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I have searched for answers/help, but I am not able to find specifics. I am on a "FOL for Dummies" level, I really have no clue what I'm doing. Edit: I understand most of the symbols (∀x, the backwards E meaning there exists), I just don't know how to tie them together conclusively. I understand (barely) truth tables. Very basic knowledge of the information.

Having trouble with first order translations.

Assume the universe is all human beings.

(a). Some citizens are unhappy in countries ruled by dictators.

(b). No college dean is greedy.

Attempt: For this, I have ∀x (x is a college dean -> ~(x is greedy)), but I don't think that is what I am looking for.

(c). All freshmen that drink regularly never make A's.

Attempt: x= Freshman, P(x)=drinking, z=Making A's. (∀xP(x)) -> ~Z

AND

Consider the following formulas:

(a). [(∀xPx) v (∀xRx)] -> ∀x(Px v Rx) and

(b). [∀x(Px v Rx)] -> [(∀xPx) v (∀xRx)]

Which one is true in any model? Give a model where the other one is false.

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Provide a few of your attempts at these problems. That will provide clues for ways to help you. –  Jay Apr 30 '12 at 22:50
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1 Answer

I'll get you started by working one problem of each type. In the first set I'll do (c): All freshmen that drink regularly never make A's. This sentence involves three predicates: is a freshman, drinks regularly, and never makes A's. I'll denote these by $F$, $D$, and $N$, respectively. That is, $Fx$ means that $x$ is a freshman, while $Nx$ means that $x$ never gets A's.

Clearly the sentence is a statement that something is true of all freshmen, so a good start is $$\forall x(Fx\to\text{ something})\;:$$

for all $x$, if $x$ is a freshman, then $\text{something}$ is true of $x$.

But that isn't quite right: the sentence is really a statement about all freshmen who drink regularly, so we add that specification: $$\forall x(Fx\land Dx\to\text{ something})$$

for all $x$, if $x$ is a freshman and $x$ drinks regularly, then $\text{something}$ is true of $x$.

Once you get this far, it's easy: the $\text{something}$ is clearly $Nx$, i.e., $x$ never gets A's, and we have $$\forall x(Fx\land Dx\to Nx)\;.\tag{1}$$

The trick is to recast the sentence in ways that better match the various logical connectives and quantifiers.

By the way, $(1)$ isn't the only possible answer, though it's probably the intended one. You might think about why $$\forall x\big(Fx\to(Dx\to Nx)\big)$$ is also a legitimate translation.


In the second part let's look at (b). Suppose that $Px$ means '$x$ is at least $20$ years old', and $Rx$ means '$x$ is less than $20$ years old'. Then $Px\lor Rx$ means '$x$ is at least $20$ years old or $x$ is less than $20$ years old (or both)', so $\forall x(Px\lor Rx)$ simply says that everyone is at least $20$ years old or less than $20$ years old (or both). That's pretty clearly true. Now what does $$\forall xPx\lor\forall xRx$$ say in plain English? Is it true in the real world?


Added: Your attempt at (b) in the first part is fine, though you should probably finish the job by introducing symbols for the two predicates, e.g., $\forall x(Dx\to\lnot Gx)$.

Sentence (a) in that part is difficult to translate, because it's a little ambiguous in English. Probably the most reasonable interpretation, however, is that there are countries ruled by dictators, and some of the citizens of those countries are unhappy. You came pretty close to formalizing this interpretation, but you need to quantify over the countries as well as over the citizens. Using your predicates from the comment, I'd make it $$\exists x\exists y(Dx\land Pyx\land Uy)\;.$$

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Thank you! For the first example (Some citizens are unhappy in countries ruled by dictators), could it be: Pxy = x is a citizen of country y Ux = x is unhappy Dy = y is a country ruled by a dictator Ex(Pxy ∧ Dy -> Ux)? Sorry for formatting! –  Caley Apr 30 '12 at 23:01
    
And for the second part, would the second equation be saying that everybody is at least 20 years old or everybody is less than 20, but NOT both? –  Caley Apr 30 '12 at 23:06
    
@Caley (second comment): Simple English is a little a ambiguous here, so you might or might not be saying the right thing. (I suspect that your are.) It means that at least one of the following two things is true: (1) everyone is at least $20$ years old; (2) everyone is less than $20$ years old. –  Brian M. Scott Apr 30 '12 at 23:10
    
Am I understanding correctly: it is false in the real world because it is not true that everyone is at least 20 (because they do not both have to be true)? Thank you so much for your help--it is really, really appreciated! –  Caley Apr 30 '12 at 23:13
    
@Caley: That's right: it's false in the real world, because if it were true, either there would be no children, or there would be no elderly! –  Brian M. Scott Apr 30 '12 at 23:14
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