Logic and Probability

Argument Strength

Consider the following two arguments:

  1. If Lauren lived in Amsterdam, then Lauren speaks Dutch. Lauren did not live in Amsterdam. So, Lauren does not speak Dutch.
  2. If Lauren lived in Amsterdam, then Lauren speaks Dutch. Lauren does not speak Dutch. So, Lauren did not live in Amsterdam.

Which argument is more compelling? (Try answering this before reading further.)

Argument 1 is not a good argument. One reason to claim that argument 1 is a bad argument is because the premises are false. That is, perhaps Lauren did live in Amsterdam or it is not the case that if someone lives in Amsterdam, then that person speaks Dutch. However, argument 1 can be criticized without knowing anything about where Lauren lived or what languages are spoken in Amsterdam. The key observation is that there are situations in which both premises are true while the conclusion is false. That is, one can imagine a situation in which it is true that if Lauren lived in Amsterdam, then she speaks Dutch, Lauren did not live in Amsterdam, and that Lauren does speak Dutch. Perhaps Lauren learned Dutch in school or lived in another Dutch city, such as Groningen.

Argument 2 does not suffer from the same problem as argument 1: There is no way that both premises can be true while the conclusion is false. That is, it is not possible that Lauren lived in Amsterdam, does not speak Dutch and the statement "if she lived in Amsterdam, then she speaks Dutch" is true.

Argument 2 is an example of a valid argument. The informal definition we have been using so far is that an argument is valid if it is impossible that the premises are true while the conclusion is false. One complication with this definition is that there are different "levels" of impossibility:

  1. "It is impossible for an object to travel faster than the speed of light." or "It is impossible for me to hold my breath for longer than 15 minutes." These are physical impossibilities that depend on physical and/or biological facts.
  2. "It is impossible that 2+2=52+2=5." This is an arithmetic impossibility that depends on number theoretic facts, such as the definition of addition.
  3. "It is impossible that Bob is both married and a bachelor." The impossibility here depends on the meanings of the properties involved (e.g., "being married" and "being a bachelor").

In each of these cases, the notion of impossibility depends on the content (i.e., on which objects are being talked about, or which properties are involved). For validity, the notion of impossibility depends only on the logical form of the statements involved. For example, any claim of the form "φ\varphi and not φ\varphi" is (logically) impossible. It does not matter which statement is substituted for φ\varphi (or which objects φ\varphi is about). If φ\varphi is true, then "not φ\varphi" is false and if "not φ\varphi" is true, then φ\varphi is false. There is no situation in which both φ\varphi and "not φ\varphi" are both true. This suggests the following definition of a valid argument. We will give a more precise definition of validity later in this book.

Validity (informally)

An argument, or reasoning pattern, is valid when there is no situation in which all the premises are true and the conclusion is false.

There is an equivalent ways to define validity:

  • An argument or reasoning pattern is valid if there are no counter-examples, where a counter-example to an argument is a situation in which the premises are all true and the conclusion is false.
Challenge

Is the following argument valid? If no, explain why it is not.

If Lauren lived in Amsterdam, then Lauren speaks Dutch. Lauren does speak Dutch. So, Lauren lived in Amsterdam.

There are two ways you can use a valid argument:

  1. Infer the conclusion is true: If all of the premises are true, then the conclusion is also true.
  2. Refute one of the premises: If the conclusion is false, then some premise is also false.

Note that a valid argument does not require that all of the premises are true. For example, the following argument is valid:

UMD has a campus in College Park or it has a campus on the Moon. UMD does not have a campus in College Park. So, UMD has a campus on the Moon.

Of course, the conclusion is false. Since the argument is valid, one of the premises must be false. In this case, the second premise is false. If all the premises of a valid argument are true, then the conclusion must be true.

Sound Argument

An argument, or reasoning pattern, is sound when it is valid and all the premises are true.

In these notes, we are focused on validity. While it is clearly very important to determine when an argument is sound, the truth or falsity of a premise is often a question of science or history.

The arguments and inferences we have discussed so far direct the reasoner to a certain conclusion. If you are certain that your keys are either in your office, your bedroom or locked in your car, and you learn that your keys are not in your office and not in your bedroom, then you can be certain that they are locked in your car. But many inferences we draw are not so certain. Suppose that 7 weeks into the semester, you note that your classmate, Ann, always had her laptop. It is natural to infer that she will bring her laptop to the next class. But you can't be certain of this conclusion. After all, she might forget her laptop, or she may be sick and not come to class. Nonetheless, observing that Ann brought her laptop to past classes does support the conclusion that she will bring her laptop to the next class.

In this course, we will discuss both types of inferences. Deductive logic is the branch of logic that studies what follows with certainty. Inductive logic deals with uncertainty, things that only follow with high probability.

Practice Questions

1. If an argument has a false conclusion, then it is invalid.

2. If the conclusion of a valid argument is false, then at least one of its premises is false.

3. Every invalid argument has a false conclusion.

4. If the conclusion of a valid argument is true, then all the premises must be true as well.

5. If the conclusion of a valid argument is false, then all of its premises are false as well.


Make up an argument with the following properties, or explain why such an argument is impossible.

  1. A valid argument with a false premise and a true conclusion.
  1. A valid argument with a false premise and a false conclusion.
  1. A valid argument in which all the premises are true and the conclusion is false.
  1. A invalid argument with a true premises and a true conclusion.
  1. A invalid argument with a true premises and a false conclusion.

Consider the following arguments:

Argument 1

  1. John saw a man on the hill with a telescope.
  2. So, the man on the hill has a telescope.

Argument 2

  1. Ann bumped into a man with an umbrella.
  2. So, Ann was holding an umbrella.

Are these good arguments? That is, do the premises support the conclusion?