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Section 2.1 Deductive Reasoning

Deductive or direct reasoning is a process of reaching a conclusion from one (or more) statements, called the hypothesis (or hypotheses). This somewhat informal definition can be rephrased using the language and symbolism of the preceding sections. An argument is a set of statements in which one of the statements is called the conclusion and the rest make up the hypothesis. A valid argument is an argument in which the conclusion must be true whenever the hypotheses are true. In the case of a valid argument we say the conclusion follows from the hypothesis. For example, consider the following argument: “If it is snowing, then it is cold. It is snowing. Therefore, it is cold.” In this argument, when the two statements in the hypothesis, namely, “if it is snowing, then it is cold” and “It is snowing” are both true, then one can conclude that “It is cold.” That is, this argument is valid since the conclusion follows necessarily from the hypotheses.

It is important to distinguish between the notions of truth and validity. While individual statements may be either true or false, arguments cannot. Similarly, arguments may be described as valid or invalid, but statements cannot. An argument is said to be an invalid argument if its conclusion can be false when its hypothesis is true. An example of an invalid argument is the following: “If it is raining, then the streets are wet. The streets are wet. Therefore, it is raining.” For convenience, we will represent this argument symbolically as \([(p \rightarrow q) \wedge p] \rightarrow p\text{.}\) This is an invalid argument since the streets could be wet from a variety of causes (e.g., fire hydrant open, sprinkler system malfunction, etc.) without having had any rain. It is possible for valid arguments to contain either true or false hypotheses, as indicated in the two valid arguments in Example 2.1.1.

  • Arguement 1:

    • All counting numbers are positive.
    • All positive numbers are larger than negative \(2\text{.}\)
    • Therefore, all counting numbers are larger than negative \(2\text{.}\)
  • Arguement 2:

    • All numbers are positive.
    • All positive numbers are larger than \(5\text{.}\)
    • Therefore, all numbers are larger than \(5\text{.}\)

Note that in both Arguments 1 and 2, the conclusions follow necessarily from the hypotheses. Thus, Argument 2 is considered valid even though both hypotheses are false. It should also be noted that an argument may be invalid even though the hypotheses and the conclusion are true. In Argument 3 below, even though both hypotheses may be true, it is possible for the conclusion to be either true or false; thus, the argument is invalid.

  • Arguement 3:

    • If it is raining outside, then the lawn gets wet.
    • It is not raining outside.
    • Therefore, the lawn is not wet.

The truth table below (Table 2.1.2) shows that Arguement 3 is invalid, since it is possible to have the hypotheses, \((p \rightarrow q) \wedge \negate p\text{,}\) true with the conclusion, \(\negate q\text{,}\) false. This situation, of course, makes the statement \([(p \rightarrow q) \wedge \negate p] \rightarrow \negate q\) false, and the argument is invalid.

\(p\) \(q\) \(\negate p\) \(\negate q\) \(p \rightarrow q\) \((p \rightarrow q) \wedge \negate p\) \([(p \rightarrow q) \wedge \negate p] \rightarrow \negate q\)
T T F F T F T
T F F T F F T
F T T F T T F
F F T T T T T
Table 2.1.2. Truth table for \([(p \rightarrow q) \wedge \negate p] \rightarrow \negate q\)
TAKS CONNECTION.

How might a student apply deductive reasoning to answer the following question taken from the 2006 Texas Assessment of Knowledge and Skills (TAKS) Grade 5 Mathematics test?

Sue is taller than Bianca and shorter than Colette. If Colette is shorter than Dora, who is the shortest person?

  • F. Sue
  • G. Bianca
  • H. Colette
  • J. Dora

Charles Dodgson (1832–1898) was an English mathematician who taught logic at Oxford University. As a teacher of logic and a lover of nonsense, he designed entertaining puzzles to train people in systematic reasoning. In these puzzles he would string together a list of implications, purposefully nonsensical so that his students would not influenced by any preconceived opinions. The task presented to the student was to use all the listed implications to arrive at an inescapable conclusion. You may know Charles Dodgson better by his pen name Lewis Carroll, author of Alice's Adventures in Wonderland and Through the Looking-Glass.

For example, consider the following statements.

  1. All babies are illogical.
  2. Nobody is despised who can manage a crocodile.
  3. Illogical persons are despised.

Let

\begin{align*} p & \text{ it is a baby}\\ q & \text{ it is logical}\\ r & \text{ it can manage a crocodile}\\ s & \text{ it is despised}. \end{align*}

The statements now translate to

  1. \(p \rightarrow \negate q\) (All babies are illogical.)
  2. \(r \rightarrow \negate s\) or \(s \rightarrow \negate r\) (Nobody is despised who can manage a crocodile.)
  3. \(\negate q \rightarrow s\) (Illogical persons are despised.)

Linking these statements together, we see that \(p \rightarrow \negate q \rightarrow s \rightarrow \negate r\text{.}\) In other words, \(p \rightarrow \negate r\) or “babies cannot manage crocodiles.”

Translating a conditional statement into “if-then” form can be quite confusing. The statement “All babies are illogical” is not in a very useful form; however, we can write an equivalent “if-then” statement: “If it is a baby, then it is not logical.” Consider the following examples.

  • “It rains only if I carry an umbrella” can be rewritten as “If it rains, then I carry an umbrella.”
  • “All citizens of Egypt speak Arabic.” can be rewritten as “If someone is a citizen of Egypt, then they speak Arabic.”
  • “Unless it is sunny, I carry an umbrella.” can be rewritten as “If it is not sunny, I carry an umbrella.”
  • “No one in MTH 300 speaks Chinese.” can be rewritten as “If you are in MTH 300, then you do not speak Chinese.”
  • “For cows to fly it is sufficient that 3 + 4 = 8.” can be rewritten as “If 3 + 4 = 8, then cows fly.”
  • “For cows to fly it is necessary that 3 + 4 = 8.” can be rewritten as “If cows fly, then 3 + 4 = 8.”
  • “When it rains, I carry an umbrella.” can be rewritten as “If it rains, I carry an umbrella.”

Exercises 2.1.1 Exercises

1.

Rewrite the following conditional statements as “if-then” statements.

  1. All citizens of Egypt speak Arabic.
  2. Dallas is the capital of Texas only if \(2 + 3 \neq 7\text{.}\)
  3. Nacogdoches is the oldest city in Texas unless mermaids exist.
  4. No resident of Boston likes hot peppers.
  5. For \(3 + 7 = 10\) it is necessary that cows fly.
  6. For \(3 + 7 = 10\) it is sufficient that cows fly.
  7. I carry and umbrella when it rains.
  8. I carry an umbrella only if it rains.

See how many of the following Lewis Carroll puzzles you can solve.

2.
  • All babies are illogical.
  • Nobody is despised who can manage a crocodile.
  • Illogical persons are dispised.
3.
  • None of the unnoticed things, met with at sea, are mermaids.
  • Things entered in the log, as met with at sea, are sure to be worth remembering.
  • I have never met with anything worth remembering, when on a voyage.
  • Things met with at sea, that are noticed, are sure to be recorded in the log.
4.
  • No ducks waltz.
  • No officers ever decline to waltz.
  • All my poultry are ducks.
5.
  • No birds, except ostriches, are 9 feet high.
  • There are no birds in this aviary that belong to anyone but me.
  • No ostrich lives on mince pies.
  • I have no birds less than 9 feet high.
6.
  • All writers, who understand human nature, are clever.
  • No one is a true poet unless he can stir the hearts of men.
  • Shakespeare wrote “Hamlet.”
  • No writer, who does not understand human nature, can stir the hearts of men.
  • None but a true poet could have written “Hamlet.”
7.
  • No kitten, that loves fish, is unteachable.
  • No kitten without a tail will play with a gorilla.
  • Kittens with whiskers always love fish.
  • No teachable kitten has green eyes.
  • No kittens have tails unless they have whiskers
8.
  • No shark ever doubts that he is well fitted out.
  • A fish, that cannot dance a minuet, is contemptible.
  • No fish is quite certain that it is well fitted out, unless it has three rows of teeth.
  • All fishes, except sharks, are kind to children.
  • No heavy fish can dance a minuet.
  • A fish with three rows of teeth is not to be despised.