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Valid and Invalid Valid and Invalid argumentsarguments

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Definition of Definition of ArgumentArgument• Sequence of statements:

Statement 1; Statement 2;Therefore, Statement 3.

• Statements 1 and 2 are called premises.

• Statement 3 is called conclusion.

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Examples of ArgumentsExamples of Arguments• It is raining or it is snowing; It is not snowing;

Therefore, it is raining.

• If x=2 then x<5; x<5; x is an even integer;Therefore, x=2.

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Argument FormArgument 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 FormValidity 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 Checking the validity of an argument formof an argument form

1) Construct truth table for the premises and the conclusion;

2) Find the rows in which all the premises are true (critical rows);

3) 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 Example of valid argument formform

p and q; if p then q; q. premises conclusion

Critical row

p q p and q if p then q q

T T T T T

T F F

F T F

F F F

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Example of invalid argument Example of invalid argument formform

p or q; if p then q; p. premises conclusion

Critical row

Critical row

p q p or q if p then q p

T T T T T

T F T F

F T T T F

F F F

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Valid Argument FormsValid Argument Forms• Modus ponens: If p then q; p;

q.

• Modus tollens: If p then q;~q;~p.

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Valid Argument FormsValid 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 FormsValid 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 deductionA 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?