You can use both medicine and pill and both can mean that you are taking legitimate therapy. However, sometimes these terms are interchangeable, and sometimes not.
- Historically, a pill was a dosage form made manually, using the active substance, a sugar and a liquid binding agent :
small, round, solid pharmaceutical oral dosage form of medication that was in use before the advent of tablets and capsules. Pills were made by mixing the active ingredients with an excipient such as glucose syrup in a mortar and pestle to form a paste, then rolling the mass into a long cylindrical shape (called a "pipe"), and dividing it into equal portions, which were then rolled into balls, and often coated with sugar to make them more palatable. (Wikipedia)
Today a pill refers to all oral solid dosage forms that are intended to be swallowed as such (note that an effervescent tablet for example, most likely won't be referred to as a pill).
The pill or capitalized The Pill (definition 2) refers specifically to birth control pills (i.e. oral contraceptives).
In everyday (colloquial) usage to use the term pill meaning medicine taken orally in a solid dosage form is fine, but as a technical term it would be incorrect (unless of course you are making an exhibition about the history of pharmacy).
But what if you are taking a medicine in a different dosage form?
You might need to take a syrup, an injection (insulin e.g.), use an inhaler etc. In this case pill doesn't work and you have to use the term medicine or use the term for a specific dosage form.
To take a medicine is used less often than to take a pill. But that doesn't mean that the word medicine is used less common in this sense than the word pill. Medicine is more often used in different constructions:
You need to take this medicine three times a day; this one you need to take twice a day.
Make sure that the children take their medicine.
or
Kanga replied, “Roo, dear, you must take your medicine" [...]
“I liked it better when Tigger ate my medicine for me,” said Roo. (House at Pooh Corner, by A. A. Milne)
The word medicine can be both countable and uncountable, according to LDOCE.
This Google Ngram shows that in fact, medicine (green) is used more often than pill (bright blue and dark red) except when they are used with the indefinite article:
A side note: medicines can also be administered and (especially for topical medicines) applied. (See more in this post. You definitely don't eat a medicine, but this doesn't apply to Tiggers, because "that's what they like best".
No particular meaning is implied. This is a set phrase, and it is marked as such in "Collins Cobuild":
high/great hopes
phrase (!!!)
If you have high hopes or great hopes that something will happen, you are confident that it will happen.
I had high hopes that Derek Randall might play an important part.
Britain's three-day event team has high hopes of winning the Olympic gold medal.
He had no great hopes for the success of his undertaking.
[Also + of/for]
Set phrases just come to be a certain way over time. You just need to learn their "overall" meaning and simply parrot them.
That it makes no sense to try to analyze a phrase for literal meaning of its components is also evident in the case of phrasal verbs. As Oxford Dictionaries grammar section says, "A phrasal verb is a verb that is made up of a main verb together with an adverb or a preposition, or both. Typically, their meaning is not obvious from the meanings of the individual words themselves."
In the same way, analyzing this particular phrase (which seems to have emerged at the beginning of the 19th century) for countability or uncountability makes no sense. It just is what it is.
Best Answer
Here, some means
So there isn't an issue regarding plurality: justification can be singular.
You can use justification as a plural like you would with good reason.
Here is an example in the same sense as your example. I don't know why there were budget cuts. There must be some good reason for the recent budget cuts →
There can also be multiple undetermined justifications. I don't know why there were budget cuts. There must be some good reasons for the recent budget cuts →
If there is just one justification for a particular action, then justification (singular) is correct.
If there are more than one reasons for a particular action, then justifications (plural) is correct. For example, imagine that there are ten reasons for the budget cuts, but only three are presented: here are some reasons for the recent budget cuts→
Here's a headline I found online that uses justifications.