Entailment can be taken in a broad sense and in a narrower one.
One thing that is uncontroversial is that ' Daisie has four legs"' does not entail ' Daisie can run' for it is possible to have four legs and, for some reason ( having broken legs, having a disease affecting muscles, etc), not being able to run.
So, the minimal condition in order proposition P to entail proposition Q is that :
if P is true , Q cannot be false,
necessarily ( if P is true, Q is true).
Note : asserting that P entails Q is stronger than asserting that the material conditional ( P --> Q) is true. "2+2 = 4" materially implies "Churchill smoke cigars" ( since both sentences are actually true) , but the first sentence does not entail the second, for there is a possible case in which the first is true while the second is false , namely the possible case in which Churchill did not smoke cigars.
- So entailment requires necessity, and more precisely, logical necessity. We may therefore say that, in the broad sense :
P entails Q means ( synonymously ) : it is logically necessary that if
P is true, Q is also true.
In this broad sense of entailment or logical implication, it is perfectly correct to say that having four legs entails having more than two legs.
In the same way , " Peter is a pianist" entails " Peter is a musician", or " Mary is John's wife" entails " Mary is a woman".
- Now there is a sense of entailment or logical implicaton that is narrower . It relates to the idea of a reasoning that is valid " vi formae" , in virtue of its form alone. In his sense " Daisy has 4 legs" does not entail " Daisy has more that two legs". Suppose you did not understand english, except its logical vocabulary ( 'for some', 'for all', 'is', 'not' ) , in that case the first sentence would reduce for you to its form
"individual d is Predicate_1"
and the second would reduce to
"individual d is Predicate_2"
in such a way that you wouldn't be able to conclude from the first sentence to the second. ( The reason is that there is a possible interpretation - in fact infinitely many possible interpretations - of 'd' , ' Predicate_1 ' and ' Predicate_2 ' that make the first sentence true and the second false).
As I said there are infinitely many interpretations in which the first sentence is true and the second false. You only have to find an individual that actually satisfies a first predicate and that does not satisfy a second one. Having an actual case in which the premise is false and the conclusion false is enough to prove that it is possible to have such a case. And nothing else is required to give a couterexample.
Let's say that ' d' denotes the Earth
Let's say that 'Predicate_1' is ' x is a planet'
Let's say that 'Predicate _2' is : ' x has 2 natural satellites'
In that case, sentence (1) becomes : The Earth is a planet.
And sentence (2) becomes : The Earth has 2 natural satellites.
The premise is true and the conclusion is false. This is enough to prove
that the form of the reasoning is not truth preserving. When a form
isvalid, it has no possible instance with true premises and a false
Compare this with the following form :
(1) ' d ' is Predicate _1
(2) For all x, if x is predcate_1 then x is Predicate_2
(3) Therefore, 'd' is Predicate_2.
in that case, is it possible to find an interpretation ( even arbitrary or fanciful) ) that makes the premises true and the conclusion false?
- We may therefore define ths strong sense of entailment as
P entails Q iff (a) necessarily , (b) in virtue of the sole forms of
propositions P and Q, (c) if P is true, Q is true.
the form of the sentences rules out the possibility of having an
interpretation in which the first sentence is true and the second
Note : Entailment in this narrow and strong sense can be seen as the converse of the relation of logical consequence. There are 3 excellent entries by Mc Keon on this topic in the Internet Encyclopedia Of Philosophy (https://www.iep.utm.edu/logcon/#H2)
Also, on the distinction between material and formal consequence ( by Restall, SEP) : https://plato.stanford.edu/entries/logical-consequence/#FormMateCons