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In the book Discrete Mathematical Structures with Applications to Computer Science by J.P. Tremblay and R. Manohar, there's an example question in the section on Predicate Calculus.

Symbolize the expression "All the world loves a lover".

My solution is:

Rewrite it in the logical form as:

Given any $x$ and $y$, if $x$ is in the world and $y$ is a lover, then $x$ loves $y$.

Then denoting predicates as:

$W(x)$ : $x$ is in the world
$L(x)$ : $x$ is a lover
$R(x,y)$ : $x$ loves y

the above statement could be written as:

$(x)(y)((W(x) \wedge L(y)) \to R(x,y)) $

Is this approach and solution correct? If so, is this solution unique, or are there other "correct" solutions?

I am self-learning Discrete Mathematics using this book, and do not have any "professor" or "mentor" to seek guidance. The below solution given in the book confused me due to the reason mentioned further down, so I am hoping to get some guidance here.

Book solution

$(x)(P(x)\to (y)(P(y) \wedge L(y) \to R(x,y))$

wherein:

$P(x)$ : $x$ is a person
$L(x)$ : $x$ is a lover
$R(x,y)$ : $x$ loves $y$

I hate to put it this way, but I am not convinced the book is right. Logical form of the statement shouldn't be biased by real world experiences. It is nowhere given or implied that "all the world" refers to people only (for example, it could include computers, robots, animals, plants, aliens - logic doesn't care), and logic-wise, it is even less convincing to assume that "a lover is (necessarily) a person".

Masked Man
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    I agree. The authors are inflicting their anthropomorphic biases on others. – Prahlad Vaidyanathan Dec 30 '13 at 10:37
  • I disagree with your critique of the book. Switching between natural language and formal language is very important for mathematics and this is a good exercise as long as it is used as motivation and not as an exam exercise with an expected answer. However, I am very disappointed that the book did not try to express "lover" with the "x loves y" relation which seemed to me the whole point of the exercise. So I suspect that the copied the question from somewhere else. – Phira Dec 30 '13 at 21:06
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    I have voted to close because the main point of the question IS opinion-based. Learning formal logic notation SHOULD teach the connection between formal expressions and natural language as this exercise tries to do. I am aware that other people will disagree. This makes it "opinion-based" and "likely to incite discussion". – Phira Dec 30 '13 at 21:09
  • @Phira I only wanted to know the right method, if there is one, to solve such questions, because I do have to answer an exam. There could be multiple correct solutions is good enough of an answer for me. – Masked Man Dec 31 '13 at 02:15
  • @Happy Yes, there are multiple correct solutions and there is not one right method. Despite my close-vote, I sincerely hope that the answer and the comments have been helpful to you. – Phira Dec 31 '13 at 09:58
  • @Phira I have reframed the question to make it less "opinion based" (I hope). – Masked Man Dec 31 '13 at 10:57

1 Answers1

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Formal logic does not (or should not) depend on knowledge of the real world.

Natural language intrinsically does depend on a lot of knowledge of the real world in order to be understood.

When you're translating a natural-language claim to logic, it is part of the task to identify world knowledge that the original claim implicitly depends on, and decide on a reasonable way to represent that explicitly in the logical formalism.

Threre's not necessarily a unique correct way to do this.

hmakholm left over Monica
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  • Yeah I agree, but it is rather odd in the book because they wrote two paragraphs on the previous page explaining how the truth value of a statement shouldn't depend on our real world experience and so on, and then go on to do exactly that multiple times in the examples, so I got thoroughly confused as to whether their symbolization is indeed the only "right" way of doing it. – Masked Man Dec 30 '13 at 12:31
  • @Happy But aren't you also injecting your own experiences, by presuming that "the world" consists of things, and that something is true of "all the world" precisely when it is true of each thing within the world? – Andrew Dudzik Dec 30 '13 at 13:10
  • @User-33433 No, I don't think so, I worded it that way only to show one counterexample why I believe the authors' interpretation is incorrect. My statement says "given any x, if x is in the world". If the world doesn't consist of anything, then the antecedent of the implication would be false, and the statement would still be true. Note also that I could have posted only my answer and skipped the book answer entirely, in which case, I would have avoided making any reference to the world consisting of things. – Masked Man Dec 30 '13 at 15:02
  • @Happy You are assuming that the world behaves like the topos of sets, in which the vanishing of a sheaf may be checked on elements. But this fails in many other category—"is zero everywhere" and "is zero at each point" may not coincide. (an extreme example would be the geometry of pointless spaces...) – Andrew Dudzik Dec 31 '13 at 00:22