Is "William only eats icecream when the sun is shining" a biimplication?

Question

William only eats icecream when the sun is shining

Let $P(t)$ be the sun is shining at time $t.$
Let $Q(t)$ be William is eating an icecream at time $t.$

Which implication is there between $P(t)$ and $Q(t)?$

My friend has suggested that it is a biimplication, but I disagree, since it would then mean that every time the sun is shining William would be eating an icecream.

I have suggested that it is instead $Q(t)→P(t),$ since we know that if William is eating icecream then the sun will be shining.

Answer 1

Indeed you are right.

The sun might be shining without William eating an icecream.

Answer 2

@b00n Het’s answer is short and sweet and correct; but I’ll expand on it a bit, to back it up and consider the complications raised in @ryang’s answer.

The key issue is that only in natural English is a bit subtle in several ways, and sometimes ambiguous. The main ambiguity is that only doesn’t have to directly precede the specific item it’s governing — it usually precedes the main verb. So for instance William only mows the lawn on Mondays can potentially be read as:

1. [only governs Mondays] It’s only on Mondays that William mows the lawn (not on other days).
2. [only governs the lawn] It’s only the lawn that William mows on Mondays (not the field beyond).
3. [only governs mows] The only thing William does to the lawn on Mondays is mowing it (not raking it).
4. [only governs mows the lawn] The only thing William does on Mondays is mows the lawn (then he goes straight back to bed).

In practice, usually it’s clear from context which reading is the intended one — and when it wouldn’t be clear, then speakers do place only directly with the phrase it governs, and say for instance William mows the lawn only on Mondays. However, as long the intended reading is clear from context, only prefers to precede the main verb — and so when only is in this position, it should be read as governing whichever subsequent part of the phrase is semantically most natural. So in William only eats ice cream on Mondays, other readings are in principle possible, but it’s semantically clear that the intended reading is William eats ice cream only on Mondays.

The other is to what extent only entails that the thing in question happens at all. If I say “I’ve only been to Brazil twice”, you will rightly conclude that I have been to Brazil twice. If I say “I only visit democratic countries”, you will conclude that I have probably visited some democratic countries — but not that I’ve visited all of them. So this positive entailment is present but subtle; as such, it’s hard to judge whether it’s part of the logical content of the only sentence, or just implicature. In formal usage (mathematical and some legal), it’s well-established that the positive entailment is not taken as part of the logical content; in other contexts, the positive entailment is usually analysed as implicature, but is at least arguably part of the logical content. But in any case, the positive claim entailed is only existential, not universal.

So William only eats ice cream on Mondays has a couple of reasonable readings; writing $P(t)$ for “William is eating ice cream at time $t$ and $Q(t)$ for “$t$ is a Monday” as you suggest, these are:

1. $\forall t\ P(t) \Rightarrow Q(t)$, “the only time William may eat ice cream is on Mondays”

2. $(\forall t\ P(t) \Rightarrow Q(t)) \wedge (\exists t\ Q(t) \wedge P(t))$ “the only time William may eat ice cream is on Mondays; and he sometimes does eat it on Mondays”

The first of these is the more standard reading, especially in formal or semi-formal contexts. The second reading is also defensible in natural contexts. But neither of these is a bi-implication; the potential bi-implicative reading $(\forall t\ P(t) \Leftrightarrow Q(t))$, “at all times, William is eating ice cream precisely if it’s Monday”, is not justified either by natural usage or formal conventions.

Answer 3

Let $P(t)$ be the sun is shining at time $t.$
Let $Q(t)$ be William is eating an icecream at time $t.$

• William eats icecream only when the sun is shining

  $\forall t\big( Q(t)→P(t)\big).$

• William eats icecream precisely when the sun is shining

  $\forall t\big(Q(t)\leftrightarrow P(t)\big).$

William only eats icecream when the sun is shining

The sentence $$\text{William eats icecream when the sun is shining}$$ contains 9 possible positions to insert the modifier ‘only’, all legitimate, with no pair of resulting sentences having exactly the same set of possible readings (to disambiguate, we typically rely on context, common sense, word stress and pausing).

Every technically correct reading of the quoted sentence gives that if the sun is shining, then William eats ice cream but doesn't, say, donate ice cream. None is equivalent to either of the above formulae or even $\forall t\big( P(t)→Q(t)\big).$

Now, when perfect grammar is awkward, or at least pedantic, in conversation, we sometimes idiomatically sacrifice accuracy for rhythm. Intending the quoted sentence to mean the first bulleted sentence is arguably such a case.

In any case, this excerpt from The Grammar According to West is apropos:

Live mathematical conversations use many shortcuts that are inappropriate in precise mathematical writing. The context is known by all participants, and shortcuts evolve to save time. Furthermore, the speaker can immediately clarify ambiguity. Without immediate access to the author, written mathematics must use language more carefully. Also, mathematical concepts are abstract, without context from everyday experience, so the writing must be more consistent to make the meaning clear. Outside mathematics, imprecise writing can still be understood because the objects and concepts discussed are familiar.

P.S. The Word ONLY Can’t Go Just Anywhere

Answer 4

1 - William only eats icecream when the sun is shining

I want you to contrast this sentence with this one:

2 - William eats only icecream when the sun is shining

In example 1, what's being conveyed is that a specific scenario (eating ice cream) is restricted to a particular precondition (when the sun is shining). In other words, when the condition is not met, there's no ice cream being eaten either.

In example 2, what's being conveyed is that William eats nothing but icecream when the sun is shining. He won't eat any other food when the sun's out. If there is no ice cream around he will starve because he will refuse everything else.

I strongly suspect that your friend is parsing the sentence closer to example 2 than example 1. While this is still not a perfect match to the logical claim your friend then goes on to make, it's within the reasonable ballpark of what a human might infer from such a statement being made in an informal conversation.

but I disagree, since it would then mean that every time the sun is shining William would be eating an icecream.

You are logically correct. But I want to tackle this not from a point of logical accuracy, but rather what a human might understand in a casual conversation.

If I tell you that whenever the sun shines, William refuses to eat any food unless it is ice cream (which is what example 2 tells you); then it's reasonable enough for your friend to interpret this as "William always eats ice cream when the sun is shining", in the sense that they're not saying that he always eats ice cream (i.e. he constantly ingests ice cream) but that he always chooses to eat ice cream (i.e. he constantly makes the same choice to ingest ice cream when he makes a choice to ingest something).

I suspect your friend is not a logician and is approaching this from a point of semantical intuition. When presented with example 2, their logical inference may not be pedantically correct but it's a reasonable understanding in an informal conversation.

However, in this scenario we're dealing with example 1, not 2. And in example 1, your friend's interpretation is not correct. You are correct in your interpretation of the implication here.

Answer 5

English is ambiguous. Consider:

'A person is a bachelor only if that person is unmarried.'

Here, being unmarried is merely a necessary condition for being a bachelor, not sufficient. So we clearly do not want to use a biconditional: just because someone is unmarried doesn't make them a bachelor. Of course, we can say that if someone is a bachelor, then they are unmarried. Or, equivalently, that if someone is not unmarried, then they cannot be a bachelor.

But now consider:

'You can take Calculus II only if you have taken Calculus I.'

Here, one can totally see a context where it is clear that this doesn't just mean that having taken Calculus I is necessary for taking Calculus II, but that it is also sufficient: e.g. this statement was given in response to a question as to what the prerequisities are or taking Calculus II .... in which case one would assume that had there been other requirements to taking Calculus II pother than Calculus I, those would have been spelled out. So if that is clear, use a bi-conditional.

When in doubt, though, I also choose the weaker reading: any 'P only if Q' at a minimum says that Q is a necessary condition for P. So, if you don't know anything more, then I would just work with that one-way conditional, just to be safe. For the same reason, when someone uses an 'or', and you don;t know whether it was meant as an inclusive or exclusive or, I always use the inclusive or, which is the weaker of the two, so I make sure that I don't assume too much.