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I'm trying to understand first-order-logic and have this simple question.

Given the following predicates:

$Thing(t)$, which states that $t$ is a thing;

$Word(w)$, which states that $w$ is a word; and

$HurtsYouMoreThan(x,y)$, which states that $x$ hurts you more than $y$,

I need to create a first-order-logic statement that says "There is nothing that hurts you more than words do.

Victor
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3 Answers3

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Revised to match new question.

HINT: There is nothing that hurts you more than words do can be understood as follows:

If $x$ is anything other than a word, and $y$ is any word, then $x$ does not hurt you more than $y$ does.

From here the translation into a sentence of first-order logic using your three predicates should be pretty straightforward. Note that you will have a couple of quantifiers.

Brian M. Scott
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  • I updated the statement to be less slangy. – Victor Jul 25 '13 at 20:18
  • @Victor The tricky part is, it could be argued that $\operatorname{word}(w)\to\operatorname{thing}(t)$ as Brian M. Scott's answer suggests, so it really all depends on how rigorously these predicates are defined. Just something to be mindful of. – rurouniwallace Jul 25 '13 at 20:23
  • Can you please look at related problem http://math.stackexchange.com/questions/452231/how-to-use-first-order-logic-with-both-constants-and-predicates – Victor Jul 25 '13 at 21:48
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It will help to re-word it:

No thing hurts more than words do.

In other words:

For all $w$ and for all $t$, if $w$ is a word and $t$ is a thing, it is not true that $t$ hurts you more than $w$.

$$(\forall{w})(\forall{t}){(\operatorname{word}(w)\land\operatorname{thing}(t))\to\lnot\operatorname{HurtsYouMoreThan}(t,w)}$$

rurouniwallace
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  • I updated the statement to be less slangy. – Victor Jul 25 '13 at 20:17
  • @Victor: This is arguably not quite correct. There is nothing in the statement of the problem that rules out the possibility that a word is a thing, and it is not explicitly stated that $\forall x(\neg\text{HurtsYouMoreThan}(x,x))$. If you use this translation, you should say that you’re assuming that $\forall x(\neg\text{HurtsYouMoreThan}(x,x))$. – Brian M. Scott Jul 25 '13 at 20:28
  • Can you please look at related problem http://math.stackexchange.com/questions/452231/how-to-use-first-order-logic-with-both-constants-and-predicates – Victor Jul 25 '13 at 21:46
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Hint: If I'm interpreting the given sentence correctly, you should be able to think about this as a sentence more like $$\text{Given any thing $$X which isn't a word and any thing $Y$ which is a word, ...}$$

not all wrong
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