Rewrite in predicate calculus

Question

Represent the following sentences in predicate calculus:

1. If a computer passes Turing test, then it is intelligent.
2. Everybody needs somebody and is needed by somebody.

Answer 1

Introduce a new symbol for all the concepts involved in the questions: e.g. introduce unary predicate symbols for 'being computer', 'passes Turing test' and 'is intelligent'. Then formalize:

"For all $x$, if $x$ is computer and passes Turing test, then it is intelligent."

...

$\forall x \left(\,(C(x)\land T(x))\to I(x)\,\right)$

Similarly, for 2. we need a binary predicate symbol of '$X$ needs $Y$' (perhaps besides the unary predicate symbol 'body').

Answer 2

(2): Everybody needs somebody and is needed by somebody.

• let's use the unary predicate $P(x):$ "x is a person" to stand for the "body" in "everybody" and "somebody".

• Put $N(x, y):$ to denote "x needs y".

Then we want to say the following:

For all $x\,\Big($ if $x$ is a person, then [there exists $y$ such that $y$ is a person and $x$ needs $y$, and there exists $z$ such that $z$ is a person and $z$ needs $x]\Big)$.

Spoiler:

$$\forall x \Big(P(x) \rightarrow [\exists y(P(y) \land N(x, y)) \land \exists z( P(z) \land N(z, x))]\Big)$$