# Why does not 'the x dislikes the y and the y is a fox' = 'the x dislikes the fox'?

Source: Understanding Semantics (2 ed, 2013) by Sebastian Löbner

[p 333:] In the following, the Individual Variables are: x, y.
The 1-place Predicate Constant is: fox. The 2-place Predicate Constant is: dislike.

[pp 335-336:] It is important to note that free variables are interpreted on a par with individual constants. They stand for a particular individual, functioning roughly like ‘the x’. Existentially bound variables, however, achieve the reading of ‘some x’. Therefore, steps 4 and 5 are interpreted as follows:

The last paragraph above asserts that in 14a, the blue cannot replace the red.
But why not? Why cannot [14.1] be rewritten as [14.4] below?

[14.4] the x dislikes the fox named y.

• You need to give us the natural language phrase which the formal analysis is notating. Commented May 28, 2016 at 5:55
• @curiousdanii Does my edit resolve the problem?
– user5306
Commented May 28, 2016 at 6:07
• Does the book not actually give a natural language sentence behind 14a and b? Commented May 28, 2016 at 6:59
• On your edit: because y is not a name. Commented Jun 13, 2016 at 7:45

I'm not sure I entirely got the point (I find it weird to say that the variable can be eliminated in natural language, because 1. you don't explicitly use variables in natural language anyway and 2. "there is some fox..." is not a variable, so I find that wording misleading), but I think it's important for you to have the the basic definitions in mind (sorry for the rudimentary formatting and sloppy wording):

• [[P(x)]]^g is true iff g(x) is in the denotation of P
• [[ExP(x)]]^g is true iff there is an x-alternative g' of g such that g'(x) is in the denotation of P

where g is an assignment function mapping from variables to individuals in the domain, and an x-alternative of g is a variable assignment that is just like g except for possibly the value it assigns to x.

This means that "fox(y)" without a quantifier is true only if the current variable assignment happens to assign to y an individual which is in the denotation of the predicate "fox" in the current model, i.e. if THAT VERY y stands for an individual which is a fox (and is disliked by x). Since we stick to one particular variable assignment and don't quantify over other possible variable assingments, the free variable y behaves just like an individual constant.

On the other hand, with the existential quantifier, the formlula is true if there is SOME individual in the domain which is able to fulfill the fox predicate, i.e. if there is some way of mapping the variable y to some element of the actual world which is also a fox in the actual world, i.e. if there is at least one fox (and that fox is disliked by x).

What I think the paragprah wants to say is that you don't need to explicitly spell out the quantifier in natural language (i.e. not necessarily "There is some y that is a fox and x dislikes it" but rather "x dislikes a fox"), while you can't eliminate the constant (you need to say "x dislikes THE FOX", where "the fox" is referring to the denotation of g(y), the sentence behaves just like "x dislikes John", where "John" also refers to an individual constant).

This is also why in the first example (without the quantifier), you have the definite article "the", because it refers to a single individual, while the second example (with the quantifier) ranges over possibly several individuals where you need to use the indefinte article "a" (which obvisoulsy in natural language usually has the meaning of "only one", which with the existential quantifier would have to be stated explicitly though; and of course, the set of foxes disliked by x might happen to be a singleton, but the logical type is still a different one).

Regarding your comment: "the fox named y" doesn't make sense anyway. y is not the name of the fox, but a variable we use in the formalisation of the sentence in order to represent the logical entities involved and map placeholders for individuals to their "real-life counterparts" of our domain. "y" behaves somewhat like a name in the formalisation of a sentence in referring to a certain individual, but it is certainly not the name of the actual fox in the extension the sentence the notes. You need to be carful with mixing up different language levels.

I hope this at least in some aspect an answer to your question; as I said, I'm not sure I got the point of it without more context and am little confused by the wording.

The complex sentence without quantifiers cannot be replaced by "x dislikes the fox" because the article "the" (and the logic of the short sentence) would implicitly say that there only exists a single particular "fox". But that isn't necessarily the case. y was a free variable and it must be possible to incorporate it in many ways; while "the fox" is a name for a particular entity, so it can't be embedded in sentences with quantifiers.

For example, you may add both "for every y" or "there exists some y" in front of the sentence "the x dislikes y and y is a fox" – but not in front of "the x dislikes the fox". Only with the existential quantifier, you may simplify "there exists y such that y is a fox and..." by "there exists a fox that...", other propositions cannot be simplified.