Completing Truth Tables

Truth tables are used to assess deductive arguments as valid or invalid by showing the “inner workings” of an argument. This worksheet will help you complete and evaluate truth tables.

Complete numbers 1 to 6 and submit your work as a Word document. Before starting, review the Lecture Video – Logical Connectives, Lecture Video – Truth Tables, and related handouts.

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Note that different students have different arguments to use for this worksheet. Use the arguments that appear for you below.

 

Comprehension

  1. What are the four logical connectives? How can they be used to translate sentences into symbolic form and evaluate complex propositions as true or false? Answer in 6-8 sentences with two direct quotes from the text by Van Cleave. Quotes go in quotation marks with the in-text citation (Van Cleave, 2016, p. ___). (15 points)
  2. What pattern of T’s and F’s in a truth table shows that an argument is invalid? (This is what I call the “toxic pattern” in my video.) Why is it impossible for an argument to have this pattern and still be valid? (15 points)

Translating Arguments

Translate the following arguments into symbolic form. Use the Handout – Meaning of the Logical Connectives for help.

  1. If Honda releases a new truck, then both Honda and Nissan will release a new truck. Honda or Nissan will release a new truck. So, we can conclude that Nissan will release a new truck. (15 points)
  2. The pig ate apples or corn. If the pig ate corn, then it did not eat apples. Hence, it is not the case that the pig ate apples. (15 points)

Handout – Meaning of the Logical Connectives

Basic meaning of the logical connectives
Name Symbol Meaning Example Translation
Negation Tilde (~) Not ~A Apples are not red.
Conjunction Dot (⋅) And A ⋅ B Apples are red and bananas are yellow.
Disjunction Wedge (v) Or A v B Apples are red or bananas are yellow.
Conditional Horseshoe (⊃) If… then A ⊃ B If apples are red, then bananas are yellow.
Therefore (∴) Therefore [Placed before the conclusion of an argument]

Some connectives also have additional meanings not given in the table. For example, conjunction can also be translated as “but.” These additional meanings are explained in the course text.

The conditional connective is covered in the next lesson (Truth Tables), but I include it here so that all four logical connectives are together.

Completing Truth Tables

Use the Handout – Completing a Truth Table for help in creating your own truth tables and an example of what to do. This task can be challenging, so look over the lecture video and practice exercises in the lesson for further review. You will also need the Handout – Truth Tables for the Logical Connectives for reference as you complete your truth tables.

  1. Create a truth table for the argument in question 3. Then, use the truth table to assess the argument as valid or invalid. (20 points)
  2. Create a truth table for the argument in question 4. Then, use the truth table to assess the argument as valid or invalid. (20 points)

Handout – Completing a Truth Table

There are five steps for completing a truth table:

  1. Symbolize the argument. If you receive an argument in English, translate it into symbolic form using variables and logical connectives.
  2. Create the first row of the truth table. Moving from left to right, the first row should include one column for each variable in the argument, one column for each premise of the argument, and finally a column for the conclusion of the argument.
  3. Complete the first two columns. For an argument with two variables, the same pattern of T’s and F’s always goes in the first two columns. Under the first variable, put T T F F. Then under the second variable, put T F T F.
  4. Complete the remaining columns. This is the hardest part. Each time a variable appears in the premise or conclusion columns, put the same pattern of four T’s and F’s under it as found in its variable column on the left. Then, use the truth tables for each logical connective to compute the final value of each truth table box. Parentheses can guide you through which parts of a box to solve first. Put the final value for each box in bold.
  5. Assess for validity. Once the truth table is complete, use the final bold value in each premise and conclusion box to check for validity. Look for the toxic pattern of a row with all true premises and a false conclusion, which can never occur in a valid argument. If this pattern occurs, the argument is invalid. If it does not occur in any row, then the argument is valid.

Example Truth Table

Here is an example argument to illustrate the five steps.

If Adele performs in New York, then she will perform in New York and Boston. Adele will perform in Boston. So, she will also perform in New York.

      1. Symbolize the argument.

        N (N B)
        B
        N

      2. Create the first row of the truth table.
        N B N ⊃ (N B) B N
      3. Complete the first two columns.
        N B N ⊃ (N B) B N
        T T
        T F
        F T
        F F
      4. Complete the remaining columns.
        N B N ⊃ (N B) B N
        T T T T T T T T T
        T F T F T F F F T
        F T F T F F T T F
        F F F T F F F F F
      5. Assess for validity.

        The argument is invalid.

        In the second row from the bottom, we see the toxic pattern of T T F where both premises are true and the conclusion is false. Since that pattern appears in at least one row, the argument is invalid.

Handout – Truth Tables for the Logical Connectives

The left columns of each table show the input for p and q. Then the right column shows the truth value of the expression as a whole.

For example, consider the statement: “Dolphins can swim and whales can fly.” The connective used is conjunction (and), so we use the truth table for conjunction. The first part about dolphins is true (that is p) and the second part about whales is false (that is q). Accordingly, we use the row of the truth table for conjunction that has T under p and F under q. The value in the right column of that row is F, so the expression as a whole is false.

Truth table for negation
p ~p
T F
F T
Truth table for conjunction
p q p ⋅ q
T T T
T F F
F T F
F F F
Truth table for disjunction
p q p v q
T T T
T F T
F T T
F F F
Truth table for conditional
p q p q
T T T
T F F
F T T
F F T

Example Solution

Here is an example solution to illustrate the kind of work you will be doing.

If Adele performs in New York, then she will perform in New York and Boston. Adele will perform in Boston. So, she will also perform in New York.

  • Translate the argument into symbolic form.

    N ⊃ (N ⋅B)
    B
    ∴ N

  • Create a truth table for the argument and use the truth table to assess the argument as valid or invalid.
    N B N ⊃ (N ⋅ B) B N
    T T T T T T T T T
    T F T F T F F F T
    F T F T F F T T F
    F F F T F F F F F

    The argument is invalid. In the second row from the bottom, we see the toxic pattern of T T F which makes the argument invalid.

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