Info

Solution

1. This statement is (relatively) unambiguous.

2. There are at least two ways of reading this:

   • I see her perform the action of ducking (lowering the head or body quickly).

   • I see an animal of the species duck , which belongs to her.

   On the first reading, you might for example infer (deductively) that she is now in a crouching position or (inductively) that she tried to avoid getting hit by something. On the second reading, you could infer (deductively) that she owns an animal. Neither inference is valid on the other reading.

3. Many people consider this ambiguous between:

   • He thanked her two children, called “Alan” and “Ada”.

   • He thanked her children, whoever they are, and two people called “Alan” and “Ada”.

   On the first reading, you can infer that Alan is her child, while you cannot n the second reading.

4. This statement instead uses the serial comma to avoid the previous ambiguity.

5. This statement is ambiguous between:

   • I see a man who has my binoculars.

   • I see a man through my binoculars.

   On the first reading, you can infer that the speaker does not currently have their binoculars in their possession, while on the second reading you can.

6. This statement is (relatively) unambiguous.

7. This phrase has the following two readings:

   • Someone who walks dogs and is friendly.

   • Someone who walks (only?) friendly dogs.

   On the first reading, you can infer that the person is friendly, on the second you can’t.

8. There are two readings:

   • I am happy that I am here and she is happy that I’m here.

   • I am happy that I’m here and she is also here.

   On the second reading you can, for example, infer that she is here, while on the second one you can’t.

9. This statement is ambiguous between:

   • The priest officiated my uncle’s wedding.

   • The priest got married to my uncle.

   On the second reading, you can infer that the priest is now the speakers uncle in law, while on the first one you can’t.

10. Also this statement is ambiguous:

• They fed her food for dogs

• They fed food to her dog.

On the second reading, you can infer that she owns a dog, while on the first one you can’t.

Solution

We don’t know whether John smokes or not, but we can consider a situation, where he never smoked. In such a situation, he never stopped smoking, so the premise is true. But he’s not a smoker, so the conclusion isn’t true. This makes the inference invalid.

In natural language, we often convey meaning that goes beyond the literal meaning of statements, for example by means of implicature . In this case, we’re dealing with a seeming presupposition , where the first statement seems to have the implicit presupposition that John did smoke. This might make the inference seem valid, even though it isn’t.

Solution

1. { }, with nothing written between the curly brackets. There is also the set-theoretic symbol ∅.

2. { IA , UU, {IA } }

3. { 2, 4, 6, 8 } or { n : n is an even number between 1 and 10}

4. { n : n is an even natural number }

5. { {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} } or:

   { x : x is a non-empty set with only member from {1, 2, 3} }

Solutions

1. Here’s the parse tree:

   

2. Here’s the parse tree:

   

3. Here’s the parse tree:

   

4. Here’s the parse tree:

   

Solution

1. I’m using ϕ as the variable in the form, rather than A as before to avoid confusion with the symbol for disjunction:

2. Here are the rewrite rules:

And here’s the tree:

3. pCpqNqCpr can’t be a formula in Polish notation since there are only two options: either a formula starts with an operator from N,K,A,C,B or it is a propositional variable. But this formula starts with a propositional variable and then continues. There is no rule, which can generate such a formula, so it isn’t one.

4. The formula was . If we write this in Polish notation, it becomes NKpq. But this formula now has a unique reading, given by the following parse tree:

   

   If we wanted to give it “the other” reading, we would need to write KNpq.

Solution

1. A straight-forward representation is . But an equally valid alternative is: .
2. This one is trickier than it might seem at first. I’ve attached a paraphrase to facilitate reading:

3. This one has at least two readings in natural language. The most straight-forwardly intended one is , which reads as if the treasure is not in the lower right and not in the lower left, then it’s in the upper left. But if we take the “only if” parlance seriously, and read the statement as “if the treasure is not in the lower right, then it’s only in the upper left, if it’s not in the lower left,” we get this formula: .
4. This is then just the consequence of the second reading from 3.:
5. 
6. A very straight-forward reading is this: (“the treasure is not here, not here, not here, and not here”), but an equivalent statement is just the negation of 6.: .
7. 
8. 

As you can see, in knowledge representation, we have to handle natural language ambiguity, there are often more than one solution, and things that sound easy in natural language, can be hard to formalize.