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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:

  • For any houses/each house in the area, it's not the case that food can be found in them
  • There does not exist a house such that food can be found in it.

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:

  • There are houses in this area and there is no food that can be found in any of them.

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).

  • I marked this down for reasons set out in my answer below. The answer confuses formal renderings of ordinary language statements, with the ordinary language rendering. – quis est ille Dec 22 '14 at 12:31
  • I would disagree. You are right to point out the difference between what is asserted and what is presupposed. However, I don't think that makes my answer misleading. The natural language sentence offered does not offer any reason to think that a house's existence is being supposed (indeed unlike some similar sentences do). “Leprechauns cannot be found by any pots of gold in this area” similarly does not suppose Leprechauns. Also, the tags indicate that it is a question about logic and quantification. – ChristopherE Dec 22 '14 at 14:20
  • (a) There is logic and quantification in ordinary language too! And (b) I agree there is an interpretation of your first sentence that is equivalent to the second, but the most natural reading IMO involves presupposition. – quis est ille Dec 22 '14 at 14:31
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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.

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"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.

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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.

  • This answer isn't wrong but could be worded more clearly. If the OP understands "material conditional", it's doubtful OP would have trouble knowing the answer. Without it, you should explain more thoroughly... – virmaior Dec 21 '14 at 20:52
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You cannot infer that from its explicit first-order logical structure, but you can given context of utterance: I don't think anyone would say utter that sentence if there is no house around.

  • Mathematicians would say this sentence without the slightest hesitation if there were no houses around. Just to have a bit of fun and confuse everyone. – gnasher729 Dec 23 '14 at 14:53
  • I think what you might be getting at in this answer is what is expressed in quis est ille's answer... but the difference is that you don't explain it. – virmaior Dec 24 '14 at 6:04

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