### Truth Table

Definition of 'truth' in Sentential logic:

- Truth only applies to propositions. When I assert: "There is a computer in my office." And indeed there is a computer in my office. The proposition that I assert above is true. The computer is not true and the office is not true. It makes no sense to say of computers or chairs that they are either true or false. Truth can only be attached to propositions.

This concept has an arithmetic analog:

- The numbers 2 or 3 are neither true nor false.

But the arithmetic proposition:

- 2 + 3 = 5 is true, and 2 + 3 = 4 is false.

Every proposition is either true or false. No proposition can be both true and false or neither true nor false. Consider the simple, atomic proposition: "Four is even." We can ask of it two questions:

- (a)What is the truth value of, "Four is even."?
- (b) What are the possible truth values for, "Four is even."?

The answer (a) is a single value: true. This is the case because four is indeed an even number. The answer to (b) is two values: true or false. Since, "Four is even." is a proposition, it can have two values. Now let's consider the compound, molecular proposition: "Four is even and three is odd." What are the possible truth values of this proposition? A compound proposition with two components generates four possibilities. To see this let's follow the logical convention of replacing simple propositions with a single letter: proposition with a single letter:

- E = "Four is even."
- O = "Three is odd."

And replace the conjunction 'and' with the symbol: '&' The compound proposition can be symbolized:

- (E . O)

The four possible truth value combinations are:

- E could be true and O could be true.
- E could be true and O could be false.
- E could be false and O could be true.
- E could be false and O could be false.

A truth table is a diagrammatic device to show all the possible truth values of any given proposition. As we saw above a simple, atomic proposition has only two possible values: true or false. Let us use the convention:

- T = true
- F = false

The truth table for (E . O) is:

CONJUNCTION

INPUT | OUTPUT
--- | ---
E......O | E & O
T......T |
T......F |
F......T |
F......F |

We still have to derive a truth value for (E & O). Conjunction is very intuitive: a conjunction is only true when the components to the left and right of '&' are both true. It is false under any other condition. So (E & O) is true only under truth value combination one above. Conjunction, then, can be thought of as function since it outputs a truth value for '&' based on the inputs, the truth values of its right and left components. In logic this is characterized by a truth table:

CONJUNCTION

INPUT | OUTPUT
--- | ---
E......O | E & O
T......T | T
T......F | F
F......T | F
F......F | F