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Discrete Mathematics with Applications | 4th Edition | ISBN: 9780495391326 | Authors: Susanna S. Epp
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Some of the arguments are valid by universal

Chapter 3, Problem 8E

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QUESTION:

Problem 8E

Some of the arguments are valid by universal modus ponens or universal modus tollens; others are invalid and exhibit the converse or the inverse error. State which are valid and which are invalid. Justify your answers.

Universal Modus Ponens

The rule of universal instantiation can be combined with modus ponens to obtain the valid form of argument called universal modus ponens.

Universal Modus Ponens

Formal Version

Informal Version

∀x, if P(x) then Q(x).

If x makes P(x) true, then x makes Q(x) true.

P(a) for a particular a.

a makes P(x) true.

 ∴ Q(a).

 ∴a makes Q(x) true.

Universal ModusTollens

Another crucially important rule of inference is universal modus tollens.Its validity results from combining universal instantiation with modus tollens. Universal modus tollens is the heart of proof of contradiction, which is one of the most important methods of mathematical argument.

Universal Modus Tollens

Formal Version

Informal Version

∀x, if P(x) then Q(x).

If xmakes P(x) true, then xmakes Q(x) true.

~Q(a), for a particular a

a does not make Q(x) true.

∴ ~P(a).

∴ a does not make P(x) true.

Exercise

All freshmen must take writing.

Caroline is a freshman.

∴ Caroline must take writing.

Questions & Answers

QUESTION:

Problem 8E

Some of the arguments are valid by universal modus ponens or universal modus tollens; others are invalid and exhibit the converse or the inverse error. State which are valid and which are invalid. Justify your answers.

Universal Modus Ponens

The rule of universal instantiation can be combined with modus ponens to obtain the valid form of argument called universal modus ponens.

Universal Modus Ponens

Formal Version

Informal Version

∀x, if P(x) then Q(x).

If x makes P(x) true, then x makes Q(x) true.

P(a) for a particular a.

a makes P(x) true.

 ∴ Q(a).

 ∴a makes Q(x) true.

Universal ModusTollens

Another crucially important rule of inference is universal modus tollens.Its validity results from combining universal instantiation with modus tollens. Universal modus tollens is the heart of proof of contradiction, which is one of the most important methods of mathematical argument.

Universal Modus Tollens

Formal Version

Informal Version

∀x, if P(x) then Q(x).

If xmakes P(x) true, then xmakes Q(x) true.

~Q(a), for a particular a

a does not make Q(x) true.

∴ ~P(a).

∴ a does not make P(x) true.

Exercise

All freshmen must take writing.

Caroline is a freshman.

∴ Caroline must take writing.

ANSWER:

Solution:

Step 1:

In this problem we need to state that which are valid statements and which are invalid statements and we need to justify the answer.

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