Week 2: Discussion 1 and 2 03/02/2017

Congratulations for making it through the first week! There is no paper Assignment this week, but there is a lot of material to cover. Here are the topics in this week's guidance:

1. What is Deductive Reasoning?

2. How to Make Arguments Valid

3. A Common Way to Make a Valid Moral Argument

4. Categorical Argument Forms

5. The Counterexample Method of Showing that an Argument Form is Invalid

6. Propositional Logic

7. Things to Do this Week

In week 1, we covered arguments, how to put them into standard form, and the basic criteria for evaluating them. In particular, we mentioned that the following are necessary for an argument to be good:

1. The premises should be true.

2. The conclusion should follow logically from the premises.

What distinguishes deductive reasoning from inductive reasoning is the strength in which the conclusion is supposed to follow from the premises. In a deductive argument, the premises are supposed to guarantee the truth of the conclusion. An argument that succeeds in this intention is called valid. A valid argument is one in which the truth of the conclusion follows with absolute certainty from the truth of the premises. Another way to put it is that there is no possible way for the premises to be true and the conclusion false. In sum:

For an argument to be valid:

If the premises are true then the conclusion must be true

By contrast, in an inductive argument, the premises are just supposed to supply a high probability that the conclusion is true, not absolute certainty.

Here is an example of valid reasoning:

Premise 1: Mike is taller than Lucy.

Premise 2: Lucy is taller than Joe.

Conclusion: Therefore, Mike is taller than Joe.

There is no way that both premises could be true and the conclusion false.

The following reasoning, by contrast, is invalid:

Premise 1: Mike is taller than Lucy.

Premise 2: Joe is taller than Lucy.

Conclusion: Therefore, Mike is taller than Joe.

Though this reasoning may seem very similar, it is actually invalid because it is possible for the premises to be true and the conclusion false. Take a look at this possibility:

Though it is still possible that the conclusion is true, what makes it invalid is the mere possibility that the conclusion could be false while the premises are true. Notice that validity does not depend upon the actual truth of the premises. An argument with the right form would be valid even if the premises were false. Here is an example:

Premise 1: Some dogs have feathers.

Premise 2: All things with feathers are birds.

Conclusion: Therefore, some dogs are birds.

This argument is valid even though one premise and the conclusion are false. It is valid because if the premises were true, then the conclusion would have to be true As well.

However, As mentioned in our two criteria, we also want our arguments to have true premises. An argument that is both valid and that has all true premises is called sound. A sound argument will always have a true conclusion, because the premises are true and the conclusion follows logically from them. In sum:

Validity: If the premises are true then the conclusion must be true as well.

Soundness: It is valid and all of the premises are true.

It is frequently difficult to judge whether the premises of an argument are true. Validity, on the other hand, is generally a formal matter, and can be determined with precise methods. Due to the complexity of this process, we will only briefly touch on a couple of such methods this week. We will focus instead on creating valid arguments.

A very important skill within logic (and one that will help you with your papers for this course) is to be able to construct valid arguments. It is, however, more difficult than it may sound. A great place to start is simply to work on filling in missing premises within arguments to make them valid. Here would be an example: “I can’t eat Nutty Bars because I am allergic to peanuts.” Putting this into standard form, we get:

Premise 1: I am allergic to peanuts

Conclusion: Therefore, I can’t eat Nutty Bars

This argument is not valid As stated; we need to fill in the suppressed premise in order to make it valid. What premise might complete this argument? Perhaps it is the fact that Nutty bars contain peanuts:

Premise 1: I am allergic to peanuts

Premise 2: Nutty bars contain peanuts

Conclusion: Therefore, I can’t eat Nutty Bars

This argument looks better, but is it valid? The answer is that it is still not valid (validity is a very strict notion)! The problem is that there is nothing in the premises that states that one cannot eat something to which one is allergic. There are many possible ways to fix the argument, but here is one way to represent the full intended argument:

Premise 1: I am allergic to peanuts.

Premise 1: My peanut allergy makes it so that eating anything that contains peanuts causes severe reactions.

Premise 1: It would be unwise for me to eat anything that causes severe reactions.

Premise 1: Nutty Bars contain peanuts.

Conclusion: Therefore it would be unwise for me to eat Nutty Bars.

All of this may have been implied within the original brief argument. In real life, it is not always easy, or even desirable, to have to state arguments in their full glory, as with this one. We generally save time (and prevent potentially insulting the intelligence of our listeners) by giving much abbreviated versions and let our audience fill in the gaps. However, when your task is to give deductively valid reasoning, as it some points in this course, you should not suppress any premises but provide all of the logical links necessary to get from your premises to the conclusion.

Here is another worked example of the tricky process of making an argument valid, for those who want more instruction: An Example of Making an Argument Valid by Virtue of its Form

As noted in the linked example, the best way to make an argument valid is by using a valid form. This process can be useful for writing your arguments for your papers, so this section includes some specific instruction on that topic.

Suppose you are arguing that something (X) is wrong. Furthermore, suppose that it is based on one particular reason, like that it does A (e.g. kills people), then you can construct your argument as follows:

Premise 1: X does A.

Conclusion: Therefore, X is wrong.

This argument, however, is invalid unless we add a premise. What premise would make this one valid? Good guess! Here is the revised argument:

Premise 1: X does A.

Premise 2: Anything that does A is wrong.

Conclusion: Therefore, X is wrong.

What premise do we need to add to make this argument form valid? The second premise tells us that X will be wrong since it does A unless it is C, meaning that in all other cAses besides C, things that do A will be wrong. Therefore, the premise needed to make it valid is the one added here:

Premise 1: X does A.

Premise 2: Anything that does A is wrong, unless it is C.

Premise 3: X is not C.

Conclusion: Therefore, X is wrong.

A video providing an example of how to do this type of thing on a specific moral topic is linked in the video Constructing Valid Arguments (if the link to this doesn't work here use the link under the Week 3 assignment instructions). For further instruction in constructing arguments, see also the document How to Construct a Valid Main Argument.

Now that we know how strict the concept of validity is, see if you can tell whether the following arguments are valid (note: this is a bit of comic relief and not mandatory):

You may have noticed that the validity of an argument often has to do with its form. Take, for example, the following two arguments:

Premise 1: All dogs are mammals.

Premise 2: No mammals are cold blooded.

Conclusion: Therefore, no dogs are cold blooded.

Premise 1: All investment strategies carry risk.

Premise 2: Nothing that carries risk should be done thoughtlessly.

Conclusion: Therefore, no investment strategies should be done thoughtlessly.

These arguments are about totally different topics, but they are both valid in virtue of the same logical form:

Premise 1: All As are Bs

Premise 2: No Bs are Cs

Conclusion: Therefore, no As are Cs

This argument form is valid because of the meaning of the word “all” and because of the way that the letters A, B, and C link up to lead to that conclusion. No matter what you put in for W, W, and W, the resulting argument will be valid.

There are many argument forms, many of which are valid and many of which are invalid. There are methods for testing the validity of argument forms, depending upon the type of argument. Because the letters A, B, and C represent general categories of things, the type of logic that deals with these types of forms is called categorical logic (also known as syllogisitic logic).

There are precise methods of determining whether arguments of this type are valid. Two such methods are Venn Diagrams and the categorical rules for syllogisms. We will not cover those methods here, but they are addressed in our book (Chapter 3).

Here are some more valid categorical logic forms, both valid and invalid. See if you can tell why they are valid or invalid, using Venn diagrams, the rules for validity, or plain old careful thought:

There are, of course, many, many more possible categorical argument forms like these (both valid and invalid). One relatively simple way of showing that these types of forms are invalid is known as the counterexample method:

One relatively easy way to show than an argument form is invalid is to find another instance of the exact same form but with true premises and a false conclusion. If there is an argument of the exact same form but in which the premises are true and the conclusion false, then we know that the argument’s form is invalid. This provides a refutation of that argument form’s validity.

Here are refutations of each of the invalid argument forms (on the right) in the table above:

This method is useful, in order to persuade others clearly that an argument has an invalid form. Coming up with the examples can be a nice intellectual puzzle as well. Now we move on to one more major category of logic: propositional logic.

Categorical argument forms are only one type of form. There is another important type of argument whose validity/invalidity is not a matter of terms like “all” and “some,” but of words like “if … then,” “and,” “or,” and “not.” Take a look at this argument, for example:

Premise 1: We are either eating out or eating at home

Premise 2: We cannot eat out

Conclusion: Therefore we are eating at home

This argument’s form is valid even though it does not contain categorical words like “all” and “some.” In fact, it has the following form:

Premise 1: Either P or Q

Premise 2: Not P (meaning that P is false)

Conclusion: Therefore Q

In the example, “P” is replaced with “we are eating out,” and “Q” is replaced with “we are eating at home.” Because the letters (called variables or logical operators) are replaced, not with categories (like dogs) but with whole sentences (called propositions), this type of logic is called propositional logic.

Many famous propositional forms involve the use of “If … then” statements. Such statements are called conditionals and are written “If P, then Q.” In such sentences, P is called the antecedent, and Q is called the consequent (there are more details about all of this in Chapter 4 of our book). Here are some famous propositional argument forms involving conditionals and their names:

As with categorical forms, there are many, many valid propositional forms and many, many invalid ones. Here is a sample of a few of each. See if you can tell which ones are valid and which ones are not:

There is a method in general for determining the validity of propositional forms in general. It is known as the truth-table method. It is covered in the book, but goes beyond the scope of this guidance. For instruction on truth tables take a careful look at the method in Chapter 4 of our book and in the additional recommended resources for this week. We have only covered the very general idea of propositional logic here.

It is hoped that learning deductive reasoning, and the concept of valid and invalid argument forms, will help us to think more rigorously about our reasoning and to be able to make our reasoning as sound as possible.

Finally, to go out on a fun note, perhaps you have heard this puzzle: Suppose you are at a gate at which there are two brothers, one of whom always tells the truth and one of whom always lies. You get to ask them just one question to determine which way to go. One way leads to where you are going; the other leads to certain death. What question would you ask? The answer uses deductive reasoning. First see if you can figure it out, then check the answer:

1. Read the required materials for the week, including this guidance and chapters 3 & 4 from the textbook. Note that the following sections of the text are optional: “Drawing Immediate Inferences” from section 3.5, “Distribution,” “Rules for Validity” from section 3.6, “Truth Tables with Three Letters” from section 4.3, and “Examples with Arguments with Three Letters” from section 4.4.

2. Watch the weekly intro video and all of the videos under the “Lectures” tab for this week of the course and view all other required materials.

3. Post a timely (initial post by day 3) and thorough response to the first discussion forum as well as substantive replies to peers. This discussion prompt has you present a valid argument on the topic of your choice. Prior to posting, carefully review the above guidance as well as the accompanying links on constructing arguments for more instruction on how to make arguments valid. You can use this discussion to get practice and valuable feedback on your argument.

4. Respond to the second discussion prompt as well (initial post by day 3) as well as responses to peers. This discussion topic is up to your instructor. The specific prompt will be posted by your instructor as the first response in the discussion forum. Make sure to follow carefully all elements of the discussion prompt.

5. Take the Quiz for the week (by day 7). It covers the central concepts of the course as covered in the textbook, this guidance, and the lecture videos for this week.

If you have any questions, make sure to let your instructor know, either via email or in the Ask Your Instructor forum.

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