This chapter introduces the truth functions of logic. You will learn what these functions represent, how you can use them (likely you have already used some of them thoughtless in your everyday life) and in which way they are characterized. Starting with an overview of all considered connectives everyone will be discussed later more in detail ...

Table1: Logic functions

Shortcut	Symbol	Description
AND		Conjunction
OR		Disjunction
XOR		Exclusive Disjunction
NOT		Negation
NAND		Negated Conjunction
NOR		Negated Disjunction
		Material Implication
		Material Equivalence

First of all we want to have a look at all logic functions considered in this chapter. In Table 1 you can see a list of all usual connectives used in logic. The red indicated ones are the basic functions, the remaining ones are special combinations compiled of the basic functions and simply replaced by a single truth-functional connective (because of its importance). Knowldeges about the logic functions are necessary in many fields of science e.g. in the Design of digital circuits.

One possible description of a logic function is a definition with words, like "If ... then ..." or something like that. You will encounter some definitions of that kind later. Another way of presenting this rule is by means of a truth table. We don't need to subscribe the function in words, we can do it the way most of the tables show. Each row of the truth table corresponds to one of the four possible combinations of truth values which may be possessed by interpretations of the schematic letters a and b. The last entry gives the truth value of the corresponding interpretations of a and b. As an example: The schema ab is called the conjunction of the two schemas a and b. Conversely, a and b are called the conjuncts of a and b.

But now we will start with the detailed function explanations ...

Let's have a look at the following sentence:

Everyday we use such expressions. The sentence expresses two main ideas. We could divide it into seperate statements, the phrase "Yesterday you went to the post office" and "I called Anne at the phone box". We join together two smaller statements by means of the link-word "and". By using different link-words we can construct a series of new statements such as:

1. Yesterday you went to the post office and I called Anne at the phone box.
2. Yesterday you went to the post office because I called Anne at the phone box.
3. Yesterday you went to the post office but I called Anne at the phone box.
4. Yesterday you went to the post office while I called Anne at the phone box.
5. Yesterday you went to the post office although I called Anne at the phone box.
6. Yesterday you went to the post office or I called Anne at the phone box.
7. Yesterday you went to the post office whenever I called Anne at the phone box.
8. Yesterday you went to the post office if I called Anne at the phone box.

"Link-words" are said to be truth-functional whenever the truth value of the statement obtained by using it to link two given statements is uniquely determined by the truth values of the two statements themselves. Let's try to solve a simple exercise ...

Hopefully you didn't find this exercise somewhat problematic. All answers could be solved in common sense without any special knowledge of the "technical logic".

The first truth function we already know: the conjunction.

"An interpretation satisfies the statement ab if and only if both a and b are satisfied."

Table2: The Conjunction

a	b	ab
false	false	false
false	true	false
true	false	false
true	true	true

Table 2 illustrates the truth table of the conjunction. As expected the compound statement is only true if both of the input variables (in our case a and b) are true.

The link-word "or" may cause slight complications in the meaning of the english language. In practice, the sense of "or" most often encountered in logic is the inclusive "or". That means: "a or b or both". To demonstrate this we will have a look at the statement: "Mike is in Paris or in Berlin". Only one of both phrases could be true, they exclude each other. Thus, this is the exclusive version of "or" which is different to the logical "or". It should just demonstrate that there is a lot of differences between the language and the logical meaning, so don't mix them up.

The connective for "or" is called disjunction and the function can be explained as follows:

"An interpretation satisfies the statement ab if and only if at least one of a and b is satisfied."

Table3: The Disjunction

a	b	ab
false	false	false
false	true	true
true	false	true
true	true	true

In other words we can say: "An interpretation falsifies ab if and only if it falsifies both a and b".

The statement "ab" is called the disjunction of a and b. The truth table corresponding to this rule is shown in Table 3.

As an example we might have a look at the phrase "Peter ate fish or Susan ate steak". Both expressions could be true at the same time. If we apply the truth values of the "or" function to the sentence, we notice that the whole statement is true in 3 cases: At first if Peter actually ate fish and Susan didn't had steak, furthermore if Peter didn't ate fish and Susan had a steak and finally if both is true, Peter really ate fish and Susan had actually steak to dinner.

Now we will have a look the next connective, a slightly changed version of the disjunction ...

Table4: The Exclusive Disjunction

a	b	a XOR b
false	false	false
false	true	true
true	false	true
true	true	false

If we consider the compound statement "The earth is a cube or it is a sphere.", we notice the quite different meaning to the sentence considered in the last paragraph. It is straigthforward that the earth can't be both a cube and a sphere. So this sentence expresses the "exclusive or". It excludes the case that both phrases in the statement could be true at the same time.

Table 4 illustrates the truth table of the XOR function. "XOR" is used as an abbrevation of the "eXclusive OR". (Hint: We will encounter all "short names" of all truth functions later at the "Overall Exercise". There you can select the desired truth function by the symbolic name.)

For the exclusive "or" we use the abbrevation XOR (for "exclusive or") and define it as follows:

Let's continue with the next connective ...

The negation has much the same mening as the english word "not". For example, "John is not married".
Look at the following two statements:

1. John is married.
2. John is not married.

Table5: The Negation

a	a
false	true
true	false

Note that "not" is not a link-word in the sense of joining together two statements; rather it acts on a single statement to produce a new one. A single statement with "not" in is, usually, completely determined by truth value of the statement obtained by leaving out the "not" ---- namely, it is the opposite truth value.
A more strictly truth-functional expression is "It is not the case that" ,which behaves very much as we want negation to behave in the prospositional calculus. Thus instead of "John is not married." we could say "It is not the case that John is married. " , in which there is a clear seperation between the negation and what is negated. Sometimes, it is necessary to use "It is not the case that"-style to express the negation of a sentence, since simply inserting "not" has a different effect.

Example:
The negation of The shop is open until midday. is not The shop is not open until midday since both these statements could be false(e.g. if the shop is now shut, but will open at 11 a.m.).
The true negation can only readily the expressed as It is not the case that the shop is open until midday.

The Negation is one of the tree basic logical operations(the others are the Conjunction and Disjunction), with them all statements are explainable.

"An interpretation satisfies the statement a if the statement a is not satisfied."

Exercise(4)

Unfortunately your browser doesn't support Java applets, so you can't see the application here ...

Table6: The Negated Conjunction

a	b	a NAND b
false	false	true
false	true	true
true	false	true
true	true	false

The negated conjunction is a combination of conjunction and negation. At first the statements a and b would be used with the conjunction. Then the result of this logical operation would be the parameter of the negation. The result of the negation is also the result of the whole negated conjunction. Sometimes would be used a special abbrevation NAND(consist of NOT and AND) for the negated conjunction. The a NAND b (NAND is the shortcut for the negated conjunction) can be explained by NOT(a AND b) or (ab).

Table7: The Negated Disjunction

a	b	a NOR b
false	false	true
false	true	false
true	false	false
true	true	false

The negated conjunction is a combination of disjunction and negation. The a NOR b (NOR is the shortcut for the negated disjunction) can be explained by (ab).

Table8: The Marterial Implication

a	b	a b
false	false	true
false	true	true
true	false	false
true	true	true

The marterial implication's are typicall if-statements.

Look at the following statements:

1. If I reach the train then I arrive at 8 p.m. .
2. I arrive at 8 p.m. if I reach the train.
3. When I reach the train then I arrive at 8 p.m. .

"An interpretation satisfies the statement a b if if the statement a is satisfied then b is also satisfied else it's always satisfied."

Table9: The Marterial Equivalence

a	b	a b
false	false	true
false	true	false
true	false	false
true	true	true

The marterial equivalence is a marterial implication in both directions. So you can a b explained by (a b)(b a). So the marterial equivalence means that the two statements, which are connected, have the same truth status. Some examples:

1. Tom go to the party if and only if Jane goes also to the party.
2. If Bill work then Mike also work and If Mike work then Bill also work.

"An interpretation satisfies the statement a b if and only if the statement's a and b has the same truth status."

So far we got to know all usual functions and should be able to apply them in an easy exercise. Just select the desired function and then fill out the result row of the table by clicking anywhere within the cells (the value will alter if you click again in the cell). If you think it is correct press the "Evaluation" button and see what happens. In the case of an error you may select "Verification" to see the correct values.