Proof by Deduction


















What's an Argument?

	A list of premise statements followed by a conclusion

	premises:
	    P -> Q
	    P and Q
	    Q or not R

	conclusion:
	    Q or R


What's a Valid (or Sound) Argument?

	An argument is valid when,
	if all the premises are true, the conclusion is true.

	When an argument is valid,
	the following statement has to be a tautology:
	(the conjunction of the premises) -> (conclusion)


What's a Proof by Truth Table?
Give a 'Proof by Truth Table' that the argument is valid.

	premises:
	    P
	    P -> Q

	conclusion:
	    Q


What's a Fallacy?

	an invalid (or unsound) argument

Give a 'Proof by Truth Table' that the argument is a fallacy.

	premises:
	    not P
	    P -> Q

	conclusion:
	    not Q


















Why is 'Proof by Truth Table' not always practical?

How do you show an argument is valid without using truth tables?

	1. write each premise on a line
	2. use inference rules to write new lines
	3. eventually write the conclusion on a line


What Inference Rules can you use? (page 76)

	A, A -> B	=>  B		Modus Ponens
	not B, A -> B	=>  not A	Modus Tollens
	A -> B, B -> C	=>  A -> C	Hypothetical Syllogism
	A or B, not A	=>  B		Disjunctive Syllogism
	A		=>  A or B	Addition
	A and B		=>  A		Simplification
	A, B		=>  A and B	Conjunction


How are Inference Rules different from Logical Equivalences?

How do you use Logical Equivalences as Inference Rules?

	For an equivalence P <=> Q
	you have two inference rules: P => Q, Q => P


















Give a deductive proof for the argument.

	There is an undeclared variable or
	there is a syntax error.
	If there is a syntax error,
	there is a missing semicolon or
	a variable name is misspelled.
	There is not a missing semicolon.
	There is not a misspelled variable name.
	Therefore there is an undeclared variable.

	U  undeclared variable
	E  syntax error
	S  missing semicolon
	M  misspelled variable name


Does this mean you have proven that there is an undeclared variable?


















Classwork
You may work with a partner.

Give a deductive proof for the argument.

	If the house is next to a lake,
	the treasure is not in the kitchen.
	If the tree in the yard is an elm,
	the treasure is in the kitchen.
	The house is next to a lake.
	The tree in the yard is an elm or
	the treasure is buried under the flagpole.
	Therefore the treasure is under the flagpole.

	L  house is next to lake
	K  treasure is in kitchen
	E  tree is an elm
	P  treasure is under flagpole