Give this statement "Food cannot be found in any house (in this area)"
Can you correctly infer that "There is AT LEAST a house (in this area)"
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No. On what I take to be the most natural interpretation, two logically-equivalent ways of rendering the first statement in formal logic would be:
Neither sentence entails that a house exists.
However, user “quis est ille” reminds us that, as with nearly all ordinary language sentences, we can imagine contexts in which this sentence could be taken to imply the existence of houses. If the sentence is asserted in a neighborhood of houses, then its meaning might be just about whether there is food in them, and rendered in formal logic as:
Yet, I think when there are houses around, we don't need to assert their existence, even if we acknowledge it. So while we might suppose that houses exist, it does not logically follow from this comment alone that houses exist, because we could equally well use the same comment, with the same meaning (a) in a scenario where there are questions about whether there are houses (e.g. most have been bombed), and (b) in a scenario where there are no houses (e,g, when we're search in the desert for any source of food).
Yes, you can you correctly infer from the ordinary language sentence "Food cannot be found in any house (in this area)" that "There is AT LEAST [one] house (in this area)". This is because the existence of houses in the area is presupposed, not asserted.
Note that the first reply confuses the way the English sentences would be rendered formally, with the sentences themselves. It is of course correct that the two formal renderings are equivalent:
(x) [x is a house in this area -> not: food can be found in x ]
Not for some x, [x is a house in this area and food can be found in x]
But the first English sentence is not equivalent to the first formal rendering. “Food cannot be found in any house in this area” presupposes the existence of houses in this area. The classic example, discussed in Strawson’s Introduction to Logical Theory (?) is “all of John’s children are asleep”. This presupposes the existence of John’s children, and is clearly not equivalent to “it is not the case that John has some children who are not asleep”, since the latter is true when John has no children.
"Christmas trees are not carried by any pink elephants (in this area). Can you correctly infer that "there is at least one pink elephant (in this area)"?
Your example is badly chosen, because (most likely) there are houses in that area, which leads you to incorrectly infer that their existence follows from the statement you gave.
You could also write this statement as a material conditional: if there is a house in this area, then there is no food to be found in it. If the antecedent of a material conditional is false, the statement always evaluates to true. The only way to evaluate a material conditional as false is if the antecedent is true and the consequent is false
After all, if there are no houses, then there certainly no houses with food in them.