Symbolic Logic, Part II: The “And” Operator

In the last post, we discussed the statement P, which we defined as “The car is green,” and how the statement is either True, or it is False.  We also talked about “not P”, which we symbolized as ~P.

Now, let’s say we have another statement:

The car has four doors.

We’re going to call this statement Q. It also has two possible states: True and False. Here is its truth table:

Q

True

False

And here is the truth table for Q and ~Q:

Q	~Q
True	False
False	True

Now we can combine P and Q to create a compound statement thusly:

The car is green and it has four doors.

We are going to symbolize this conjunction like so:

P ∧ Q

That little upside-down “V” between P and Q represents the operator “and,” and the statement as a whole would be read “P and Q.” Now, P∧Q also evaluates as a whole statement to either True or False, but this depends on the values of both P and Q. In the case of the ∧ operator, both P and Q must be True for P∧Q to be True. Here is the truth table for this:

P	Q	P∧Q
True	True	True
True	False	False
False	True	False
False	False	False

So, if we say P∧Q, but the car actually has two doors, then it doesn’t really matter what color the car is, the entire statement is false.  The rule of the “And” operator is this:

A compound statement using the “And” operator evaluates to True if and only if both component statements are True; otherwise, the compound statement is False.

If we were going to say that:

The car is green and does not have four doors.

We would symbolize it thusly:

P ∧ ~Q

And the truth table would look like this:

 

P	Q	~Q	P ∧ ~Q
True	True	False	False
True	False	True	True
False	True	False	False
False	False	True	False

So, in the case of our hypothetical two-door, green car, P ∧ ~Q will evaluate to True, since it is true that the car is green and it does not have four doors.  Note that this statement is also True if the car has five doors, or three doors, or anything other than four doors.

Next: The “Or” Operator