Math 3000 Lectures 2 & 3

III. Elementary Logic - Propositions and Connectives

A Proposition (or statement) is a sentence that is either true or false (without additional information).

The logical connectives are defined by truth tables (but have English language counterparts) .

Logic Math English
Conjunction and
Disjunction or (inclusive)
Negation ~ not
Conditional if ... then ...
Biconditional if and only if 

Two statements are equivalent (logically equivalent) if they have the same truth table values.

A denial is a statement equivalent to the negation of a statement.

Work out the truth table for ~ (P (Q P)).

For 2 logical variables there are 16 ways to combine them, but we can get all of them with simple combinations of conjunction, disjunction and negation. In fact, we can get them all with just one symbol (nand | ).

P Q ~ (P ~ Q)

P Q (P Q) (Q P)

On a real mythical island ...

NB: Truth values of these logical forms have nothing to do with meaning!!!! (content) ... that is a philosophical concept, not a mathematical one.

Problem:   1    C    B    2

You see the four cards above. Every card has a number on one side and a letter on the other side. Which of these cards must be turned over to determine the truth of the statement:
If a card has a B on one side, then it has a 2 on the other side. [First and third]

Tautologies and Contradictions:

A tautology is a statement which is always true. A contradiction is a statement which is always false.

A (B C) (A B) (A C)

~ (A B) ~A ~B

P ~(~ P)

(A ~ A) (B ~B)

(P (P Q)) Q   modus ponens

P (Q R) (P Q) R

Contrapositive

The contrapositive of the statement if P then Q is if ~Q then ~P. An implication and its contrapositive are logically equivalent, so one can always be used in place of the other.

(P Q) (Q R) (P R)

Reading from Martin Gardner if time permits.

A predicate (open sentence) is a sentence containing one or more variables which becomes a proposition upon replacement of the variables.

Universal and Existential quantifiers

A predicate is not a proposition, it does not have a truth value. One can however use quantifiers to make propositions about predicates. For instance, the universal quantifier () is used to say that a given predicate is true for all possible values of its variables. This is a proposition, since it is either true or false. Similarly, the existential quantifier () is used to say that there is some value of the variables which makes the predicate a true statement.

Denials of quantified statements

A denial of a quantified statement is obtained by switching the quantifier (changing for all to there exists, or vice versa) and negating the predicate.

A sequence {xn} is Cauchy iff for each > 0 there is a natural number N so that if n,m > N then | xn - xm | < .

What does it mean for a sequence not to be Cauchy?

Counterexample: To prove a universally quantified statement to be false, all that is needed is one set of values for the variables which makes the predicate a false proposition. Such an example is called a counterexample.

Other examples of quantified statements:

  1. x y (x + y = 0)
  2. x y (x + y = 0)
  3. x y ( x + y = 0)
  4. x y (x + y = 0)
  5. All poobahs are zots. x (P(x)Z(x))
  6. Some poobahs are zots. x (P(x)Z(x))
  7. Everybody has a friend who is always honest. x y (F(x,y) t H(y,t))
  8. negate above