The difference between validity and entailment in first order logic I'm a newbie to logic so please forgive me if this is a basic question. I've searched the web and stack exchange but can't seem to find an answer.
The book I'm reading on predicate logic (forallx) says that although the argument '${\rm A}$ therefore ${\rm B}$' may be valid, this doesn't mean that ${\rm A}$ necessarily entails ${\rm B}$.
To me, validity and entailment both seem to have the same definition, namely that ${\rm A}$ entails ${\rm B}$ (or an argument is valid) if it is not possible for the premise to be true and the conclusion false.
Clearly my understanding of the above definition is wrong if entailment and validity are in fact different things, but I can't see the difference between them.
Can anyone enlighten me please? And are the definitions different in propositional logic versus predicate logic?
Many thanks
Max
 A: *

*OP: "It's the example regarding foxes and vixens on p. 281 of
forallx.openlogicproject.org/forallxyyc.pdf "

The author is ostensibly cautioning about the relationship between
the validity of an argument and FOL-entailment (first-order-logic
entailment), yet their given example is making a different point,
which is that one must be careful to retain all pertinent information
when symbolising an argument (in the example, the original argument
contains an implicit/unstated premise "Every vixen is a fox" that
became lost in translation).


*As for what the author says the relationship between argument
validity and FOL-entailment is, from a quick keyword-search of the
book, I believe it is this: "In FOL, an argument is valid iff its
conclusion is a FOL-entailment of its premises."
A: Consider the following instance:
 (0 != 0) therefore (God exists)

Since 0 = 0 the premise is false, hence the implication is valid.  Does this prove God exists?
[note: If you happen to believe in God, change the above conclusion to 'God does not exist'.  The argument still works]
