Subsection 7.2.11 Ambiguity of Some Logical Operators - IMPLIES
The logical operator → (implies) has a single clear meaning. If you’re not sure you remember our discussion of that, you may want to rewatch our Implies video.
It’s much less clear, however, what we mean when we say, in English, “implies” or “if” or “if/then”.
There are at least three common meanings of those terms (and the many other ways there are of saying the same thing):
Material implication. This is the meaning of → that we have been using. So, in particular, the meaning of p → q is given by this truth table:
p | q | p → q |
---|---|---|
T | T | T |
T | F | F |
F | T | T |
F | F | T |
If Drew comes, there will be ice cream.
If Drew comes then there will be ice cream.
Drew coming definitely implies there will be ice cream.
In all three of these cases, we’re saying that Drew’s coming means there will be ice cream. We haven’t said anything about what will happen if Drew doesn’t come. Possibly there are other things that might cause ice cream to appear.
Equivalence (if and only if). Sometimes,“if” actually means “if and only if”, whose meaning is given by this truth table:
p | q | p ↔ q |
---|---|---|
T | T | T |
T | F | F |
F | T | F |
F | F | T |
In other words, you might say p → q but mean (p → q) ∧ (¬p → ¬q). Notice that (¬p → ¬q) is the converse of your actual claim. The converse of a statement P is not necessarily true whenever P is. So I (the listener) can’t logically conclude ¬p → ¬q. But it may nevertheless part of the meaning that you intended. How can I know that? I will generally assume that you are trying to communicate effectively. So you’ll make the strongest claim you reasonably can. If q is true regardless of p, you would just have asserted q. We’ll say more about this idea, called conversational implicature, later.
If it rains, we’ll move the picnic indoors.
But what if it doesn’t rain? On hearing this sentence, most of us would conclude that, unless it rains, we’ll have the picnic outside. Why? Partly because we can’t imagine why the picnic would be moved unless there is rain. But also because, if the picnic is going to be moved regardless of the weather, why would you not simply have said, “We’re going to move the picnic inside”?
By the way, it is common for mathematicians (and others) to say “if” in definitions, when they actually mean “if and only if”.
An integer greater than 1 is prime if it has no divisors other than itself and 1.
Causality.
If the sweater gets wet, the colors will run. (The color run will be caused by the water.).
If you eat too much, you’ll get sick. (Eating too much causes one to be sick.)
Each of these sentences is making a claim of material implication and an additional claim about causality. That claim must be represented separately in logic if we want to reason with it.
Exercises Exercises
Exercise Group.
Indicate for each of the following sentences, which meaning of if/implies is most likely intended by the speaker. (Do not stress if your answer to some of these differs from ours. These sorts of sentences can be ambiguous and open to misinterpretation.)
Part 1.
If Koko is tired, she’ll be grumpy.
Material implication.
Equivalence.
Part 2.
Morgan will sing if Casey does.
Material implication.
Equivalence.
Part 3.
We run out of hot dogs whenever the Astros play.
Material implication.
Equivalence.
Correct answer is A.
Explanation: Another way to say this is, “If the Astros play, we run out of hot dogs.” (There are many ways to say “if” in English.) But there’s no claim that the two things are equivalent. It’s possible that other situations also result in a lack of hot dogs, but, for the moment, we are discussing Astros games.
Part 4.
You are a senior if you have at least 90 credits.
Material implication.
Equivalence.
Part 5.
On Fridays, we go to Torchy’s. (Hint: start by rewriting this to make the “if” explicit.)
Material implication.
Equivalence.
Correct answer is A.
Explanation: There’s no reason to believe that we never go to Torchy’s on any other days, particularly if the current discussion is about what will happen on a particular day that happens to be a Friday. However, if we go to Torchy’s every day we’d probably say, “We go to Torchy’s every day.”