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Example 1:

No intelligent person who drinks to excess also eats to excess.

I am stuck on deciding whether this means

a) $\forall x(Ix \implies -(Dx \lor Ex)$

or

b) $\forall x(Ix \land Dx \implies -Ex).$

Example 2:

None of the paintings is valuable except the battle pieces.

I think that what this is saying (using intuition) is that, if you give me a Painting then it is not Valuable unless you give me a Battle piece in which case it is Valuable; thus, in symbols:

c) $\forall x(Px \implies -Vx) \lor \forall x(Bx \implies Vx).$

Alternatively, it could be closer to Example 1; thus, in symbols:

d) $\forall x(Px \implies -(Vx \lor Bx)).$

Or I could very well be out to lunch on all of these translations.

ryang
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JCAL
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    Example 1: second one; equivalently: $\lnot \exists x (Ix \land Dx \land Ex)$ – Mauro ALLEGRANZA Apr 04 '23 at 06:45
  • In the second example your (d) version is almost correct. It is saying that any painting is either not valuable or else a battle painting and valuable (as you said), so we could write it as $\forall x (Px \implies (\neg Vx \vee (Vx \wedge Bx))$. But now $\vee$ distributes over $\wedge$ so that becomes $(\neg Vx \vee Vx)\wedge (\neg Vx \vee Bx)$ which is $\neg Vx \vee Bx$. – thebogatron Apr 04 '23 at 07:03
  • Jeez, my c) option was out to lunch, I felt so confident about it too! – JCAL Apr 04 '23 at 07:09
  • @thebogatron The parenthetical "as you said" in your comment is wrong: your translation does not actually correspond to what the OP said ("What this is saying (using intuition) is that, if you give me a Painting then it is not Valuable unless you give me a Battle piece in which case it is Valuable"), but to my answer's Suggestion (2) below. $\quad$ Since the word 'except' is slightly ambiguous, especially in technical writing, it should probably be used judiciously. – ryang Apr 09 '23 at 10:02

1 Answers1

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Example 2:

None of the paintings is valuable except the battle pieces.

I think that this is saying that if you give me a painting then it is not valuable unless you give me a battle piece, in which case it is valuable.

I agree, so: $$\forall x\big(Px\to(Vx\leftrightarrow Bx)\big).\tag1$$

But look out for another interpretation of the word 'except': $$\forall x\big(Px\to(Vx\to Bx)\big).\tag2$$

  1. Everyone except Sue attended the event.

    Reading (1): Sue did not attend the event.
    Reading (2): It is not stated whether Sue attended the event.

  2. I won't take an umbrealla, except when it rains.

    Reading (1): When it rains, I will take an umbrella.
    Reading (2): When it rains, I may still not take an umbrella.

On the other hand, your suggested translations are both incorrect.

Example 1:

No intelligent person who drinks to excess also eats to excess.

$$\lnot \exists x (Ix \land Dx \land Ex).\tag3$$

b) $\forall x\;(Ix \land Dx \implies -Ex).\tag4$

This means that every person who is intelligent and drinks to excess does not eat to excess.

Statements (3) and (4) are equivalent to each other.

a) $\forall x(Ix \implies -(Dx \lor Ex))\tag5$

This translation makes a stronger assertion than required: it says that every intelligent person neither drinks to excess nor eats to excess.

ryang
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