How to say “everyone respects someone” and “someone respects everyone” in quantified logic?

I’m having a hard time with this logic obviously isn’t my thing

• You need a two-valued predicate like RESPECTS(x,y) which is TRUE if x respects y, FALSE if x does not respect y. See the similar functions on p. 4 of the PDF The Syntax of Predicate Logic, along with subsequent discussion of how to use the universal quantifier and existential quantifier to express statements like "everything chased Socrates" or "Kermit kissed something" (there's an editing error in the PDF where they keep writing "Kermit chased" rather than "Kermit kissed" when using the predicate "KISS") Nov 29, 2021 at 19:56
• Also note in particular the two propositions 20a and 20b at the bottom of page 5 of the pdf, and their English translations--they both use a combination of the universal and existential quantifier with statements involving the two-valued predicate LOVE(x,y), you'll need to do something similar with RESPECTS(x,y). Nov 29, 2021 at 20:06
• Consider the binary relation Is_Son_of(x,y): what do you think about "everyone Is_Son_of someone" vs "someone Is_Son_of everyone" ??? Nov 30, 2021 at 9:12
• Thanks to @Hypnosifl link you can look at expression 33 and 35 therein. And then my comment below the answer below Dec 1, 2021 at 13:19
• I've changed 'some respects' to 'someone respects', which I take to be your meaning. If I'm wrong, change the text back - except that 'some [people]' would need 'respect', not 'respects'. Dec 2, 2021 at 9:52

Assuming our quantifier ranges over human beings. 1:

``````∀∃xyRxy
∃∀xyRxy
``````

or 2:

``````∀x∃yR(x,y)
∃x∀yR(x,y)
``````

or 3:

Without assuming domain

``````∀x∃y((H(x) ∧ H(y)) → R(x,y))
∃x∀y((H(x) ∧ H(y)) → R(x,y))
``````

or 4:

``````∀x∃y((H(x) ∧ H(y)) ∧  R(x,y))
∃x∀y((H(x) ∧ H(y)) ∧  R(x,y))
``````

or 5:

if you want it as set−relations

``````R={(x,y): x ∈ H, y ∈ H* : H* ⊆ H}
``````

or 6:

An even absurd version

``````H×H*, H* ⊆ H
``````

or 7:

The absurd version of the even absurd version

``````((H×H*)^c)^c
``````

or 8:

The absurd version of the vacous conditional

``````∀x∃y□((H(x) ∧ H(y)) → R(x,y))
∃x∀y□((H(x) ∧ H(y)) → R(x,y))
``````

@Rusi-packing-up I hope this clears it up for you.

1. The first one says everything relates to something.
2. The second one says what the first says but in conventional notation.
3. The third one says, for everything there is something, if the two are H, then the first R−relates to the second.
4. The fourth one, the one YOU SUGGESTED, says for everything there is something, where the two are H, and the first R−relates to the second. If you think that's wrong, that's the one YOU suggested.
5. The fifth is as trivial as could be, if you can't read that, then I can't help you.
6. Sixth one is fifth but in the most compact sense.
7. Seventh just says to take the complement of the complement of what was stated in sixth.
8. Eightth is the fish−hook version of 3 since you happen to have a problem with "vacously true" conditionals.
• If the H(uman) set is empty, the last line will be vacuously true. Intended? See plato.stanford.edu/entries/quantification «Frege and Russell... formalized sentences of the form "Some A is B" and "Every A is B" as ∃x(Ax∧Bx) and ∀x(Ax→Bx), respectively.» Note this also respects de Morgan correctly generalizing ∧∨ to ∀∃ since Ax→Bx negates as Ax∧¬Bx Dec 1, 2021 at 2:34
• The issue is that the → in the last line should be an ∧ following SEP. Also "some As are Bs" and "there exist A such that B" are synonymous. See more details Wikipedia Or see the @hypnosifl link examples I've highlighted below the question above. May help to see it as Venn diagrams which would help show "some As are Bs" as more symmetric than "all As are Bs". Dec 2, 2021 at 3:26