# Re-writing “Everybody loves somebody” and variants in symbolic logic.

Right now I'm working on a set of questions and I came across a two-parter that confused me a bit with the way it was asked. The question is simply put as such:

Write the statements below in symbolic logic.
a. Everybody loves somebody.
b. Somebody loves everybody.


Would the way to rewrite this be a simple as using existential and universal quantifiers? Such that a. would be translated to

"∀x, where x is a person, ∃ person x they love."

And b. would be translated to

"∃x such that x loves ∀x."

Is this a proper way to answer the question? Any help is appreciated.

• You might want to define $L(x,y)$ to mean that $x$ loves $y$ – Mark Bennet Dec 31 '14 at 23:42
• Another problem that been hasn't mentioned with your formulation: you reuse variables, which is a Very Bad Idea. (In particular, in general your sentences wouldn't be considered syntactically correct.) – Steven Stadnicki Dec 31 '14 at 23:47
• Here's an obligatory cartoon on this very issue: dilbert.com/strips/comic/2001-04-19 – KCd Jan 1 '15 at 0:42

You have the right ideas, basically, but there is some more work to get the statements into symbolic logic.

Let $L(x,y)$ be the relation "$x$ loves $y$". Then we get:

$$a.\quad ∀x \, ∃y \, L(x,y)$$

$$b.\quad ∃x \, ∀y \, L(x,y)$$

(The exact punctuation depends on the logical system.)

• Ah so by defining a variable L(x,y) I can essentially skip the unneeded wordy answer I originally came up with? – Johnathan Scott Dec 31 '14 at 23:44
• @JohnathanScott: Yes. In fact, such a relation (not a "variable": $x$ and $y$ are variables) is needed to use actual symbolic logic. The "wordy" forms are only half-way there. – Rory Daulton Dec 31 '14 at 23:45
• Wonderful. Thanks so much for your help! – Johnathan Scott Dec 31 '14 at 23:46

The phrase "everybody loves somebody" can be translated to "every person x loves some person y" but notice that I said "Person" therefore you have to declare that x and y are both a person. The correct translation would be: ∀𝑥∃𝑦(P(x)^P(y))->L(x,y) And if you were to translate this to a language that makes more sense to a computer you would say "for every x and some y, such that x and y are a person (not saying they are equal or different) there is an x that loves y"