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LOGIC MAIN PAGE

LOGICAL OPERATORS

Sample sentences
AND operator
IF/THEN operator
NOT operator
OR operator
XOR operator

RULES OF LOGIC

Chain Rule
Conjunctive Addition
Contrapositive
DeMorgan's Law
Disjunctive Addition
Disjunctive Inference
Disjunctive Infer. (XOR)
Double Negation
Modus Ponens
Modus Tollens
Mutual Exclusion
Simplification

VALIDITY PROOFS

2-step
3-step
4-step
5-step or more
Bad Argument

AND Operator

The logical operator AND (symbolized ^) is called the conjunction.

The use of AND in formal logic is the same as its most common use in everyday language. It joins two statements asserting they are both true. Here is an example:

Manhattan is an island, and North America is a continent.

In formal logic, we can replace the two statements with the letters p and q (we can use any other letters that we want to). So, let the statement "Manhattan is an island" be p and let the statement "North America is a continent" be q.

Now we can rewrite our example in the formal way:

p ^ q: "Manhattan is an island, and North America is a continent."

This is a logical expression. In this expression, p and q are called conjuncts.

 

Truth table for AND

A truth table lists truth values (that is, either true or false) for every possible combination of component values in a logical expression.

In order to build a truth table for AND, we need to consider the simplest example of a conjunction:

p ^ q: "Manhattan is an island, and North America is a continent."

In formal logic, p and q (called conjuncts) can take on two possible values: true or false. How many combinations of p and q can we come up with? The answer is four. Let us list all the possible combinations:

1. p is true and q is true
2. p is true and q is false
3. p is false and q is true
4. p is false and q is false

We know that a conjunction is true only when both conjuncts are true. There is only one case in which both p and q are true -- it's the first case. Now we can build a truth table. We are using T for true and F for false.

p
q
p ^ q
T
T
T
T
F
F
F
T
F
F
F
F

As you read the table, you will notice that when p is true and q is true, the entire expression is true. When p is true and q is false, the entire expression is false, and so on.

 

Commutativity of Conjunction

Commutativity of Conjunction means that conjuncts can be interchanged without altering the result.

We can write p ^ q or q ^ p -- these two expressions are equivalent. The phrase "He's hot and he's tired" is completely equivalent to "He's tired and he's hot."

 

Rules of Conjunction

These are links to specific rules pertinent to the AND operator.

Conjunctive Addition
Simplification
DeMorgan's Law