# Predicate Is Before The Subject Is This The Correct Translation?

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?

Domain:
O(x) - x is Orange Basketballs
R(x) - x is Round

OR

Domain:
O(x) - x is Orange
R(x) - x is Round

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

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 $$a \to (b\to c)$$:

• 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! – wolfeyes90 Sep 22 '18 at 22:02
• 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. – Namaste Sep 22 '18 at 22:05
• Excellent, I'll write out a truth table to compare these two. This makes much better sense. Thank you so much for the assistance. – wolfeyes90 Sep 22 '18 at 22:08
• @ruwadidr Glad to help! – Namaste Sep 22 '18 at 22:09
• 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. – Namaste Sep 22 '18 at 22:14

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.