"I have found a dead body on my car." Given a statement "I have found a dead body on my car", and considering the fact that I do not own any car, is this statement true?
If so, is this a special case of false implies anything?
 A: Well, Roy seemed to disappear, so I'll add my comment as an answer: Your statement is just false if you include as true the fact that "you do not own any car". No further implications seem important
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A: The statement could be true or false.
You HAVE found a dead body on MY car
implies you found it in the past.  The fact that you do not own a car only implies to be true NOW.
So without more information we can not presume anything.  My next question would be "Have you ever owned a car?" and then "When did you find the body?".
A: Ignoring matters of ambiguity in natural language (since this does not seem relevant to what you are asking), your sentence could be rephrased as:
$$\text{'I have found a dead body on a car that I own'}$$
where 'a car that I own' is an indefinite description according to Russell's theory of descriptions. The whole sentence may be formalised as follows:
$$\exists x,y : \mathrm{car}(x) \wedge \mathrm{body}(y) \wedge \mathrm{dead}(y) \wedge \mathrm{foundOnTopOf}(i,y,x) \wedge \mathrm{owns}(i,x)$$
If you don't own a car, then the statement is false, since you are in part asserting that there exists a car that is yours.
A: I would take a different approach and say you can not determine if this statement is false or true by the information provided.
While you do not OWN a car, you could have a rental car (so not owned) but still would qualify as being "my car".  
A: There is no implication in your statement. A natural translation into first-order logic would be something like "There exist a car and a person such that: the car is mine and the person is dead and the person is on top of the car."
If you do not own a car, then the statement is false since "the car is mine" is always false.
A: If you do not own any cars, the following statement will be true: "I have found a dead body on each of my cars."
A: In English "my car" can mean any car you are currently using, whether it be borrowed, rented, stolen, or a chauffeured vehicle like a taxi or limousine. Moreover, "car" can also mean "truck" or "SUV" etc. Therefore we cannot determine if the statement is false just because we are told that you "do not own any car"; although the word "any" would strongly imply that trucks and SUVs are off the plate, cars you don't own can still be "your car" in many contexts.
A: I think the statement that acts the way you are thinking is the statement "If I have a car, then I found a dead body on it." This statement is a case of false implies anything is true.
A: Is true the statement?.
Answer:

STATEMENT equals CAR implies FOUND-DEADBODY

If you own a car then you found a dead body, hence it is true even if you don't own a car
or you could say

STATEMENT equals  FOUND-DEADBODY and OWN-A-CAR

However, you found a dead body only if you own a car and since you don't own one then, the statement is false
I.e. the solution is how you interpret the statement.
A: "considering the fact that I do not own any car" is a red herring -- people use the phrase "my car" or "my house" all the time and none of them require actual ownership.
