Worksheet – Completing Logical Proofs

Logical proofs are a method for proving arguments valid. They use eight valid forms of inference that are often common in everyday life and thought.

Complete numbers 1 to 6 and submit your work as a Word document. Before starting, review the Lecture Video – Valid Forms of Inference, Lecture Video – Logical Proofs, 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. How is a logical proof constructed? Why can a proof show that an argument is valid but not that an argument is invalid? Answer in 6-8 sentences with two direct quotes from the course text by Van Cleave. Put quotes in quotation marks with the in-text citation (Van Cleave, 2016, p. ___). (15 points)

  2. What is the difference between modus ponens and modus tollens? As part of your answer, compose and explain an example of each concerning your favorite animal. (15 points)

Translating Arguments

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

  1. If Bella visited China or Japan, then she was in Asia. Bella visited China. So, she was in Asia. (15 points)
  2. McDonald’s does not sell broccoli, but if it sells French fries, then it sells hamburgers. If McDonald’s sells hamburgers, then it sells cheeseburgers. Hence, if McDonald’s sells French fries, then it sells cheeseburgers (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 Logical Proofs

Use the Handout – Completing Logical Proofs for help in creating your own logical proofs 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 – Eight Valid Forms of Inference for reference as you complete your truth tables.

  1. Create a logical proof for the argument in question 3. List all lines needed to derive the conclusion as well as the valid form and line number(s) used to derive each line. (20 points)
  2. Create a logical proof for the argument in question 4. List all lines needed to derive the conclusion as well as the valid form and line number(s) used to derive each line. (20 points)

Handout – Completing a Logical Proof

There are three steps for completing a logical proof:

  1. List the premises and conclusion of the argument in symbolic form. Put each premise on its own numbered line of the proof. Do not put the conclusion on its own line. Instead, place it to the right of the final premise line.
  2. Generate additional lines of the proof. Beneath the premises, add further numbered lines using the eight valid forms of inference. Existing lines of the proof will be the premises of a valid form and the new line you create will be the conclusion of the form. After writing a new line, put the name of the valid form used to generate it and the earlier numbered lines which served as premises to the right of the new line of the proof.
  3. Continue until you reach the conclusion. Keep adding new lines by the above method until you generate the conclusion of the argument on its own line. Think creatively about which valid forms of inference will help you get to the conclusion. Once you reach the conclusion, you have proven the argument valid.

Example Logical Proof

Here is an example argument to illustrate the three steps.

If Alberto or Barbara invest in the company, then Clarissa will not invest in it. Either Clarissa or David will invest in the company. Alberto will invest in the company. Therefore, David will invest in the company.

  • List the premises and conclusion of the argument in symbolic form.
    1. (A v B) ⊃ ~C
    2. C v D
    3. A /∴ D
  • Generate additional lines of the proof.
    1. (A v B) ⊃ ~C
    2. C v D
    3. A /∴ D
    4. A v B Addition, 3
    5. ~C Modus ponens, 1, 4
  • Continue until you reach the conclusion.
    1. (A v B) ⊃ ~C
    2. C v D
    3. A /∴ D
    4. A v B Addition, 3
    5. ~C Modus ponens, 1, 4
    6. D Disjunctive syllogism, 2, 5

Handout – Eight Valid Forms of Inference

Eight Valid Forms of Inference
Name Symbols Example
Modus ponens p q
p
q
If it snows, then the flight is cancelled.
It snows.
Therefore, the flight is cancelled.
Modus tollens p ⊃ q
~q
~p
If it snows, then the flight is cancelled.
The flight is not cancelled.
Therefore, it is not snowing.
Hypothetical syllogism p q
q r
p r
If it snows, then the flight is cancelled.
If the flight is cancelled, then the airport will close.
Therefore, if it snows, then the airport will close.
Simplification p q
p
The ocean is large and made of water.
Therefore, the ocean is large.
Conjunction p
q
p q
The ocean is large.
The ocean is made of water.
Therefore, the ocean is large and made of water.
Disjunctive syllogism p v q
~p
∴ q
Snakes are birds or reptiles.
Snakes are not birds.
Therefore, snakes are reptiles.
Addition p
p v q
Snakes are reptiles.
Therefore, snakes are reptiles or birds.
Constructive dilemma p v q
p r
q s
r v s
Snakes are birds or reptiles.
If snakes are birds, then they have wings.
If snakes are reptiles, then they have scales.
Therefore, snakes have wings or scales.

Example Solution

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

If Alberto or Barbara invest in the company, then Clarissa will not invest in it. Either Clarissa or David will invest in the company. Alberto will invest in the company. Therefore, David will invest in the company.

  • Translate the argument into symbolic form.
    1. (A v B) ⊃ ~C
    2. C v D
    3. A /∴ D
  • Create a logical proof listing all lines needed to derive the conclusion as well as the valid form and line number(s) used to derive each line.
    1. (A v B) ⊃ ~C
    2. C v D
    3. A /∴ D
    4. A v B Addition, 3
    5. ~C Modus ponens, 1, 4
    6. D Disjunctive syllogism, 2, 5

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