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I am currently reading from multiple books and I am finding seeming contradictions regarding the rules by which HOW to use UI to a given of the form ∀x(P(x) → Q(x)).

From "How to Prove it" by Daniel Velleman. "you won’t be able to apply universal instantiation to a given of the form ∀x P(x) unless you have a particular value a to plug in for x, so you might want to wait until a likely choice for a pops up in the proof. For example, consider a given of the form ∀x(P(x) → Q(x)). You can use this given to conclude that P(a) → Q(a) for any a, but according to our rule for using givens that are conditional statements,this conclusion probably won’t be very useful unless you know either P(a) or ¬Q(a)."

From "Symbolic Logic" by Hardegree. "Example 3: ∀x(Fx → Gx) To obtain a substitution instance, we replace both occurrences of x by a name, the same name for both occurrences. Thus, all of the following follow by ∀O. Fa → Ga, Fb → Gb, Fc → Gc, etc."

what am I not understanding here? thanks!

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    Why do you think they are contradictory? – 2017-02-10
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    with velleman's book. i interpreted what he said as.. in order to instantiate P(a) → Q(a). i first must have information of whether P(a) is true or ¬Q(a) is true. On the other hand, hardegree says that i can immediately instantiate P(a) → Q(a) and not need to know about the truth value of P(a) and ¬Q(a). – 2017-02-10
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    No, Velleman claims that you can indeed instantiate $P(a)\to Q(a)$, but this on itself probably isn't useful. In mathematical practice you'll most likely be wanting to prove $Q(a)$ or $\neg P(a)$ and to prove either of this, you'll either $P(a)$ or $\neg Q(a)$. Be careful, though. Velleman teaches you how to write informally about mathematics, the other book, on the other hand, is completely formal (I'm just guessing, I don't know the second book, but it seems like it). Being completely formal, it might not mention the sort of things Velleman does, exactly like this one. – 2017-02-10
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    i would also like to know what you mean by writing math informally? i thought all these proof writing is to make everything formal. – 2017-02-10
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    @TheLastCipher: To add to what Git Gud said, Velleman didn't say that you can't instantiate the universal statement for any particular instance. He merely said that it **probably won't be very useful** if you don't have some way to use what you get. – 2017-02-10
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    understood. thank you guys – 2017-02-10
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    @TheLastCipher Velleman is for rigor, not for formalism. Formalism is reducing mathematics to symbols ($\to, \land, \lor, \forall, \exists, p,q,r,$ etc) and rules. If you want a book comparable to Velleman's, perhaps try Krantz' [Elements of Advanced Mathematics](https://www.amazon.com/Elements-Advanced-Mathematics-Steven-Krantz/dp/1439898340/ref=sr_1_1?s=books&ie=UTF8&qid=1486757334&sr=1-1&keywords=elements+of+advanced+mathematics). – 2017-02-10
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    @Git Gud: will check out your suggestion! :) – 2017-02-11

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See Velleman, page 55:

the statement $\forall x P(x)$ says that $P(x)$ is universally true.

Thus $P(x)$ must be true for every value of $x$ in the universe $U$.

Regarding universal instantiation (page 155) :

You can plug in any value, say $a$, for $x$ and use this given to conclude that $P(a)$ is true.

What is $a$ ? It seems that a "value" for a variable must be an object of the domain $U$: e.g. a number.

All this is quite "informal". In a formal settings, see some mathematical logic textbook, the universal instantiation rule "operates" with substitution of a term in place of the quantified variable $x$.

A term can be:

a variable, a constant, a "complex" term built up from variables and constants with function symbols.

In this case, the rule licenses the derivation of $P(t)$ from $\forall x P(x)$, for every term $t$ of the language.

The same in Velleman; see page 118:

if you were proving a goal of the form $\forall x \in A \ P(x)$, you would probably start by introducing a variable $x$ to stand for an arbitrary element of $A$.

Thus, if no "already given" (name of) objects are available, we can use variables.

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    got it! also what do you mean informal? – 2017-02-10
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    @TheLastCipher - "informal" is exactly the treatment of logic by Velleman, compared to that used in a math log textbook. In Velleman, it seems to me, there is no e.g. any formnal specification of *terms*, *formulae*, *derivations*, etc. – 2017-02-10
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    ohh that is why i felt a bit alien when reading other math logic text since i started with velleman. – 2017-02-10
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    @TheLastCipher - as already noted by Git Gud above, there is nothing "wrong" in Velleman's book: it is aimed at "using" logical tools in mathematical proofs. Thus, he decided to avoid spending time on formal specifications. In a mathematical logic textbook, the formal specifications (of e.g. syntax and semantics, proof, etc.) are unavoidable: they are the *definitions* of the "objects" (the "logical" objects) to wich the theory (the "mathematical" one) will aplly. In e.g. a *topology* textbook we start with the def of *topological space*... – 2017-02-10
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    probably got too confuse from extracting information from two books and multiple videos that are saying the same thing in a different way. thank you for the clarification! – 2017-02-10