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Valid and Invalid arguments. 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 . Examples of Arguments. It is raining or it is snowing;

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Presentation Transcript
definition of argument
Definition of Argument
  • Sequence of statements:

Statement 1;

Statement 2;

Therefore, Statement 3.

  • Statements 1 and 2 are called premises.
  • Statement 3 is calledconclusion.
examples of arguments
Examples 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.

argument form
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.

validity of argument form
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.

checking the validity of an argument form
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.

example of valid argument form
Example of valid argument form

p and q;

if p then q;

q. premisesconclusion

Critical row

example of invalid argument form
Example of invalid argument form

p or q;

if p then q;

p. premisesconclusion

Critical row

Critical row

valid argument forms
Valid Argument Forms
  • Modus ponens: If p then q;

p;

q.

  • Modus tollens: If p then q;

~q;

~p.

valid argument forms1
Valid Argument Forms
  • Disjunctive addition: p;

p or q.

  • Conjunctive simplification: p and q;

p.

  • Disjunctive Syllogism: p or q;

~q;

p.

  • Hypothetical Syllogism: pq;

q r;

pr.

valid argument forms2
Valid Argument Forms
  • Proof by division into cases:

p or q

p r

qr

r

  • Rule of contradiction:

~p c

p

a more complex deduction
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?