Introduction to Logic

by
Stefan Waner and Steven R. Costenoble

1. Statements and Logical Operators

This on-line text is, for the most part, devoted to the study of so-called Propositional Calculus. Contrary to what the name suggests, this has nothing to do with the subject most people associate with the word "calculus." Actually, the term "calculus" is a generic name for any area of mathematics that concerns itself with calculating. For example, arithmetic could be called the calculus of numbers. Propositional Calculus is then the calculus of propositions. A proposition, or statement, is any declarative sentence which is either true (T) or false (F). We refer to T or F as the truth value of the statement.


Example 1 Propositions


Example 1B Self-Referential Sentences

Before we go on...


We shall use the letters p, q, r, s and so on to stand for propositions. Thus, for example, we might decide that p should stand for the proposition "the moon is round." We shall write

to express this. We read this as

We can form new propositions from old ones in several different ways. For example, starting with p: "I am an Anchovian," we can form the negation of p: "It is not the case that I am an Anchovian" or simply "I am not an Anchovian." We denote the negation of p by ~p, read "not p." What we mean by this is that, if p is true, then ~p is false, and vice-versa. We can show this in the form of a truth table:

p
~p
T
F
F
T

On the left are the two possible truth values of p, with the corresponding truth values of ~p on the right. The symbol ~ is our first example of a logical operator.

Following is a more formal definition.

Negation

The negation of p is the statement ~p, which we read "not p." Its truth value is defined by the following truth table.

p
~p
T
F
F
T

The negation symbol "~" is an example of a unary logical operator (the term "unary" indicates that the operator acts on a single proposition).


Example 2 Negating Statements

Solution

Before we go on...

Notice that ~p is false, because p is true. However, ~q is true, because q is false. A statement of the form ~q can very well be true; it is a common mistake to think it must be false.

To say that diamonds are not a pearl's best friend is not to say that diamonds are a pearl's worst enemy. The negation is not the polar opposite, but whatever would deny the truth of the original statement. Similarly, saying that not all politicians are crooks is not the same as saying that no politicians are crooks, but is the same as saying that some politicians are not crooks. Negations of statements involving the quantifiers "all" or "some" are tricky. We'll study quantifiers in more depth when we discuss the predicate calculus.


Here are some for you to try:

Example 2P Practice in Negating Statements


Here is another way we can form a new proposition from old ones. Starting with p: "I am clever," and q: "You are strong," we can form the statement "I am clever and you are strong." We denote this new statement by pq, read "p and q." In order for pq to be true, both p and q must be true. Thus, for example, if I am indeed clever, but you are not strong, then pq is false.

The symbol is another logical operator. The statement pq is called the conjunction of p and q.

Conjunction

The conjunction of p and q is the statement pq, which we read "p and q." Its truth value is defined by the following truth table.

p
q
pq
T
T
T
T
F
F
F
T
F
F
F
F

In the p and q columns are listed all four possible combinations of truth values for p and q, and in the pq column we find the associated truth value for pq. For example, reading across the third row tells us that, if p is false and q is true, then pq is false. In fact, the only way we can get a T in the pq column is if both p and q are true, as the table shows.

The conjunction symbol "" is an example of a binary logical operator (the term "binary" indicates that the operator acts on a pair of propositions).

In the following examples, we begin to see the way in which the rich color and innuendo of language is stripped to the bare essentials by logical symbolism.


Example 3 Conjunction


Example 4 Combining Conjunction with Negation


Example 4P Practice with Conjunction


Example 5 Words to Symbols


Example 6 Combining Three Statements


As we've just seen, there are many ways of expressing a conjunction in English. For example, if

and the following are all ways of saying pq:

Any sentence that suggests that two things are both true is a conjunction. The use of symbolic logic strips away the elements of surprise or judgement that can also be expressed in an English sentence.

We now introduce a third logical operator. Starting once again with p: "I am clever," and q: "You are strong," we can form the statement "I am clever or you are strong," which we write symbolically as pq, read "p or q." Now in English the word "or" has several possible meanings, so we have to agree on which one we want here. Mathematicians have settled on the inclusive or: pq means p is true or q is true or both are true. 

With p and q as above, pq stands for "I am clever or you are atrong, or both." We shall sometimes include the phrase "or both" for emphasis, but even if we do not that is what we mean. We call pq the disjunction of p and q.

Disjunction

The disjunction of p and q is the statement pq, which we read "p or q." Its truth value is defined by the following truth table.

p
q
pq
T
T
T
T
F
T
F
T
T
F
F
F

This is the inclusive or, so pq is true when p is true or q is true or both are true.

Notice that the only way for the whole statement to be false is for both p and q to be false. For this reason we can say that pq also means "p and q are not both false." We'll say more about this in the next section.

The disjunctgion symbol "" is our second example of a binary logical operator.

Example 7 Disjunction

Solution


Example 7P Practice with Disjunction


Example 8 Words to Symbols

Solution

Before We Go On ...


We end this section with a little terminology: A compound statement is a statement formed from simpler statements via the use of logical operators. Examples are ~p, (~p)(qr) and p(~p). A statement that cannot be expressed as a compound statement is called an atomic statement. For example, "I am clever" is an atomic statement. In a compound statement such as (~p)(qr), we shall refer to p, q and r as the variables of the statement. Thus, for example, ~p is a compound statement in the single variable p.

Summary: Negation, Conjunction and Disjunction

Symbolic Form
Truth Table
In Words
Negation
~p
p
~p
T
F
F T
not p
Conjunction
pq
p
q
pq
T
T
T
T
F
F
F
T
F
F
F
F
p and q
Disjunction
pq
p
q
pq
T
T
T
T
F
T
F
T
T
F
F
F
p or q

Last Updated: August, 2001
Copyright © 1996 StefanWaner and Steven R. Costenoble

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