In Introduction to Symbolic Logic by Virginia Klenk, I am confused on a question about proving validity with Venn Diagrams. The problem is:
Everybody loves a lover.
No lovers are happy.
Therefore nobody loves a happy person.
I am confused because the chapter says that all these problems are capable of being translated into categorical propositions with only 3 terms (3 propositional functions). But I do not see how I can translate this argument in english into 3 categorical propositions of one variable because there seems to be a relation used in the argument. I have an idea of how the final result is supposed to look in terms of shading of the venn diagram to demonstrate the apparent validity of this argument but I get stuck. The solution is supposed to show three overlapping circles shaded in such a way to represent the first two statements in the argument. The circles represent classes of objects referred to in the argument. And the composite shading from the first two statements must demonstrate how the third statement follows from the first two. The book's convention is that shading represents nothing exists in the shaded region. An "x" represents the existence of at least one object in that region.
My attempt at a solution is:
Px: x is a person
Lx: x is a lover
Hx: x is happy
My venn diagram solution attemmpt
My thinking was that if everyone loves a lover, then everyone loves, which means that everyone is a lover, which I represent by the shading the Px circle. "No lovers are happy is simple enough to translate into shading. But am I justified in rephrasing that first argument in the way that I did?
Your attempt to put it in a Venn diagram doesn’t work as the diagram ends up saying nothing about who or what is being loved. Indeed, the only interesting thing you can conclude from the diagram would be that the intersection between People and Happy people is empty, i,e that no person can be a happy person.
Now, instead of having a general ‘Person’ category, you could try to have ‘people that are loved’ as a category. So then you have ‘All lovers are people that are loved. No lovers are happy people. Therefore, no happy people are loved people’. But, you will find that argument to be invalid:
The problem is, as you say, that there is s relationship involved, and that we are not looking at a normal categorical syllogism, even as it looks like one on the surface. In fact, in order to see that the original argument is valid, you really need to employ some non-trivial (and non-categorical) logic: if everyone loves a lover, then everyone becomes a lover themselves, and hence everyone becomes loved by everyone as well: the class of lovers and the class of loved people thus become one and the same. And therefore, since no lovers are happy, no loved people are happy either, i.e. no happy people are being loved.
I suppose one could do a Venn diagram with the 3 categories as I proposed, but now symbolizing the first claim not just be shading the part that is inside the ‘lovers’ circle but outside the ‘loved people’ circle … but also by shading the part inside the ‘loved people’ circle but outside the ‘lovers’ circle. And now, when adding the second premise, you will find the conclusion to be true:
But again, in doing so you would already need to have figured out that the class of ‘lovers’ and the class of ‘loved people’ coincide … and it will take more than just categorical logic to see that. So, frankly, I disagree with the book’s claim that you can see the validity of this argument through a basic Venn diagram analysis of a categorical syllogism.