It is my first time that I met the word "entails". In mathematical texts, one usually sees "if and only if", "implies" or "iff" which bear no ambiguity. In the following definition:
Does "entails" mean "implies"? It certainly seems so. Also In one PHD thesis I am reading, the author quotes this definition using the word "implies" instead of the word "entails". So far so good, but next comes a corollary:
But if "entails" means "implies" then this corollary is not true. Since function
$$ f_1 \left( x \right) := x $$
clearly supports
$$ f_2 \left( x \right) := 1 $$
according to the definition above, but not the other way around.
This is because $x' \geq x \Rightarrow 1 \geq 1 $ but $ 1 \geq 1 $ does not imply $x' \geq x $ for all $x', x \in \mathbb{R}$.
What am I missing here?
Best Answer
I'm 100% British (English, even), and I think this is strange and confusing usage.
According to my (American) thesaurus, some synonyms of "entails" are "brings about", "calls for", "demands", "causes", "gives rise to", "leads to", "necessitates", "requires".
From a logic/mathematics point of view, some of these terms mean "if" and some mean "only if".
I have never seen "entails" used to mean "equivalent to", and the thesaurus doesn't list this as a synonym, but who knows what these authors had in mind. Pretty poor writing, in my opinion.
Clarification of Synonyms
Let's split the terms into two groups:
I think it's clear that "brings about", "causes", "gives rise to", "leads to" all mean the same thing. Let's represent this group by "causes".
Similarly, I think "demands", "necessitates", "requires" all mean the same thing. Let's represent this group by "requires".
"A causes B" means that A implies B. In other words, "B is true if A is true".
"A requires B" means that "A can be true only if B is true"
So, in my view, the list of synonyms contains terms with two quite different meanings, which I (rather sloppily) characterised as "if" and "only if" terms.