Module 07 : Direct Proofs

What does universal instantiation state?

If a property is true of everything in a set, then it is true of any particular thing in the set.

Provide an example of universal instantiation.

If 'All men are mortal' and 'Socrates is a man', then 'Socrates is mortal'.

What is universal modus ponens? Explain in propositional logic.

If for all x, if P(x) then Q(x) and P(a) for a particular a, then Q(a)

How is universal modus tollens defined? Explain in propositional logic

If for all x, if P(x) then Q(x) and ~P(a) for a particular a, then ~Q(a)

What is the inverse error?

If for all x, if P(x) then Q(x), and ~P(a) for a particular a, then ~Q(a) ; this argument is invalid.

What is the converse error?

If for all x, if P(x) then Q(x), and Q(a) for a particular a, then P(a) ; this argument is invalid.

How do we use diagrams in testing the validity of arguments?

Making a Venn diagram for each that represents relationships between variables

How do we prove the validity of an argument? List the steps

1. Express the statement to be proved in the form “For every x belonging to D, if P(x) then Q(x).”

2. Suppose x is an arbitrarily chosen element of D
   for which P(x) is true. “Suppose
   x belongs to D and P(x).”

3. Show that the conclusion Q(x) is true by using definitions, previously established results, and the rules for logical inference.
   
   

How to we disprove an argument using counterexample?

Find a specific instance showing that a universal statement is false where the hypothesis true and the conclusion is false.

What is the definition of an even integer?

An integer n is even if and only if n equals twice some integer.