Express the following statements in predicate logic.

Question

Let the predicate $ E(x,y) $ represent the statement "Person x eats food y". Let the predicate $ M(y) $ represent the statement "Food y is a meat product".

(a) Express the following statement in predicate logic: "Someone is a vegetarian". (b) Express the following statement in predicate logic: "Nobody (except maybe John) eats lasagna."

How could I write these statements in predicate logic ?

Could I do the following for the first one ?

$ \exists x\; E(x, beans) $

I don't know about the second one.

Answer 1

What you wrote was: "There is a person who eats beans." But that's not quite equivalent to "Someone is a vegetarian.", since it's possible that all people on earth eat both beans and meat. Instead, try something like: $$ \exists x ~ \forall y ~ [M(y) \to \neg E(x,y)] $$ which reads to be something like: "There is a person named $x$ such that for any meat product $y$, $x$ doesn't eat $y$."

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For the second statement, try convincing yourself that it's equivalent to: "The only person who might eat lasagna is John." Then we get something like: $$ \forall x ~ [E(x, \text{lasagna}) \to x = \text{John}] $$

Answer 2

Vegetatians eat more than just beans. What we do know is that if $x$ is a vegetarian, then whatever he/she eats must not be meat. $$\exists x\forall y\Big(\underbrace{E(x, y) \implies \lnot M(y)}_{*\,\text{see note}}\Big)$$

• Note: Because of the equivalence of an implication and its contrapositive, we can write, equivalently, $$\exists x \forall y \Big(M(y) \implies \lnot E(x, y)\Big)$$

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Second: We need a constant to represent "lasagna". We could use just $\mathcal l$ to denote this, or we can simply use "lasagna". "There does not exist any x who is not John and who eats lasagna." $$\lnot \exists x\Big(( x\neq \text{ John }) \land E(x, \text{ lasagna}) \Big)\tag{2}$$

Of course, by pushing negation inwards, we can write, equivalently, $$\begin{align} \lnot \exists x\Big(( x\neq \text{ John }) \land E(x, \text{ lasagna}) \Big)&\equiv \forall x\Big(\lnot(x \neq j) \lor \lnot E(x, \text{ lasagna})\Big) \\ &\equiv \forall x \Big(x\neq j \rightarrow \lnot E(x, \text{ lasagna})\Big)\\ & \equiv \forall x \Big(E(x, \text{ lasagna}) \rightarrow x = j\Big)\end{align}$$

This is indifferent to either case the ase that John eats lasagna, or the case that John doesn't eat lasagna. He either does, or doesn't. But whatever the case, it is irrelevant to the translations, since we are addressing the non-existence of any lasagna-eaters who aren't John.