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I have it as:

((x)(Px&Lxx))>(($y)(x)(Py&(Pz>Lyz))

Where P is a philosopher L is loves

(Predicate logic)

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  • Sorry brackets might be in the wrong places
    – JayneEyre
    Commented May 10, 2015 at 16:21
  • Not enough detail of self effort Commented May 10, 2015 at 17:28
  • How about : (there exists x) (for all y) xLy --> xLx . Here, xLy is the "x loves y" relation and x, y are philosophers. The statement seems to be tautological.
    – nwr
    Commented May 10, 2015 at 18:13
  • 2
    "They love themselves" in modern English should mean "Each loves himself", or "The group of philosophers loves itself" or "Each member of the group of philosophers loves all of the others", because the second 'philosophers' is a more immediate antecedent. But the comma isolates the pronoun from its logical antecedent, leading us to guess that "they" property binds to the first instance of 'philosophers' and the implication might lie within the quantification. So this can be a bunch of different logical statements, none quite clear.
    – user9166
    Commented May 10, 2015 at 18:44
  • 1
    If your question is incorrect, then edit it.
    – user9166
    Commented May 11, 2015 at 1:10

4 Answers 4

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I wouldn't even try to translate it, because it is grammatically unclear, and the meaning of the word "they" is unclear, so I wouldn't think I can make any reasonable assumption what it is supposed to mean.

Two reasonable ways to clarify: "Some philosophers love all those philosophers who love themselves, but not those who don't love themselves" and "Some philosophers love all philosophers, but any philosopher can only love all philosophers if that particular philosopher loves himself or herself".

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Let us reformulate the proposition first for the clarity's sake:

  1. Some philosophers love all philosophers, only if they love themselves
  2. If any philosopher loves herself, then there exists at least one philosopher who would love her too.

Let P(x) = x is a philosopher and L(xy) = x loves y.

Then it seems that we have:

(x)([P(x) -> L(xx)] -> ∃y [P(y) & L(yx)])

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  • apparently this assumes a single specific meaning where many meanings are possible? Commented May 10, 2015 at 23:53
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So in the comment you change your question. Now your question is how to formalize:

(1) Some philosophers love all philosophers only if some philosophers love themselves.

In fact this is also ambiguous, it has two interpretations:

(2a) (Some philosophers love all philosophers) only if (some philosophers love themselves)

or

(2b) Some philosophers are such that ((they love all philosophers) only if (some philosophers love themselves))

In the first interpretation, the implication has wider scope.

These are not equivalent. It's easy to see that (2a) is a logically true: Assume that some philosophers love all philosophers, therefore they love themselves, therefore some philosophers love themselves. On the other hand (2b) is not logically true: just take a model where there are no philosophers, now (2b) is false.

Little more formally (2a) and (2b) are (assuming that "some philosophers" means "there exists at least one philosopher"):

(3a) (There exists at least one philosopher who loves all philosophers) only if (there exists at least one philosopher who loves itself)

(3b) There exists at leas one philosopher such that ((it loves all philosophers) only if (there exists at least one philosopher who loves itself)

Formalizing these:

(3a)* {Ex[P(x) & (y)(P(y)-->L(x,y))]}-->{Ex(P(x) & L(x,x))}

(3b)* Ex{(P(x)) & [(y)(P(y)-->L(x,y))-->Ez(P(z) & L(z,z))]}

If we had taken "some philosophers" to mean that "there exist at least two philosophers" this would have to be modified.

In speech those two interpretations of (1) could be represented by different speech patterns: e.g. in the first interpretation you put a long pause before and after the "only if", in the second interpretation you put a long pause after the first "some philosophers".

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I'm not going to try translating the formula; as there have been good efforts by others; but I want to point out that some could be formalised by what is called a plural quantifier; and this might allow keeping some of the ambiguities in the phrase; which after all, it can be argued, is part of its meaning

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