# Translate the following into predicate calculus. State assumed universe of discourse

This is my first assignment on these, so I would greatly appreciate your help.

Translate the following into predicate calculus. For each answer, also state the assumed universe of discourse.

a) "Anyone who was an ancient Roman citizen and tried to kill Caesar was not loyal to Caesar."

My attempt:

Ex(R(x) & k(x)) -> ~L(x)

R(x) = x is a Roman citizen, k(x) = x is tried to kill Caesar, L(x) = x is loyal to Caesar

b) "All cats which are calico, are female."

My attempt:

Ax(C(x) & O(x)) -> F(x)

C(x) = x is a cat, O(x) = x is calico, F(x) = x is female

c) "Some Texans have never left the state of Texas."

My attempt:

~Ax(T(x) & L(x))

T(x) = x is a Texan, L(x) = x is left Texas

Is the universe of discourse kind of like the "key" for the variables, as I included?

• By "Ex" and "Ax" do you mean the quantifiers $\forall x$, $\exists x$? Feb 9 '16 at 4:24
• @DanSimon it might be helpful to mention that these can be typeset with \exists and \forall respectively, between dollar signs, e.g., $\exists x (R(x) \land k(x) \implies \neg L(x))$ is produced by $\exists x (R(x) \land k(x) \implies \neg L(x))$. Feb 9 '16 at 4:43
• Yes, that's what I meant. Thank you for the tip, @pjs36. I tried looking up "Ajax code" on Google to find a list of symbols & their associated code--as I knew Ajax was the preferred syntax on here--but I couldn't find one. Do you know of one? Feb 9 '16 at 8:01
• @wad11656 Yeah, the Mathjax tutorial thread is here, and bullet point 12 contains a few special characters, and links to many, many more. Note that Mathjax is $\LaTeX$-based, so searching for latex symbols is the more standard thing to look up. Feb 10 '16 at 0:26

## 1 Answer

a), b) are correct, although I don't see that you have stated the universe of discourse for any of the three sentences.

However, but c) is not correct. The sentence you have for c) is close but not quite right: moving the negation inward shows that it's equivalent to: \begin{align} \neg\forall x\,(T(x)\land L(x)) &\equiv \exists x\,\neg\,(T(x)\land L(x)) \\ &\equiv \exists x\,(T(x)\to \neg L(x)) \\ &\equiv \exists x\,(\neg T(x)\lor \neg L(x)), \end{align} which says "Some people either aren't Texans or have never left Texas", assuming $x$ ranges over only people (i.e. the universe consists only of people).

c) should be: $$\exists x\, (T(x) \land \neg L(x)),$$ where $T(x) := \text{$x$is a Texan}$, $L(x) := \text{$x$has left Texas (at some point)}$.

The universe of discourse for c) is all people (all living people, let's say).

In a), your predicates seem to be set in the present tense, and you do filter for "is a(n ancient) Roman citizen", so your universe of discourse can be all contemporaries of Caesar. In fact, it could even be *anything contemporaneous with Caesar", since presumably contemporaneous cats, kitchen utensils, and so on were not Roman citizens.

In b), few restrictions apply. $x$ can range over everything — cats, dogs, people, hydrogen atoms, the integers, etc. — and the statement remains true because $C(x)$ is in the antecedent.

• I didn't include the universe of discourse because I didn't know what it was. It means whatever x is/can be defined as? Is it flexible? So how's this: Universe of Discourse: a) Living people during Caesar's life. b) Cats. c) All living people Feb 9 '16 at 8:13
• Yes: the universe is all the possible values that variables range over. In this problem, it's flexible, because you're the one coming up with both the formal sentence and the universe in which it's to be true. Those all work, but b) is overly restrictive. E.g. in b), if the universe is already only all cats, then you don't need a "$C(x)$" predicate. Better the universe be all living things, or better still every object, animate or inanimate (even both physical and abstract, if you like). But not just cats. Feb 9 '16 at 12:19
• Thank you for the explanation!! Feb 10 '16 at 2:10
• You very welcome. Feb 10 '16 at 2:11