# 1.3: Truth Tables and the Meaning of '~', '&', and 'v'

We have said that '~A' means not A, 'A&B' means A and B, and 'AvB' means A or B in the inclusive sense. This should give you a pretty good idea of what the connectives '~', '&', and 'v' mean. But logicians need to be as exact as possible. So we need to specify how we should understand the connectives even more exactly. Moreover, the method which we will use to do this will prove very useful for all sorts of other things.

To get the idea, we start with the very easy case of the negation sign, '~'. The sentence 'A' is either true or it is false. If 'A' is true, then '~A' is false. If 'A' is false, then '~A' is true. And that is everything you need to know about the meaning of '~'. We can say this more concisely with a table, called a Truth Table:

Truth table definition of '~'

 Case A ~A 1 t f 2 f t

The column under 'A' lists all the possible cases involving the truth and falsity of 'A'. We do this by describing the cases in terms of what we call Truth Values. The case in which A is true is described by saying that A has the truth value t. The case in which A is false is described by saying that A has the truth value f. Because A can only be true or false, we have only these two cases. We explain how to understand '~' by saying what the truth value of '~A' is in each case. In case 1, '~A' has the truth value f; that is, it is false. In case 2, '~A' has the truth value t; that is, it is true. Although what we have done seems trivial in this simple case, you will see very soon that truth tables are extremely useful.

Let us see how to use truth tables to explain '&'. A conjunction has two atomic sentences, so we have four cases to consider:

 Case A B A&B 1 t t 2 t f 3 f t 4 f f

When 'A' is true, 'B' can be true or false. When 'A' is false, again 'B' can be true or false. The above truth table gives all possible combinations of truth values which 'A' and 'B' can have together.

We now specify how '&' should be understood by specifying the truth value for each case for the compound 'A&B':

Truth Table definition of '&'

 Case A B A&B 1 t t t 2 t f f 3 f t f 4 f f f

In other words, 'A&B' is true when the conjuncts 'A' and 'B' are both true. 'A&B' is false in all other cases, that is, when one or both of the conjuncts are false.

A word about the order in which I have listed the cases. If you are curious, you might try to guess the recipe I used to order the cases. (If you try, also look at the more complicated example in Section 1.5.) But I won't pause to explain, because all that is important about the order is that we don't leave any cases out and all of us list them in the same order, so that we can easily compare answers. So just list the cases as I do.

We follow the same method in specifying how to understand 'V'. The disjunction 'AvB' is true when either or both of the disjuncts 'A' and 'B' are true. 'AvB' is false only when 'A' and 'B' are both false:

truth table definition of 'v'

 Case A B AvB 1 t t t 2 t f t 3 f t t 4 f f f

We have defined the connectives '~', '&', and t' using truth tables for the special case of sentence letters 'A' and 'B'. But obviously nothing will change if we use some other pair of sentences, such as 'H' and 'D'.

This section has focused on the truth table definitions of '~', '&' and 'v'. But along the way I have introduced two auxiliary notions about which you need to be very clear. First, by a Truth Value Assignment of Truth Values to Sentence Letters, I mean, roughly, a line of a truth table, and a Truth Table is a list of all the possible truth values assignments for the sentence letters in a sentence:

An Assignment of Truth Values to a collection of atomic sentence letters is a specification, for each of the sentence letters, whether the letter is (for this assignment) to be taken as true or as false. The word Case will also be used for 'assignment of truth values'.

A Truth Table for a Sentence is a specification of all possible truth values assignments to the sentence letters which occur in the sentence, and a specification of the truth value of the sentence for each of these assignments.