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