Either P or Q is true. Therefore, both R and S are true." The form is still that of Modus Ponens, if you look at the superstructure. Similarly, this is still the form of Hypothetical Syllogism: "If (P and Q) then (R or S). If (R or S) then (T and not U). Therefore, If (P and Q) then (T and not U).

3.Validity and Soundness

A valid argument is an argument in which the premises provide sufficient support for the drawing of the conclusion. That is, if we assume the premises were true and the conclusion could not be false, then the argument is valid. This has to do with the relationship between the premises and the conclusion. Validity a question of structure--namely, whether the premises either separately or in combination sufficiently support the conclusion. The key is that the connection entails certainty: If true premises force the conclusion to be true, then the conclusion certainly follows from those premises. If we could have true premises and a false conclusion, then the argument is invalid

EXAMPLE OF A VALID ARGUMENT:

All peanuts are legumes.
Some children are fond of peanuts.
Therefore, some children are fond of legumes.

If the two premises were true then the conclusion could not be false. Remember, it is not important for validity whether or not the premises are actually true. The issue is the connection between the premises and the conclusion.

An invalid argument is an argument in which the premises fail to adequately support the conclusion. We can tell an argument is invalid when the premises could be true and the conclusion false.

EXAMPLE OF AN INVALID ARGUMENT:

Some peanuts are chocolate-covered treats.
Some children are fond of chocolate-covered treats.
Therefore, some children are fond of peanuts.

If we assumed the two premises were true, it would not necessarily follow that "Some children are fond of peanuts." The conclusion could be false while the premises were true. The conclusion, therefore, simply doesn't follow and, thus, the argument is invalid.

Once validity is determined, we can assess the soundness of an argument. Now's the time we ask whether or not the evidence is actually true and not just if it were true would the conclusion follow. If the argument is valid and the premises are really true, the argument is sound.

An argument is unsound whenever either or both of these conditions are met: (1) the argument is invalid or (2) the premises are not all true. The odds are that an argument will be unsound, because many arguments are invalid and often one or more of the premises are false. For example:

EXAMPLE OF A SOUND ARGUMENT

All raccoons are mammals.
All mammals are warm-blooded animals.
So, all raccoons are warm-blooded animals.

Note: The argument is valid (if the premises were true, the conclusion could not be false) and the premises really are true. Thus ii is sound.

EXAMPLE OF AN UNSOUND ARGUMENT

All raccoons are birds.
All birds are rodents.
Therefore, all raccoons are rodents.

Note: The argument is valid (if the premises were true, the conclusion could not be false), but the premises are not true. Thus, it is unsound.

4. Inductive Reasoning

An inductive argument is an argument in which the premises only provide some support for the drawing of the conclusion, but not sufficient support. In that sense, an inductive argument is like a puzzle with some missing pieces. So, there will always be an element of doubt in the argument. The conclusion can only be said to follow with likelihood or probability--never with certainty. In that sense, the conclusion goes beyond what is contained in the premises.

There are five major kinds of inductive arguments:

1. Predictions: In predictions, an argument is made about the future based on past or present evidence.
   
   
2. Arguments about the past based on present evidence: In these arguments, an inference is drawn about what happened at some earlier point in time based on current evidence.
   
   
3. Cause and effect reasoning: Here it is claimed that an event (effect) is based on one or more causal factors. Given the existence, then, of the causal factor(s), the effect should follow.
   
   
4. Arguments based on analogy: This argument rests on a comparison, from which it is claimed that a characteristic true of the one term in the equation will also be true of the other. In law this usually involves the application of a precedent or legal principle.
   
   
5. Statistical reasoning. These arguments draw from sample studies or statistical reasoning, from which an inference then is drawn about either all or part of the targeted population.
   
   

→ The Wedge of Doubt in Inductive Arguments: In an inductive argument the premises could be true, but the conclusion will never follow with certainty. Remember, there is always some wedge of doubt between the premises and the conclusion.

Assessing Inductive Arguments

Inductive arguments can never be said to be valid--they are assessed in terms of how strong or weak they are. Because the premises of inductive arguments never supply enough evidence to force the conclusion to be true, there is always an element of uncertainty, or probability, to inductive reasoning. Even if the premises are all true, the conclusion might still be false.

There is no one specific method of assessing the strength of inductive arguments. Considerations vary according to the type of inductive argument. But one thing is common: There will always be a degree of probability involved. In all inductive arguments there exists a fundamental uncertainty about whether or not the conclusion follows from the premises. However, each inductive argument can be evaluated in terms of how strong or weak it is. This, then, means an inductive argument is neither sound nor unsound. You cannot talk about validity or soundness with regard to inductive arguments. So never say an inductive argument is valid. Never say it is invalid. Never say it is sound. Never say it is unsound--these terms just do not apply.