Universal instantiation allows us to infer an instance of the object,how can we ensure that the object is arbitrary,please give a mathematical example,example of people satisfies the above statement.But when we call people by a name ,are we still preserving arbitrariness?
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$\begingroup$An "arbitrary" object is an object of the domain for which nothing is presupposed.
If we choose a number $n$ whatever and we prove that $P(n)$ holds, then we are licensed to deduce: $\forall x P(x)$.
If we choose a multiple of four and we prove that it is even, we cannot deduce that every number is even, because in the purported proof we have to use some fact that is not a property of a number whatever.
Another simple example of the fallacy is: Socrates is a Philosopher. Therefore, every Greek is a Philosopher.
$\endgroup$ $\begingroup$First, it isn't universal instantiation that requires us to work with a notion of arbitrary instance, but the inference in the reverse direction, universal generalization or universal quantifier introduction.
To amplify @MauroAllegranza's answer on that just a bit, think about the following argument:
Everyone likes pizza. Whoever likes pizza likes ice cream. So everyone likes ice cream.
Obviously valid! But how can we derive the universally generalized conclusion from the premisses with an informal proof? The trick is to consider an arbitrary representative from the domain.
Regimenting the premisses using informal quantier prefixes, the idea is that we can argue like this:
(1) (Everyone x is such that) x likes pizza.
(2) (Everyone x is such that) if x likes pizza, then x likes ice cream.
Now pick any person in the domain as an arbitrary representative, temporarily dub them 'Alex'. Then:
(3) Alex likes pizza (from 1)
(4) If Alex likes pizza, Alex likes ice cream (from 2)
(5) Alex likes ice cream. (from 3, 4)
But Alex was arbitrarily chosen, and we have appealed to no special facts about them; so what we can deduce about them applies to everyone:
(6) (Everyone x is such that) x likes ice cream. (from 5)
Now, the final step here is not relying on the hopeless idea that whatever is true of some individual in a domain is true of everyone/everything. Rather, the principle at stake is this:
Suppose, given some background assumptions, we can infer that an arbitrary representative member a of the relevant domain is F. Then, from the same background assumptions, we can infer that everything in the domain is F (where the conclusion no longer mentions the arbitrary representative a).
But when can we treat some individual as an arbitrary representative of the domain? When we rely on no special distinguishing facts about that individual. In other words,
An individual counts as an arbitrary representative for the purposes of our inference rule when that individual is not specifically mentioned in any premisses or additional assumptions -- so we can only draw on general knowledge about the domain in establishing that the individual in question is F.
This informal (and intuitively compelling) line of thought is what then gets regimented into a formal natural deduction system for the universal quantifier, and the condition on "arbitrariness" gets built into the universal quantifier introduction rule.
For more on this, see Chapter 31 (and the following short chapters) of my CUP book Introduction to Formal Logic -- which you can now freely download from .
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