Predicate Logic and Inference

first-order-logicformal-proofslogicnatural-deductionpredicate-logic

Assume that given three predicates are presented below:

$H(x)$: $x$ is a horse

$A(x)$: $x$ is an animal

$T(x,y)$: $x$ is a tail of $y$

Then, translate the following inference into an inference using predicate logic expressions and prove whether inference is valid or not (for instance, using natural deduction):

Horses are animals.

Horses' tails are tails of animals.

My thoughts: I am quite good at translating predicate logic expressions, but here I struggled to come up with formula for Horses' tails. My initial idea was to consider similar sentence such as "w is a tail of a horse" to form required inference, but it was not successful. Would be welcomed to hear your ideas about this task.

Best Answer

Hints:

"$x$ is a $P$'s tail" means that $x$ is a tail of $y$ and $y$ is a $P$.

"Horses' tails are tails of animals" means that for all tails $x$ and tail-bearers $y$, the tail being a horse's tail implies the tail being an animal's tail (where for "being a $P$'s tail" insert the above definition).

With the appropriate formalization of this paraphrase, it is possible to find a formal proof of the inference.

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