Trying to understand the logic behind this sentence: The lecture will be given if at least ten people are there

I am not really sure whether my question is suitable here, but I will give it a try.

Consider the following sentence:

The lecture will be given if at least ten people are there.

From the perspective of everyday speech, which statement below will be equivalent to the sentence above:

1. "At least ten people" is a sufficient condition for the "lecture being given". (In other words, if there are at least ten people, the lecture will be given; if there are fewer than ten people, there is a possibility that the lecture will be given.)
2. "At least ten people" is a sufficient and necessary condition for the "lecture being given". (In other words, if there are at least ten people, the lecture will be given; if there are fewer than ten people, the lecture will definitely not be given.)

In math, the distinction is pretty clear:

1. “The lecture will be given if at least ten people are there” is equivalent to the first statement.
2. “The lecture will be given if and only if at least ten people are there” is equivalent to the second statement.

But as the author of How to Prove It suggests, in everyday speech, the situation is different:

One of the reasons it’s so easy to confuse a conditional statement with its converse is that in everyday speech we sometimes use a conditional statement when what we mean to convey is actually a biconditional. For example, you probably wouldn’t say “The lecture will be given if at least ten people are there” unless it was also the case that if there were fewer than ten people, the lecture wouldn’t be given. After all, why mention the number ten at all if it’s not the minimum number of people required? Thus, the statement actually suggests that the lecture will be given iff (iff is a abbrevation for "if and only if") there are at least ten people there.

Velleman, Daniel J.. How to Prove It (p. 53). Cambridge University Press. Kindle Edition.

So my question is, if someone were to say “The lecture will be given if at least ten people are there”, would it be safe to assume that if there are fewer than ten people, the lecture won't be given?

Solution 1:

Welcome to ELU and thank you for this interesting question.

First of all, you are entirely right that, for the purpose of mathematics (and, indeed, of logic, which Bertrand Russel identified as essentially mathematical), the statement

If at least ten people are there, the lecture will be given

means no more than it says. It does not follow that, if there are fewer than ten people there, the lecture will not be given.

In other words, you are correctly saying that the protasis (the 'if clause') sets a sufficient condition for "the lecture will be given", but that it is not (in strict logic) a necessary condition. That is, strictly, it does not state / logically imply that if the fewer than ten people are there the lecture will not take place. That would require a statement that this would happen only if the quorum was satisfied.

But what would be the point of someone's promising to give a lecture if the stated number was present, unless this were the minimum without which the lecture could not be given? It would be pointless and misleading. Surely to make a statement like this the speaker must mean us to understand that the ten is a necessary condition for the lecture. The speaker must mean that it will take place if and only if (which logicians write as "iff) at least ten people show up.

So what is going on? Well, with something like this sort of situation in mind, the philosopher and logician H.P.Grice coined the word implicature. There is a useful discussion of the concept in the Stanford Encyclopaedia of Philosophy:- https://plato.stanford.edu/entries/implicature/.

“Implicature” denotes either (i) the act of meaning or implying one thing by saying something else, or (ii) the object of that act. Implicatures can be part of sentence meaning or dependent on conversational context, and can be conventional (in different senses) or unconventional. Figures of speech such as metaphor, irony, and understatement provide familiar examples. Implicature serves a variety of goals beyond communication: maintaining good social relations, misleading without lying, style, and verbal efficiency.

So, according to Grice, there is a kind of everyday logic that lies outside the confines of strict mathematical / propositional logic.

This is, for example, a standard feature of conditional promises.

If you eat your greens, you can have some dessert.

I’ll wash the car if you let me go to the cinema.

These are, of course, conditions as well as conditionals, and they are meant as necessary as well as sufficient conditionals. Oddly, what is really meant from a logical point of view is

If you don’t let me go to the cinema, I won’t wash the car.

But that, of course sounds like a threat. So does

If you don’t eat your greens, you won’t have any dessert.

So in the end, promises and threats work in this strictly ‘illogical’ way. Our usage allows us all usually to understand the embedded implicatures without difficulty, and sometimes to be misled by them.