Translation logic to English

Given these conditions... $P(x) = x$ is a cow, $Q(x) = x$ makes milk, $R(x,y) =$ both $x$ and $y$ are the same object.

This expression says the following.. $$(\exists x)[P(x) \wedge Q(x)]$$ and another $$(\exists x)[P(x)] \wedge (\exists x)[Q(x)]$$

I translated those to the following... 1. Some cows produce milk. 2. There are some cows that produce milk.

Are these correct?

• Your $R(x,y)$ is never used in your question. Is this correct? Commented Feb 1, 2015 at 20:58
• Second one is wrong. It should be - There are some cows, and there are some creatures that produce milk. You cannot infer from this logical statement that those creatures are indeed cows. Commented Feb 1, 2015 at 21:04
• Thank you Dillion I'm new to this Commented Feb 1, 2015 at 21:16
• @MD_90 What do you mean the course doesn't allow for that symbol? The symbol $\exists !$? Do you know what it means? Commented Feb 1, 2015 at 21:16
• Sadly I don't @induktio :( the symbol was not covered in class therefore they don't allow us to use it in the expression notation Commented Feb 1, 2015 at 21:18

I would answer it as follows:

$$(\exists x)[P(x)\land Q(x)]$$

means, in the context of your statements, that (note this is a very strict interpretation) there exists a cow that makes milk (maybe not more than one but perhaps).

Now,

$$(\exists)[P(x)]\land (\exists)[Q(x)]$$

means that there exists a cow and there exists a milk maker.

The first is basically correct, though I would prefer "A cow makes milk." Your formulation implies that there are multiple cows, i.e. at least two cows, that make milk, while the logic statement says at least one cow. So I prefer my formulation. Perhaps "There is at least one cow that makes milk" is the most clear, though somewhat clumsy.

For (2): There is a cow and something makes milk.

• Is the first one really correct though? Some cows is a lot different than a cow, of course. Nitpicky, but I think it's more than a matter of preference in this context. Commented Feb 1, 2015 at 20:59
• @induktio: It is largely a matter of preference. But the OP's formulation implies that there are multiple cows, i.e. at least two cows, that make milk, while the logic statement says at least one cow. That's why I prefer my formulation. I have edited my answer to make that more clear. Commented Feb 1, 2015 at 21:00
• Interesting. I guess I take the $\exists$ to be very literal in the sense that it does not guarantee more than one cow (even though it obviously shouldn't be interpreted as $\exists !$). Commented Feb 1, 2015 at 21:03
• I'm currently taking a Discrete Mathematics course for my computer science major. It's hard for me to see how to translate between the expression form, and English form. I am trying though but any insight that will help makes these easier would be very helpful :) Commented Feb 1, 2015 at 21:13
• course doesn't allow that symbol @induktio Commented Feb 1, 2015 at 21:14