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.
The sentence "1 = 0" is also a statement, but its truth value is F.
"It will rain tomorrow" is a proposition. For its truth value we shall have to wait for tomorrow.
The following statement might well be uttered by a Zen Master to a puzzled disciple: "If I am Buddha, then I am not Buddha." This is a statement which, we shall see later on, really amounts to the simpler statement "I am not Buddha." As long as the speaker is not Buddha, this is true.
"Solve the following equation for x" is not a statement, as it cannot be assigned any truth value whatsoever. (It is an imperative, or command, rather than a declarative sentence.)
"The number 5" is not a proposition, since it is not even a complete sentence.
"This statement is true" may seem like a statement, but there is no way that its truth value can ever be determined: is it true, or is it false? We thus disqualify it as well. (In fact, it is the negation of the liar's paradox; see below for a discussion of negation.)
Sentences such as these are called self-referential sentences, since they refer to themselves.
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
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.
The negation of p is the statement ~p, which we read "not p." Its truth value is defined by the following truth table.
The negation symbol "~" is an example of a unary logical operator (the term "unary" indicates that the operator acts on a single proposition).
|(a)||p: "2+2 = 4"|
|(b)||q: "1 = 0"|
|(c)||r: "Diamonds are a pearl's best friend."|
|(d)||s: "All the politicians in this town are crooks."|
(b) ~q: "1 0."
(c) ~r: "Diamonds are not a pearl's best friend."
(d) ~s: " Not all the politicians in this town are crooks."
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:
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.
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.
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.
pq: "This galaxy will ultimately disappear into a black hole and 2+2=4," or the more astonishing statement: "Not only will this galaxy ultimately disappear into a black hole, but 2+2 = 4!"
q is true, so if p is true then the whole statement pq will be true. On the other hand, if p is false, then the whole statement pq will be false.
p(~q) says: "This galaxy will ultimately disappear into a black hole and 2+2 4," or "Contrary to your hopes and aspirations, this galaxy is doomed to eventually disappear into a black hole; moreover, two plus two is decidedly different from four!"
Since ~q is false, the whole statement p(~q) is false (regardless of whether p is true or not).
The first clause is the negation of p, so is ~p. The second clause is simply stating the (false) claim that logic is a boring subject, and thus amounts to q. The phrase "even though" is a colorful way of saying that both clauses are true, and so the whole statement is just (~p)q.
The statement is asserting that all three statements p, q and r are true. (Note that "but" is simply an emphatic form of "and.") Now we can combine them all in two steps: Firstly, we can combine p and q to get pq, meaning "This topic is boring and this web site is boring." We can then conjoin this with r to get: (pq)r. This says: "This topic is boring, this web site is boring and life is boring." On the other hand, we could equally well have done it the other way around: conjoining q and r gives "This web site is boring and life is boring." We then conjoin p to get p(qr), which again says: "This topic is boring, this web site is boring and life is boring." We shall soon see that
As we've just seen, there are many ways of expressing a conjunction in English. For example, if
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.
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.
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.
|(a)||What does pq say?|
|(b)||What does (pq)(~r) say?|
(Remember that this does not exclude the possibility that the butler and cook both did it-or that they were in fact the same person! The only way that pq could be false is if neither the butler nor the cook did it. )
(b) (pq)(~r) says "either the butler or the cook did it, but not the lawyer." .
(a) "Either 55 is not divisible by 11 or 676 is not divisible by 11."
(b) "Either 55 is divisible by either 5 or 11, or 676 is divisible by 11."
(b) This is the disjunction of all three statements, and is thus (pr)q, or, equivalently, p(rq). It is also equivalent to (pq)r and p(qr).
(a) is true because ~q is true. (b) is true because p is true. Notice that r is also true. If at least one of p, q, or r is true, the whole statement (pq)r will be true.
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.