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Conditionals and Biconditionals: → and ↔
Chapter 7 adds two new connectives to our collection of treasures: → and ↔. We can read "P → Q" as "if P, then Q" and we can read "P ↔ Q" as "P if and only if Q." (Sometimes philosophers abbreviate "if and only if" to "iff.")
A sentence of the form "P → Q" is called a conditional. The antecedent of the conditional is P, and its consequent is Q. The converse of the conditional P -> Q is the conditional Q -> P. Note that the converse of a conditional is not equivalent to the conditional! They are independent: either can be true without the other being true. The contrapositive of P -> Q is ~Q -> ~P; the contrapositive of a conditional is tautologically equivalent to the conditional itself.
A sentence fo the form "P ↔ Q" is called a biconditional. Unfortunately the terminology of "antecedent" and "consequent" aren't helpful for biconditionals, where both sides are in effect both antecedent and consequent. So with biconditionals we will usually just refer to the two component sentences as the "left-hand side" and the "right-hand side" of the biconditional.
As with our other connectives, we can give the semantics of the conditional and biconditional by using truth tables:
P |
Q |
P → Q |
P ↔ Q |
T | T | T | T |
T | F | F | F |
F | T | T | F |
F | F | T | T |
There are a few tricky points to keep in mind in doing translations involving these connectives. Let's consider a few of the many English phrases that can be translated using conditionals. Some of the following sentences are frequently stumbling blocks for students doing translations, so if you have trouble doing the translation exercises, you might want to look over this table again!
English Sentence | Translation into Symbolic Notation | Comments |
If P then Q P is sufficient for Q |
P → Q | |
P only if Q | P → Q | note: "P only if Q" means the same thing as "if P then Q"! |
P is necessary for Q | Q → P | Q doesn't occur without P also occurring |
P if Q | Q → P | except in the case of "only if," whatever follows the "if" is usually the antecedent of the conditional |
P unless Q | ¬Q → P or P ∨ Q | an easy way to remember how to translate "unless" is to think of it as "if not" |
P if and only if Q P just in case Q |
P ↔ Q |
Sometimes people think that "P only if Q" should be symbolized as Q →
P or P
↔ Q instead of as P
→ Q.
A good cure for this temptation is to think about the following example. Let
W mean that I win the lottery, and let B mean that I buy a ticket. How
should we translate "I will win the lottery only if I buy a ticket," that
is, "W only if B"?
Clearly, it should not be translated by B → W or W ↔ B. The first means that if I buy a ticket, I will win, which is clearly not necessarily the case. The second says that I will win if and only if I buy a ticket, which implies that if I buy a ticket, I will win, and again, this isn't true. The translation we want is W → B, which says that if I win, then I bought a ticket.
Some conditional equivalences:
equivalent sentences | |
P -> Q | ~P v Q ~Q -> ~P (this is known as the "contrapositive" of P -> Q) |
P <-> Q | (P -> Q) & (Q -> P) (P & Q) v (~P & ~Q) |