Predicate Calculus English Translation

Question

I'm having difficulty translating the following English sentences into predicate logic. Any help would be greatly appreciated.

$B:\qquad$_ is a book
$A:\qquad$_ is an author
$H:\qquad$_ is a publishing house (a publisher)
W:$\qquad$_ is one (possibly the only) author of book _
P:$\qquad$_ publishes _
a:$\qquad$author Alex
s:$\qquad$author Sonia
c:$\qquad$publisher Carlson

a. Every author has written some book.
b. Every book has at least one author.
c. Some author has no book published.
d. Some publisher publishes all the books of at least one author.
e. Every author has at least one book published by Carlson.
f. No publisher publishes a book by Alex and a book by Sonia.
g. Carlson publishes a book whose authors include Alex and Sonia.

For a) I have:

$$ \forall x(xA \land \exists y(yB \land xyW)) $$

and for b) I have:

$$ \forall x(xB \land \exists y(yA \land yxW)) $$

but am having trouble with the rest. I think c) involves 3 variables somehow but I don't know how to put them together. Any help would be greatly appreciated.

Thanks.

Answer 1

For a) and b), see comments above :

"all men are mortal" is "formalized" as : $\forall x (Man(x) \rightarrow Mortal(x))$

and:

"there is a number that is greater than zero" as : $\exists x (Number(x) \land Greater-than-zero(x))$.

For c): "Some author has no book published", we proceed by steps :

i) "an author has written a book" is $(A(x) \land B(y) \land xWy)$

ii) "a book is published" is $(B(y) \land H(z) \land zPy)$

Now :

$\exists x [A(x) \land \forall y (B(y) \rightarrow \lnot xWy) \lor (\exists y B(y) \land xWy \land \forall z (H(z) \rightarrow \lnot zPy))]$.

For d): "Some publisher publishes all the books of at least one author" , we may try with :

$\exists x \exists y [P(x) \land A(y) \land \forall z ((B(z) \land yWz) \rightarrow xPz)]$.

For e): "Every author has at least one book published by Carlson", we have :

$\forall x \exists y ((A(x) \land B(y) \land xWy) \rightarrow cPy)$.

And so on ...