Have students think of another quantity implicature not discussed here and figure out what alternative it is based on, and how that alternative is stronger.
As we saw in the module on weak and strong implicatures, a certain kind of implicature (specifically, implicatures based on the maxim of quantity) is based on reasoning about "stronger alternatives" a person could have said. For example, "Josh is smart" may be taken to imply that Josh is not brilliant, because "Josh is brilliant" is a stronger alternative that the speaker could have said but chose not to say.
But what exactly counts as a "stronger alternative"?
In the abovementioned example, the "alternative" is an utterance we get by replacing one word. The only difference between "Josh is smart" and "Josh is brilliant" is a single word. So, the alternative is an utterance that's as similar as possible to the actual utterance, just with one word changed.
But alternatives don't have to be like that. An alternative could be much shorter than the actual utterance. For example, as we saw in the previous module, "There's a prize behind Door #1 or Door #2" implies that the speaker does not know there's a prize behind Door #1. So in this case, the alternative (There's a prize behind Door #1) is shorter than the actual utterance (There's a prize behind Door #1 or Door #2).
An alternative can also be much longer than the actual utterance. For example, if you ask me what countries I have lived in, and I say "I've lived in the USA and the UAE", I imply that I haven't lived in any other countries. In this case, the alternatives that are ruled out by the quantity implicature are hundreds of possible utterances like "I've lived in the USA, the UAE, and France", "I've lived in the USA, the UAE, and Ghana", "I've lived in the USA, the UAE, France, and Ghana", and so on and so forth. (In fact, assuming there are 195 countries in the world, then the number of possible alternatives for this sentence is about ten octodecillion—that's a one with 58 zeroes after it.)
So it's not clear that we can systematically figure out what the possible alternatives are just by their surface similarity to the actual utterance. As we've seen above, alternatives can be shorter than, longer than, or about the same length as the actual utterance.
To have a more formal idea of what counts as a stronger alternative, we need to use the concept of entailment.
We briefly discussed entailment at the end of the "Truth-conditional semantics" module. Recall that one proposition entails another if the first proposition being true guarantees that the second is true. For example, "That's a cat" entails "That's an animal", because if the first one is true then the second one is definitely also true (because all cats are animals).
With the concept of entailment, we can formally define what makes an alternative "stronger". For some utterance A, B is a "stronger" (or "more informative") alternative if B entails A and A does not entail B. That probably sounds a little abstract, so let's apply it to the examples we saw above.
It should be clear that each of the B propositions above entails the corresponding A proposition, but not vice versa. If Josh is brilliant, it's guaranteed that he is smart (under the literal, logical interpretation of smart as meaning something like "at least smart, and maybe even brilliant"). But if Josh is smart, it's not guaranteed that he's brilliant (maybe he's just somewhat smart but not a super genius). Likewise, if it's true that there's a prize behind Door #1, then it's certainly true that "there's a prize behind Door #1 or there's a prize behind Door #2". (This is the rule of logical disjunction, which you have probably learned about in math or philosophy classes before; logically, "P or Q" is true as long as either [or both] P is true or Q is true.) And if if it's true that I've lived in the USA, UAE, and France, then it's certainly true that I've lived in the USA and UAE; but if it's true that I've lived in the USA and UAE then there's no guarantee that I've also lived in France.
So on this basis we can formally identify what counts as a stronger (or more informative) alternative. Another way to think about this same concept is to think about what situations each proposition is true in. For example, the set of situations in which "That's a cat" is true is a smaller subset of [i.e., completely enclosed within] the set of situations in which "That's an animal" is true, as shown in the picture below:
Something that's true in a smaller set of situations is stronger (more informative) than something that's true in a bigger set of situations, because it provides the hearer with a more specific view of what's going on.
(This is in fact the same thing as entailment, it's just another way to understand the same concept.)
This conception also helps explain why implicatures very similar to the examples above do not arise in certain contexts, such as downward-entailing environments. (A detailed explanation of downward entailment is something you'll find in a semantics class, but here's a quick one: an upward-entailing environment is one where specific subsets entail more general sets, and a downward-entailing environment is one where general sets entail specific subsets. For instance, "That's a cat" entails "That's an animal" [upward entailment]; but "That's not an animal" entails "That's not a cat" [downward entailment]). Compare the two examples below:
For the first example, the alternative ("Josh is brilliant") entails the original utterance ("Josh is smart"); the set of brilliant people is smaller than (and is completely contained within) the set of smart people. Hence, the implicature goes through. On the other hand, for the second example, the apparent alternative ("Josh is not brilliant") actually is entailed by the original utterance ("Josh is not smart"); the set of people who are not brilliant is bigger than (and completely contains) the set of people who are not smart. (The set of people who are not brilliant contains both average people and dummies; the set of people who are not even smart contains just the dummies.) Hence, "Josh is not brilliant" is not actually a stronger alternative, and thus it doesn't trigger any quantity implicature.
Our analysis of alternatives so far still leaves some issues unsolved. Here we'll discuss two.
First of all, so far we have been looking at examples where a quantity implicature involves denying a stronger alternative. But there are many cases in which we implicate that a stronger alternative is true. Levinson (chapter 3.2.4) discusses several examples; here we'll just review the case of generalized conversational implicatures with "and".
As we've seen in previous modules, X and Y generally implicates X and then Y (e.g., "Sammy and Chris fell in love and got married" suggests that they fell in love first and got married second). We could go even further; "X and Y" often implies that X caused Y to happen.
Consider, for instance, a sentence like "I slammed the door and the picture fell off the wall". There are several alternatives that the speaker could have uttered:
These alternatives are progressively stronger. (1) entails the original utterance and the original utterance doesn't entail it, so (1) is a stronger alternative. Furthermore, (2) entails (1) and (1) does not entail (2), so (2) is even stronger than (1). According to the analysis we've done so far, a person uttering "I slammed the door and the picture fell off the wall" should be understood as implying that his slamming the door did not cause the picture to fall off the wall. But that analysis is clearly wrong; in fact we will usually interpret the utterance as meaning (2). So, there must be some other mechanism that causes the implicature in (2) to arise; furthermore, there must be some explanation of why a quantity implicature doesn't arise in this example.
Competence-based strong implicatures, which we discussed in the "Weak and strong implicatures" module, are a similar example. Just like the previous example, these implicatures involve going from a weaker proposition ("the speaker doesn't believe P") to a stronger one ("the speaker believes ¬P"), rather than denying the stronger proposition.
The second problem is how to decide which alternatives are relevant to the conversation. As we've seen above, an utterance can have many alternatives. For an utterance like "I have lived in the USA and the UAE", there are nearly 200 other countries I could have mentioned (depending on what we count as a "country"), plus over 18,000 possible combinations of two other countries, plus over a million possible combinations of three other countries, etc., such that the total number of alternatives is over 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. (Geurts, chapter 6, suggests a way to avoid this problem by considering speakers' "intentional state" rather than specific alternatives they could have said, but I am not convinced that his approach solves the problem.)
In fact, for any given utterance or proposition, there are an infinite number of propositions that entail it (for example, "That's a cat" is entailed by "That's a cat and I like ice cream", "That's a cat and I like ice cream and snails are slow", etc.). Therefore, any utterance has an infinite number of alternatives. But it is clear that uttering that utterance does not imply the denial of most of these alternatives; if I point at a cat and say "That's a cat" then no one would think I am implying "It is not true that [that's a cat and I like ice cream]". So, people must have some way of deciding which alternatives are relevant (and thus might be denied, via quantity implicatures) and which are not. We will turn to this issue in the next module.
Have students think of another quantity implicature not discussed here and figure out what alternative it is based on, and how that alternative is stronger.
Saying someone "tried" to do something often implicates that they didn't succeed at doing it; for example, "I tried to save him" suggests that the speaker didn't succeed at saving the person he was trying to save.
At first this seems like a scalar implicature, just like how "Josh is smart" can suggest that Josh is not brilliant. It seems like succeeded is a stronger alternative to tried, so a speaker uttering "I tried to X" conversationally implicates that they don't believe they succeeded at X.
There is a problem, though. Remember that in the module we said a "stronger alternative" is one that entails the expression which was uttered (e.g., brilliant is a stronger alternative for smart because brilliant entails smart). succeed, however, does not seem to entail try; as Huang (2014, section 2.2.2) points out, we can say "He succeeded without even trying!" and it's not a contradiction. This would not be possible if succeed entailed try (compare to, e.g., "He saw a black cat without seeing any cats", which is self-contradictory).
Have students discuss how the implicature (i.e., the interpretation of tried as meaning "did not succeed") arises. Off the top of my head we have a few possibilities:
by Stephen Politzer-Ahles. Last modified on 2022-03-24. CC-BY-4.0.