Tigers and lions attack if they are hungry or threatened.

Question

Which one of the first order predicate calculus statements given below correctly expresses the following English statement?

"Tigers and lions attack if they are hungry or threatened".

1. $∀x[(tiger(x) ∧ lion(x)) → \{(hungry(x) ∨ threatened(x)) → attacks(x)\}]$
2. $∀x[(tiger(x) ∨ lion(x)) → \{(hungry(x) ∨ threatened(x)) ∧ attacks(x)\}]$
3. $∀x[(tiger(x) ∨ lion(x)) → \{attacks(x) → (hungry(x) ∨ threatened(x))\}]$
4. $∀x[(tiger(x) ∨ lion(x)) → \{(hungry(x) ∨ threatened(x)) → attacks(x)\}]$

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My attempt :

The statement "Tigers and lions attack if they are hungry or threatened" means that if an animal is either tiger or lion at the same time.

"Tigers or lions if they are hungry or threatened then attacked "

So, option $(4)$ is correct.

Can you explain in English statements for each option?

Answer 1

$1$. Tigons and ligers attack if they are hungry or threatened

$2$. Tigers and lions are always hungry or threatened and always attack

$3$. Tigers and lions are hungry or threatened if they attack

$4$. Tigers and lions attack if they are hungry or threatened

Answer 2

In English, "and" can be moved into the antecedent of a conditional when the logical connective that renders the statement is really "or". It's short for the conjunction of two conditionals.

For example, given $\forall x(Boy(x)\to Happy(x))$ ("[all] boys are happy") and $\forall x(Girl(x)\to Happy(x))$ ("[all] girls are happy"), we have $\forall x(Boy(x)\to Happy(x)) \land \forall x(Girl(x)\to Happy(x))$, which can be said as "all boys and all girls are happy", or just "[all]boys and girls are happy". Note, though, that the logical equivalence used is: $$ ((p\to r)\land (q\to r))\iff ((p\lor q)\to r). $$ The formula equivalent to the conjunction of the statements about girls and boys is: $\forall x((Boy(x)\lor Girl(x))\to Happy(x))$.

1. means: Anything $x$ that is both a tiger and a lion attacks if it is either hungry or threatened.

2. All lions and tigers are hungry or threatened, and attacks. Or: every lion or tiger is hungry or threatened, and attacks.

3. Every lion or tiger that attacks is hungry or threatened.

4. Every lion or tiger that is hungry or threatened, attacks. Or: lions and tigers attack when hungry or threatened.

I see that I wrote "lions and tigers", not "tigers and lions". Fortunately, they're equivalent! I couldn't help it: because of the Wizard of Oz ("Lions and tigers and bears!") that's more idiomatic. Similarly, "it's raining cats and dogs!", but it never rains dogs and cats :)