Valid and Invalid arguments

1
Valid and Invalid arguments
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Definition of Argument

• Sequence of statements
• Statement 1
• Statement 2
• Therefore, Statement 3.
• Statements 1 and 2 are called premises.
• Statement 3 is called conclusion.

3
Examples of Arguments

• It is raining or it is snowing
• It is not snowing
• Therefore, it is raining.
• If x2 then xlt5
• xlt5
• x is an even integer
• Therefore, x2.

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Argument Form

• If the premises and the conclusion
• are statement forms
• instead of statements,
• then the resulting form is called
• argument form.
• Ex If p then q
• p
• q.

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Validity of Argument Form

• Argument form is valid means that
• for any substitution of statement variables,
• if the premises are true,
• then the conclusion is also true.
• The example of previous slide is a
• valid argument form.

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Checking the validity of an argument form

• Construct truth table for the premises and the
  conclusion
• Find the rows in which all the premises are true
  (critical rows)
• a. If in each critical row the conclusion
• is true
• then the argument form is valid
• b. If there is a row in which conclusion
  is false
• then the argument form is invalid.

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Example of valid argument form

• p and q
• if p then q
• q. premises conclusion
• Critical row

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Example of invalid argument form

• p or q
• if p then q
• p. premises conclusion
• Critical row
• Critical row

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Valid Argument Forms

• Modus ponens If p then q
• p
• q.
• Modus tollens If p then q
• q
• p.

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Valid Argument Forms

• Disjunctive addition p
• p or q.
• Conjunctive simplification p and q
• p.
• Disjunctive Syllogism p or q
• q
• p.
• Hypothetical Syllogism p q
• q r
• p r.

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Valid Argument Forms

• Proof by division into cases
• p or q
• p r
• q r
• r
• Rule of contradiction
• p c
• p

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A more complex deduction

• Knights always tell the truth,
• and knaves always lie.
• U says None of us is a knight.
• V says At least three of us are knights.
• W says At most three of us are knights.
• X says Exactly five of us are knights.
• Y says Exactly two of us are knights.
• Z says Exactly one of us is a knight.
• Which are knights and which are knaves?