Logical ambiguity formula for "I have not had one of them"

Question

The full sentence is "There exist 50 different types of sushi, and I have not had one of them."

I understand that the two different interpretations of this sentence are:

1. There exist 50 different types of sushi, and I have had all but one of them.
2. There exist 50 different types of sushi, and I have not had any of them.

How would I write two different formulas of propositional logic?

The first propositional variable would be:

a: There exist 50 different types of sushi.

Some ideas for other propositional variables are:

• I have had all of them
• I have had only one of them

HOWEVER, the propositional variable needs to be the same for the different formulas of propositional logic. Any ideas?

Answer 1

Define $S$ to be the set of all types of sushi. For any $x\in S$, define $H(x)$ to be the statement "I have had sushi type $x$." (As a side note, you're saying $|S|=50$.)

• "I have not had any of them" $\iff$ $\forall x\in S.\; \neg H(x)$
• "I have had all but one of them" $\iff$ $\exists y\in S.\; \left[\left(\neg H(y)\right)\wedge \left(\forall x\in S-\{y\}.\; H(x)\right)\right]$

Answer 2

I would say:

1. "I have not had any of them": $\forall x. \neg(\text{I have had }x)$.

2. "I have not had all of them": $\neg \forall x . (\text{I have had }x)$.

3. "There's one of them I haven't had": $\exists x.\neg(\text{I have had }x)$.

4. "There's exactly one of them I haven't had": $\exists! x.\neg(\text{I have had }x)$.

Of these, (2) and (3) are logically equivalent.

For (4), in mathematical logic people usually prefer to consider $\exists!$ to be an abbreviation of something more complex, but doing so is probably not productive if your goal is to model natural language.

Answer 3

First: you really need to use a first-order logic formula for this, not a propositional logic formula.

Now, using:

$S(x): x$ is a type of Sushi

$H(x,y): x$ has had $y$

$i: I$

to say that you haven't had any type of sushi would be:

$$\forall x (S(x) \rightarrow \neg H(i,x))$$

and to say that there is (exactly) one of the types of sushi's you haven;t had you can use:

$$\exists x (S(x) \land \forall y (\neg H(i,x) \leftrightarrow y=x))$$