Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 3.5 Symbolic Arguments.

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 3.5

Symbolic Arguments


What You Will Learn

Symbolic arguments

Standard forms of arguments

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Symbolic Arguments

A symbolic argument consists of a set of premises and a conclusion.It is called a symbolic argument because we generally write it in symbolic form to determine its validity.

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Symbolic Arguments

An argument is valid when its conclusion necessarily follows from a given set of premises.An argument is invalid or a fallacy when the conclusion does not necessarily follow from the given set of premises.

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Law of DetachmentAlso called modus ponens.Symbolically, the argument is written:Premise

1:Premise 2:Conclusion:

If [premise 1 and premise 2] then conclusion[(p → q) ⋀ p ] → q

p → q

p

∴ q

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To Determine Whether an Argument is Valid1. Write the argument in symbolic

form.2. Compare the form of the

argument with forms that are known to be either valid or invalid. If there are no known forms to compare it with, or you do not remember the forms, go to step 3.

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To Determine Whether an Argument is Valid3. If the argument contains two

premises, write a conditional statement of the form

[(premise 1) ⋀ (premise 2)] →

conclusion

4. Construct a truth table for the statement above.

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To Determine Whether an Argument is Valid5. If the answer column of the

truth table has all trues, the statement is a tautology, and the argument is valid. If the answer column of the table does not have all trues, the argument is invalid.

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Example 2: Determining the Validity of an Argument with a Truth Table Determine whether the following argument is valid or invalid.

If you watch Good Morning America, then you see Robin Roberts.

You did not see Robin Roberts.

∴ You did not watch Good Morning America.

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Example 2: Determining the Validity of an Argument with a Truth Table SolutionLetp: You watch Good Morning America.q: You see Robin Roberts.In symbolic form, the argument is

p → q~p

∴ ~pThe argument is [(p → q) ~⋀ q] → ~p.

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p q [(p → q)

⋀ ~ q] → ~p

TTFF

TFTF

FFTT

TFTT

FFFT

FTFT

Solution Construct a truth table.

Since the answer, column 5, has all T’s, the argument is valid.

231 4

TTTT5

Example 2: Determining the Validity of an Argument with a Truth Table

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Standard Forms of Valid ArgumentsLaw of Detachment Law of

Contraposition

p q

q r

p r

p q

~q

~ p

p q

~ p

q

Law of Syllogism Disjunctive Syllogism

p q

p

q

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Standard Forms of Invalid ArgumentsFallacy of the Converse

Fallacy of the Inverse

p q

q

p

p q

~ p

~q3.5-13


Example 4: Identifying a Standard ArgumentDetermine whether the following argument is valid or invalid.

If you are on Facebook, then you see my pictures.If you see my pictures, then you know I have a dog.

∴ If you are on Facebook, then you know I have a dog.

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Example 4: Identifying a Standard ArgumentSolutionLetp: You are on Facebook.q: You see my pictures.r: You know I have a dog.In symbolic form, the argument is

p → qq → r

∴ p→ rIt is the law of syllogism and is valid.

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Example 5: Identifying Common Fallacies in ArgumentsDetermine whether the following argument is valid or invalid.

If it is snowing, then we put salt on the driveway.We put salt on the driveway.

∴ It is snowing.

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Example 5: Identifying Common Fallacies in ArgumentsSolutionLetp: It is snowing.q: We put salt on the driveway.In symbolic form, the argument is

p → qq

∴ pIt is in the form of the fallacy of the converse and it is a fallacy, or invalid.

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Example 5: Identifying Common Fallacies in ArgumentsDetermine whether the following argument is valid or invalid.

If it is snowing, then we put salt on the driveway.It is not snowing.

∴We do not put salt on the driveway.

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Example 5: Identifying Common Fallacies in ArgumentsSolutionLetp: It is snowing.q: We put salt on the driveway.In symbolic form, the argument is

p → q~p

∴ ~qIt is in the form of the fallacy of the inverse and it is a fallacy, or invalid.

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Example 6: An Argument with Three PremisesUse a truth table to determine whether the following argument is valid or invalid.

If my cell phone company is Verizon, then I can call you free of charge.I can call you free of charge or I can send you a text message.I can send you a text message or my cell phone company is Verizon.∴ My cell phone company is Verizon.

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Example 6: An Argument with Three PremisesSolutionLetp: My cell phone company is Verizon.q: I can call you free of charge.r: I can send you a text message.In symbolic form, the argument is

p → qq ⋁ rr ⋁ p

∴ p3.5-21


Example 6: An Argument with Three PremisesSolution

Write the argument in the form

(p → q) (⋀ q ⋁ r) (⋀ r ⋁ p)] → p.

Construct a truth table.

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The answer, column 7, is not true in every case. Thus, the argument is a fallacy, or invalid.

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 3.5 Symbolic Arguments.

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Transcript of Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 3.5 Symbolic Arguments.