Introduction to Logic
by
Stefan Waner and Steven R. Costenoble

This Section: 3. The Conditional and the Biconditional

2. Logical Equivalence, Tautologies and Contradictions Section 3 Exercises 4. Tautological Implications and Tautological Equivalences Main Logic Page "Real World" Page

3. The Conditional and the Biconditional

Consider the following statement: "If you earn an A in logic, then I'll buy you a Corvette." It seems to be made up out of two simpler statements:

What the original statement is then saying is this: if p is true, then q is true, or, more simply, if p, then q. We can also phrase this as p implies q, and we write pq.

Now let us suppose for the sake of argument that the original statement: "If you earn an A in logic, then I'll buy you a Corvette," is true. This does not mean that you will earn an A in logic; all it says is that if you do so, then I will buy you that Corvette. Thinking of this as a promise, the only way that it can be broken is if you do earn an A and I do not buy you a Corvette. In general, we ue this idea to define the statement pq.

Conditional

The conditional pq, which we read "if p, then q" or "p implies q," is defined by the following truth table.

    p
    q
    pq
    T
    T
    T
    T
    F
    F
    F
    T
    T
    F
    F
    T
The arrow "" is the conditional operator, and in pq the statement p is classed the antecedent, or hypothesis, and q is called the consequent, or conclusion.


Here are some examples that will help to explain each line in the truth table.





Looking at the truth table once more, notice that pq is true if either p is false or q is true (or both). Once more, the only way the implication can be false is for p to be true and q to be false. In other words, pq is logically equivalent to (~p)q. The following examples demonstrate this fact.




For lack of a better name, we shall call the equivalence in Example 5 the "Switcheroo" law.

Switcheroo Law

The Switcheroo law is the logical equivalence

    pq(~p)q.

In words, it expresses the equivalence between saying "if p is true, then q must be true" and saying "either p is not true, or else q must be true."


We have already seen how colorful language can be. Not surprisingly, it turns out that there are a great variety of different ways of saying that p implies q. Here are some of the most common:

Some Phrasings of the Conditional

Each of the following is equivalent to the conditional pq.

    If p, then q. p implies q.
    q follows from p. Not p unless q.
    q if p. p only if q.
    Whenever p, q. q whenever p.
    p is sufficient for q. q is necessary for p.
    p is a sufficient condition for q. q is a necessary condition for p.

Notice the difference between "if" and "only if." We say that "p only if q" means pq since, assuming that pq is true, p can be true only if q is also. In other words, the only line of the truth table that has pq true and p true also has q true. The phrasing "p is a sufficient condition for q" says that it suffices to know that p is true to be able to conclude that q is true. For example, it is sufficient that you get an A in logic for me to buy you a Corvette. Other things might induce me to buy you the car, but an A in logic would suffice. The phrasing "q is necessary for p" we'll come back to later (see Example 11).



In the exercises for Section 2, we saw that the commutative laws hold for both conjunction and disjunction: pqqp, and pqqp.

We call the statement qp the converse of pq. Again, a conditional and its converse are not equivalent. This fact hardly deters the sales pitches used in the promotions of certain products. For example, the slogan "Drink Boors, the designated beverage of the US Olympic Team" suggests that all US Olympic athletes drink Boors (i.e., if you are a US Olympic athlete, then you drink Boors). What it is trying to insinuate at the same time is the converse: that all drinkers of Boors become US Olympic athletes (if you drink Boors then you are a US Olympic athlete, or: it is sufficient to drink Boors to become a US Olympic athlete).

Although the conditional pq is not the same as its converse, it is the same as its so-called contrapositive, (~q)(~p). While this could easily be shown with a truth table (which you will be asked to do in an exercise) we can show this equivalence by using the equivalences we already know:






The Biconditional We already saw that pq is not the same as qp. It may happen, however, that both pq and qp are true. For example, if p: "0 = 1" and q: "1 = 2," then pq and qp are both true because p and q are both true. The statement pq is defined to be the statement (pq)(qp). For this reason, the double headed arrow is called the biconditional. We get the truth table for pq by constructing the table for (pq)(qp), which gives us the following.

Biconditional

The biconditional pq, which we read "p if and only if q" or "p is equivalent to q," is defined by the following truth table.

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

The arrow "" is the biconditional operator.

Note

From the truth table, we see that, for pq to be true, both p and q must have the same truth values; otherwise it is false.

Some Phrasings of the Bionditional

Each of the following is equivalent to the biconditional pq.

    p if and only if q.

    p is necessary and sufficient for q.

    p is equivalent to q.

Notice that pq is logically equivalent to qp (you are asked to show this as an exercise), so we can reverse p and q in the phrasings above.

For the phrasing "p if and only if q,", remember that "p if q" means qp while "p only if q" means pq. For the phrasing "p is equivalent to q," the statements A and B are logically equivalent if and only if the statement AB is a tautology (why?). We'll return to that in the next section.



2. Logical Equivalence, Tautologies and Contradictions Section 3 Exercises 4. Tautological Implications and Tautological Equivalences Main Logic Page "Real World" Page

We would welcome comments and suggestions for improving this resource. Mail us at:
Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: July, 1996
Copyright © 1996 StefanWaner and Steven R. Costenoble