Logic of Compound Statements: Valid and Invalid Arguments, Slides of Discrete Mathematics

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Chapter 1

The Logic of Compound Statements

Section 1.

Valid & Invalid Arguments

Testing Argument Form

• Identify the premises and conclusion of the

argument form.

• Construct a truth table showing the truth

values of all the premises and the conclusion.

• If the truth table reveals all TRUE premises

and a FALSE conclusion, then the argument

form is invalid. Otherwise, when all premises

are TRUE and the conclusion is TRUE, then the

argument is valid.

Example

• If Socrates is a man, then Socrates is mortal.

• Socrates is a man.

• :. Socrates is mortal.

• Syllogism is an argument form with two premises

and a conclusion. Example Modus Ponens form:

• If p then q.
• p
• :. q

Example Invalid Form

• p -> q v ~r

• q -> p ^ r

• :. p -> r

Modus Tollens

• If p then q.
• ~q
• :. ~p
• Proves it case with “proof by contradiction”
• Example:
• if Zeus is human, then Zeus is mortal.
• Zeus is not mortal.
• :. Zeus is not human.

Rules of Inference

• Rule of inference is a form of argument that is

valid.

• Modus Ponens, Modus Tollens
• Generalization, Specialization, Elimination, Transitivity, Proof by Division, etc.

Rules of Inference

• Generalization

• p :. p v q
• q :. p v q

• Specialization

• p ^ q :. p
• p ^ q :. q
• Example:
  • Karl knows how to build a computer and Karl knows how to program a computer
  • :. Karl knows how to program a computer

Rules of Inference

• Transitivity (Chain Rule)

• p -> q, q -> r, :. p -> r
• Example
  • If 18,486 is divisible by 18, then 18486 is divisible by 9.
  • If 18,486 is divisible by 9, then the sum of the digits of 18,486 is divisible by 9.
  • :. 18,486 is divisible by 18, then the sum of the digits 18,486 is divisible by 9.

Rules of Inference

• Proof by Division

• p v q, p->r, q->r, :.r
• Example
  • x is positive or x is negative.
  • If x is positive, then x 2 > 0.
  • If x is negative, then x 2 > 0.
  • :. x 2 > 0

Contradictions and Valid Arguments

• Contradiction Rule – If you can show that the

supposition that statement p is false leads logically to a contradiction, then you can conclude that p is true.

• ~p -> c, :. p