For propositions \(P\) and \(Q\text{,}\) the conditional sentence \(P \implies Q\) is the proposition “If \(P\text{,}\) then \(Q\text{.}\)” The proposition \(P\) is called the antecedent, \(Q\) the consequent. The conditional sentence \(P \implies Q\) is true if and only if \(P\) is false or \(Q\) is true. In other words, \(P \implies Q\) is equivalent to \((\neg P) \vee Q\text{.}\)
A conditional is meant to make precise the standard language construct “If …, then …”, but it is has some seemingly counterintuitive properties. For example, do you think the statement “if the moon is made of green cheese, then it is tasty,” is true or false? What about the statement, “if the moon is made of green cheese, then the Red Sox will win the world series,” is it true or false? In fact, both statement are true because the antecedent, “the moon is made of green cheese”, is false. Note, in each case, we are not asking about the truth of the atomic propositions, but rather the statement as a whole. Moreover, there is no reason the antecedent and consequent need to be logically connected, which violates our intuition.
Suppose you are a waiter in a restaurant and you want to make sure that everyone at the table is obeying the law: the drinking age is 21. You know some information about who ordered what to drink and their ages which is indicated in the table below. What is the minimal additional information you need to determine if the law is obeyed?
| Person | Age | Drink |
| A | 33 | — |
| B | — | Beer |
| C | 15 | — |
| D | — | Coke |
B's age and C's drink. You can think of obeying the law as making “If under 21, then no alcohol,” a true statement. Then the statement is true whenever each person is either 21 and up or did not order alcohol. A is above 21, so he is obeying the law no matter what he ordered. B ordered alcohol, so we must check how old he is to determine if the law is obeyed. C is under 21, so we must check what he ordered to determine if the law is obeyed. D order coke, so he is obeying the law regardless of his age.
Let \(P\) and \(Q\) be propositions and consider the conditional \(P \implies Q\text{.}\) Then the
Make the truth table.
For propositions \(P\) and \(Q\text{,}\) the biconditional sentence \(P \iff Q\) is the proposition “\(P\) if and only if \(Q\text{.}\)” \(P \iff Q\) is true exactly when \(P\) and \(Q\) have the same truth value.
For propositions \(P\) and \(Q\text{,}\)
These can be read in English as “the negation of a conjunction is the disjunction of the negations,” and “the negation of a disjunction is the conjunction of the negations.”
Make a truth table.
For propositions \(P\) and \(Q\text{,}\)
So there is no ambiguity when we say “the conjunction of \(P\) and \(Q\text{,}\)” or “the disjunction of \(P\) and \(Q\text{,}\)”
Make a truth table.
For propositions \(P\text{,}\) \(Q\) and \(R\text{,}\)
So there is no ambiguity in the propositions \(P \wedge Q \wedge R\) or \(P \vee Q \vee R\text{.}\)
Make a truth table.
For propositions \(P\text{,}\) \(Q\) and \(R\text{,}\)
You should interpret this as indicating that conjunction and disjunction distribute over each other.
Make a truth table.
For propositions \(P\) and \(Q\text{,}\)
Make a truth table.
The logical conditional \(P \implies Q\) has several translations into English which include:
Similarly, the biconditional \(P \iff Q\) translates into a few of English phrases including: