The expression "otherwise" in predicate logic

Question

I'm currently working on the following problems, and wondering how I can express "otherwise" in predicate logic in a sentence like (d) below.

A hunted animal is called game. 
The definition of game is that everything that is either a big game or small game is a game. 
Examples of big games are moose, deer, boar and capercaillie.  
Examples of small games are fox, rabbit and bird. 

Write the following in predicate logic: 

My answers...

(a) Write the definition for game in predicate logic 
∀x (Game(x) ↔      BigGame(x) V SmallGame(x))           


(b) ”If there is a fox or rabbit, there is a small game” 
∀x (fox(x) V rabbit(x) →      SmallGame(x) )       


(c) ”If there are both rabbit and moose, there are both small game and big game”
∀x (rabbit(x) ∧ moose(x) →  SmallGame(x) ∧ BigGame(x) )          


(d) ”If there are moose, deer, boar and capercaillie, there are big games, otherwise there are just small games”
∀x (moose(x) ∧ deer(x) ∧ boar(x) ∧ capercaillie(x) →  BigGame(x) )

Answer 1

(d) ”If there are moose, deer, boar and capercaillie, there are big games, otherwise there are just small games”

$\forall x (moose(x)\wedge deer(x) \wedge boar(x) \wedge capercaillie(x)\Rightarrow BigGame(x) \wedge (\neg[moose(x)\wedge deer(x) \wedge boar(x) \wedge capercaillie(x)]\Rightarrow SmallGame(x))$.

Answer 2

(b) ”If there is a fox or rabbit, there is a small game”

$∀x~(\operatorname{fox}(x) \lor\operatorname{rabbit}(x) \to \operatorname{SmallGame}(x)) $

While fox and rabbit are examples of small game, that is not what you were asked to state.

$$\exists x~(\operatorname{fox}(x)\lor\operatorname{rabbit}(x))\to\exists x~\operatorname{SmallGame}(x)$$

(These statements are not equivalent.)

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So (d) ”If there are moose, deer, boar and capercaillie, there are big games, otherwise there are just small games”

$$\small(\exists x~\operatorname{moose}(x)\land\exists x~\operatorname{deer}(x)\land\exists x~\operatorname{boar}(x)\land\exists x~\operatorname{capercaillie}(x))\to(\exists x~\operatorname{BigGame}(x)))\land(\lnot \exists x~\operatorname{moose}(x)\lor\lnot \exists x~\operatorname{deer}(x)\lor\lnot \exists x~\operatorname{boar}(x)\lor\lnot \exists x~\operatorname{capercaillie}(x))\to(\exists x~\operatorname{SmallGame}(x)))$$