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Let $\mathit{L}$ be the set of Lions, $\mathit{M}$ the set of Mammals and $\mathit{A}$ the set of all Animals. Use rules of inference with quantifiers to formalize the following statement:

$\mathit{S}$: "If an animal is a Lion, then, they must also be a Mamal"

Here is my answer to this problem:

Universe of discourse is defined as $\mathit{A}$, then:

$\forall x(x \in L \to x \in M)$

Now, my issue is that I've never seen sets used in predicate logic like this, usually I am given predicate functions instead, such as $\mathit{L(x)}$: "x is a Lion". Then an answer will look more like this :

$\forall x(L(x) \to M(x))$

I've been told that I don't need to create my own predicate functions to solve the exercise so my question is. Is my answer above using sets correct? Is that the correct syntax for using set notation with quantifiers and rules of inference ? Is it equivalent to the answer using predicate functions?

Thanks in advance

Andrés E. Caicedo
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デリエゴくん
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  • The two notations are "conceptually" equivalent. In "pure" predicate logic there is no $\in$ symbol; thus you have to use predicates $L(x)$ to specify sub-domains. If the textbook that you are using introduces $\in$ from the start into the syntax, then it will be Ok to use it. – Mauro ALLEGRANZA Dec 10 '19 at 10:58
  • Thanks. Am I correct in thinking that answering $\forall x (L(x) \to M(x))$ would be incorrect in this case since L and M are sets rather than predicate functions? – デリエゴくん Dec 10 '19 at 11:11
  • Maybe... but the rigorous answer will depend on the detailed syntactical specifications of the language. – Mauro ALLEGRANZA Dec 10 '19 at 11:52

1 Answers1

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$\forall x(x \in L \to x \in M)$ seems to be the correct answer here, since the exercise doesn't require you to write a formula in a given language.

Olivier Roche
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