# Horses have heads Symbolic Logic

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In summary, the conversation is about representing the sentence "Horses have heads" using first-order predicate calculus. Two possible formulas were discussed, but it was determined that the second formula is not correct because it states that a single horse has all heads. Instead, the correct formula would be (Ax)[HORSEx --> (Ey)[HEADy & HASxy]]. The conversation also clarified that HORSE and HEAD are one-place predicates and HAS is a two-place predicate.
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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"

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

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.

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.

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".

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]].

## 1. What is the symbolic logic behind the statement "Horses have heads"?

The symbolic logic behind this statement is a simple categorical syllogism, which follows the form of "All A are B". In this case, the categorical statement is "All horses are animals", and the symbolic logic representation would be "A --> B".

## 2. How does the statement "Horses have heads" relate to the Law of Identity in symbolic logic?

The statement "Horses have heads" directly relates to the Law of Identity in symbolic logic, which states that "A is A". In this case, "horses" is the subject and "heads" is the predicate, and the statement affirms that the subject (horses) is identical to itself (having heads).

## 3. Can the statement "Horses have heads" be translated into a logical proposition?

Yes, the statement "Horses have heads" can be translated into a logical proposition using the form of "All A are B" or "A --> B". This proposition can then be analyzed using logical rules and principles.

## 4. Is the statement "Horses have heads" a valid argument in symbolic logic?

No, the statement "Horses have heads" is not a valid argument in symbolic logic because it only presents a single categorical statement and does not include a logical conclusion.

## 5. How does the statement "Horses have heads" illustrate the use of symbolic logic in everyday life?

The statement "Horses have heads" may seem like a simple and obvious statement, but it actually illustrates the use of symbolic logic in everyday life. By representing the statement in symbolic logic, we can analyze its validity and determine if it follows logical rules. This can also be applied to other statements and arguments in real-life situations.

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