# The expression “otherwise” in predicate logic

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:


(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) )

• What is the intended natural language interpretation of e.g. $\operatorname{fox}(x)$? – Hagen von Eitzen Apr 30 at 12:55
• fox(x) is read as "there is a fox", I suppose. – Shinichi Takagi Apr 30 at 18:17

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

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

• Thank you! It was simpler than I thought it would be. – Shinichi Takagi Apr 30 at 12:43

(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.)

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

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

• "As long as either fox or rabbit is there, you can always say that there is at least one small game" That was my understanding, and in that case ∀ is to be used, I thought. Could you explain how I misunderstood? – Shinichi Takagi May 2 at 9:54
• You can only say that because you are have extra information (that foxes and rabbits are examples of small game) that is not given in the sentence you were translating. Try to translate the sentence "If there are wolves, there is small game," without making the claim that all wolves are small game. @ShinichiTakagi – Graham Kemp May 2 at 10:13