# Logic disambiguation in composed sentences with negative predicates and "or" conjunction

Today, I've attended a psychological test for the master thesis of a friend of mine. The target is children and adults. So don't scare if you see in the following lines that I speak about animals.

I ask you a minute of patience to read fully my question.

The test is composed by two phases:

1. You see animals (on a paper) that may eat two foods, it's choosen by the examiner if the animal eat one food, two foods or none of them. If the answer is two you attach to the animal a gold medal, if it's one a bronze medal, if it's zero a sad face.

2. The examiner (using a frog) states a sentence and you have to say if it's true or false.The disambiguation emerges at this phase with a particular type of sentences.

Example. In the phase 1 the hippo ate one food (a carrot) and it has a bronze medal.The frog say

The hippo didn't eat a carrot or a cucumber.

Is it true or false? I answered that's true. Why? Because I transformed the sentence to

The hippo didn't eat a carrot or it didn't eat a cucumber

The first part is false but the second one is true, with an or this means true. Anyway, it's possible another transformation

The hippo did not ( eat a carrot or eat a cucmuber)

Which thanks to De Morgan's laws become

The hippo did not eat a carrot and did not eat a cucumber

This is obviously false.

I transformed the sentence beacuse in my opinion conjuction like or/and "must" connect sentences, I.e "the carpet is red or heavy" is not valid, a right one may be "the carpet is red or the carpet is heavy".

Probably, you have already understood that the issue is related the negation, logic operator precedence, sentences truncation and De Morgan's laws and how the latter is felt by people (i.e. embedded in natural language). I want to deepen in this topic and see what you expert say. In addition feel free to correct my answer and to say that I'm wrong with my test answer.

So my question are:

• What's the correct transformation to eliminate the disambiguation?

• Does the transformation depend on language (e.g english and russian use the same transformation)?

• If no transformation is needed how do you face with this kind of sentences?does a way exist or may be only interpreted?

These articles are helping me to answer to the question:

(there is a third, but I've to wait for 10 points so it is in the comments)

• Natural language is unavoidably ambiguous, so there is usually no single correct transformation. But, it may indeed be the case that speakers of different languages have different distributions for their understanding of conjunctions/disjunctions in sentences. Nov 5 '15 at 18:40
• What you ran into is known as De Morgan's Law. It says that "Not (A Or B) is Equivalent to (Not A) And (Not B)". If he is not sick and he is not absent, then he is not sick or absent. We can eliminate one ambiguity by saying neither sick nor absent, or in this case did not eat a carrot nor a cucumber. Nov 5 '15 at 19:05
• And in English negation scopes over disjunction (but it varies in other languages, e.g., Mandarin has reverse scope), so yes, you apply De Morgan's law here. Nov 5 '15 at 21:19
• @JeremyNeedle, thanks for your comment, I agree with you. Anyway, personally how do you interpretate the frog question? Nov 6 '15 at 9:22
• the third article nyu.edu/projects/szabolcsi/… Nov 6 '15 at 9:40