# Critical Thinking: A User’s Manual

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Critical Thinking: A User’s Manual
Chapter 8 Evaluating Truth-Functional Arguments

Truth-Functional Arguments
A truth-functional argument is a deductive argument that contains truth-functional claims.

Truth-Functional Claims
Simple claim I have a cat. Negation I do not have cat. Disjunction I have a cat or I have a dog. Conjunction I have a cat and I have a dog. Conditional If I have a cat, then I have a dog.

Operator Symbols not ~ or  and • if, then  therefore 

Symbolizing Truth-Functional Claims
I have a cat. C I do not have a cat. ~ C I have a cat or I have a dog. C  D I have a cat and I have a dog. C • D If I have a cat, then I have a dog. C  D

Translating Conditional Claims
___  ___ Antecedent Consequent

Anatomy of Conditional Claims
If __________, then __________. Antecedent Consequent

Your turn! What is the antecedent of the following conditional claim? What is the consequent? If it is raining, then there are clouds in the sky.

Anatomy of Conditional Claims
__________ only if __________. Antecedent Consequent

Your turn! What is the antecedent of the following conditional claim? What is the consequent? It is raining only if there are clouds in the sky.

Anatomy of Conditional Claims
__________ if __________. Consequent Antecedent

Your turn! What is the antecedent of the following conditional claim? What is the consequent? There are clouds in the sky if it is raining.

“unless” = “if not” X unless Y = X if not Y X unless Y = ~ Y  X
Translating “Unless” “unless” = “if not” X unless Y = X if not Y X unless Y = ~ Y  X

Using Proper Punctuation
I don’t have a cat but I have a dog. ~ C • D It is not the case that I have both a cat and a dog. ~ (C • D) If I have a cat, then either I have a dog or a fish. C  (D  F) Either I have a cat and a dog, or I have a fish. (C • D)  F

Your turn! Translate the following compound claim into symbolic form. It is not the case that we live in Canada and in South America.

DeMorgan’s Law ~ ( X • Y )  ~ X • ~ Y ~ ( X • Y ) = ~ X  ~ Y ~ ( X  Y )  ~ X  ~ Y ~ ( X  Y) = ~ X • ~ Y

Your turn! Apply DeMorgan’s Law to the following claim. ~ (C • S)

Evaluating Truth-Functional Arguments
Truth-functional arguments may be valid or invalid. Demonstrate by identifying argument forms. Demonstrate using Truth Table Method. Demonstrate using Shortcut Method.

Anatomy of Conditional Claims
If __________, then __________. The antecedent is a sufficient condition for the consequent. The consequent is a necessary condition for the antecedent.

Valid Argument Forms X  Y X  Y Modus Ponens
X is a sufficient condition for Y. X is true. Thus, Y must be true.

Valid Argument Forms X  Y ~ Y  ~ X Modus Tollens
Y is a necessary condition for X. Y is false. Thus, X must be false.

Invalid Argument Forms
X  Y Y  X Affirming the Consequent Y is a necessary condition for X. Y is true. Yet, we cannot conclude that X is true; Y is necessary, not sufficient for X.

Invalid Argument Forms
X  Y ~ X  ~ Y Denying the Antecedent X is a sufficient condition for Y. X is false. Yet, we cannot conclude that Y is false; X is sufficient, not necessary for Y.

Your turn! Identify the argument form. P  ~ C C ~ P

Your turn! Identify the argument form. ~ P  C ~ P C

Truth-Functional Definitions
A truth-functional definition specifies when a compound claim is true and when it is false. Used to complete Truth Tables

Simple Claim I have a cat. C T F

Negation I do not have a cat. ~ C T F

Negation I do not have a cat. ~ C F T T F

I have a cat and I have a dog.
Conjunction I have a cat and I have a dog. C • D T T T F F T F F

Conjunction I have a cat and I have a dog. C • D T T T T F F F F T
The only time a conjunction is true is when both conjuncts are true.

I have a cat or I have a dog.
Disjunction I have a cat or I have a dog. C  D T T T F F T F F

Disjunction I have a cat or I have a dog. C  D T T T T T F F T T
F F F The only time a disjunction is false is when both disjuncts are false.

If I have a cat, then I have a dog.
Conditional If I have a cat, then I have a dog. C  D T T T F F T F F

If I have a cat, then I have a dog.
Conditional If I have a cat, then I have a dog. C  D T T T T F F F T T F T F The only time a conditional is false is when the antecedent is true and the consequent is false.

Using the Truth Table Method
Step 1: Translate the argument into symbolic form. Step 2: Write the argument horizontally, using / to separate premises and // in front of the conclusion. Step 3: Calculate the number of lines in the truth table using the formula, L = 2n.

C  ~ R / C // R

Using the Truth Table Method
Step 1: Translate the argument into symbolic form. Step 2: Write the argument horizontally, using / to separate premises and // in front of the conclusion. Step 3: Calculate the number of lines in the truth table using the formula, L = 2n. Step 4: Assign truth-values to each simple claim in the argument.

C R C  ~ R / C // R T T T F F T F F

C R C  ~ R / C // R T T T T T T T F T F T F F T F T F T F F F F F F

Using the Truth Table Method
Step 5: Determine the truth-values of each premise and conclusion using the appropriate truth-functional definitions. You may find it helpful to highlight or draw a box around these final values.

C R C  ~ R / C // R T T T F T T T T F T T F T F F T F F T F T F F F T F F F

C R C  ~ R / C // R T T T T F T T T T F T T T F T F F T F F F T F T F F F T T F F F

Using the Truth Table Method
Step 5: Determine the truth-values of each premise and conclusion using the appropriate truth-functional definitions. You may find it helpful to highlight or draw a box around these final values. Step 6: Evaluate whether the argument is valid or invalid by looking for any row with all true premises and a false conclusion. If you find such a row, then the argument is invalid. Otherwise, the argument is valid.

C R C  ~ R / C // R T T T T F T T T T F T T T F T F F T F F F T F T F F F T T F F F
Both premises are true, and the conclusion is false, so the argument is invalid.

Your turn! How many lines would your truth table have if there were 3 simple claims in the argument?

Using the Shortcut Method
Step 1: Translate the argument into symbolic form. Step 2: Write the argument horizontally, using / to separate premises and // in front of the conclusion. Step 3: Write out the goal truth-values for each premise and conclusion, i.e. all true premises and false conclusion.

C  ~ R / C // R T T F

Using the Shortcut Method
Step 1: Translate the argument into symbolic form. Step 2: Write the argument horizontally, using / to separate premises and // in front of the conclusion. Step 3: Write out the goal truth-values for each premise and conclusion, i.e. all true premises and false conclusion. Step 4: Assign the truth-values to simple claims for which there is only one possible truth-value assignment that would result in a true premise or false conclusion.

C  ~ R / C // R T F T T F

Using the Shortcut Method
Step 5: Insert truth-values throughout the argument for any simple claim whose value was determined in Step 4.

C  ~ R / C // R T F T F T T F

Using the Shortcut Method
Step 5: Insert truth-values throughout the argument for any simple claim whose value was determined in Step 4. Step 6: Determine the truth-values of any remaining premise or conclusion.

C  ~ R / C // R T T F T F T T F

C  ~ R / C // R T T T F T F T T F

Using the Shortcut Method
Step 5: Insert truth-values throughout the argument for any simple claim whose value was determined in Step 4. Step 6: Determine the truth-values of any remaining premise or conclusion. Step 7: Evaluate whether the argument is valid or invalid by checking whether the truth-values of the premises and conclusion achieve the goal of all true premises and a false conclusion. If they do, then the argument is invalid. If they do not, then the argument is valid.

Both premises are true, and the conclusion is false, so the argument is invalid.
C  ~ R / C // R T T T F T F T T F

Complete Analysis plus Evaluation
Step 1: Write a Basic Analysis of the passage. Identify the passage. Analyze the passage. Step 2: If it is an argument, determine whether it commits a fallacy. Identify the fallacy, and explain how it is committed. Step 3: If it is a nonfallacious argument, diagram it. Verify that your diagram is consistent with your Basic Analysis.

Complete Analysis plus Evaluation
Step 4: Identify the kind of argument. If the argument is deductive, identify it as a categorical argument or a truth-functional argument. If the argument is inductive, identify it as an analogical argument, an inductive generalization, or a causal argument.

Complete Analysis plus Evaluation
Step 5: Evaluate the argument. If the argument is categorical, state the syllogism in standard form, and demonstrate whether the argument is valid or invalid using either a Venn diagram or the rules for valid syllogisms. If the argument is truth-functional, translate the argument, and demonstrate whether the argument is valid or invalid by identifying the argument form, using the truth table method, or using the shortcut method.

The Mona Lisa was painted using either acrylic or oil
The Mona Lisa was painted using either acrylic or oil. There’s no way that it could have been painted using acrylic since acrylic paints were available only after the 1940s, and the Mona Lisa was painted in the sixteenth century. So, the material used is obvious.

This passage contains an argument
This passage contains an argument. The issue is whether the Mona Lisa was painted using oil. The conclusion is that the Mona Lisa was painted using oil. The first premise is that the Mona Lisa was painted using either acrylic or oil. The second premise is that the Mona Lisa was not painted using acrylic. The passage contains a subargument. The intermediate conclusion is that the Mona Lisa was not painted using acrylic. The first premise is that acrylic paints were available only after the 1940s. The second premise is that the Mona Lisa was painted in the sixteenth century.

 The Mona Lisa was painted using either acrylic or oil
 The Mona Lisa was painted using either acrylic or oil.  There’s no way that it could have been painted using acrylic since  acrylic paints were available only after the 1940s, and  the Mona Lisa was painted in the sixteenth century. So,  the material used is obvious.  The Mona Lisa was painted using oil.  +    +  

This passage contains an argument
This passage contains an argument. The issue is whether the Mona Lisa was painted using oil. The conclusion is that the Mona Lisa was painted using oil. The first premise is that the Mona Lisa was painted using either acrylic or oil. The second premise is that the Mona Lisa was not painted using acrylic. The passage contains a subargument. The intermediate conclusion is that the Mona Lisa was not painted using acrylic. The first premise is that acrylic paints were available only after the 1940s. The second premise is that the Mona Lisa was painted in the sixteenth century. This passage contains a deductive truth-functional argument.

A = The Mona Lisa was painted using acrylic
A = The Mona Lisa was painted using acrylic. O = The Mona Lisa was painted using oil. P1: A  O P2: ~A___  O

A  O / ~ A // O T T T T T F T F F T F T F F F F

A  O / ~ A // O T T T F T T T T F F T F F T T T F T F F F T F F

This passage contains an argument
This passage contains an argument. The issue is whether the Mona Lisa was painted using oil. The conclusion is that the Mona Lisa was painted using oil. The first premise is that the Mona Lisa was painted using either acrylic or oil. The second premise is that the Mona Lisa was not painted using acrylic. The passage contains a subargument. The intermediate conclusion is that the Mona Lisa was not painted using acrylic. The first premise is that acrylic paints were available only after the 1940s. The second premise is that the Mona Lisa was painted in the sixteenth century. This passage contains a deductive truth-functional argument. The argument is valid, as is shown using the truth table method.