Source: A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley

[p 464:] Since, however, the first eight of these rules [of Inference] [hereafter ROI] are applicable only to whole lines in an argument, as long as the quantifier is attached to a line these rules of inference cannot be applied—at least not to the kind of arguments we are about to consider. To provide for their application, four additional rules are required to remove quantifiers at the beginning of a proof sequence and to introduce them, when needed, at the end of the sequence. These four rules are called universal instantiation, universal generalization, existential instantiation, and existential generalization. The first two are used to remove and introduce universal quantifiers, respectively, and the second two to remove and introduce existential quantifiers.

[p 466:] As the two previous examples illustrate, we have two ways of performing universal instantiation [hereafter abbreviated to UI]. On the one hand, we may instantiate with respect to a constant, such as a or b, and on the other, with respect to a variable, such as x or y. The exact way in which this operation is to be performed depends on the kind of result intended. If we want some part of a universal statement to match a singular statement on another line, as in the first example, we instantiate with respect to a constant. But if, at the end of the proof, we want to perform universal generalization over some part of the statement we are instantiating, then we must instantiate by using a variable. This latter point leads to an important restriction governing universal generalization—namely, that we cannot perform this operation when the instantial letter is a constant. [...]

[p 469:] In the formulation that follows, the symbols ℱx and ℱy represent any statement function—that is, any symbolic arrangement containing individual variables, such as Ax ⊃ Bx, Cy ⊃ (Dy ⋁ Ey), or Gz ● Hz. The symbol ℱa represents any statement; that is, any symbolic arrangement containing individual constants (or names), such as Ac ⊃ Bc, Cm ⊃ (Dm ⋁ Em), or Gw ● Hw. And the symbol ℱ is a predicate variable that represents any predicate such as F, G, or H.* (*Some textbooks use Greek letters such as φ (phi) χ (chi) and ψ (psi) in the place of ℱ to express these and other rules.)

  1. Universal instantiation (UI):
    (x)ℱx [...]


I ask about only UI of a Variable, and not of a Constant.
Besides the notation, does (x)ℱx differ from ℱy? If so, how?

I wish to verify my inference that rewriting (x)ℱx as ℱy accomplishes only one objective: to enable the application of the ROI to arguments, because (per p 464 above) ROI cannot be applied to Statements with Quantifiers (eg: (x)ℱx), but only to Statements without Quantifiers (eg: ℱy).

  • All the elephants in my left front pocket are purple. But no such purple elephant exists. The inference fails if the domain is empty. – user4894 Dec 23 '20 at 20:57

An expression such as Fx, with no quantifier, contains an unbound variable, and so it is not a sentence and does not have a determinate meaning. The following are sentences: "everyone is happy", "someone is happy", "Fred is happy", but this one is not "___ is happy". That is a just a fragment of a sentence with a place-holder where something ought to be. So if H is the predicate "happy", (x)Hx is a sentence, as is (Ex)Hx and Hfred, but not Hx.

So the question is, why does your logic book bother to introduce statement functions at all? What is the point of writing expressions with unbound variables? The answer is that if you are going to use the rules of inference that you have learned within the propositional calculus, the presence of quantifiers within a sentence gets in the way and obstructs the straightforward operation of the rules. I'm not familiar with Hurley's book, but a common method for eliminating the quantifiers is this: 1. Convert the sentence to prenex normal form; 2. Eliminate the existential quantifiers by skolemisation; 3. Eliminate the universal quantifiers and operate with the assumption that any unbound variable is universally quantified.

It is with reference to this third rule, that Fx may be described as a surrogate for (x)Fx "with certain liberties".

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