I am confronted with two (at first glance) different definitions on closure under logical consequence and would like to know whether they are equivalent and, if not, which is the one that would match the intuitive notion of what closure under logical consequence should mean better.
Note: The problem originates from linguistics/natural languages semantics and involves the notion of propositions being sets of possible worlds; it would blow up this already very long post to go into detail about this concept, so I hope the problem is more or less understandable without too much of the philosophical backgrounds.
I encountered the problem in the context of Angelika Kratzer's "Lumps of thought", a semantic theory which was meant to support appropriate truth conditions for counterfactual statements (such as "If I won the lottery, I would buy a house" or "If I won the lottery, I might buy a house"), where she requires premise sets relevant to the truth of counterfactual propositions to be closed under her so-called "lumping relationship" and under logical consequence.
The first definition is the one in the original paper ($\mathscr{P}(S)$ is meant to be the powerset of the set of situations, which in this framework simply equates to the set of propositions; and a proposition is, for our purposes, to be associated with a set of possible worlds in which that proposition is true):
A set of propositions $\mathbb{A}$ is (strongly) closd under logical consequence if and only if the following condition is satisfied for all $\mathbb{B} \subseteq \mathbb{A}$ and all $q \in \mathscr{P}(S)$:
If $\mathbb{B}$ logically implies $q$, then $q \in \mathbb{A}$.
Since this is unfortunately a bit unformal, my translation into a predicate statement would be:
$\forall \mathbb{B} \in \mathscr{P}(\mathbb{A}) \forall q \in \mathscr{P}(S) [\bigcap \mathbb{B} \subseteq q \to q \in \mathbb{A}]$
(BTW, could you instead simply say $\forall \mathbb{B} \subseteq \mathbb{A}$? – having a subset relation in a quantificational phrase instead of an element-of-relation looks somehow weird to me.)
Now the other paper (I'm sorry I couldn't find a free link to it; it's "On the lumping semantics of counterfactuals" by M. Kanazawa et. al., 2005), which provides several formal proofs for essential parts of Kratzer's theory being conflictual or trivial, defines closure under logical consequence differently (adopted slightly since they make use of some additional notions which are not relevant here):
A set of propositions $\mathbb{A} \in \mathscr{P}(S)$ is closed under logical consequence if and only if for all $p \in \mathscr{P}(S)$, if $p$ logically follows from $\mathbb{A}$, then $p \in \mathbb{A}$.
Formally, this looks like this (again, I had to slightly adapt the notation):
$\forall \mathbb{A} \subseteq \mathscr{P}(S) \forall p \in \mathscr{P}(S) [\bigcap \mathbb{A} \subseteq p \to p \in \mathbb{A}]$
They note that there is also a definition for weak (instead of strong) closure under logical consequence, which is
$\forall p \in \mathbb{A} \forall q \in \mathscr{P}(S) [p \subseteq q \to q \in \mathbb{A}]$
but I think this one is not relevant here.
No my problem is that I think the two definitions express something different:
While the original definition requires every proposition which follows from any subset of $\mathbb{A}$ to be an element of $\mathbb{A}$, the second definition only requires this for those propositions which logically follow from the intersection over $\mathbb{A}$ (where $\mathbb{A}$ is a set of propositions and thus a set of sets of possible worlds), i.e. the set of possible worlds that are an element of every proposition in $\mathbb{A}$, i.e. the set of worlds of which every proposition in $\mathbb{A}$ is true.
The first definition now involves every possible subset of $\mathbb{A}$ (including the empty set and $\mathbb{A}$ itself), so everything which follows from any subset of propositions, even if that proposition just contains one world of which it is true, should also be in $\mathbb{A}$.
On the other hand, the second definition should involve significantly less propositions, since it only considers those worlds which are an element of every proposition in $\mathbb{A}$ (i.e. those propositions in $\mathbb{A}$ which are true in every world), excluding those which contain e.g. just one world which is not part of every proposition in $\mathbb{A}$.
So my first thought was that the two definitions should actually differ.
But then again, if we also allow more "minimal" propositions (as in the case of the first definition, where we consider every possible subset), wouldn't this actually yield the same results since those propositions are part of the ones holding in every world anyway, so the reulting set of propositions logically following from $\mathbb{A}$ would be the same since all of them already follow from the larger set and the smaller subsets have no effect?
And in case they do differ, which should be the definition that actually matches what one would intuitively understand under "closure under logical consequence"?
To me, the second variant seems to yield the propositions following from $\mathbb{A}$ in a more direct way than first ranging over every subset of $\mathbb{A}$, but it might be that I am wrong.
I thought about the definitions for quite a long time, but I'm still not sure whether they are actually equivalent (although they should be, as the two papers refer to the same theory).
I also tried looking for referrable standard defintions on closure under logical consequence, but I unfortunately could find none that is appropriate for my problem (especially in being defined on sets of possible worlds).
Thanks already for reading up to this point. Since this topic is probably a very specific in being related to linguistics, I would already be thankful for general notes on the formulae or hints on literature that might discuss closure under logical consequence in more detail.
Best Answer
I think you have things turned around a bit. As I read them, the two definitions are equivalent as long as the underlying logic is monotonic (that is, $\Gamma\vdash \varphi$ and $\Gamma\subseteq \Gamma'$ implies $\Gamma'\vdash \varphi$).
The first definition says $\mathbb{A}$ is strongly closed if:
The second definition says $\mathbb{A}$ is closed if
Since $\mathbb{A}$ is a subset of itself, clearly "strongly closed" imlies "closed."
For the converse, suppose $\mathbb{A}$ is closed and $\mathbb{B}$ implies $p$ for some $\mathbb{B}\subseteq\mathbb{A}$. Then - as long as the underlying logic is monotonic - $\mathbb{A}$ also implies $p$. So, since $\mathbb{A}$ is closed, $p\in\mathbb{A}$.