5

Given

  • $C(x)$ is “$x$ is a cat”
  • $S(x)$ is “$x$ is shy”
  • $T(x)$ is “$x$ is happy”

translate the following sentence to predicate logic

If any cat is shy, then it is not happy.

This is what I came up with

$$\forall x \left[(C(x) \land S(x)) \to \neg T(x)\right]$$

Seems correct to me but can anyone else offer some insight? Thanks!

ryang
  • 44,428
fluffy dog
  • 145
  • 5
    Your translation is a great translation of "shy cats are unhappy", which is what I think your instructor meant. However, "If any cat is shy, then it is not happy" is not clear English (the referent for "it" is unclear - any cat? or the one that's not shy? Or just some impersonal "it" as in "if any cat is shy, then it is not raining"?). So I think you should moan at your instructor, if the problem was presented in English like this. – Rob Arthan Jul 23 '16 at 00:03
  • 1
    The problem was in fact presented like this Rob! Perhaps to confuse us! Thanks for your input though, I feel more comfortable with the answer now. – fluffy dog Jul 23 '16 at 06:23
  • ... and I should have said that your instructor could (rightly) reply that he/she chose a vague English phrasing deliberately to highlight how logical notation forces you to be precise. – Rob Arthan Jul 23 '16 at 22:39
  • 2
    The word 'any' is ambiguous. the clause "If any cat is shy" could mean either "If every cat is shy" or else "If some cat is shy". – murray Jul 06 '17 at 18:48

2 Answers2

1

$\text{If $\textbf{any}$ cat is shy, then $\color\red{\text{it}}$ is not happy.}\tag*{}$

A more natural wording:

  • $\text{If a cat is shy, then it is not happy.}\quad✓\tag*{}$ $\forall x \:\Big(C(x)\land S(x)\to \lnot T(x)\Big)\tag2$

Mechanistically formalising the original sentence (under the understanding that the collocation "if any" idiomatically means "if some" rather than "if every"): $$\forall \color\red y \:\Big(\exists x \,\big(C(x)\land S(x)\land \color\red y=x\big)\to \lnot T(\color\red y)\Big).\tag1$$

Even if, as Rob Arthan remarked in the comments above, the original sentence isn't particularly clear English, its formalisation Formula $1$ turns out to be logically equivalent to Formula $2$:

\begin{align}&\forall \color\red y \:\Big(\exists x \,\big(C(x)\land S(x)\land \color\red y=x\big)\to \lnot T(\color\red y)\Big)\\ \equiv{}&\forall x \:\Big(\exists y \,\big(C(y)\land S(y)\land x=y\big)\to \lnot T(x)\Big)\\ \equiv{}&\forall x \:\Big(C(x)\land S(x)\to \lnot T(x)\Big)\end{align} (we first interchange the bound variables, then notice that $$\exists y \,\big(C(y)\land S(y)\land x=y\big)\;\equiv\; C(x)\land S(x)\quad).$$

ryang
  • 44,428
0

To translate this sentence, you can write ∀x ∈ C(x), S(x) → ¬T(x). I believe that what you have written is correct.

Discrete Math
  • 186