EF Standard arguments

Section 5.2 Standard arguments

Subsection 5.2.1 Modus ponens

modus ponens

standard argument with form

p \to q

p

q

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Worked Example 5.2.1.

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Verify the validity of the modus ponens standard argument.

Solution.

Verify the validity by ensuring that each row in the truth table with premises all true also has the conclusion true.

(pr)	(c)	(pr)
$p$	$q$	$p \to q$
$T$	$T$	$T$	$✓$ argument is valid
$T$	$F$	$F$
$F$	$T$	$*$
$F$	$F$	$*$

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Example 5.2.2.

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The argument in Example 5.1.2 has modus ponens form. So it is valid, even though the first premise and the conclusion are not actually true.

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Subsection 5.2.2 Modus tollens

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modus tollens

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standard argument with form

p \to q

\neg q

\neg p

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Worked Example 5.2.3.

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Verify the validity of the modus tollens standard argument.

Solution.

Verify the validity by ensuring that each row in the truth table with premises all true also has the conclusion true.

		(pr)	(pr)	(c)
$p$	$q$	$p \to q$	$\neg q$	$\neg p$
$T$	$T$	$T$	$F$	$*$
$T$	$F$	$F$	$*$	$*$
$F$	$T$	$T$	$F$	$*$
$F$	$F$	$T$	$T$	$T$	$✓$ argument is valid

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Example 5.2.4.

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The argument in Example 5.1.1 has modus tollens form.

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Subsection 5.2.3 Law of Syllogism

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Law of Syllogism

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standard argument with form

p \to q

q \to r

p \to r

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Note 5.2.5.

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We already verified that the Law of Syllogism is valid in Worked Example 5.1.7.

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The Law of Syllogism may be extended to chains of conditionals of arbitrary (finite) length.

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Extended Law of Syllogism

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standard argument with form

p_{1} \to p_{2}

p_{2} \to p_{3}

⋮

p_{n - 1} \to p_{n}

p_{1} \to p_{n}

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Note 5.2.6.

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We will verify that the extended Law of Syllogism is a valid argument using mathematical induction in Section 7.2.

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Example 5.2.7. A syllogistic argument in English.

If I don’t study hard this term, I won’t master the course material.

If I don’t master the course material, I will fail the course.

If I fail the course, I will have to take it again next year.

If I take it again next year, I will have to study harder.

Therefore, if I don’t study hard this term, I will have to study harder next year.

Elementary Foundations: An introduction to topics in discrete mathematics

Search Results:

Section 5.2 Standard arguments

Subsection 5.2.1 Modus ponens

Worked Example 5.2.1.

Example 5.2.2.

Subsection 5.2.2 Modus tollens

Worked Example 5.2.3.

Example 5.2.4.

Subsection 5.2.3 Law of Syllogism

Note 5.2.5.

Note 5.2.6.

Example 5.2.7. A syllogistic argument in English.