It seems you are hitting against the expressive limits of Existential Positive First-Order Logic. It seems that they are exponentially less powerful than first order logic, in the following sense:
every existential positive first-order sentence can be transformed in an equivalent one in prenex normal form without an exponential blowup, thanks to the absence of universal quantifiers and negation symbols.
Bodirsky, Manuel, Miki Hermann, and Florian Richoux. “Complexity of existential positive first-order logic.” Journal of Logic and Computation 23.4 (2013): 753-760.
Yeah, I’m not surprised that English-positive is a lot weaker than English (in terms of short statements), I’m just surprised that the weakness turns up so quickly with such natural-seeming sentences rather than pathological or complex statements. You see an example like “the sky is not green”, and it’s so easy to turn into “the sky is blue” or “grass is inedible” to “grass provides 0 calories”, and you start to think maybe most or all normal natural statements have feasible rewrites, and maybe this is like constructive mathematics and actually works for most things if you think about them a little harder—and then you hit “I didn’t see John.”
“Is this gluten-free?” (If we allow “gluten-free” we would allow “Every room is John-free.” and of course “Grass is edibility-free.” and very quickly Abs-E is trivial.)
Attempt: “This product contains rice flour, corn starch, tapioca flour, and salt.” but that just prompts the further question “Does any of those contain gluten?” …
Wittgenstein interrupted: “What can be said at all can be said clearly, and what we...”
“I think not all swans are white, and if we look for it we will find one that is not white.”
Attempt: “There exists a swan that is …” Blue? Green? Red? I can’t say “non-white”. I also can’t just list every color.
“I don’t believe in magic.”
I don’t even know how to start converting this to a positive statement.
It seems you are hitting against the expressive limits of Existential Positive First-Order Logic. It seems that they are exponentially less powerful than first order logic, in the following sense:
Bodirsky, Manuel, Miki Hermann, and Florian Richoux. “Complexity of existential positive first-order logic.” Journal of Logic and Computation 23.4 (2013): 753-760.
Yeah, I’m not surprised that English-positive is a lot weaker than English (in terms of short statements), I’m just surprised that the weakness turns up so quickly with such natural-seeming sentences rather than pathological or complex statements. You see an example like “the sky is not green”, and it’s so easy to turn into “the sky is blue” or “grass is inedible” to “grass provides 0 calories”, and you start to think maybe most or all normal natural statements have feasible rewrites, and maybe this is like constructive mathematics and actually works for most things if you think about them a little harder—and then you hit “I didn’t see John.”
A few more examples:
“Is this gluten-free?” (If we allow “gluten-free” we would allow “Every room is John-free.” and of course “Grass is edibility-free.” and very quickly Abs-E is trivial.)
Attempt: “This product contains rice flour, corn starch, tapioca flour, and salt.” but that just prompts the further question “Does any of those contain gluten?” …
Wittgenstein interrupted: “What can be said at all can be said clearly, and what we...”
“I think not all swans are white, and if we look for it we will find one that is not white.”
Attempt: “There exists a swan that is …” Blue? Green? Red? I can’t say “non-white”. I also can’t just list every color.
“I don’t believe in magic.”
I don’t even know how to start converting this to a positive statement.