Universal Instantiation in logic

Question

I have a simple question.

Suppose we have a universally quantified statement $\forall xP(x)$. And we want to use universal instantiation rule.

And when we instantiate $P(c)$, this $c$ must be specific or can also be arbitrary?

I am asking this because in discrete mathematics book I have read that for Universal Instantiation, that $c$ is a particular member of the domain

Fragment from the Discrete Mathematics and Its Applications by Kenneth Rosen (on the first page of the fragment)

I have also an example. Could you help me how to use rules of inference here

“All hummingbirds are richly colored.” - $\forall x(H(x)→R(x))$;

“No large birds live on honey.” - $\forall x(L(x)→¬Y(x))$;

“Birds that do not live on honey are dull in color.” - $\forall x(¬Y(x)→¬R(x))$;

to prove this:

“Hummingbirds are small.” - $\forall x(H(x)→¬L(x))$

(domain is all the birds)

We should write universally quantified statements for those propositions. However, then to infer the conclusion I don't see any way to use universal instantiation.

Answer 1

When you apply Universal Instantiation, the placeholder $c$ that is introduced can be any particular member of the discourse universe. So, this $c$ can be arbitrary, and it also can be as specific as you want.

On the other hand, Universal Introduction can be applied only when the $c$ is an arbitrary member of the discourse universe.

Answer 2

When you instantiate a universal statement $\forall x P(x)$ to obtain $P(c)$ on a new line of proof, it is true that $c$ is a particular member of the domain. However, you could also perform instantiation with $d, e, f$ or any particular member of the domain to obtain statements like $P(d), P(e), P(f), ... $. In other words, the choice of the specific member you instantiate with is completely arbitrary, but $c$ itself is not arbitrary.

This becomes more clear when we consider the meaning of $\forall x P(x)$, which states

"For every $x$ in the domain, $x$ is $P$."

The universal quantifier $\forall$ implies that the formula $P$ is true for each and every member $x$ of the domain. If this is indeed true, then certainly I can infer $P(a)$ or $P(b)$ or $P(c)$ on any line of proof. In fact, you can think of the universal statement $\forall x P(x)$ as a concise way of writing a long chain of conjunctions

$$P(a) \wedge P(b) \wedge P(c) \wedge P(d) \text{ } \wedge \text{ } ...$$

where the domain $D = \{a,b,c,d,...\}$. Hence, when you apply universal instantiation, you are actually applying the inference rule known as conjunction elimination $\wedge E$ to obtain any of the above conjuncts on a new line of proof.

Answer 3

There are two schools of thought on this matter:

(1) In mathematical proofs, each universal quantifier is restricted to a specified domain, e.g. the set of real numbers. Different quantifiers even within the same statement may be restricted to different domains. These domains may be empty. Single free variables or expressions with multiple free variables (e.g. $x+y$) may be specified provided no new variables are introduced.

(2) In more philosophically oriented proofs (e.g. in standard FOL), every quantifier in a given proof is assumed to be restricted to the same non-empty, often unnamed domain. Every free variable is assumed to be an element of that domain. There is no restriction on the free variables that can be specified using universal instantiation.