Horses have heads Symbolic Logic

"Horses have heads" Symbolic Logic

I was given this sentence to represent in first-order predicate calculus.
The formula must use the following terms--horse, has, head--where:

"horse" represents "x is a horse"
"has" represents "x has a head"
"head" represents "x is a head"


Are these possibilities?
1) (x)(horsex-->hasxhead) which means(?) "For all x, if x is a horse then x has a
head"

2) (x)(y)((horsex & heady)-->hasxy) which means(?) "For all x and for all y, if x is a horse and y is a head, then x has y"


If not, how can "Horses have heads" be represented using these specification? Thank you.
 

HallsofIvy

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The second is not true- a horse does not have all heads!
2) says "every horse has every head".
The first looks to me like a correct statement.
 
Thank you. I am not positive that (1) correctly represents the sentence.

Regarding (2): you said the formula is stating "a horse has all heads" Does it really?
If so, I don't understand how it says that. If it's saying a single horse has all entities (which is plural) that are heads, why wouldn't it say "all horses have all heads". That is, I don't understand why it would say a single horse has multiple heads as opposed to saying multiple, i.e. all, horses have multiple, i.e. all, heads.
I suppose i'm asking if you can explain how (2) says what you claimed; how (2) is incorrect. Thank you.
 

HallsofIvy

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You said:
2) (x)(y)((horsex & heady)-->hasxy) which means(?) "For all x and for all y, if x is a horse and y is a head, then x has y"
Okay, here is a horse, x, standing just over that fence, and here is a head, y, between my shoulders. Does x have y? You did say "for all x and for all y".
 
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Just to get things straight, HORSE and HEAD are one-place predicates, and HAS is a two-place predicate, right?

I think what you want is (Ax)[HORSEx --> (Ey)[HEADy & HASxy]].
 

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