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I'm trying to figure out what the proper translation in predicate logic would be for the example below, I'm confused because the predicate comes before the subject. So i'm wondering if I need to include it into the domain, or make it a separate domain predicate.

Example: "All orange basketballs are round."

I was thinking that I could translate this in one of two ways, which one of these would be correct?

"All orange basketballs are round."
Domain:
O(x) - x is Orange Basketballs
R(x) - x is Round

Answer 1- (∀x)(O(x)-->R(x)

OR

"All orange basketballs are round."
Domain:
B(x) - x is Basketball
O(x) - x is Orange
R(x) - x is Round

Answer 2- (∀x)(B(x)-->O(x)-->R(x))

I'm under the assumption that answer 1 seems more sound, but any advice is appreciated!

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Just an adjustment on your (2), which is correct, if you parenthesize properly:

How about $$\forall x\Big((B(x)\land O(x))\to R(x)\Big)\tag{1}$$

which is equivalent to $$\forall x\Big(B(x)\to \big(O(x) \to R(x)\big)\Big)\tag{2}$$


Below, you'll see the results when entering $(a \land b)\to c$, followed by the truth table resulting from $a \to (b\to c)$:

enter image description here

enter image description here

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  • $\begingroup$ I was under the impression that the universal quantifier had a rule that stated a implication was necessary after the subject. Edit: Ah I see, thank you for the edit! $\endgroup$ – wolfeyes90 Sep 22 '18 at 22:02
  • $\begingroup$ Implication is used in the first expression. For all x, if x is a basketball and x is orange (i.e. if x is an orange basketball), then x is round. The primary connective is implication in the first case. The primary connective is implication in the equivalent form (2). Try comparing the two expressions, I wrote, using a truth table, or prove they are equivalent by laws of equivalency. $\endgroup$ – Namaste Sep 22 '18 at 22:05
  • $\begingroup$ Excellent, I'll write out a truth table to compare these two. This makes much better sense. Thank you so much for the assistance. $\endgroup$ – wolfeyes90 Sep 22 '18 at 22:08
  • $\begingroup$ @ruwadidr Glad to help! $\endgroup$ – Namaste Sep 22 '18 at 22:09
  • $\begingroup$ Note the first would technically be correct if $O(x)$ means x is an orange basketball, with $R(x)$ meaning "x is round," except you left out a required parenthesis on the right: you'd need $\forall x\big(O(x) \to R(x)\big)$. But I think the second version, and/or my suggestion, is richer. $\endgroup$ – Namaste Sep 22 '18 at 22:14
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Each of those representations will work. Note, however, that you seem to neglect to specify the domain in each of the cases -- you write "Domain:" but nothing follows the colon.

In the first proposal, your domain would presumably be all basketballs.

In the second, your domain would could be everything in the world, or all sports paraphernalia.

Which of the possibilities you choose in a particular situation is not really a question of being correct, but depends on what is the most convenient and useful way to model the reasoning about balls, colors, and roundness you have in mind to use it for.

Of course, if you're just given the sentence as a homework exercise without any context, there's no actual reasoning to have in mind, so you can't use convenience and utility to make your choice (as you would in any real situation). Instead you'll have to guess at what the teacher would most like to see. If you're in a classroom situation where it is usual to specify a domain for the variables, choosing the first representation would give you a chance to show that in action.

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  • $\begingroup$ Thanks for the comment Henning, from my class I was under the impression that the domain would be "O(x) - x is Orange Basketballs & R(x) - x is Round " for example, since we are specifying the domain of each one. I figured that it would be read as Universal Quantifier = All and then "Orange Basketballs Are Round". In a classroom setting would I need to add something else to the domain to make this correct? $\endgroup$ – wolfeyes90 Sep 22 '18 at 23:00
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    $\begingroup$ @ruwadidr: It is possible that your course is using "domain" in a different sense than I'm used to, but to me, those would be interpretations of the predicate, and the "domain" of discourse should be a description of which things it is possible for a variable to take as values. $\endgroup$ – Henning Makholm Sep 22 '18 at 23:04
  • $\begingroup$ @henning-markholm Got it, I'll investigate that moving forward, but your explanation on that makes sense. $\endgroup$ – wolfeyes90 Sep 22 '18 at 23:08

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