OCR Document - Jeff's Readings



P-reading: Deductive Arguments: Validity & Soundness

(Moore & Parker, 44-46; Guttenplan, 17-19, 28-31; Johnson, 4-31)

Earlier in the course we defined a deductive argument as one in which the premises are claimed to support the conclusion in such a way that it is impossible for the premises to be true and the conclusion false. If the premises do in fact support the conclusion in this way, the argument is said to be valid. Thus, a valid deductive argument is an argument such that it is impossible for the premises to be true and the conclusion false. In these arguments the conclusion follows with strict necessity from the premises. Conversely, an invalid deductive argument is a deductive argument such that it is possible for the premises to be true and the conclusion false. In invalid arguments the conclusion does not follow with strict necessity from the premises, even though it is claimed to.

An immediate consequence of these definitions is that there is no middle ground between valid and invalid. There are no arguments that are "almost" valid and "almost" invalid. If the conclusion follows with strict necessity from the premises, the argument is valid: if not, it is invalid. To test an argument for validity we begin by assuming that all premises are true, and then we determine if it is possible, in light of that assumption, for the conclusion to be false. Consider our earlier argument again.

1. Jeff and his next-door neighbor live a few blocks from the campus of State University.

2. Jeff's next-door neighbor looks about twenty years old.

3. Jeff's next-door neighbor often wears State University T-shirts and sweatshirts.

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Therefore, Jeff's next-door neighbor attends State University

Notice that even if the three premises are true, there is still a chance that the conclusion will be false. The next-door neighbor might live there and wear State University clothing, but not attend the nearby university. Nonetheless, if the premises are true, then there is a high probability that Jeff's next-door neighbor does attend State University.

1. All prisons in Albuquerque are public institutions.

2. CNM is a public institution in Albuquerque.

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Therefore, CNM is a prison.

In this case, we don't have to just imagine if the premises are true, because these premises are, in fact, true. Next we determine, in light of this assumption, if it is possible for the conclusion to be false. In this case it is possible. In other words, assuming the premises true and the conclusion false does not involve any contradiction, and so the argument is invalid. The question is not whether premises and conclusion are true or false, but whether the premises necessarily, logically produce the conclusion.

1. Mary is either in the library or in the cafeteria.

2. She is not in the library.

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Therefore, she is in the cafeteria.

1. All men are mortal.

2. Socrates is a man.

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Therefore, Socrates is mortal.

These two arguments have a feature that the CNM argument lacks. For these two, if the premises are true, then there is no way that the conclusion can be false. … The reason these conclusions are certain is because all of the information in the conclusions is contained in the premises. The information in the premises is simply moved around in order to get a particular conclusion. Moving this information around has to follow certain rules, but as long as these rules are followed, the conclusion is guaranteed to be true (if the premises are true).

An argument is deductively valid when it is the case that if the premises are true, then the conclusion has to be true. (Or, a deductively valid argument is an argument in which the truth of the premises guarantees the truth of the conclusion.)

1.4 Some Argument Forms

Many arguments take a precise form or structure. For such arguments, the form stays the same, or basically the same, no matter what the argument is about. The purpose of this section is, first, to introduce the idea of an argument's form, and, second, to begin explaining how specific types of arguments work - and how it is that arguments with the same form, but with different content, still work the same way.

The two forms of arguments introduced in this section are called modus ponens and modus tollens. Any argument that has one of these two forms is deductively valid. …

Modus Ponens

1. If today is Thursday, then Sam is in court.

2. Today is Thursday.

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Therefore, Sam is in court.

To understand this type of argument and why it is always deductively valid, it is useful to consider what if A, then B means. One way to think about a conditional statement is as a rule that does not get broken. As such, it is a rule stating that if the antecedent happens, then the consequent has to happen. … In other words, it has this form:

1. If A, then B.

2. A.

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Therefore, B.

Modus Tollens

… Like modus ponens, modus tollens is always deductively valid. It also has a premise containing a conditional statement, and it exploits the same simple feature of conditional statements, namely, if the antecedent happens, then the consequent has to happen. For modus tollens, however, the second premise is a denial of the consequent. For instance, Sam is not in court. And when the consequent has not happened, it can safely be concluded that the antecedent has not happened either: today is not Thursday.

1. If today is Thursday, then Sam is in court.

2. Sam is not in court.

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Therefore, today is not Thursday.

Remember, if the antecedent happens, then the consequent has to happen. Hence, if the consequent didn't happen, then the antecedent couldn't have either. The form of modus tollens, then, is this:

1. If A, then B.

2. not B.

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Therefore, not A.

Let's discuss some examples.

1. All men are mortal.

2. Herb is not a man.

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Therefore, Herb is not mortal.

One way to figure out if an argument is valid or invalid is to try to imagine a

scenario in which the premises are true while the conclusion is false. If there is such a scenario, then the argument is invalid. For this argument, it's not too difficult to imagine that the first premise is true (since it is true). For the second premise, imagine that Herb is a dog. If Herb is a dog, he's mortal, and so the conclusion is false.

1. If the shape is a square, then the shape has four sides.

2. The shape has four sides.

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Therefore, the shape is a square.

From the first premise we know that a square has four sides. Premise two only tells is that this shape has four sides. But we can imagine that this new shape could be a rectangle, so it does not necessarily follow that this new four-sided shape is a square. Therefore, the argument is invalid.

1. If the shape is a square, then the shape has four sides.

2. The shape does not have four sides.

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Therefore, the shape is not a square.

The above argument is valid, and it's form is modus tollens.

Soundness

At times, however, it is desirable to know if an argument has premises that are actually true. If an argument is deductively valid, and the premises are true, then the conclusion has to be a true statement.

A sound argument is a deductive argument that is valid and has all true premises.

Both conditions must be met for an argument to be sound, and if either is missing the argument is unsound. Because a valid argument is one such that it is impossible for the premises to be true and the conclusion false, and because a sound argument does in fact have true premises, it follows that every sound argument, by definition, will have a true conclusion as well. A sound argument, therefore, is what is meant by a perfect deductive argument in the fullest sense of the term. Sometimes they are even called “proofs”. This is an improvement over just being able to say that if the premises are true, then the conclusion will be true. That said, in philosophy the main focus is generally on the form of arguments and how premises support a conclusion. Philosophers usually leave it to others - scientists, journalists, Donald Trump's advisers (zing) - to determine if premises are actually true.

1. New York City is in Maryland.

2. Claire is in New York City.

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Therefore, Claire is in Maryland.

1. George H. W. Bush's oldest son was the forty-third president of the United States.

2. George H. W. Bush's oldest son is George W. Bush.

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Therefore, George W. Bush was the forty-third president of the United States.

These are both valid arguments, but only the last one is sound.

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