Let L (x, y) be the statement “x loves y” where the domain for both x and y consists of all people in the world. Use quantifiers to express “Joy is loved by everyone”.
∀x L(x, Joy)
In predicate logic the order of arguments matters: L(x, y) means x loves y, so the lover is the first argument and the loved person is the second. A universal quantifier ∀ states that the property holds for every value in the domain.
Step 1 — Joy is loved by everyone means: for every person x, x loves Joy.
Step 2 — Joy is the loved one, so Joy goes in the second argument: L(x, Joy).
Step 3 — Quantify over all people: ∀x L(x, Joy).
∀y L(Joy, y) puts Joy first and says Joy loves everyone. ∃y∀x L(x, y) only says that someone is loved by everyone, without naming Joy. ∃x¬ L(Joy, x) says there is someone Joy does not love.
∀x L(x, Joy) option is correct.
Similar Questions
Equivalent logical expression for the Well Formed Formula (WFF),
\(\sim(\forall x)\,F[x]\)
is
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Codes :
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A. Constant
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Important Questions from First Order Logic - Teaching
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\(\sim(\forall x)\,F[x]\)
is
Let P(m, n) be the statement “m divides n” where the Universe of discourse for both the variables is the set of positive integers. Determine the truth values of the following propositions.
(a) \(\exists m \forall n P(m, n)\) (b) \(\forall n P(1, n)\) (c) \(\forall m \forall n P(m, n)\)
Codes :
